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[ "Expected Sensitivity to Invisible Higgs Boson Decays at the ILC with the SiD Detector", "Expected Sensitivity to Invisible Higgs Boson Decays at the ILC with the SiD Detector" ]	[ "Chris Potter \nDepartment of Physics\nUniversity of Oregon\n\n\nInstitute for Fundamental Science\nUniversity of Oregon\n\n", "Amanda Steinhebel \nNASA Goddard Space Flight Center\n\n", "Jim Brau \nDepartment of Physics\nUniversity of Oregon\n\n\nInstitute for Fundamental Science\nUniversity of Oregon\n\n", "Austin Pryor \nDepartment of Physics\nUniversity of Texas at Arlington\n\n", "Andy White \nDepartment of Physics\nUniversity of Texas at Arlington\n\n" ]	[ "Department of Physics\nUniversity of Oregon\n", "Institute for Fundamental Science\nUniversity of Oregon\n", "NASA Goddard Space Flight Center\n", "Department of Physics\nUniversity of Oregon\n", "Institute for Fundamental Science\nUniversity of Oregon\n", "Department of Physics\nUniversity of Texas at Arlington\n", "Department of Physics\nUniversity of Texas at Arlington\n" ]	[]	In the Standard Model (SM) of particle physics, the branching ratio for Higgs boson decays to a final state which is invisible to collider detectors, H → ZZ → νννν, is order 0.10%. In theories beyond the SM (BSM), this branching ratio can be enhanced by decays to undiscovered particles like dark matter (DM). At the Large Hadron Collider (LHC), the current best upper limit on the branching ratio of invisible Higgs boson decays is 11% at 95% confidence level. We investigate the expected sensitivity to invisible Higgs decays with the Silicon Detector (SiD) at the International Linear Collider (ILC). We conclude that at √ s = 250 GeV with 900 fb −1 integrated luminosity each for e − L e + R and e − R e + L at nominal beam polarization fractions, the expected upper limit is 0.16% at 95% confidence level.	null	[ "https://arxiv.org/pdf/2203.08330v2.pdf" ]	247,476,118	2203.08330	acf6ef6b3b00f030e5b9b4290dba7e25461f4685	Expected Sensitivity to Invisible Higgs Boson Decays at the ILC with the SiD Detector March 30, 2022 Chris Potter Department of Physics University of Oregon Institute for Fundamental Science University of Oregon Amanda Steinhebel NASA Goddard Space Flight Center Jim Brau Department of Physics University of Oregon Institute for Fundamental Science University of Oregon Austin Pryor Department of Physics University of Texas at Arlington Andy White Department of Physics University of Texas at Arlington Expected Sensitivity to Invisible Higgs Boson Decays at the ILC with the SiD Detector March 30, 2022 In the Standard Model (SM) of particle physics, the branching ratio for Higgs boson decays to a final state which is invisible to collider detectors, H → ZZ → νννν, is order 0.10%. In theories beyond the SM (BSM), this branching ratio can be enhanced by decays to undiscovered particles like dark matter (DM). At the Large Hadron Collider (LHC), the current best upper limit on the branching ratio of invisible Higgs boson decays is 11% at 95% confidence level. We investigate the expected sensitivity to invisible Higgs decays with the Silicon Detector (SiD) at the International Linear Collider (ILC). We conclude that at √ s = 250 GeV with 900 fb −1 integrated luminosity each for e − L e + R and e − R e + L at nominal beam polarization fractions, the expected upper limit is 0.16% at 95% confidence level. Introduction Invisible Higgs at the ILC The discovery of the Higgs boson in 2021 at the Large Hadron Collider (LHC) [1,2] provides a new window into particle physics. The International Linear Collider (ILC) [3,4,5,6] is an e + e − collider proposed by the international community to exploit that new window with precision measurements of the Higgs boson properties. Current community planning efforts in the field of particle physics include the ILC as a viable and potentially richly rewarding next step for the field [7,8]. In the Standard Model (SM) the properties of this particle are predicted with high precision. Any measured deviation from these properties suggests new physics beyond the SM (BSM). Collider invisible decay is decay to particles which do not interact with the detector material and are either stable, or unstable but decay outside the effective sensitive volume of the detector. Thus some examples of invisible Higgs boson decays are Higgs to dark matter (DM) particles, Higgs to unknown longlived particles (LLP), and Higgs to neutrinos. In the SM the branching ratio of H → ZZ → νννν is approximately 0.10%. Thus the SM invisible Higgs decay is out of range for the LHC, which expects even at the high luminosity LHC (HL-LHC) to reach an upper limit of 2.5% at 95% confidence level [9]. The current limits from the LHC are 11% (18%) at 95% confidence level from ATLAS (CMS) [10,11]. The ILC may improve on the HL-LHC expected limit by an order of magnitude or more [12]. In the Higgstrahlung process e + e − → ZH with invisible Higgs boson decay, several channels are defined by the Z decay. In the hadron channel Z → qq, accounting for 70% of signal events, Table 1: Cross sections for signal and background processes at √ s = 250 GeV. Electron beams are 80% polarized and positron beams are 30% polarized. The effect of initial state radiation (ISR) is included but beamstrahlung is not. Obtained with Whizard 2.6.4 [13]. producing missing energy and jets with high particle multiplicity after hadronization of the quarks. In the electron channel Z → e + e − , 3.4% of signal events, the signature is missing energy and an e + e − pair reconstructing to the Z mass. In the muon channel Z → µ + µ − , also 3.4% of signal events, the signature is missing energy and a µ + µ − pair reconstructing to the Z mass. The tau channel Z → τ + τ − accounts for 3.4% of signal events but is not considered here. The neutrino channels Z → ν eνe , ν µνµ , ν τντ , account for 20% of signal events and are also not considered here. Below, the lepton channel refers only to the electron and muon channels. ILC Beams and SiD Detector The ILC design is detailed comprehensively in the ILC Technical Design Report (TDR) Volume 3 [5]. Electron and positron beams are accelerated to high energy in linacs made up of superconducting RF cavities. The nominal center-of-mass energy in the TDR is √ s = 500 GeV, but this reverts to √ s = 250 GeV in the ILC Machine Staging Report [14] with possible upgrade to √ s = 500 GeV by extension of the linacs. The maximum cross section for the Higgstrahlung process e + e − → ZH occurs near √ s = 250 GeV. See Table 1 for the cross sections of processes relevant to this study. One important design feature of the ILC is the ability to produce polarized electron and positron beams. The composition of beams with fraction P e − electron polarization and P e + positron polarization is 1 4 (1 ∓ P e − )(1 ± P e + ) for opposite polarization cases e − L e + R and e − R e + L , and 1 4 (1 ± P e − )(1 ± P e + ) for same polarization cases e − R e + R and e − L e + L . The nominal assumption for polarization fraction in the TDR is P e − =80% polarized electrons and P e + =30% polarized positrons, though it is hoped that the positron polarization fraction can be made higher. The ILC instantaneous luminosity at √ s = 250 GeV will depend critically on the beam parameters, but L = 1.8 × 10 34 cm −2 s −1 is expected. Beamstrahlung, the radiation of photons from electrons or positrons in one colliding bunch due to the field produced by the oncoming colliding bunch, also depends critically on the beam parameters. The luminosity sharing between polarization cases in the staging report is assumed to split equally between e − L e + R and e − R e + L for an integrated luminosity of 900 fb −1 each, with 100 fb −1 each for e − R e + R and e − L e + L . The total integrated luminosity in this scenario, which we assume for this study, is then dtL =2ab −1 . The Silicon Detector (SiD), one of two detectors proposed for the ILC, is described in detail in the ILC TDR Volume 4 [6]. The other detector is the International Large Detector (ILD), also documented in [6]. The SiD barrel comprises a five-layer Silicon pixel Vertex Detector, a fivelayer Silicon strip Tracker, a dodecahedral electromagnetic calorimeter (ECal) with Lead absorber layers and 20 (20) sensitive thin (thick) layers of sensitive Tungsten, a hadronic calorimeter (HCal) with 11 Steel absorber layers and scintillator sensitive layers, a 5T solenoid, and a dodecahedral muon detector of Iron layers alternating with scintillator sensitive layers. The barrel radii for these subdetectors are r in =1. 4 SiD was designed to take advantage of the particle flow technique for particle identification. Tracks reconstructed in the Vertex Detector and Tracker are extrapolated through the magnetic field produced by the solenoid to the ECal and the HCal and associated to nearby calorimeter clusters. Those tracks associated to an ECal cluster are assumed to be electrons while those associated to an HCal cluster are assumed to be charged hadrons, and those matching to hits in the muon detector are assumed to be muons. Clusters in the ECal unassociated to a track are assumed to be photons while clusters in the HCal unassociated to a track are assumed to be neutral hadrons. Signal and Background Simulation The event generation, full simulation of the SiD detector, and object reconstruction for the simulated data samples in this study are documented in [15], but are briefly summarized below. See Appendix A for a complete list of generator samples and brief descriptions of how they have been used in this study. The signal samples are generated with Whizard 2.6.4 with Pythia6 for hadronization and decay. The Higgstrahlung process e + e − → ZH with fully inclusive Z decays and SM invisible Higgs decay H → ZZ → νννν is specified. The beams are polarized according the nominal fractions for the ILC, and initial state radiation is turned on. Beamstrahlung is not. Background samples were generated with Whizard 1.4 during the Detailed Baseline Design (DBD) exercise, which was incorporated into and described in the TDR. These represent a full set of SM backgrounds with pure polarized beams, which can be mixed to reproduce any required polarization fractions. In the all SM background samples produced by SiD, they were mixed weighted by cross section and the required polarization fractions for the nominal ILC design. For the final sensitivity evaluation, however, dedicated background samples were newly generated with Whizard 2.6.4 with the same conditions as with signal: beam polarization and ISR but no beamstrahlung. Broadly, the backgrounds have final states e + e − νν (electron channel), µ + µ − νν (muon channel), and qqνν, qq ν, eqq, νqq (hadron channel). Requirements are imposed on the Z and H candidate masses and, for the three-fermion processes, the p T of the Z candidate: • candidate Z mass: 60 ≤ m ff ≤ 120 GeV (4f e + e − → e + e − νν, µ + µ − νν, qqνν) • candidate H mass: 90 ≤ m νν ≤ 170 GeV (4f e + e − → e + e − νν, µ + µ − νν, qqνν) • candidate Z p T : 20 ≤ p qq T ≤ 60 GeV (3f e + e − → eqq, νqq ) For the e + e − νν samples the m ff requirement is tightened by 10 GeV, and for the three-fermion processes it is tightened by 15 GeV. For the e + e − νν samples the m νν requirement is tightened by 10 GeV. See Table 2 for a summary of these samples. Figure 1: Track multiplicity before any selection requirements are imposed. Signal selection requires N trk = 2 (6 ≤ N trk ≤ 32) for the lepton (hadron) channels. Background all SM background (blue) is stacked on top of signal (green). The signal branching ratio is assumed to be 10%. The integrated luminosity is 900 fb −1 for each polarization case. Signal and background samples were then fully simulated in ILCSoft v02-00-02 using the compact SiD description SiD o2 v03.xml and reconstructed with Marlin and PandoraPFA for particle flow. 2 Cut-Based Analysis Lepton Channel For the electron and muon channels the signal signature is similar and therefore the signal selection is similar. An e + e − or µ + µ − pair is selected and required to be consistent with the signal Z decay opening angle, momentum and mass. For both cases the momentum is measured from the tracking rather than the calorimetry or muon detector. The mass in recoil from the candidate Z → + − , m 2 rec = s − 2 √ sE + − − m 2 Z(1) must be consistent with the Higgs boson mass. The lepton channel signal selection is as follows: • Exactly two reconstructed tracks and exactly two PFO leptons = e, µ. (N trk = N = 2) • Lepton pair signal consistency: same flavor, opposite sign leptons (N e = 2 or N µ = 2 and q 1 trk + q 2 trk = 0) separation consistent with production from signal Z decay. (−0.9 ≤ cos θ + − ≤ −0.2) Table 3 for signal and background yields, together with signal significance, after each lepton channel requirement above is imposed. Table 3: Signal yields S, background yields B, and significance S/ √ S + B in the electron, muon, and hadron channel selections. The assumed signal branching ratio is 10%. The integrated luminosity is 900 fb −1 for each polarization case. In the final two rows of each table B is estimated from the dedicated samples described in Table 2. In all other rows B is estimated from the all SM background sample. Statistical uncertainties are suppressed except for the loose BDT selection yields. Electron Channel 80% e − L e + R 30% 80% e − R e + L 30% Requirement Signal Background S √ S+B Signal Background S √ S+B All Hadron Channel For the hadron channel the signal signature is missing energy and two hadronic jets from quark pair hadronization. The jets are found with the Durham algorithm as implemented in the LCD Physics Tools 1 . The event is forced to two jets by varying the y cut jetfinding parameter, which is an effective threshold for jet separation. The jet pair momentum is required to be consistent with the signal Z decay opening angle, momentum and mass. The momentum of jet constituents is measured from the tracking for charged particles and the calorimetry for neutral particles. Then the mass in recoil from the candidate Z → jj, m 2 rec = s − 2 √ sE jj − m 2 Z(2) must be consistent with the Higgs boson mass. The hadron channel signal selection is as follows: • Lepton = e, µ veto. (N = 0) • Track and PFO multiplicity consistent with signal. Table 3 for signal and background yields, together with signal significance, after each hadron channel requirement above is imposed. Additional backgrounds in the hadron channel νqq (red) and qqlν (cyan) are also stacked. The signal branching ratio is assumed to be 10%. The integrated luminosity is 900 fb −1 for each polarization case. Electron Channel Electron Channel Muon Channel Hadron Channel Figure 4: Candidate Z mass m vis and recoil mass m rec in the signal samples after full selection up to and including the loose BDT requirement for the electron, muon and hadron channels. The signal branching ratio is assumed to be 10%. The integrated luminosity is 900 fb −1 for each polarization case. Process Electron Channel Muon Chanel Hadron Chanel e − L e + R e − R e + L e − L e + R e − R e + L e − L e + R e − R e + L 4f e − e + → W W 36% 11% 61% 20% 27% 3% 4f e − e + → e ± νW ∓ 23% 23% 0% 0% 2% 1% 4f e − e + → e + e − Z 13% 23% 0% 0% 0% 0% 4f e − e + → ZZ 6% 16% 9% 30% 12% 33% 4f e − e + → ννZ 22% 27% 27% 50% 21% 18% 3f eγ → eZ, νW 0% 0% 0% 0% 36% 39% 2f e − e + → ff 1% 0% 2% 0% 2% 1% Background Processes See Table 4 for the background composition in the all SM background sample after full signal selection up to and including the loose BDT requirement. Two-fermion (e + e − → ff ) constitute a background at the 1% level in electron, muon and hadron channels. Three-fermion processes (eγ → eZ, νW ), initiated by processes with ISR in the initial state, constitute a substantial background for the hadron channel but not the electron or muon channel. Four-fermion backgrounds constitute the dominant background for all three channels considered here. For the electron and muon channels, the dominant background is e + e − → W W , with both W → ν . Subdominant backgrounds are e + e − → ZZ with one Z → νν and the other Z → + − , and e + e − → Zνν with Z → + − . For the electron channel, substantial backgrounds are e + e − → e + e − Z with invisible Z → νν and e + e − → W eν with leptonic W → eν. These W W -and W Z-fusion processes, with only one boson on-shell, are not open to the muon channel and therefore partially explain why sensitivity in the electron channel is lower than in the muon channel. For the hadron channel, the backgrounds are democratic, shared almost equally between threefermion eγ → eZ, νW with hadronic Z → qq and W → qq decays, e + e − → W W with one leptonic W → ν and one hadronic W → qq decay, e + e − → ZZ with one invisible Z → νν decay and one hadronic Z → qq decay, and e + e − → Zνν with hadronic Z → qq decay. In the backgrounds with one electron or muon, the lepton is misidentified in the reconstruction or lost outside of the sensitive detector volume. In both the lepton and hadron channels, the order of magnitude difference between polarization cases for the process e + e − → W W accounts for the stronger sensitivity in the e − R e + L case. Finally, Higgstrahlung itself e + e − → ZH with invisible Z → νν decay and hadronic Higgs boson decays also presents a minor background in the hadron channel but is negligible for the lepton channel. In this case the reconstructed mass of the Higgs boson is low enough to mimic the Z boson. The recoil mass is correspondingly high enough to mimic the Higgs boson. These backgrounds are omitted here because dedicated analyses for each Higgs decay channel are expected to identify and reject them from this search. Multivariate Analysis In order to further improve signal sensitivity, a multivariate technique is employed to exploit differences in correlations between event parameters in signal and background events. A boosted decision tree (BDT) is used with supervised training on separate signal and background samples of events which have survived all of the cut-based requirements. The inputs to the BDT, which feature a single output, are described below. For both the lepton and hadron channels, separate BDTs are trained for each main background against signal. For the hadron channel, each polarization case of a given process is considered a distinct background, so there are ten BDTs for four-fermion backgrounds plus two BDTs for threefermion backgrounds, for a total of twelve background BDTs. For the lepton channel the polarization cases are combined and considered a single background. Therefore for the lepton channel there are five BDTs for four-fermion backgrounds and none for three-fermion backgrounds. The background BDT outputs are then used as inputs to a new BDT (combined BDT of BDTs) with a single output which is trained on signal and background composed of all major backgrounds weighted by cross section. The structure and training of the individual background BDTs as well as the BDT of BDTs are identical and are described below. Lepton Channel The lepton channels have very clean signatures with a lepton pair and nothing else in the event. Moreover the kinematics of the Z candidate have a distinct signature in signal events so the kinematic parameters from the cut-based selection are included as inputs. Because the muon backgrounds are distinct kinematically from the electron backgrounds, an additional input flags events as either the electron channel or the muon channel, thus allowing a different optimization for each. Finally, while the cut-based selection vetos extra tracks, it allows extra neutrals like bremstrahlung photons. Therefore the PFO multiplicity is also included as an input. The input parameters are as follows for the lepton channel: • Parameters from the cut-based analysis: lepton pair separation cos θ + − , Z candidate transverse momentum p vis T , Z candidate mass m vis , recoil mass m rec . • Electron multiplicity N e , either N e = 0 or N e = 2. This parameter flags either the muon channel N e = 0 or the electron channel N e = 2. • PFO multiplicity N pf o . This parameter flags events where residual neutral energy deposits suggest the event may not be signal-like. After BDT training (see below), the improvement in signal sensitivity is loosely (tightly) optimized by requiring the BDT output to be larger than 0 (X opt ). See Figure 5 for the BDT output distributions in the lepton channel after all cut-based requirements are imposed. See the final rows in Table 3 for the impact on signal and background yields and sensitivities in the lepton channels. Hadron Channel The hadron channel kinematics of the Z candidate have a distinct signature in signal events so the kinematic parameters from the cut-based selection are included as inputs to the BDTs. Moreover the track and PFO multiplicity are also included. In addition to the Z candidate transverse momentum p T , the longitudinal momentum p z also provides some discrimination from backgrounds and is included as an input to the BDT. The spatial distribution of PFOs in hadron channel events can be characterized by event shape variables. In this analysis we use thrust T and oblateness O, defined by T = max i p i ·n i \| p i \|(3)O = max i p i ·m i \| p i \| − min i p i ·m i \| p i \|(4) where i indexes the PFOs and for T the maximum is taken over variations over unit vectorsn. The thrust axisn max maximizes T , and for O the minimum and maximum are taken over variations over unit vectorsm wherem·n max = 0. Thrust T and oblateness O are calculated with the LCD package Physics Tools. We also use the y cut parameter used in the Durham jetfinder. This parameter is a distance measure which determines when two PFO groupings can be considered distinct jets. The y cut value required to force events from four to three jets (y 43 ) and from three to two jets (y 32 ) are included as inputs. Finally, the angular separation between the thrust axis calculated in the Z candidate frame and the Z candidate momentum is found to provide additional separation and is included as an input to the BDT. The BDT inputs are therefore as follows for the hadron channel: • Parameters from the cut-based analysis: track and PFO multiplicity N trk and N pf o , jet pair separation cos θ jj , Z candidate transverse momentum p vis T , Z candidate mass m vis , recoil mass m rec . • Z candidate longitudinal momentum p vis z along the beamline. • Event shape variables thrust T and oblateness O calculated in the laboratory frame. • Cosine of angle between the Z candidate momentum in the laboratory frame and the thrust calculated in the frame of the Z candidate, cos θ ZT . • Durham y cut parameters necessary for forcing the event from four to three jets and from three to two jets, y 43 and y 32 . See Table 5 for the signal and background separation S 2 of each input variable y, defined by S 2 = 1 2 i (ŷ S (y i ) −ŷ B (y i )) 2 y S (y i ) +ŷ B (y i )(5) For identical signal and background distributions, S 2 = 0, whereas for disjoint distributions S 2 = 1. Note that S 2 is independent of classification method. After BDT training (see below), the improvement in signal sensitivity is loosely (tightly) optimized by requiring the BDT output to be larger than 0 (X opt ). See Figure 5 for the BDT output distributions in the hadron channel after all cut-based requirements are imposed. See the final rows in Table 3 for the impact on signal and background yields and sensitivities in the hadron channel. Figure 5: Combined BDT outputs for the electron, muon, and hadron channels after all requirements in the cut-based selection are imposed. Backgrounds eeνν, µµνν, qqνν (blue) are stacked on top of signal (green). Additional backgrounds in the hadron channel νqq (red) and qqlν (cyan) are also stacked. The signal branching ratio is assumed to be 10%. The integrated luminosity is 900 fb −1 for each polarization case. These distributions are used to evaluate the expected signal upper limits. BDT Training A decision tree is a binary tree constructed iteratively to optimally separate signal from background events at each binary branching from a node to two new nodes. The root node contains all events, both signal and background. For any node with N sig signal events and N bkg background events, the signal purity p = N sig /(N sig + N bkg ) quantifies the separation. Various measures based on purity can quantify the separation, but for any such measure the input distribution and cut value on that distribution are chosen which maximally increase the separation measure. Branching to new nodes stops when a minimum number of events is reached in a node or a maximum number of layers is reached. Then these leaf nodes are labeled signal (p > 0.5) or background (p < 0.5) based on the purity of the node. The process of boosting a decision tree repeats the construction of the tree many times but with events slightly reweighted. The resulting forest of similar trees is optimally combined into a single output, which is the majority vote of the forest, and this boosted decision tree is then robust against statistical fluctuations in signal and background. The BDTs were trained using the Root Toolkit for Multivariate Analysis (TMVA) 2 . The separation measure is p(1 − p), the Gini index, which is optimized over twenty possible cuts on each input distribution. The minimum node size is five and the maximum number of layers is ten. The boost produces a forest of one thousand trees. For each background process identified in the all SM background sample, dedicated samples with higher statistics produced during the DBD are used to train independent background BDTs against signal. See Appendix B for some BDT evaluation plots. These plots demonstrate that, with the possible exception of the 2f samples, the training and test sample distributions match reasonably well and therefore that the BDTs have not been overtrained. Systematic Uncertainties Beam Parameters For the cross sections used in this study, Whizard 2.6.4 is required to iterate until the theoretical uncertainty is well below the percent level. We conservatively estimate all of these uncertainties at 0.5%. The ILC beam parameters which yield uncertainties are √ s, e + and e − polarization fractions, ISR and beamstrahlung. The expected experimental precision on √ s and polarization at the ILC are 0.01% and 0.25% respectively [16,17], yielding cross section uncertainties well below 0.5%. We therefore treat these as negligible. Remaining uncertainties are due to the theoretical treatment of ISR and beamstrahlung in Whizard 2.6.4, as well as uncertainty on the choice of beam parameters which determine the beamstrahlung. In Whizard 2.6.4 ISR is estimated using the Equivalent Photon Approximation (EPA). To put a conservative upper bound on the cross section uncertainty due to ISR, we compare cross sections with ISR turned off with those with ISR turned on for each important signal and background process. The differences are typically of order 5%, so we estimate a conservative 2% uncertainty on all processes due to ISR treatment. In addition to the uncertainty due to the theoretical treatment of beamstrahlung in Whizard 2.6.4, there is an uncertainty due to the choice of beam parameters. In this study we used the staged ILC250 beam parameters [14] as input, although the parameters at runtime will certainly be different. Therefore we estimate the effect of beamstrahlung on cross sections by turning bremstrahlung on and off. The differences are typically of order 2%, so we estimate a conservative 1% uncertainty on all processes due to beamstrahlung. See Table 6 for a summary of systematic uncertainties due to beam parameters and their impact on expected upper limits. Table 6: Whizard 2.6.4 systematic uncertainties and their impact on the cross section uncertainty δσ/σ for each polarization case e − L e + R /e − R e + L . The uncertainty on the upper limit is estimated by varying signal and background process cross sections within their uncertainties over many trials. Lepton Identification Efficiency The electron identification efficiency established in the ILC TDR for SiD achieved 90% (> 95%) for electrons (muons) with E = 10 GeV and 98% (> 98%) for electrons (muons) with E = 100 GeV [6]. The particle flow algorithm PandoraPFA identifies muons from tracks extrapolated to the SiD muon detector and electrons from tracks extrapolated to the SiD ECal. The efficiency for identifying these leptons depends critically on the PandoraPFA algorithm matching parameters and the kinematic parameter space of the leptons. The efficiency can be estimated from a generator sample with exactly two leptons which has been simulated and reconstructed with two tracks and two correctly identified lepton PFOs. From the background e + e − → e + e − νν sample the electron identification efficiency is estimated to be 87%, and from the background e + e − → µ + µ − νν sample the muon identification efficiency is estimated to be 90%. The binomial uncertainties on these are 0.24% and 0.16%, respectively, or approximately 0.2% for each. With a dedicated effort to tune PandoraPFA parameters for optimal lepton identification, these efficiencies are expected to reach the SiD design goals. Tracker Momentum Resolution The momentum resolution established in the ILC TDR for SiD [6] gave δ(1/p T ) = 2 × 10 −5 /GeV. This uncertainty in track momentum determines a mass uncertainty in the dilepton masses of the Z → e + e − and Z → µ + µ − candidates in the electron and muon channels, respectively. It also partly determines the djet mass uncertainty of the Z → qq in the hadron channel to the extent that the particle flow algorithm correctly identifies charged particle constituents of the jets. The visible mass m vis and recoil mass m rec , calculated from the lepton pair energy and √ s, provide the greatest separation power S 2 in the lepton channels, though they are highly correlated. Since the √ s uncertainty is expected to be negligible compared to the lepton pair energy measurement, we estimate the tracking momentum uncertainty based on reweighting the lepton pair mass distribution. Fitting the lepton pair mass after full selection in the electron and muon channels yields an uncertainty of order the natural Z width, 2.7%, reflecting the high precision expected from the SiD Tracker. We estimate the systematic uncertainty on the selection yields due to δp/p by reweighting the lepton pair mass distribution in the signal sample such that the pair width is increased by 1% above the natural Z width, to 3.7%, and reoptimizing the resulting reweighted lepton channel BDT distributions. The overall uncertainty on the dijet masses of the Z → qq candidates depends critically on the performance of the particle flow algorithm. In the limit of perfect track matching to calorimeter clusters, roughly 2/3 of jet constituents are charged and will have their momentum measured in the Tracker, and 1/3 are neutral and will have their energy measured in the ECal (π 0 ) or HCal (K L etc). The track multiplicity in the hadron channel (Figure 1), with a mean near 20, is dominated by charged pions. So we use the muon pair mass uncertainty impact on δUL/UL, added in quadrature for each track pair in the hadron channel, to estimate the impact of Tracker momentum uncertainty Table 7: Lepton identification uncertainty δ and tracking and calorimetry Z mass uncertainty δm/m and the corresponding uncertainties on the expected upper limit for each polarization case e − L e + R /e + R e + L . The latter is estimated by reweighting the δm/m distributions in signal and reoptimizing for best expected upper limit. on the hadron channel upper limit. Calorimeter Energy Resolution The energy resolution in the ECal and HCal established in the ILC TDR for SiD [6] are parametrized as follows: δE E = 0.01 ⊕ 0.17 √ E (6) δE E = 0.094 ⊕ 0.56 √ E(7) The energy of the Z boson in Higgstrahlung events e + e − → ZH at √ s = 250 GeV is E Z ≈ 110 GeV. Assuming invisible Higgs decay, the energy uncertainty δE/E is 2% (11%) in the limit of total energy deposition in the ECal (HCal). In this total deposition limit the visible mass uncertainties are approximately 3% (16%) for the ECal (HCal). Adding the natural Z width in quadrature yields 4% (16%) for the ECal (HCal). Fitting the jet pair mass after full selection in the hadron channel yields δm/m = 5%. We estimate the systematic uncertainty on the selection yields due to δE/E and the particle flow algorithm performance by reweighting the dijet mass distribution in the signal sample such that the pair width is increased by 2% above the nominal 5%, to 7%, and reoptimizing the resulting reweighted hadron channel BDT distribution. See Table 7 for the systematic uncertainties due to lepton identification efficiency, tracker resolution, and calorimeter resolution and their impact on expected upper limits. Results and Conclusion We calculate the expected 68% (95%) confidence level upper limits on the invisible Higgs branching ratio with the TLimit 3 class in Root using the combined BDT output distributions ( Figure 5). TLimit employs the CL s+b technique with a Bayesian approach. We calculate the expected limits for each channel separately for 900 fb −1 integrated luminosity for each polarization case. Then the channels and polarization case samples are combined. The combined expected upper limit for all channels and polarization cases is 0.16% at 95% confidence level. See Table 8. The SM invisible branching ratio for H → ZZ → νννν is approximately 0.10%, which lies just below the expected upper limits at 95% confidence level For enhanced branching ratios in BSM models, the yields in Table 3 Table 8: Expected upper limits at 95% confidence level on the invisible Higgs branching ratio. The integrated luminosity is 900 fb −1 for each polarization case. Systematic uncertainties are not included, but their impact is estimated in Tables 6 and 7. (discovery) is expected for invisible Higgs branching ratios of 0.50% (0.83%) or higher with this dataset. We do not include systematic uncertainties in calculating these limits. The impacts on the expected upper limits for these has been estimated in the previous section and are summarized in Tables 6 and 7. We neglect these uncertainties because they are provisional and expected to reduce significantly as ISR and beamstrahlung are better understood and detector reconstruction is tuned to optimal performance. These limits should be regarded as realistic and achievable with the current SiD design. Moreover the sensitivity is expected to improve with improved detector performance. The SiD design is being carefully reconsidered in light of recent advancements in subdetector design [18]. For the Tracker and ECal the MAPS technology promises significant improvement in measurement precision [19]. The ECal and HCal energy resolution is expected to improve significantly when machine learning techniques are employed to recover calorimeter energy leakage [20]. In conclusion, we expect that the SiD detector at the ILC will allow a precision measurement of the invisible Higgs branching ratio. The expected limit is 0.16% at 95% confidence level with data samples of 900fb −1 at √ s = 250 GeV for each polarization case. The SM invisible branching ratio lies just below this expected limit. For BSM enhanced invisible Higgs, evidence or discovery is expected above the half-percent level. Electron and Muon Channel BDTs: All Backgrounds • Transverse momentum of Z → + − candidate consistent with signal. (20 ≤ p vis T ≤ 70 GeV) • Mass of Z → + − candidate consistent with Z mass. (75 ≤ m vis ≤ 105 GeV) • Recoil mass consistent with Higgs boson mass. (110 ≤ m rec ≤ 150 GeV) See Figure 1 for the track multiplicity in signal and background prior to signal selection requirements. See Figure 2 for the lepton multiplicity in signal and background prior to signal selection requirements. See Figures 3 and 4 for the m rec and m vis distributions after full selection. See Figure 2 : 2Lepton multiplicity N = N e + N µ before any selection requirements are imposed. Signal selection requires N = 2 (N = 0) for the lepton (hadron) channels. Background all SM background (blue) is stacked on top of signal (green). The signal branching ratio is assumed to be 10%. The integrated luminosity is 900 fb −1 for each polarization case. ( 6 ≤ 6N trk ≤ 32 and 12 ≤ N pf o ≤ 70) • Transverse momentum of Z → qq candidate consistent with signal. (20 ≤ p vis T ≤ 70 GeV) • Mass of Z → qq candidate consistent with Z mass. (75 ≤ m vis ≤ 105 GeV) • Jet multiplicity: successful force to two jets by varying y cut . (N jet = 2) • Jet pair consistency. Jet separation consistent with signal Z decay. (−0.9 ≤ cos θ jj ≤ −0.2) • Recoil mass consistent with Higgs boson mass. (110 ≤ m rec ≤ 150 GeV) See Figure 1 for the track multiplicity in signal and background prior to signal selection requirements. See Figure 2 for the lepton multiplicity in signal and background prior to signal selection requirements. See Figures 3 and 4 for the m rec and m vis distributions after full selection. See Figure 3 : 3Recoil mass m rec after selection up to the recoil mass selection for the electron, muon and hadron channels. Backgrounds eeνν, µµνν, qqνν (blue) are stacked on top of signal (green). Figure 7 :Figure 8 : 78BDT signal (left) and background (middle) inputs and correlations and outputs (right) for the lepton channels. From top to bottom are e − e + → + − (106605/106606), e − e + → eνW (106586/106588), e − e + → W W (106581/106582), single e − e + → Z/W (106568), and e − e + → ννZ (106589) background samples. In the BDT output distributions, test samples and training samples are plotted separately and show no evidence of overtraining. Hadron Channel BDTs: 2f and 3f Backgrounds BDT signal (left) and background (middle) inputs and correlations and outputs (right) for the hadron channel. From top to bottom are e − L e + R → qq (106607), e − R e + L → qq (106608) , e L γ → eZ, νW (37785) and e R γ → eZ, νW (37786). In the BDT output distributions, test samples and training samples are plotted separately and show no evidence of overtraining. Hadron Channel BDTs: 4f eνW and W W Backgrounds Figure 9 : 9BDT signal (left) and background (middle) BDT inputs and correlations and outputs (right) for the hadron channel. From top to bottom are e − L e + R → eνW (106564), e − R e + L → eνW (106565), e − L e + R → W W (106577), and e − R e + L → W W (106578). In the BDT output distributions, test samples and training samples are plotted separately and show no evidence of overtraining. Hadron Channel BDTs: 4f ννZ and ZZ Backgrounds Figure 10: BDT signal (left) and background (middle) inputs and correlations and outputs (right) for the hadron channel. From top to bottom are e − L e + R → ννZ (106571), e − R e + L → ννZ (106572), e − L e + R → ZZ (106575), and e − R e + L → ZZ (106576). In the BDT output distributions, test samples and training samples are plotted separately and show no evidence of overtraining. Process σ ProcessLR [pb] σ RL [pb] e + e − → W W37.5 2.58 e + e − → e ± νW ∓ 10.2 1.09 e + e − → e + e − Z 3.17 2.00 e + e − → ZZ 1.80 0.827 e + e − → ννZ 0.220 0.013 e + e − → ZH 0.313 0.211 , 21.7,126.5,141.7,259.1, and 340.2 cm. The barrel is capped by endcaps Process Intermediate States N ev /σ LR [fb −1 ] N ev /σ RL [fb −1 ] e + e − → e + e − νν eeZ, ννZ, eνW, ZZ, W W 1000 1000 e + e − → µ + µ− νν ννZ, ZZ, W W 1000 1000 e + e − → qqνν ννZ, ZZ 1000 1000 e + e − → qq eν eνW, W W 100 800 e + e − → qq µ/τν W W 100 800 eγ → eqq, νqq eZ, νW 1800 2000 e + e − → ff νννν ZH 10000 10000 Table 2 : 2Signal and background samples generated at √ s = 250 GeV for this study and their equivalent integrated luminosities. All samples are normalized to 900 fb −1 for analysis. See the text for the generator level requirements. A sum over lepton neutrino flavors is implied if allowed.with similar subdetector technology. Table 4 : 4Background composition after full electron, muon and hadron channel selections, up to and including the loose BDT requirement, determined by the all SM background samples. Table 5 : 5Hadron channel BDT input variables signal separation power S The background samples are the DBD samples with pure beam polarization and hadronic Z and W decays.2 for each background Parameter δ δ δElec. δUL/UL Muon δUL/UL Had. δUL/ULLepton ID δ ±0.2% 0.3%/0.2% 0.3%/0.2% 0/0 Tracker δm/m +1% 4.9%/0.6% 1.9%/0.2% 5.7%/0.6% ECal/HCal δm/m +2% 0/0 0/0 8.2%/7.7% can be easily scaled down from 10% to arbitrary levels. EvidenceChannel 80% e − L e + R 30% 80% e − R e + L 30% Combined Electron 1.12% 0.35% 0.33% Muon 0.77% 0.29% 0.27% Hadron 0.42% 0.31% 0.25% Combined 0.35% 0.18% 0.16% ftp://ftp.slac.stanford.edu/groups/lcd/Physics_tools/ https://root.cern/manual/tmva/ https://root.cern/doc/master/classTLimit.html AcknowledgmentsWe thank our colleagues on ILD for useful discussions of backgrounds to this important channel, the ILC International Development Team (IDT) for organizing the global effort toward realization of the ILC, and the organizers of the DPF Snowmass 2021/2022 community planning exercise for coordinating this important community effort. We thankfully acknowledge support from the US Department of Energy grant DE-SC0017996.Appendix A: Monte Carlo Generator Samples Appendix B: BDT Evaluation PlotsCollected in this appendix are the BDT signal and background input variable correlations and the BDT output distributions.Figure 6shows these for the BDT of BDTs described in the text,Figure 7shows these for the electron and muon channel BDT, andFigures 8, 9 and 10show these for the hadron channel BDT.Electron Channel BDT of BDTsMuon Channel BDT of BDTsHadron Channel BDT of BDTs Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Georges Aad, arXiv:1207.7214Phys.Lett. 716Georges Aad et al. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys.Lett., B716:1-29, 2012, arXiv:1207.7214. Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. 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Signal used in training BDTs, signal and background used in evaluating signal sensitivity. -(sidmc20) ilc250 eLpR 2f1hinv..whizard 2 6 4.stdhep -(sidmc20) ilc250 eRpL 2f1hinv..whizard 2 6 4.stdhep -sidmc21a ilc250 eLpR ap3f.stdhep, sidmc21a ilc250 eLpR ap3f.stdhep -sidmc21a ilc250 eLpR ea3f.stdhep, sidmc21a ilc250 eLpR ea3f.stdhep -sidmc21a ilc250 eLpR eevv.stdhep, sidmc21a ilc250 eRpL eevv.stdhep -sidmc21a ilc250 eLpR mumuvv.stdhep, sidmc21a ilc250 eRpL mumuvv.stdhep -sidmc21a ilc250 eLpR qqvv.stdhep, sidmc21a ilc250 eRpL qqvv.stdhep -sidmc21a ilc250 eLpR qqev.stdhep. • SiD MC20/21 4 (80% e − , 30% e + polarization. sidmc21a ilc250 eRpL qqev.stdhep -sidmc21a ilc250 eLpR qqlv.stdhep, sidmc21a ilc250 eRpL qqlv.stdhep• SiD MC20/21 4 (80% e − , 30% e + polarization). Signal used in training BDTs, signal and background used in evaluating signal sensitivity. -(sidmc20) ilc250 eLpR 2f1hinv..whizard 2 6 4.stdhep -(sidmc20) ilc250 eRpL 2f1hinv..whizard 2 6 4.stdhep -sidmc21a ilc250 eLpR ap3f.stdhep, sidmc21a ilc250 eLpR ap3f.stdhep -sidmc21a ilc250 eLpR ea3f.stdhep, sidmc21a ilc250 eLpR ea3f.stdhep -sidmc21a ilc250 eLpR eevv.stdhep, sidmc21a ilc250 eRpL eevv.stdhep -sidmc21a ilc250 eLpR mumuvv.stdhep, sidmc21a ilc250 eRpL mumuvv.stdhep -sidmc21a ilc250 eLpR qqvv.stdhep, sidmc21a ilc250 eRpL qqvv.stdhep -sidmc21a ilc250 eLpR qqev.stdhep, sidmc21a ilc250 eRpL qqev.stdhep -sidmc21a ilc250 eLpR qqlv.stdhep, sidmc21a ilc250 eRpL qqlv.stdhep Used only in estimating background yields and distributions at all stages in the analysis chain. Not used in training BDTs or the final sensitivity estimate. Sid • Barklow, SM background -80e-+30e+ .stdhep -all SM background. DBD Mixed Samples (80% e − , 30% e + polarization)• Barklow SiD DBD Mixed Samples (80% e − , 30% e + polarization). Used only in esti- mating background yields and distributions at all stages in the analysis chain. Not used in training BDTs or the final sensitivity estimate. -all SM background -80e-+30e+ .stdhep -all SM background +80e--30e+ .stdhep • Dbd Two, Fermion Samples (100% e − , 100% e + polarization). Used only in training BDTs. -E250-TDR ws.P2f z l.Gwhizard-1 95.eL.pR.I106605..stdhep -E250-TDR ws.P2f z l.Gwhizard-1 95.eR.pL.I106606..stdhep -E250-TDR ws.P2f z h.Gwhizard-1 95.eL.pR.I106607..stdhep -E250-TDR ws. P2f z h.Gwhizard-1 95.eR.pL.I106608..stdhep• DBD Two-Fermion Samples (100% e − , 100% e + polarization). Used only in training BDTs. -E250-TDR ws.P2f z l.Gwhizard-1 95.eL.pR.I106605..stdhep -E250-TDR ws.P2f z l.Gwhizard-1 95.eR.pL.I106606..stdhep -E250-TDR ws.P2f z h.Gwhizard-1 95.eL.pR.I106607..stdhep -E250-TDR ws.P2f z h.Gwhizard-1 95.eR.pL.I106608..stdhep Used only in training BDTs. -E0250-TDR ws. • Dbd Three, Pea vxy.Gwhizard-1.95.eL.pW.I37785..stdhep -E0250-TDR ws.Pea vxy.Gwhizard-1.95.eL.pB.I37786..stdhep -E0250-TDR ws.Pae vxy.Gwhizard-1.95.eW.pR.I37815..stdhep -E0250-TDR ws. Fermion Samples (100% e − , 100% e + polarization). Pae vxy.Gwhizard-1.95.eB.pR.I37816..stdhep• DBD Three-Fermion Samples (100% e − , 100% e + polarization). Used only in training BDTs. -E0250-TDR ws.Pea vxy.Gwhizard-1.95.eL.pW.I37785..stdhep -E0250-TDR ws.Pea vxy.Gwhizard-1.95.eL.pB.I37786..stdhep -E0250-TDR ws.Pae vxy.Gwhizard-1.95.eW.pR.I37815..stdhep -E0250-TDR ws.Pae vxy.Gwhizard-1.95.eB.pR.I37816..stdhep Used only in training BDTs. • Dbd Four, E250-TDR ws.P4f sw sl.Gwhizard-1 95.eL.pR.I106564..stdhep -E250-TDR ws.P4f sw sl.Gwhizard-1 95.eR.pL.I106566..stdhep -E250-TDR ws.P4f sw l.Gwhizard-1 95.eL.pR.I106586..stdhep -E250-TDR ws.P4f sw l.Gwhizard-1 95.eR.pL.I106588..stdhep -E250-TDR ws. Fermion Samples (100% e − , 100% e + polarization). P4f szeorsw l.Gwhizard-1 95.eL.pR.I106568..stdhep• DBD Four-Fermion Samples (100% e − , 100% e + polarization). Used only in training BDTs. -E250-TDR ws.P4f sw sl.Gwhizard-1 95.eL.pR.I106564..stdhep -E250-TDR ws.P4f sw sl.Gwhizard-1 95.eR.pL.I106566..stdhep -E250-TDR ws.P4f sw l.Gwhizard-1 95.eL.pR.I106586..stdhep -E250-TDR ws.P4f sw l.Gwhizard-1 95.eR.pL.I106588..stdhep -E250-TDR ws.P4f szeorsw l.Gwhizard-1 95.eL.pR.I106568..stdhep P4f sznu sl.Gwhizard-1 95.eL.pR.I106571..stdhep -E250-TDR ws.P4f sznu sl.Gwhizard-1 95.eR.pL.I106572..stdhep -E250-TDR ws.P4f sznu l.Gwhizard-1 95.eL.pR.I106589..stdhep -E250-TDR ws.P4f zz sl.Gwhizard-1 95.eL.pR.I106575..stdhep -E250-TDR ws.P4f zz sl.Gwhizard-1 95.eR.pL.I106576..stdhep -E250-TDR ws.P4f ww sl.Gwhizard-1 95.eL.pR.I106577..stdhep -E250-TDR ws.P4f ww sl.Gwhizard-1 95.eR.pL.I106578..stdhep -E250-TDR ws.P4f ww l.Gwhizard-1 95.eL.pR.I106581..stdhep -E250-TDR ws. P4f ww l.Gwhizard-1 95.eR.pL.I106582..stdhephttps://pages.uoregon.edu/ctp/SiD_private.html -E250-TDR ws.P4f sznu sl.Gwhizard-1 95.eL.pR.I106571..stdhep -E250-TDR ws.P4f sznu sl.Gwhizard-1 95.eR.pL.I106572..stdhep -E250-TDR ws.P4f sznu l.Gwhizard-1 95.eL.pR.I106589..stdhep -E250-TDR ws.P4f zz sl.Gwhizard-1 95.eL.pR.I106575..stdhep -E250-TDR ws.P4f zz sl.Gwhizard-1 95.eR.pL.I106576..stdhep -E250-TDR ws.P4f ww sl.Gwhizard-1 95.eL.pR.I106577..stdhep -E250-TDR ws.P4f ww sl.Gwhizard-1 95.eR.pL.I106578..stdhep -E250-TDR ws.P4f ww l.Gwhizard-1 95.eL.pR.I106581..stdhep -E250-TDR ws.P4f ww l.Gwhizard-1 95.eR.pL.I106582.*.stdhep Generator Parameters • Whizard 2.6.4 Beam Parameters -sqrts=250 GeV -beams= "e. @(+1) or beams pol density=@(+1), @(-1) -epa q min=1 GeV and epa x min=0. 1isr -beams pol fraction=80%, 30% -either beams pol density=@SiD MC20/21 Whizard 2.6.4 Generator Parameters • Whizard 2.6.4 Beam Parameters -sqrts=250 GeV -beams= "e-", "e+" => isr, isr -beams pol fraction=80%, 30% -either beams pol density=@(-1), @(+1) or beams pol density=@(+1), @(-1) -epa q min=1 GeV and epa x min=0.01 • Whizard 2.6.4 Aliases and Decays -alias nu=nue:numu:nutau and alias nubar=nuebar:numubar:nutaubar -alias q=u:d:s:c:b and alias qbar=ubar:dbar:sbar:cbar:bbar -process wpdecay=Wp => q, qbar and process wmdecay=Wm => q, qbar -unstable Wp(wpdecay) and unstable Wm. wmdecay• Whizard 2.6.4 Aliases and Decays -alias nu=nue:numu:nutau and alias nubar=nuebar:numubar:nutaubar -alias q=u:d:s:c:b and alias qbar=ubar:dbar:sbar:cbar:bbar -process wpdecay=Wp => q, qbar and process wmdecay=Wm => q, qbar -unstable Wp(wpdecay) and unstable Wm(wmdecay) => "mu+. 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GeV [nu, nubar] and all M < 170. GeV [nu, nubar] -process ilc250 eLpR qqev= "e-", "e+" => (Wp, "e-", nuebar) + ("e+", nue, Wm) and cuts=all M > 60. GeV [q,qbar] and all M < 120. GeV [q, qbar] -process ilc250 eLpR qqlv= "e-", "e+" => (q, qbar, "mu-", numubar) + ("mu+", numu, q, qbar) + (q, qbar, "tau-", nutaubar) + ("tau+", nutau, q, qbar) and cuts=all M > 60. GeV [q,qbar] and all M < 120. GeV [q, qbar] -process ilc250 eLpR ap3f= photon, "e+" => ("e+", q, qbar)+(nuebar, q, qbar) and cuts=all M > 75. GeV [q, qbar] and all M < 105. GeV [q,qbar] and all Pt > 20. GeV [q,qbar] and all Pt < 60. GeV [q,qbar] -process ilc250 eLpR ea3f= "e-", photon => ("e-", q, qbar)+(nue, q, qbar) and cuts=all M > 75. GeV [q, qbar] and all M < 105. GeV [q,qbar] and all Pt > 20. GeV [q,qbar] and all Pt < 60. GeV [q,qbar]	[]
[ "Magnetism in the three-dimensional layered Lieb lattice: Enhanced transition temperature via flat-band and Van Hove singularities", "Magnetism in the three-dimensional layered Lieb lattice: Enhanced transition temperature via flat-band and Van Hove singularities" ]	[ "Kazuto Noda \nNTT Basic Research Laboratories\nNTT Corporation\n243-0198AtsugiJapan\n", "Kensuke Inaba \nNTT Basic Research Laboratories\nNTT Corporation\n243-0198AtsugiJapan\n", "Makoto Yamashita \nNTT Basic Research Laboratories\nNTT Corporation\n243-0198AtsugiJapan\n" ]	[ "NTT Basic Research Laboratories\nNTT Corporation\n243-0198AtsugiJapan", "NTT Basic Research Laboratories\nNTT Corporation\n243-0198AtsugiJapan", "NTT Basic Research Laboratories\nNTT Corporation\n243-0198AtsugiJapan" ]	[]	We describe the enhanced magnetic transition temperatures Tc of two-component fermions in three-dimensional layered Lieb lattices, which are created in cold atom experiments. We determine the phase diagram at half-filling using the dynamical mean-field theory. The dominant mechanism of enhanced Tc gradually changes from the (delta-functional) flat-band to the (logarithmic) Van Hove singularity as the interlayer hopping increases. We elucidate that the interaction induces an effective flat-band singularity from a dispersive flat (or narrow) band. We offer a general analytical framework for investigating the singularity effects, where a singularity is treated as one parameter in the density of states. This framework provides a unified description of the singularity-induced phase transitions, such as magnetism and superconductivity, where the weight of the singularity characterizes physical quantities. This treatment of the flat-band provides the transition temperature and magnetization as a universal form (i.e., including the Lambert function). We also elucidate a specific feature of the magnetic crossover in magnetization at finite temperatures.	10.1103/physreva.91.063610	[ "https://arxiv.org/pdf/1505.06591v1.pdf" ]	117,363,450	1505.06591	85d56d8d2a967660100ad534a4114d33996e70e5	Magnetism in the three-dimensional layered Lieb lattice: Enhanced transition temperature via flat-band and Van Hove singularities 25 May 2015 Kazuto Noda NTT Basic Research Laboratories NTT Corporation 243-0198AtsugiJapan Kensuke Inaba NTT Basic Research Laboratories NTT Corporation 243-0198AtsugiJapan Makoto Yamashita NTT Basic Research Laboratories NTT Corporation 243-0198AtsugiJapan Magnetism in the three-dimensional layered Lieb lattice: Enhanced transition temperature via flat-band and Van Hove singularities 25 May 2015arXiv:1505.06591v1 [cond-mat.quant-gas]numbers: 6785-d7110Fd7127+a7510-b We describe the enhanced magnetic transition temperatures Tc of two-component fermions in three-dimensional layered Lieb lattices, which are created in cold atom experiments. We determine the phase diagram at half-filling using the dynamical mean-field theory. The dominant mechanism of enhanced Tc gradually changes from the (delta-functional) flat-band to the (logarithmic) Van Hove singularity as the interlayer hopping increases. We elucidate that the interaction induces an effective flat-band singularity from a dispersive flat (or narrow) band. We offer a general analytical framework for investigating the singularity effects, where a singularity is treated as one parameter in the density of states. This framework provides a unified description of the singularity-induced phase transitions, such as magnetism and superconductivity, where the weight of the singularity characterizes physical quantities. This treatment of the flat-band provides the transition temperature and magnetization as a universal form (i.e., including the Lambert function). We also elucidate a specific feature of the magnetic crossover in magnetization at finite temperatures. I. INTRODUCTION Phase transitions, such as magnetism and superconductivity, are of fundamental interest in lattice fermions. As a common feature, transition temperatures T c are usually given as a function of the density of states (DOS) at the Fermi energy ρ(E F ) and an interaction U , T c /W ∝ e −1/ρ(EF )U for U/W ≪ 1, where bandwidth W is a unit of energy. A singularity located on the Fermi energy [ρ(E F ) → ∞] changes this functional form, which could greatly increase the T c . For instance, the logarithmic Van Hove singularity (VHS) induces a characteristic dependence: T c /W ∝ e − √ W/U [1]. Recent studies proposed the emergence of another interesting delta-functional singularity, which we call the flat-band singularity (FBS), at the surface of a layered graphene [2] or of a topological material [3]. This FBS is also expected as the origin of the higher transition temperature: T c ∝ U . Although these singularities have attracted attention, we still lack a comprehensive understanding of these singularity effects on phase transitions. Cold atoms in an optical lattice, where we can control lattice geometry and resulting DOS singularity [4,5], provide opportunities for studying the singularity effects in bulk systems. In particular, successful creations of twodimensional (2D) optical lattices with singular DOSs, the Kagome [6] and Lieb (line-centered-square) [7] lattices, have activated theoretical studies on phenomena related to the FBS [8][9][10][11][12][13][14][15][16][17][18][19]. In general, in these 2D lattices, the Mermin-Wagner theorem states that no phase transitions occur at finite temperatures [20]. A layered structure exhibits specific features in the DOS as a remnant of 2D lattices, even though the system itself is three-dimensional (3D) [21]. Here we focus on the 3D layered Lieb lattice * [email protected] [7,21], which is a test-bed for systematic investigations of the effects of various singularities on phase transitions. In this paper, we show that the magnetic transition temperature T c is clearly enhanced by the FBS and VHS in a 3D layered Lieb lattice at half-filling using the dynamical mean-field theory (DMFT). We determine the phase diagrams with several anisotropic hoppings between the inter-and intralayer directions. It is shown that this anisotropy is a practical parameter for controlling T c /W for both weakly and strongly interacting regions. We also propose an analytical framework for dealing with the singularity effects with a single parameter called a singularity weight a. We demonstrate that T c ∝ aU for FBS and T c /W ∝ e − √ 2W/aU for VHS. These forms clearly provide a unified picture of phase transitions dominated by singular DOSs, where a large singularity-weight enhances T c . This is in stark contrast to phase transitions in nonsingular systems. We also demonstrate that, for both singularities, a conventional form e −1/ρ(EF )U can universally be reproduced for a → 0. Our approach also phenomenologically explains that the FBS induces an anomalous behavior of thermodynamic quantities even in the paramagnetic state above T c . II. MODEL AND METHOD H = −t xy l i,j σ c † liσ c ljσ −t z l,l ′ iσ c † liσ c l ′ iσ +U li n li↑ n li↓ , where c liσ (c † liσ ) is the annihilation (creation) operator of an atom with spin σ at site i on the l-th layer, and n liσ = c † liσ c liσ . The subscript i, j ( l, l ′ ) denotes the summation over the nearest neighbor sites in the xy plane (z direction). We impose periodic boundary conditions for all x, y, z directions. To investigate the magnetic properties of this model at half-filling, we use the DMFT approach [21,22] with a six-site unit cell as shown in Fig. 1 (a). The bipartite structure allows us to focus on the antiferromagnetic ordering. We employ the numerical renormalization group method (NRG) [23,24] to solve the effective impurity problem. NRG is applicable to energy scales ranging from the ground state to finite temperatures [25][26][27]. Our approach (DMFT+NRG) succeeded in studying the ground state properties of the present system [21], suggesting the validity of the application to study finite temperature properties. The numerical procedures are detailed in Ref. [21]. Here, we choose bandwidth W (= 4 √ 2t xy + 4t z ) as the unit of energy and use the notationX ≡ X/W to describe dimensionless parameters. As an exception, we uset z ≡ t z /t xy to characterize the anisotropy of intralayer (xy-plane) and interlayer (z-direction) hoppings. The DOS ρ(ω) is rescaled asρ(ω) ≡ W ρ(ω). We calculate thermodynamic quantities, i.e., magnetization m α = (n α↑ − n α↓ )/2 and double occupancy D α = n α↑ n α↓ , where α = H, A, B. Note that the lattice geometry and the half-filling condition result in symmetry e.g. m A = m B and D A = D B , and so on. III. PHASE DIAGRAM We first overview the characteristic magnetism of the present system based on the phase diagram. Figure 1 showsT c as a function ofŪ fort z = 0.1, 0.5, and 1.0. BelowT c , the antiferromagnetic insulating states appear, while aboveT c , non-magnetic (metallic or Mott insulating) states appear. From nonmonotonicT c curves, we can see that crossovers from the band picture to the Heisenberg (local) picture of magnetic transitions occur at aroundŪ ∼ 0.8-1.0 for allt z . We next discuss how anisotropyt z affectsT c . For clarity, we provide a change inT c in a weakly interacting regionŪ = 0.2 in Fig. 1 (c). ForŪ 1,T c is strongly enhanced for smallt z (= 0.1). Surprisingly,T c fort z = 0.1 shows specific behaviorT c ∝Ū , which is qualitatively distinct from the well-known conventional weak-interacting behaviorT c ∝ exp(−1/Ū ) (as found in those fort z = 0.5). We find thatT c also slightly increases fort z → 1, which is the result of another distinct behavior ofT c ∝ exp(−1/ √Ū ) [1]. These qualitative changes in the magnetism will be discussed in detail below with an analytical approach [see thick dotted lines in Fig. 1 (b)]. For the strongly interacting regionŪ 1 in Fig. 1 (b), we find thatT c is enhanced ast z decreases. In contrast to the above, this enhancement can be understood quantitatively:T c can be scaled by the effective Heisenberg parameterJ Hei . Interestingly, in this localized-spin picture region, the difference in the number of adjacent sites plays an important role, as discussed in Sec. V. IV. WEAKLY INTERACTING REGION The magnetism in this region will be characterized by the band structure, and, in particular, the DOS at Fermi energy ω = 0 is an important quantity. In Fig. 2 (a)-(c), we thus provideρ ave (ω)[= α=H,A,Bρ α (ω)/3] forŪ = 0. As is well known, the 2D Lieb lattice (t z = 0) has a flat band [10], and therefore DOS has the delta function singularity at the Fermi energy (not shown). For the present systems with a finitet z ( = 0), the flat band becomes dispersive with a width of 4t z . From Fig. 2 (a), we can see that a large DOS still appears for a smallt z . This remaining flat band structure is naively regarded as the origin of the specific behaviorT c ∝Ū . As shown in Fig. 2 (b), the DOS atω = 0 decreases with increasingt z , and then the FBS disappears, resulting in the conventionalT c ∝ e −1/Ū fort z ∼ 0.5. Figure 2 (c) shows another interesting feature of the DOS fort z = 1: a logarithmic singularity at ω = 0 like a 2D VHS, leading toT c ∝ e −1/ √Ū [1]. These results suggest that, generally, a singularity of DOS changes the functional forms of T c and then drastically enhances T c . Some previous studies have discussed logarithmic VHS effects in the 2D square lattice [1,28] and also investigated the FBS effects [2,29]. However, to the best of our knowledge, a general analytical formalism for dealing with singularity effects has not yet been established. Here, we propose a general approach for revealing the singularity effects, which can explain theŪ dependence of T c in Fig. 1 (b). We consider the mean-field gap equation 1 U = dω ρ ave (ω) 2 ω 2 + ∆ 2 ave tanh ω 2 + ∆ 2 ave 2T ,(1) where ρ ave (ω) and ∆ ave are the average DOS and spectral gap, respectively, with respect to sites A, B, and H. Gap ∆ ave can be rewritten as ∆ ave = U m ave , where the average magnetization m ave = α=A,B,H \|m α \|/3. We simplify the multiband structure and the specific lattice structure, which leads to the above site-averaged gap equation. The average DOS can be set to a simple sum of the singular and the nonsingular parts:ρ ave (ω) = aρ S (ω)+bρ NS (ω) [see insets of Fig. 2 (a)-(c)]. Here, we introduce the specific parameter a (b) defined as a weight of the singular (nonsingular) DOS with normalization condition a + b = 1. We simply setρ NS as the uniform DOS ρ uni (ω) = θ(1/2 − \|ω\|), where θ(ω) is a step function. Here, we should comment that our approach provides a general extension of the conventional forms of T c and ∆ [∝ e −1/ρ(EF )U ]. A divergent ρ(E F ) cannot parameterize T c and ∆ any more, and instead of this, the singularity weight a determines these physical quantities. Note that, generally, any singularities of ρ(ω) should disappear in an integral and a weight a is always definable: Namely, ρ(ω)dω(≡ a + b) = 1 even though ∃ ω ∈ R; ρ(ω) = ∞. Thus, our approach is applicable to any singularities on any lattice geometry. In what follows, we show that the above simplified approach with a parameterized singularity can capture the essence of the magnetic transition forŪ ≪ 1. We start with a general discussion with a of any value, which qualitatively explains theŪ dependence ofT c . After that, with a given a, we quantitatively compare the analytical form with the numerical results. We also show that the introduction of singularity weight a allows us to phenomenologically understand the anomalous behavior of some thermodynamic quantities. We first discuss the linear-Ū behavior ofT c shown in Fig. 1(b). We here consider a DOS with the FBS given byρ FBS ave (ω) = aδ(ω) + bρ uni (ω) shown in the inset of Fig. 2 (a). By solving Eq. (1) with ∆ ave = 0, we obtain the transition temperature (see Appendix A): T FBS c = a 4bW aπ 4be γ e 1/bŪ ,(2) where γ is the Euler constant and W(x) is the Lambert function defined as x = W(x)e W(x) . For a = 0, given W(x) ∼ x for x ∼ 0, Eq. (2) reproduces the conventional behavior without the singularity:T NS c = e γ−1/Ū /π. ForŪ → 0, except for a = 0, the divergent argument e 1/bŪ requires another asymptotic property W(x) ∼ ln x − ln(ln x) for x → +∞. We thus obtain T FBS c ∼ aŪ /4 + (abŪ 2 /4) ln(4e γ /aπŪ ) forŪ ∼ 0. This explainsT c ∝ aŪ shown in Fig. 1 (b) and the strong enhancement ofT c in Fig. 1 (c). Importantly, this asymptotic behavior with a divergent term indicates that, even if a singularity weight a is very small, the FBS changes the nature of the transition at aroundŪ ∼ 0 [30]. We next describe the enhancement ofT c due to the VHS. Here we consider the DOSρ VHS ave (ω) = a ln(1/2\|ω\|)θ(1/2−\|ω\|)+bρ uni (ω) [see Fig. 2 (c)], and then we obtainT VHS c ∼ e γ+ b a − b a 1+ 2a b 2Ū /π (see Appendix B). For a = 0,T VHS c also reproducesT NS c . ForŪ → 0, T VHS c ∝ e − √ 2 /aŪ , which causes the higher transition temperatures shown in Fig. 1 (b) and (c). The exponential decay forŪ → 0 suggests that the enhancement caused by the VHS is much weaker than that of the FBS (see Appendix C). We further provide the analytical form of m ave (= ∆ ave /U ) for any a. Here, we should note that m FBS ave ∼ a/2 + (abŪ/2) ln(2/aŪ ) forŪ → 0, and the constant term a/2 explains the specific feature of the flat band magnetism: m FBS ave,T =0 shows a jump at an infinites-imalŪ (∼ +0) [21]. To show the validity of the above qualitative discussions, we quantitatively compare the DMFT calculations with the analytical results. Figure 2 (d) shows the average magnetizations m ave at T = 0 calculated with the DMFT. The inset in Fig. 2 (d) shows that the analytical forms of m NS ave,T =0 and m VHS ave,T =0 with a = 1/3π agree well with the DMFT calculations. Later we will discuss the nonmonotonic behavior of magnetization fort z = 0.1 in Fig. 2 (d). Here, we should note that the DMFT does not use the simplified average DOS, suggesting the validity of our simplification with the extraction of the singularity. Here, we should discuss what determines the singularity weight a. For VHS systems, a can be obtained from the series expansion ofρ ave (ω) at aroundω ∼ 0: the present system witht z = 1 has a ∼ 1/3π. For FBS systems, the averaging assumption gives a as follows: A simple example is the 2D Lieb lattice (t z = 0) with a of 1/3, where one of the three bands is the flat band located atω = 0 [31]. On the other hand, for the 3D Lieb lattices (t z = 0), a is zero because all the bands are dispersive. However, as shown in Fig. 2 (a), there is a very narrow band at around the Fermi energy for smallt z . Within the framework of the static mean-field approximation [Eq. (1)], the narrow band can be regarded as a flat band when U becomes larger than the bandwidth of 4t z (see Appendix D). To effectively explain such phenomena, we redefine the singularity weight a as a function of the other parameters: a → a(Ū ,T ,t z ). Figure 2 (d) shows that, forŪ 0, m ave fort z = 0.1 rapidly increases from zero without a jump, which can be phenomenologically explained by an increase in a from zero to a finite value as discussed above. Then, for U 0.1, m ave increases linearly, which can be effectively explained by the analytical form of m FBS ave with a ∼ 0.14 as shown in the inset of the figure. These findings are consistent withT c behavior at aroundŪ = 0 andT c ∝ aŪ shown in Fig. 1 (b). The behavior of m ave andT c is characteristic of a flat-band magnetism in the 3D layered Lieb lattice [21]. Our DMFT calculations in Fig. 1 (b) and (c) clearly elucidate that, for smallt z 0.1, T c is strongly enhanced by the effective FBS resulting from a narrow dispersive band with the interaction effects. We should stress that this effective FBS can be seen in various systems, such as the multiorbital systems with different bandwidths [32], and may greatly enhance T c of magnetism and also superconductivity in these systems. We further demonstrate that the introduction of a(Ū ,T ,t z ) effectively explains the behavior of the thermodynamic quantities. In Fig. 3, we show the average double occupancy D ave = α=H,A,B D α /3 fort z = 0.1, 0.5, and 1.0 withŪ = 0.2. A kink in D ave clearly shows the transition between magnetic insulating and non-magnetic metallic phases. At low temperatures, D ave increases with increasingT ; ∂D ave /∂T > 0 for allt z [33]. At higher temperatures, in the metallic region, we find ∂D ave /∂T > 0 fort z = 0.1, whereas ∂D ave /∂T < 0 fort z = 0.5 and 1.0. The inset shows that, for smaller U = 0.1, the metallic region shows ∂D ave /∂T < 0 for allt z . In fact, ∂D ave /∂T < 0 can be understood from the usual Fermi liquid behavior, while ∂D ave /∂T > 0 is unusual as discussed below. The thermodynamic relation provides ∂D/∂T = −∂S/∂U , where S is entropy. The Fermi liquid obeys S ∝ M * T , where M * is the effective mass. The interaction-induced mass renormalization means ∂M * /∂U > 0, which leads to ∂D ave /∂T < 0 [22]. The FBS breaks down the above scenario. The entropy is given as ∝ a ln 2+bM * T , and thus ∂D ave /∂T ∝ −∂a/∂U at low temperatures. We conclude that the unusual behavior ∂D ave /∂T > 0 atŪ = 0.2 andt z = 0.1 results from ∂a/∂U < 0, meaning that the renormalization effects greatly reduce the weight of fermions in the flat (very narrow) band at the Fermi energy. On the other hand, as discussed above, until U becomes comparable to 4t z , we can expect ∂a/∂U > 0, which is consistent with what is shown in the inset of Fig. 3. Thus, our phenomenological approach explains the unusual D ave behavior caused by the FBS. This unusual behavior signals the onset of the specific magnetic transition with T c ∝ U . We finally discuss m ave near transition temperatures. V. STRONGLY INTERACTING REGION The magnetism in this region is effectively discussed within the local spin picture. Thus, an important quantity is the site-dependent η α , namely the number of adjacent sites (i.e., coordination number) in the xy plane of the site α (= A, B, H). Note that the following discussion will be generally applicable to bipartite lattices with different η α . Employing a simple mean-field approach, we can obtain T c = J Hei /2 and J Hei = (2 √ η A η H t 2 xy +4t 2 z )/U , which explains the quantitative change inT c in Fig. 1 (b). Here, T c obeys ∝J Hei = (4 √ 2 + 4t 2 z )/(4 √ 2 + 4t z ) 2 (1/Ū ), which takes its minimum value at aroundt z = 1.0. We also obtain m A(H) near T c with the mean-field ap- proach: m A(H) = ± 3 2 η A(H) ηA+ηH T c −T Tc , meaning that the site with a large η α shows a large \|m α \|. Namely, \|m H \| > \|m A \| forŪ 1 at aroundT c , which is confirmed by the DMFT calculations shown in Fig. 4 (a). In contrast, we find \|m H \| < \|m A \| at low temperatures. In this region, the quantum fluctuations caused by the itinerancy of electrons yield η α -dependent double occupancies D H > D A , and a large D α suppresses the development of \|m α \| [31]. This causes the crossing curves m H -T and m A -T seen in Fig. 4 (a) forŪ = 1.2. Furthermore, we can stress that the change in the sign of ∆m = \|m H \| − \|m A \| is a clear manifestation of the crossover of magnetism between band and Heisenberg pictures [see Fig. 4 (b)]. VI. SUMMARY We investigated magnetism in three-dimensional layered Lieb lattices and determined the phase diagrams using the dynamical mean-field theory. We revealed that the (delta-functional) flat-band and (logarithmic) Van Hove singularities affect phase transitions, which greatly increases the transition temperatures for weakly interacting region. We also pointed out that the effective flatband singularity emerges from a dispersive flat-band as a consequence of the interaction effects, which can appear in multiband systems. For strongly interacting region, we characterize T c /W by the number of adjacent sites. The larger Heisenberg interaction for site H triggers the onset of magnetization. Stimulated by this, we proposed a suitable quantity, the difference of magnetization, for clearly detecting the crossover from the flat-band to Heisenberg magnetism, which can be observed in cold atom experiments. We proposed a comprehensive approach for investigating the singularity effects by introducing the singularity weight a. We derived the universal forms of T c and magnetization for both singularities, which offer a remarkable statement: a large singularity-weight induces enhanced T c . Thus, we elucidated a common feature between the flat-band and Van Hove singularities, which suggests that the singularity weight a is a unified parameter for describing the singularity-induced phase transitions. 1 U = a 4T c dxδ(x) tanh x x + b 1/4Tc 0 dx tanh x x . Using y 0 dx tanh x/x = ln(4e γ y/π) for y 1, we obtain aπ 4b e 1 bŪ −γ = a 4bTc e a 4bTc , where γ is the Euler constant. Then, with the Lambert function W(x) defined as x = W(x)e W(x) , we find a 4bTc = W aπ 4b e 1 bŪ −γ . Finally, we obtainT FBS c = a 4bW aπ 4be γ e 1/bŪ ,(A1) where we assumeT c 1.This form reproducesT c ∝ aU for any a andT c ∝ e −1/Ū for a → 0 as mentioned in Sec. IV. By solving Eq. (1) with the same type of calculations as the above, we can obtain the magnetization m ave at zero temperature: m FBS ave,T =0 = a 2bŪ W a 2b e 1/bŪ ,(A2) where we assume∆ ave = m aveŪ ≪ 1. It should be noted that m FBS ave,T =0 shows a finite jump with infinitesimal U , which is a distinct feature of the flat-band magnetism. Employing Eq. (1), we can obtain magnetization m ave near the transition temperature: m FBS ave,T ∼Tc = 2 √ 3πT c U a + 4b 2T c aπ 2 + 42bζ(3)T c T c −T T c ,(A3)1 U = −a 1/4Tc 0 dx ln(4T c x) tanh x x + b 1/4Tc 0 dx tanh x x . To solve the above, we give y 0 ln x tanh x x dx = ln 2 y 2 + A for y 1, A = γ 1 − π 2 8 − ln 2 π 2 + ln 2 π 2 + γ ln( π 4 ), where γ 1 is the Stieltjes constant. Note that we use the following relation to derive the above constant A: ∞ n=1 (−1) n+1 ln 2 n − ln 2 (n + 1) = γ 1 + γ 2 2 − π 2 24 + ln 2 2 + ln(4/π) ln π 2 . The gap equation reduces to −aA + b ln(e γ 4/π) − [b + a ln(e γ /π)] ln(4T c ) + ln 2 (4T c ) = 1/Ū . Then, we obtain the analytical form T VHS c = e γ+ b a − b a 1+ 2a b 2Ū + a 2 C b 2 /π,(B1) where C = 2A + ln 2 (e γ 4/π) = γ 2 − π 2 /4 + 2 ln 2 2 + 2γ 1 . The above equation holds whenT c 1. We can obtain the magnetization m ave at zero temperature, by solving Eq. (1) similarly: m VHS ave = e b a − b a 1+ 2a b 2Ū − a 2 π 2 6b 2 /Ū ,(B2) where we assume∆ ave = m aveŪ ≪ 1. Using Eq. (1), we can obtain magnetization m ave near the transition temperature: m VBS ave,T ∼Tc = 2 √ 2πT c √ 7Ū × b + aγ − a ln(πT c ) (a + b)ζ(3) + aζ ′ (3) − aζ(3) ln(πT c /2 6/7 ) T c −T T c ,(B3) where we assume∆ ave = m aveŪ ≪ 1. With an additional assumptionT c ≪ 1, we obtain m VBS ave,T ∼Tc ∼ 2 √ 2πTc √ 7ζ(3)Ū T c−T Tc for any a. Appendix C: Universal ratio By using Eqs. (A1)-(B2), we can discuss the universal ratio R = 2∆ ave,T =0 /T c , which allows us to quantitatively evaluate the effects of the singularities as discussed below. It is known that, without the singularity, R stays constant at 2πe −γ ∼ 3.53, meaning that ∆ T =0 and T c obey a similar form. Generally speaking, the singularity yields R depending on U and also a. With the FBS, the universal ratio is given by R FBS = 2πW a 2b e 1/bŪ W aπ 4be γ e 1/bŪ . We find that R FBS ranges from 2πe −γ to 4 depending on U and a. ForŪ ≪ 1, we obtain R FBS = 2πe −γ (a = 0) and R FBS = 4 (a = 0). This jump of R FBS results from the divergent term e 1/bŪ in the arguments of Lambert functions. On the other hand, for the VHS, we obtain R VHS = 2πe −γ e b a 1+ 2a b 2Ū − a 2 π 2 6b 2 − b a 1+ 2a b 2Ū − a 2 C b 2 . ForŪ ≪ 1, we find R VHS = 2πe −γ for any a. These findings indicate that the effects of the FBS (VBS) on the magnetism are strong (weak). We should note that, forŪ ≪ 1, the universal ratio depends only on the types of singularities, and does not depend on the weight a. Thus, we can stress that the universal ratio forŪ ≪ 1 characterizes the strength of the singularity effects. Appendix D: Flat band singularity caused by a narrow but finite bandwidth In this section, we explain why the narrow bands can be regarded as the origin of the FBS within the framework of the gap equation. We now consider a twouniform-band (TUB) model, whose DOS is described as ρ TUB (ω) = (a/W nar )θ(W nar /2−\|ω\|)+bθ(1/2−\|ω\|), wherē W nar (= W nar /W < 1) is the bandwidth of the narrower band W nar scaled by that of the wider band W . This DOS is naturally regarded as the simplified DOS of the 3D layered Lieb lattice shown in Fig.2 (a) fort z = 0.1 . As an example, we consider the gap equation for T c , which is given by 1 U = ā W nar W nar /4Tc 0 dx tanh x x + b 1/4Tc 0 dx tanh x x . When bothW nar 4T c and 1 4T c are satisfied, given . This gap equation is the same as that for the FBS as shown above in Appendix A, which leads to T FBS c ∼ aU/4. Thus, when the interaction strength becomes greater than the narrower bandwidth,Ū W nar , we find the linear-U behavior of T c . We should note that here we use a relationŪ ∝ 4T c . Consequently, we can conclude that the very narrow band can be regarded as the origin of the (remaining) FBS as mentioned in Sec. IV. Here, we should comment that the above discussion is based on the static mean-field approximation. Thus, we miss some effects caused by the interactions, such as the mass renormalization and the creation of the Hubbard bands. Simply put, the above gap equation cannot be used for the strongly interacting (Heisenberg) region, and thus, this approach may overestimate the re-gion in which T c ∝ U behavior can be obtained. Note that the DMFT calculations properly capture the effects mentioned above, and the DMFT calculations confirm that T c ∝ U behavior is obtained for smallt z ( 0.1). By extending the simple gap equation, we can discuss the above phenomena effectively. Here, we redefine a as a function of other parameters: a → a(Ū ,T ,t z ). For example, we can expect that a the quasi-particle weight of the fermions in the narrow band will decrease owing to the renormalization effects. This phenomenological approach can explain the characteristic behavior of thermodynamic quantities as shown in Fig. 3 in the main text. Our model is described by the Hubbard Hamiltonian on a 3D layered Lieb lattice [seeFig. 1 (a)]: FIG . 1. (Color online) (a) 3D layered Lieb lattice. Solid lines represent the unit cell for our calculations. (b) Transition temperatureTc vs. interactionŪ fortz = 0.1, 0.5, and 1.0. The thick dotted lines are analytical forms:. See text for their full forms. (c)Tc vs.tz forŪ = 0.2. (b) (d) average magnetizations m ave vs. U fortz = 0.1, 0.5, and 1.0 at zero temperature. Insets of (a)-(c) represent the simple DOSs obtained by extracting essential structures of ρave(ω ∼ 0) with normalized condition a + b = 1:ρ FBS (ω) = aδ(ω) + bρ uni (ω),ρ uni (ω) = θ(1/2 − \|ω\|), and ρ VHS (ω) = a ln(1/2\|ω\|)θ(1/2 − \|ω\|) + bρ uni (ω), respectively. The inset of (d) is a closeup aroundŪ = 0 of m ave for all threetz with analytical results (thick dotted lines): m FBS ave ∼ a/2 + (abŪ /2) ln(2/aŪ ) with a = 0.14, m NS ave = e −1/Ū /Ū , and m VHS ave ∝ e − √ 2/aŪ /Ū with a = 1/3π. See text for the full form of m FBS ave and m VHS ave . T =0 , written as m FBS ave,T =0 = a 2bŪ W a 2b e 1/bŪ and m VHS ave,ave,T =0 and m VHS ave,T =0 reproduce the non-singular limit m NS ave,T =0 = e −1/Ū /Ū for a = 0. FIG. 4 . 4(Color online) (a) Magnetization \|mγ \| (γ = H, A) and (b) the difference of magnetization ∆m = \|mH \| − \|mA\| vs.T withtz = 0.1 forŪ = 0.2, 0.8, and 1.2. ForŪ c ≪ 1, the gap equation(1) provides m FBS ave,T ∼Tc = (2 √ 3T c /Ū ) (T c −T )/T c 1/2 except for a = 0, and m VBS ave,T ∼Tc = (2 √ 2πT c / 7ζ(3)Ū ) (T c −T )/T c 1/2 for any a, where ζ(x)is the Riemann zeta function (see Appendix A and B). The magnetization is proportional to (T c −T ) 1/2 even with the singularities, which can be confirmed by the following DMFT calculations.Figure 4(a) shows m γ (γ = H, A) forŪ = 0.2, 0.8, and 1.2 with t z = 0.1 as a function ofT . where we use∆ ave = m aveŪ ≪ 1. With an additional as-sumptionT c ≪ 1, we obtain m FBS ave,T ∼Tc ∼ Appendix B: Derivation ofT VHS c We next deriveT VHS c . Substitutingρ VHS (ω) = [a ln(1/2\|ω\|) + b] θ(1/2 − \|ω\|) and ∆ ave = 0 in Eq. (1), we obtain y 0 0dx tanh x/x = ln(4e γ y/π) for y 1, This form is qualitatively equivalent to the non-singular formT NSc ∝ e −1/ρ(EF )Ū , whereρ(E F ) = (a + bW nar )/W nar .IfW nar 4T c and 1 4T c are satisfied, by using another relationy 0 dx tanh x/x = y for y 1, we obtain aπ 4b e 1 bŪ −γ = a 4bTc e a 4bTc ACKNOWLEDGMENTSWe thank N. Kawakami and Y. Takahashi for valuable discussions and R. Peters for his support with the numerical calculations. This work was supported by JSPS KAKENHI (Grant No. 25287104). . J E Hirsch, D J Scalapino, 10.1103/PhysRevLett.56.2732Phys. Rev. Lett. 562732J. E. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 56, 2732 (1986). . N B Kopnin, T T Heikkilä, G E Volovik, http:/link.aps.org/doi/10.1103/PhysRevB.83.220503Phys. Rev. B. 83220503N. B. Kopnin, T. T. Heikkilä, and G. E. Volovik, Phys. Rev. B 83, 220503 (2011). . E Tang, L Fu, 10.1038/nphys3109Nat. Phys. 10964E. Tang and L. Fu, Nat. Phys. 10, 964 (2014). . 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[ "Dynamics of gelling liquids: a short survey", "Dynamics of gelling liquids: a short survey" ]	[ "Henning Löwe \nInstitut für Theoretische Physik\nGeorg-August-Universität\nD-37077GöttingenGermany\n", "Peter Müller \nInstitut für Theoretische Physik\nGeorg-August-Universität\nD-37077GöttingenGermany\n", "Annette Zippelius \nInstitut für Theoretische Physik\nGeorg-August-Universität\nD-37077GöttingenGermany\n" ]	[ "Institut für Theoretische Physik\nGeorg-August-Universität\nD-37077GöttingenGermany", "Institut für Theoretische Physik\nGeorg-August-Universität\nD-37077GöttingenGermany", "Institut für Theoretische Physik\nGeorg-August-Universität\nD-37077GöttingenGermany" ]	[]	The dynamics of randomly crosslinked liquids is addressed via a Rouse-and a Zimm-type model with crosslink statistics taken either from bond percolation or Erdős-Rényi random graphs. While the Rouse-type model isolates the effects of the random connectivity on the dynamics of molecular clusters, the Zimm-type model also accounts for hydrodynamic interactions on a preaveraged level. The incoherent intermediate scattering function is computed in thermal equilibrium, its critical behaviour near the sol-gel transition is analysed and related to the scaling of cluster diffusion constants at the critical point. Second, non-equilibrium dynamics is studied by looking at stress relaxation in a simple shear flow. Anomalous stress relaxation and critical rheological properties are derived. Some of the results contradict long-standing scaling arguments, which are shown to be flawed by inconsistencies.Dynamics of gelling liquids: a short survey	10.1088/0953-8984/17/20/002	[ "https://arxiv.org/pdf/cond-mat/0412101v2.pdf" ]	13,219,549	cond-mat/0412101	af81273b125df2e4c4e3f61b4faa85d1eac3f323	Dynamics of gelling liquids: a short survey 21 Dec 2004 Henning Löwe Institut für Theoretische Physik Georg-August-Universität D-37077GöttingenGermany Peter Müller Institut für Theoretische Physik Georg-August-Universität D-37077GöttingenGermany Annette Zippelius Institut für Theoretische Physik Georg-August-Universität D-37077GöttingenGermany Dynamics of gelling liquids: a short survey 21 Dec 2004TOPICAL REVIEW Version of 21 December 2004 Dedicated to Lothar Schäfer on the occasion of his 60 th birthday The dynamics of randomly crosslinked liquids is addressed via a Rouse-and a Zimm-type model with crosslink statistics taken either from bond percolation or Erdős-Rényi random graphs. While the Rouse-type model isolates the effects of the random connectivity on the dynamics of molecular clusters, the Zimm-type model also accounts for hydrodynamic interactions on a preaveraged level. The incoherent intermediate scattering function is computed in thermal equilibrium, its critical behaviour near the sol-gel transition is analysed and related to the scaling of cluster diffusion constants at the critical point. Second, non-equilibrium dynamics is studied by looking at stress relaxation in a simple shear flow. Anomalous stress relaxation and critical rheological properties are derived. Some of the results contradict long-standing scaling arguments, which are shown to be flawed by inconsistencies.Dynamics of gelling liquids: a short survey Introduction Gelling liquids are part of everyday life. One encounters them, for example, when preparing a chocolate pudding or when sticking two materials together with the help of glue. From a microscopic point of view, gelling liquids consist of irregularly structured clusters of molecules or macromolecules. The formation of these clusters is either a result of intermolecular association, produced by e.g. van der Waals forces, electrostatic attractions or hydrogen bonding, or a result of chemical reactions such as polycondensation, polymerisation or vulcanisation induced by a chemical crosslinker [1,2]. Intermolecular association, also called physical gelation, leads to weakly bound clusters, which typically form and dissolve reversibly in the course of time during an experiment. On the other hand, chemical gelation leads to permanent clusters at temperatures of interest, and it is this situation that we will exclusively consider here. When increasing the concentration of crosslinks in a liquid (sol) one observes a more and more viscous behaviour under shear stresses, until a sudden transformation to an amorphous solid state takes place at a certain critical crosslink concentration. This point marks the gelation transition or sol-gel transition. The static shear viscosity diverges at the transition, and the onset of a static shear modulus is found. Carothers [3] was the first to interpret the gelation transition as due to the formation of a macroscopic cluster of molecules in the system. His considerations were quantified and refined by Flory [4,5] and Stockmayer [6,7] to what is nowadays called "classical theory", a percolation model of tree-like structures, closely related to percolation on Bethe lattices [8]. So the classical theory arises [9] in the mean-field approximation of lattice-bond percolation [10]. Stauffer [11] and de Gennes [12] suggested the latter as a mathematical model for gelation, in particular, if caused by polycondensation. Lattice-bond-percolation clusters may also contain loops, and the spatial dimension becomes relevant, too. More importantly, upon identifying the gelation transition with the lattice-bond-percolation transition, it is revealed to be a continuous phase transition. Its driving parameter is crosslink concentration, not temperature. Within this theoretical picture, the critical behaviour at the gelation transition is dictated by scaling and universality [13,10]. The resulting predictions for static properties of gelation clusters agree well with experiments in the vicinity of the sol-gel transition [14,15]-a substantial improvement over the mean-field like classical theory. As far as dynamical phenomena are concerned, a variety of competing attempts have been made to seek an interpretation in terms of the percolation picture, see e.g. [16][17][18] for contradictory predictions concerning the shear viscosity. Yet, all of these attempts rely on more or less ad hoc assumptions needed to compensate for the lack of thermal fluctuations or any sort of dynamics in a pure percolation model. Rather, the appropriate strategy should be to start from a (semi-) microscopic dynamical model for gelation clusters, from which the desired link to quantities in percolation theory can be deduced. This route will be followed here. Other analytical approaches to gelation from a microscopic model include [19][20][21][22][23][24][25][26][27]. Among others, they describe thermostatic fluctuations in the gel phase and calculate the static shear modulus. Computer simulations of microscopic models for gelation have been done by e.g. [28][29][30][31][32][33][34]. In this survey we will concentrate on the sol phase and report on results obtained in [35][36][37][38][39][40][41][42]. The dynamics of the sol phase is characterised by strong precursors of the gelation transition, even well below it. These include anomalous, stretched-exponential decays in time of both dynamical density correlations [43] and shear-stress relaxation [44]. Both decays are characterised by typical time scales which diverge when the critical crosslink concentration is approached. Our exact results on critical rheological properties contradict long-standing scaling arguments, which are shown to be flawed by inconsistencies. The paper is organised as follows. In Section 2 we briefly lay out a suitable generalisation of the usual Rouse and Zimm model for linear polymers to describe gelling liquids. The model is then used to investigate time-dependent density fluctuations in Section 3. Section 4 deals with stress relaxation and critical rheological properties in a simple shear flow. Both Section 3 and Section 4 are subdivided in a part pertaining to the Rouse model, a part pertaining to the Zimm model and a part where the results are discussed and put in a wider perspective. Finally, Section 5 adds some closing remarks. Rouse and Zimm model for randomly crosslinked monomers In this section we give a brief description of a model which is to be considered a theoretical minimal model for the dynamics of gelling complex fluids. This model is a generalisation of one of the most fundamental models of polymer physics [45][46][47][48] to the case of randomly connected monomers. In this context, it has been discussed before by e.g. [49-56, 35, 37-39, 57-59, 40-42]. Dynamical equation We consider N point-like monomers, which are characterised by their time-dependent position vectors R i (t), i = 1, . . . , N , in three-dimensional Euclidean space Ê 3 . Permanently formed, harmonic crosslinks connect M randomly chosen pairs of particles (i e , j e ), where 1 ≤ i e = j e ≤ N for all e = 1, . . . , M . The potential energy associated with these entropic Hookean springs takes the form V := 3 2a 2 M e=1 R ie − R je 2 =: 3 2a 2 N i,j=1 R i · Γ i,j R j ,(1) where the length a > 0 plays the role of an inverse crosslink strength, and physical units have been chosen such that k B T = 1. It will be convenient to specify a given crosslink configuration G := {(i e , j e )} M e=1 in terms of its N × N -connectivity matrix Γ, which is defined by the right equality in (1). For part of what follows this setting could be generalised to the crosslinking of N identical molecular units which consist themselves of a given number of monomers that are connected in some fixed manner, such as N identical chains, rings or stars of monomers [37,38]. For the ease of presentation, however, we will not consider such a generalisation here. We study the dynamics of these harmonically crosslinked monomers in the presence of an incompressible solvent fluid, which may induce hydrodynamic interactions between them. Hydrodynamic interactions will be incorporated on a preaveraged level in the spirit of Kirkwood and Riseman [60] and Zimm [46]. This is a traditionally accepted way of doing so albeit the limitations of this approach are still not sufficiently well explored [47,48]. We also allow for the presence of an externally imposed, simple shear flow in x-direction v(r, t) :=γ(t)y e x(2) with a time-dependent shear rateγ(t). Here r = (x, y, z). A purely relaxational monomer dynamics is then described by [47,48] d dt R i (t) − v R i (t), t = − N j=1 H eq i,j ∂V ∂R j (t) + ξ i (t)(3) for i = 1, . . . , n. This is the defining equation of the Zimm model for crosslinked monomers (in solution). The rest of this subsection is devoted to a brief explanation and discussion of (3), see [41,42] for more details. The jointly Gaussian thermal noises ξ i in (3) have zero mean and covariance ξ i (t) ξ † j (t ′ ) = 2 H eq i,j δ(t − t ′ )1, as is required by the fluctuation-response theorem. As usual, the ξ i "thermalize" the system in the long-time limit. Here, the dagger denotes the transposition of a vector, δ the Dirac-delta function and 1 the 3 × 3-unit matrix. Interactions between the monomers and the solvent fluid are subsumed in the spatially isotropic and homogeneous preaveraged mobility matrix H eq i,j := 1 ζ δ i,j + (1 − δ i,j ) h κ 2 π/R i,j .(4) It emerges [41,42] from taking Oseen's expression [61,60] for the mobility tensor and averaging it with respect to the suitably normalised Boltzmann weight ∼ e −V . However, when it is indispensable to have a positive definite mobility matrix in the sequel, we will replace the Oseen tensor with the Rotne-Prager-Yamakawa tensor [62,63] in this procedure. Depending on which tensor is used, the function h in (4) is given by [64] h( x) := x/π Oseen, erf( √ x) − (1 − e −x )/ √ πx Rotne-Prager-Yamakawa.(5) The expression in the second line of (5) involves the error function erf and reduces to the expression of the Oseen case asymptotically as x ↓ 0. The diagonal term in the preaveraged mobility matrix (4), which is proportional to the Kronecker symbol δ i,j , accounts for a frictional force with friction constant ζ that acts when a monomer moves relative to the externally imposed flow field (2). The non-diagonal term reflects the solventmediated average influence of the motion of monomer j on monomer i. The parameter κ := 6/π ζ/(6πη s a) involves the solvent viscosity η s and serves as the coupling constant of the hydrodynamic interaction. Formally setting κ = 0 in (4) yields H eq i,j = ζ −1 δ i,j , and the Zimm model for crosslinked monomers reduces to the Rouse model for crosslinked monomers [35][36][37][38][39][40] d dt R i (t) − v R i (t), t = − 1 ζ ∂V ∂R i (t) + ξ i (t) ,(6) where i = 1, . . . , n and the jointly Gaussian thermal noises ξ i have zero mean and covariance ξ i (t) ξ † j (t ′ ) = (2/ζ) δ(t − t ′ )1. It is only for convenience that we introduced the Rouse model as the special case κ = 0 of the Zimm model here. Physically, it has its own standing as the minimal model for polymer melts under theta conditions, see e.g. [47,48] for the case of linear polymer chains. In particular, all the approximations that entered the derivation of the (off-diagonal part of the) preaveraged mobility matrix H eq do not affect the Rouse model, of course. It remains to explain the quantity R i,j in (4), which is simply the mean squared displacement between monomers i and j in the thermal-equilibrium state characterised by the suitably normalised Boltzmann weight ∼ e −V . In order to write down a formula for R i,j , let us remark that, by construction, the connectivity matrix Γ ≡ Γ(G) is block-diagonal with respect to the clusters of a given crosslink configuration G (which are the maximal connected components of G). Moreover, Γ(G) possesses as many zero eigenvalues as there are clusters in G. This is easily seen from the fact that the centre of mass of each cluster does not feel a force from the potential energy V . Hence, Γ cannot be inverted, but it possesses a Moore-Penrose pseudo-inverse Z [65], which is the inverse of Γ on the complement of its zero eigenspace and zero elsewhere. It can be represented as Z := (1 − E 0 )/Γ, where E 0 denotes the projector on the zero eigenspace of Γ in Ê N and 1 denotes the N × N -unit matrix. The mean-squared displacement R i,j is then given in terms of Z according to R i,j := Z i,i + Z j,j − 2Z i,j if i and j belong to the same cluster, +∞ otherwise. There is also another interpretation for R i,j , which we will use below: Viewing each monomer as an electric contact and each crosslink as a unit Ohmian resistor connecting two contacts, R i,j is the effective electric resistance between the contacts i and j of this corresponding electrical resistor network [66]. This exact correspondence between Hookean bead-spring clusters and Ohmian electrical resistor networks relies on the linearity of Hooke's and Ohm's law. Since both the connectivity matrix Γ and the preaveraged mobility matrix H eq are blockdiagonal, it follows that clusters move independently of each other in this model. The salient feature of the Zimm and Rouse equations (3) and (6) is that they are linear in the monomers' positions. Hence, they admit an explicitly known solution. The results we present in this paper rely heavily on this solution. Average over crosslink ensemble So far, everything in this section was meant for an arbitrary but fixed realisation G of M crosslinks among N monomers. For practical reasons, G can never be determined experimentally in macroscopically large gelling fluids. Neither should physically meaningful observables depend on specific microscopic details of G, but only on some macroscopic characteristics of it. Therefore, we follow the general philosophy of the theory of disordered systems and take G as an element of a statistical ensemble of crosslink configurations, within which it occurs with probability P N (G). The just made statement on physically meaningful observables A(G) now translates into a self-averaging property: the two quantities A(G) and its ensemble average G ′ P N (G ′ )A(G ′ ) coincide (with probability one) in the macroscopic limit. Therefore we will compute the macroscopic limit A := lim N →∞ G P N (G)A(G)(8) of such averages with a fixed crosslink concentration c := lim N →∞ M/N . This will be done for two different crosslink ensembles. (i) Clusters are generated according to three-dimensional continuum percolation, which is closely related to the intuitive picture of gelation, where monomers are more likely to be crosslinked when they are close to each other. Since continuum percolation and lattice percolation are believed to be in the same universality class [10], we employ the scaling description of the latter. It predicts [10] a cluster-size distribution of the form τ n ∼ n −τ exp{−n/n * }(9) for ε := (c crit − c) ≪ 1 and n → ∞ with a typical cluster size n * (ε) ∼ ε −1/σ that diverges as ε → 0. Here, σ and τ are (static) critical exponents, see Table 1 below for their numerical values. (ii) Each pair of monomers is chosen independently with equal probability c/N , corresponding to Erdős-Rényi random graphs, which are known to resemble the critical properties of mean-field percolation [9]. After performing the macroscopic limit, there is no macroscopic cluster for c < c crit = 1/2 and almost all clusters are trees [67]. Furthermore, all n n−2 trees of a given "size" n, that is, with n monomers, are equally likely. The clustersize distribution can also be cast into the scaling form (9) with the exactly known critical exponents τ and σ listed below in Table 1. Time-dependent density fluctuations In this section we address dynamical properties of gelling liquids in thermal equilibrium. Therefore we will assume throughout this section that there is no externally imposed shear flow, i.e.γ = 0. Experiments [43,68] on quasi-elastic light scattering in gelling liquids allow to measure how spatial density fluctuations of a given wave vector q are correlated to each other at different times t. This information is encoded in the incoherent intermediate scattering function S(q, t) := lim t0→−∞ 1 N N i=1 e i q·[Ri(t+t0)−Ri(t0)] .(10) The right-hand side of (10) is determined by the solution R i (t) of the dynamical equation (3) for a given crosslink realisation G and with initial conditions being imposed at time t 0 . The average over the thermal noise and the subsequent limit t 0 → −∞ in (10) ensure that the system reaches its thermal-equilibrium state. Then, for large retardation times t, one expects [69,43] that this correlation is determined by the slowest relaxation processes in the system. Due to the independent motion of different clusters in the model under consideration, the slowest relaxation processes correspond to the centre-of-mass diffusion of whole clusters of monomers. This argument can be quantified-see e.g. [35], [41] or Eq. (4.12) in [37]-and yields S(q, t) t→∞ ∼ K k=1 N k N exp{−q 2 tD(N k )} .(11) Here we have set q := \|q\| and introduced the clusters N k , k = 1, . . . , K, of the given crosslink configuration G. The number of monomers in the cluster N k is denoted by N k and D(N k ) := lim t→∞ 1 6t R CM k (t) − R CM k (0) 2 = i,j∈N k 1 H eq i,j −1(12) defines its diffusion constant in terms of the mean-square displacement of its centre of mass R CM k (t) := N −1 k i∈N k R i (t). The right equality in (12) follows from a short calculation with the exact solution of the dynamical equation (3). It was previously established in [70]. Another diffusion constant has been introduced by Kirkwood [48,47] D(N k ) := 1 N 2 k i,j∈N k H eq i,j .(13) It provides an upper bound to the former, D(N k ) ≤ D(N k ) ,(14) as can be shown by applying the Jensen-Peierls inequality, see e.g. Sect. 8c in [71], to (12). Customarily, one also defines an effective diffusion constant D eff for the whole gelling liquid by D −1 eff := lim q→0 q 2 ∞ 0 dt S(q, t) = K k=1 N k N 1 D(N k ) .(15) Since S(q, t) is expected to develop a time-persistent part in the gel phase, D eff is expected to vanish when approaching the gelation transition from the sol side. Rouse dynamics We recall from Sect. 2.1 that in the absence of hydrodynamic interactions, κ = 0, we have H eq i,j = ζ −1 δ i,j . Hence, the cluster-diffusion constant (12) and the Kirkwood diffusion constant (13) are equal D(N k ) = D(N k ) = 1 ζN k ,(16) and inversely proportional to the number of monomers in the cluster [35]. In other words, cluster topology does not influence diffusion within Rouse dynamics. Next, we discuss the long-time behaviour of the incoherent intermediate scattering function in the macroscopic limit. According to Sect. 2.2, this amounts to calculating the average of (11) S(q, t) t→∞ ∼ K k=1 N k N exp{−q 2 tD(N k )} .(17) Thanks to (16) this average is easily performed by reordering the clusters according to their size S(q, t) t→∞ ∼ ∞ n=1 nτ n e −q 2 t/(ζn) ,(18) where τ n := K k=1 1 N δ N k ,n(19) is the cluster-size distribution and (18) holds in the absence of an infinite cluster. Using the scaling form (9) of τ n , we find [35,37] S(q, t) t→∞ ∼ ζ q 2 t y 1 ε = 0 , [t/t * q (ε)] (y−1/2)/2 exp{−const. [t/t * q (ε)] 1/2 } ε > 0 .(20) At the critical point, the long-time decay is algebraic with a critical exponent y = τ − 2. In the sol phase one has a Kohlrausch or stretched-exponential behaviour with a time scale that diverges as t * q (ε) ∼ (ζ/q 2 )ε −µ with a critical exponent µ = 1/σ, when the critical point is approached. For the effective diffusion constant (15) we conclude from (20) that it vanishes like D eff ∼ lim q↓0 q 2 t q (ε) −(3−τ ) ∼ ε a with a = (3 − τ )/σ(21) as ε ↓ 0. The exponent a could have also been deduced directly from the right expression in (15). Indeed, given any cluster-additive observable, a reordering of the clusters according to their size yields A = K k=1 N k N A(N k ) = ∞ n=1 nτ n A n ,(22) where A n := 1 τ n K k=1 1 N δ N k ,n A(N k )(23) is the partial average of A over all clusters of a given size n. Now, if the partial averages exhibit the critical divergence A n := A n ε=0 ∼ n b ,(24) then A ∼ ε −u as ε ↓ 0 with u = (2 − τ + b)/σ,(25) provided that u > 0. Zimm dynamics In contrast to the free-draining limit described by Rouse dynamics in the last subsection, one expects that with hydrodynamic interactions being present, cluster topology will have an influence on the diffusion constants. For simplicity, let us start with the Kirkwood diffusion constant. In order to extract a size dependence out of D, we look at the average D n over all clusters of a given size n and study its behaviour as a function of n. More specifically, we will perform this average precisely at the critical concentration c crit , where we expect an algebraic decrease as n → ∞ due to the absence of any other length scale at criticality. Indeed, using the Oseen tensor for the hydrodynamic interactions we deduce from (13), (4) and (5) that D n := D n c=ccrit = 1 ζn + κ ζn 2 n i,j=1 i =j R −1/2 i,j n c=ccrit n→∞ ∼ 1 ζ 1 n + λκ n 1/d (G) f ,(26) where λ is some dimensionless proportionality constant. The asymptotic behaviour of the average over the resistances in (26) is derived in [41]. The derivation has to distinguish between the two different cases for the crosslink ensemble. For Erdős-Rényi random graphs the asymptotics can be deduced from the exact probability distribution of R i,j in [72]. For three-dimensional bond percolation we use the scaling form of the probability distribution, which was established within two-loop order of a renormalisation-group treatment of an associated field theory [73,74]. Equation (26) involves the fractal Hausdorff dimension d (G) f := 2d s /(2 − d s )(27) of Gaussian phantom clusters, which also determines the scaling of their radius of gyration according to [50,52,55] R gyr,n : = 1 2n 2 n i,j=1 (R i − R j ) 2 n c=ccrit 1/2 n→∞ ∼ n 1/d (G) f .(28) The other fractal dimension in (27) is the spectral dimension d s of the incipient percolating cluster [75,76]. Their numerical values are listed in Table 1. We conclude from (26) that D n shows a crossover from Rouse behaviour D n ∼ n −1 for n < n(κ ) ∼ κ −1/(1−1/d (G) f ) to Zimm behaviour D n ∼ n −1/d (G) f ∼ 1/R gyr,n(29) for asymptotically large n > n(κ). Now we turn to the averaged diffusion constant D n := D n c=ccrit n→∞ ∼ n −bD(30) of clusters of size n at the gel point, which is also expected to obey a critical scaling for large cluster sizes n. From the Jensen-Peierls inequality D n ≤ D n , see (14), we then infer the inequality b D ≥ 1/d (G) f(31) for the critical exponents. Figure 1 shows numerical data for the cluster diffusion constant D n , plotted against n, for different values of the hydrodynamic interaction strength κ. The crosslink ensemble in Fig. 1(a) corresponds to Erdős-Rényi random graphs. In Fig. 1(b) crosslinks were chosen according to three-dimensional bond-percolation. In the numerical computations we have used H eq corresponding to the Rotne-Prager-Yamakawa tensor so that a positive definite mobility matrix is always guaranteed. Like the Kirkwood diffusion constant, D n also exhibits a crossover from Rouse to Zimm behaviour at a cluster size comparable to n(κ). Figure 2 shows the exponent b D of the power-law fit (30) to the data of b D = 1/d (G) f .(32) We now turn to the long-time behaviour of the incoherent intermediate scattering function (10). The asymptotics (11), (22) and Jensen's inequality yield the lower bound [41] S(q, t) ≥ ∞ n=1 nτ n e −q 2 tDn . In fact, there is numerical evidence that this inequality actually captures the correct long-time asymptotics of S(q, t) . Evaluating the right-hand side of (33) for large times t, this then leads to the scaling form [41] ‡ S(q, t) t→∞ ∼ ζ q 2 t y 1 ε = 0 , [t/t * q (ε)] x(y−1/2) exp{−const. [t/t * q (ε)] x } ε > 0(34) with the time scale t * q (ε) ε↓0 ∼ q −2 ε −z . The exponents are given by x = (1 + b D ) −1 , y = (τ − 2)/b D , z = b D /σ(35) and are expressed in terms of b D ≈ 0.25, see (32) and Table 1. The Rouse limit (20) of (34) corresponds to setting b D = 1 in the above expressions. The critical vanishing D eff ∼ ε a with a = (2 − τ + b D )/σ(36) of the effective diffusion constant follows from directly from (22) - (25) provided that a > 0. This condition is fulfilled for three-dimensional bond percolation where a ≈ 0.16, but violated for Erdős-Rényi random graphs. Finally, we like to point out that, regardless of the cluster statistics, the ensemble averaged diffusion constant D never vanishes at the critical point. This is simply because it has non-vanishing contributions from all clusters, which add up. Fig. 1(a). (b) Same for Fig. 1(b). Discussion We have studied the critical scaling D n ∼ n −bD of the averaged cluster diffusion constants over clusters of size n and used it to obtain the scaling behaviour of the intermediate incoherent scattering function S(q, t) near criticality. The associated critical exponents are summarised in Table 2. Within Rouse dynamics cluster diffusion constants are inversely proportional to the cluster size n, irrespective of the cluster topology, that is, b D = 1. Zimm dynamics leads to b D = 1/d (G) f , see (32), and topology does play a role: Indeed, it is well known [48] that within Zimm dynamics the diffusion constant of a linear chain of n monomers decreases as n −1/2 . Since b D ≈ 0.25 < 1/2, this means that, on average, a monomer in a branched cluster feels less friction-which is intuitively appealing, because monomers in the interior of a cluster should be dragged along. Second, (28), (30) and (32) imply for Zimm dynamics that D n ∼ 1/R gyr,n . Hence, this relation does not only hold for linear chains, for which it has been well known [48], but in an average sense for all percolation clusters. Concerning the scaling exponents of the incoherent intermediate scattering function, Table 2 shows that neither Rouse nor Zimm dynamics provides even a reasonably good description of the experimental findings, despite their strong scatter. There are several reasons for the discrepancies between the model predictions and experiments. (i) Our results pertain to θ-conditions, in so far as excluded-volume interactions have been neglected in the (iv) Preaveraging of the hydrodynamic interactions is an uncontrolled approximation, and it remains to be seen what a full treatment of hydrodynamic interactions predicts for the critical dynamics of gelling solutions. Stress relaxation Gelling liquids exhibit striking rheological properties which have been continuously studied over the years by experiments [79][80][81][82][83][84][85][86], theories [11,16,50,87,88,35,89,36,37,90,38,39] and simulations [28,29,31,91,[32][33][34]. For example, when subjected to the homogeneous shear flow (2), distinct relaxation patterns are observed, which are due to the participation of many different excitation modes of all sorts of clusters. More precisely, experiments suggest the scaling form [92, 44, 80-82, 86, 87] G(t) ∼ t −∆ g(t/t) with t(ε) ∼ ε −z(37) for the macroscopic (shear-) stress-relaxation function in the sol phase for asymptotically long times t and crosslink concentrations close to the critical point, i.e. for ε ≪ 1. The typical relaxation time t diverges with a critical exponent z > 0 for ε ↓ 0. The scaling function g is of order unity for small arguments so that one finds the algebraic decay G(t) ∼ t −∆ with a critical exponent 0 < ∆ ≤ 1 for t → ∞ at the critical point. For large arguments, g decreases faster than any inverse power. Sometimes a stretched exponential has been proposed for g in this asymptotic regime [82,87]. In this section we will investigate to what extent such critical properties can be predicted by the Rouse and the Zimm model. Thus we will explore the consequences of the dynamics (3), resp. (6), in the presence of the externally applied simple shear flow (2). In reaction to the flow, the system of crosslinked monomers builds up an intrinsic shear stress. Following Kirkwood, see e.g. Chap. 3 in [48] or Chap. 16.3 in [47], this shear stress is given by the force per unit area exerted by the monomers σ(t) = lim t0→−∞ − ρ 0 N N i=1 F i (t)R † i (t).(38) Here, R i (t) is the solution of the equation of motion (3) with some initial condition at time t 0 in the distant past (so that the noise average yields a thermalized state in which all transient effects stemming from the initial condition have died out). Moreover, ρ 0 stands for the monomer concentration and F i (t) := −∂V /∂R i (t) is the net spring force acting on monomer i at time t. The explicit computation [37,38] of the right-hand side of (38) yields σ(t) = G(0) 1 + t −∞ dt ′ G(t − t ′ )γ(t ′ )   2 t t ′ dsγ(s) 1 0 1 0 0 0 0 0   (39) for arbitrary strengths of the shear rateγ(t). Here, we have defined the stress-relaxation function G(t) := ρ 0 N Tr (1 − E 0 ) exp − 6t a 2 Γ(40) as a trace over the matrix exponential of Γ := (H eq ) 1/2 Γ (H eq ) 1/2 . Due to the occurrence of the spectral projector E 0 on the kernel of Γ, this trace is effectively restricted to the subspace of non-zero eigenvalues. For a time-independent shear rateγ, the shear stress (39) is also independent of time. The viscosity η is then related to shear stress via η := σ x,ẏ γρ 0 = 1 ρ 0 ∞ 0 dt G(t) = a 2 3 1 2N Tr 1 − E 0 Γ .(41) Apparently, the viscosity is determined by the trace of the Moore-Penrose inverse of Γ. The normal stress coefficients are given by Ψ (1) := σ x,x − σ y,ẏ γ 2 ρ 0 = 2 ρ 0 ∞ 0 dt tG(t) = a 2 3 2 1 2N Tr 1 − E 0 Γ 2(42) and Ψ (2) := σ y,y − σ z,ż γ 2 ρ 0 = 0 .(43) The vanishing of Ψ (2) is typical for Rouse/Zimm-type models and has been well known for the case of linear polymers [48]. Since Γ is block-diagonal with respect to the clusters, the observables G(t), η and Ψ (1) are all cluster-additive in the sense of (22). The scaling form (37) of the macroscopic stress-relaxation function G(t) implies that the macroscopic viscosity and first normal stress coefficient exhibit a critical divergence η ∼ ε −k and Ψ (1) ∼ ε −ℓ(44) at the sol-gel transition as ε ↓ 0 with critical exponents given by the scaling relations [44,39] k = z(1 − ∆) and ℓ = z(2 − ∆) = k + z .(45) Thus, it suffices to know any two of the four critical exponents ∆, z, k and ℓ. Rouse dynamics For Rouse dynamics we have Γ = Γ/ζ so that the computation of the stress-relaxation function, the viscosity or the first normal stress coefficient requires the knowledge of spectral properties of the connectivity matrix Γ. Concerning the macroscopic viscosity η , there are several ways of calculating the critical exponent k in (44). The different ways explore connections to problems in different branches of research. Given a cluster N k , the trace of the Moore-Penrose inverse of Γ(N k ) can be expressed in terms of the resistances (7) according to [36,37] η(N k ) = ζa 2 6N k Tr 1 − E 0 (N k ) Γ(N k ) = ζa 2 12N 2 k i,j∈N k R i,j .(46) We stress that this is an exact relation [66]. It has nothing to do with electrical analogues put forward in scaling arguments [69]. For the case of Erdős-Rényi random graphs there are only tree clusters for c < c crit = 1/2. In this special case the resistance R i,j reduces to the graph distance of i and j in N k , and the right-hand side of (46) is known as the Wiener index W (N k ) in graph theory. From a graph-theoretical point of view, the right equality in (46) follows also as an application of the matrix-tree theorem, see e.g. [93], Thm. 5.5. Moreover, the partial averages W are exactly known [72], and, using (22), one finds the exact result [36,37] η = ζa 2 24c ln 1 1 − 2c − 2c .(47) It can be interpreted as a critical divergence with exponent k = 0. Alternatively, (47) can also be obtained from a replica approach [37] instead of using graph theory. The replica approach is also capable of providing us with higher inverse moments N −1 Tr [(1 − E 0 )/Γ ν ] for not too large positive integers ν [90]. Using these results for ν = 2, a (somewhat lengthy) exact expression for Ψ (1) was derived in [39] for crosslink statistics from Erdős-Rényi random graphs. It exhibits the critical behaviour Ψ (1) ∼ ε −ℓ with ℓ = 3 .(48) Now we turn to the crosslink ensemble of three-dimensional bond percolation. In order to proceed from (46) in this case, one needs to know the average resistance R i,j n between two nodes in bond-percolation clusters of size n. Luckily, random electric resistance networks have been studied extensively, and the asymptotic behaviour R i,j n ∼ n bη with b η := (2/d s ) − 1(49) can be extracted [36,37] from highly developed renormalisation-group treatments of an associated field theory [73,74]. Thus, (46), (22) and (25) lead to the critical behaviour η ∼ ε −k with k = (1 − τ + 2/d s )/σ(50) as ε ↓ 0. Of course, this exact scaling behaviour reduces to the Erdős-Rényi result k = 0 from (47), when inserting the appropriate mean-field values for the exponents. None of the above approaches is able to yield any of the other critical exponents ∆ and z-or also ℓ in the case of three-dimensional percolation statistics. Here, a connection to random walks in random environments is helpful. For the time being, let us concentrate on the case of three-dimensional percolation statistics, where the maximum number of bonds emanating from any vertex is limited to m = 6 on the simple cubic lattice. Now, consider a random walker-coined "blind ant" by de Gennes [12]-that moves along a bond from one site to another in the same cluster at discrete time steps [10,76,94,95]. If the ant happens to visit site i at time s, which is connected with m i ≤ m bonds to other sites, then it will move with equal probability 1/m along any one of the m i bonds within the next time step and stay at site i with probability 1 − m i /m. By definition of the connectivity matrix Γ of the cluster, one has Γ ii = m i for its diagonal matrix elements, Γ ij = −1 if two different sites i = j are connected by a bond and zero otherwise. Hence, the associated master equation for the ant's sojourn probability p i (s) for site i at time s reads p i (s + 1) = (1 − Γ ii /m)p i (s) + j =i (−Γ ij /m)p j (s) ,(51) which is equivalent to p i (s + 1) − p i (s) = −m −1 j Γ ij p j (s) .(52) Here the summation extends over all sites in the cluster. For long times s ≫ 1, it is legitimate to replace the difference (quotient) on the left-hand side of (52) by a derivative. This yields the solution p i (s) = e −sΓ/m ii0 , which corresponds to the initial condition p i (0) = δ i,i0 . Next we consider P (n) (s) := p i0 (s) n \| ε=0 , the mean return probability to the starting point after time s, where the average is taken over all critical percolation clusters with n sites. Clearly, these definitions are independent of the starting point i 0 , because on average there is no distinguished site by assumption. Thus we can also write P (n) (s) = 1 n Tr e −sΓ/m n ε=0(53) for finite n. The return probability behaves as [76,95,94] P (n) (s) ∼ s −ds/2 F (s/s n ) + 1/n , where s n ∼ n 2/ds and the cut-off function F (x) is of order one for x 1 and decreases rapidly to zero for x → ∞. Basically, (54) says that for times s ≫ s n the walker has no memory of where he had started from. For times s s n the fractal-like nature of a cluster at c = c crit leads to an algebraic decrease of the return probability, which involves the spectral dimension d s . Now, assuming that G(t) obeys the scaling form (37), the information provided by (54) for c = c crit is sufficient to conclude [40] the exponent relations ∆ = d s 2 (τ − 1) and z = 2 d s σ .(55) When plugging (55) into (45), we recover (50) and get the new scaling relation ℓ = (1 − τ + 4/d s )/σ .(56) Since, the critical behaviour of Erdős-Rényi random graphs coincides with that of mean-field percolation, we get the missing exponents ∆ and z for that case by inserting the mean-field values into (55). Zimm dynamics The matrix Γ, which determines stress relaxation, is by far more complicated than Γ in the presence of hydrodynamic interactions. In particular, it reflects cluster topology only in a much more subtle way than Γ. In fact, it was that apparent encoding of topology in Γ that made the analytical methods of the last subsection work. In the absence of suitable analytical tools, numerical methods remain to investigate stress relaxation in the Zimm model. Figure 3. Numerical data to determine the scaling (57) for random clusters in the case of Erdős-Rényi random graphs (left column) and three-dimensional bond percolation (right column). In each case the averaged viscosity ηn (top) and normal stress coefficient Ψ We determined the scaling as n → ∞ of the partial averages η n := η n ε=0 ∼ n bη and Ψ (1) n := Ψ (1) n ε=0 ∼ n bΨ (57) at criticality by numerically diagonalising Γ and performing the disorder average over the crosslink ensemble [42]. The critical exponents k and ℓ then follow from (25). All numerical computations were done with the Rotne-Prager-Yamakawa tensor for the hydrodynamic interactions. The reader who is interested in more details of the numerical computations is referred to [42]. In Figs. 3(a) and (b) we plot η n and Ψ n as a function of n on a double-logarithmic scale for different values of the hydrodynamic interaction parameter κ. Crosslink statistics were chosen according to Erdős-Rényi random graphs. The exponents b η and b Ψ are obtained from power-law fits in the large n-range and are displayed in Fig. 3(c). The viscosity exponent decreases from b η = 0.28 for κ = 0.05 to b η = 0.11 for κ = 0.3. We recall from (46) and (49) that the Rouse exponent for κ = 0 is exactly given by b η = 1/2. The exponent b Ψ of the normal stress coefficient ranges from b Ψ = 1.2 for κ = 0.05 to b Ψ = 0.73 for κ = 0.25. The exact Rouse value b Ψ = (4/d s ) − 1 = 2 for κ = 0 follows from (48) and (25). The same is done for three-dimensional bond percolation in the right column of Fig. 3. Figures 3(d) and (e) contain η n and Ψ (50) and (56) in the last subsection together with (25). A careful analysis of the data in [42] reveals that the true Zimm exponents b η and b Ψ are universal in κ and that their seeming dependence on κ in Figs. 3(c) and (f) is most likely due to finite-size effects. More precisely, for small κ the data suffer from a crossover to their respective Rouse values so that they come out too large. For large κ, on the other hand, the asymptotics h(x) ∼ 1 − (πx) −1/2 as x → ∞ of the lower line in (5) leads to a slower growth of b η and b Ψ at intermediate n. Hence, the exponents come out too small for larger κ. The most reliable values for the universal Zimm exponents b η and b Ψ should be obtained from around κ ≈ 0.3. It it these values which are listed in Table 3 below. The critical behaviour of the averaged viscosity η ∼ ε −k and of the averaged first normal stress coefficient Ψ (1) ∼ ε −ℓ for a polydisperse gelling solution of crosslinked monomers then follows from (25). For the viscosity this implies a finite value at the gel point for both, Erdős-Rényi random graphs and three-dimensional bond percolation. In contrast, the first normal stress coefficient is found to diverge with an exponent that depends on the cluster statistics. Choosing the cluster statistics according to Erdős-Rényi random graphs, we find ℓ ≈ 0.54. The case of three-dimensional bond percolation leads to the higher value ℓ ≈ 1.3. These exponent values are less than a third in magnitude than the corresponding exact analytical predictions of the Rouse model from (56) with the corresponding cluster statistics. All exponent values are summarised in Table 3. Discussion A fairly complete scaling picture of the gelation transition has been obtained within Rouse dynamics. All critical exponents k, ℓ, ∆ and z of the stress-relaxation function in the sol phase and at criticality could be expressed in terms of two independent static percolation exponents σ and τ plus the spectral dimension d s of the incipient percolating cluster, see the scaling relations (50), (55) and (56). These scaling relations and the resulting numerical exponent values listed in Table 3 contradict the predictions k = 2ν − β and ∆ = dν/(dν + k) of earlier scaling arguments [80,81,84,16,87,104]. What is the reason for this discrepancy? The scaling arguments involve the fractal Hausdorff dimension d f := d − β/ν of rigid percolation clusters at c crit . Rouse clusters, however, are thermally stabilised, Gaussian phantom clusters with the fractal Hausdorff dimension d (G) f , see (27) [50,52,55]. The latter is different from d f in space dimensions below the upper critical dimension d u = 6. Indeed, if one replaces d f Table 3. Summary of critical exponents for stress relaxation (see Eqs. (37), (44) and (57) for their definitions). The numerical values for Rouse dynamics are based on the scaling relations (49), (50), (55) and (56). Those for Zimm dynamics are based on the data analysis of Fig. 3. The values are listed for cluster statistics according to three-dimensional bond percolation (3D) and Erdős-Rényi random graphs (ER), and are compared to some experimental findings. in these scaling arguments, as one should consistently do within a Rouse description, the results will coincide with the ones obtained here. Since the long-standing scaling relations k = 2ν − β and ∆ = dν/(dν + k) involve the Hausdorff fractal dimension d f of rigid percolation clusters, it is sometimes argued that they describe the behaviour of a more realistic model, which, in addition to the interactions of the Rouse model, accounts for excluded-volume effects, too, see e.g. [104]. As far as we know, this claim has not been verified by analytical arguments within a microscopic model. One may even have doubts whether this claim is generally true: Extensive molecular-dynamics simulations [29] of a system of crosslinked soft spheres in three dimensions, with cluster statistics from percolation and an additional strongly repulsive interaction at short distances, yield the values k ≈ 0.7 and ∆ ≈ 0.75, which are remarkably close to the predictions of the Rouse model for randomly crosslinked monomers, see Table 3. On the other hand, simulations of the bond-fluctuation model in [28] imply k ≈ 1.3 and are thus in favour of the claim. However, the viscosity is not measured directly in these latter simulations. Rather it is derived from the scaling of diffusion constants and an additional scaling assumption that may be questioned [29]. Hence, it is an open problem to what extent the critical Rouse exponents of Table 3 are modified by excluded-volume interactions. In the context of dynamical critical phenomena, one usually expects dynamical scaling to hold. Thereby one can infer critical properties of the gel phase from those of the sol phase. In particular, the critical behaviour of the shear modulus G 0 ∼ \|ε\| µ follows from the scaling form (37) of the stress-relaxation function. The result µ = ∆z = (τ − 1)/σ involves only the two exponents σ and τ of the cluster-size distribution. Using well-known scaling relations of percolation theory, this can be rewritten as µ = dν in terms of the correlationlength exponent ν and the spatial dimension d. It is in agreement with the simple scaling argument based on dimensional analysis of the free-energy density. In a recent letter [105], the scaling of entropic shear rigidity was analysed for both phantom chains and those with excluded-volume interactions. In both cases the gel was prepared by crosslinking a melt of chains with excluded-volume interactions. Our choice of percolation statistics combined with Rouse dynamics should be comparable to phantom chains prepared in an ensemble with excluded-volume interactions. However, the results of [105] for µ disagree with the above dynamic-scaling argument. The reasons for the discrepancy are not understood. Let us return to the sol phase and discuss the Zimm results, which are based on the exact numerical determination of the scaling exponents b η and b Ψ for the fixed-size averages (57) of the viscosity and of the first normal stress coefficient. The resulting finiteness of the macroscopic viscosity η at the transition is clearly the most serious drawback of the Zimm model for randomly crosslinked monomers. Also, this failure comes unexpected, because a well-known scaling argument [14,106,50] predicts a logarithmic divergence. This scaling argument uses the (correct) scaling D n ∼ 1/R gyr,n of diffusion constants together with the Stokes-Einstein relation and yields b η = d/d f − 1. Consequently, one gets from (25) the scaling relation k = (1 − τ + d/d f )/σ. Inserting hyperscaling and the fractal dimension of rigid percolation clusters, one would get k = 0 from that, which was interpreted as a logarithmic divergence. But as we remarked already earlier on in this subsection, the correct fractal dimension d f for Gaussian phantom clusters is d (G) f . For both cluster statistics this would give an unphysical negative value around −1/4 for b η which can be definitely ruled out by our data. § Thus, we conclude that the scaling approach of [14,106,50] does not apply to the Zimm model for randomly crosslinked monomers. Another scaling approach to this model by [17] is also falsified by our data. On the other hand, Brownian-dynamics simulations of hyperbranched polymers were performed in [91]. They also account for fluctuating hydrodynamic interactions corresponding to κ = 0.35, as well as for excludedvolume interactions and lead to b η = 0.13. This result is remarkably close to our finding b η ≈ 0.11 for the highest coupling strength κ = 0.3 that we have considered, whereas experimental findings (see below) are consistently above our value. Next, we comment on how the Rouse and Zimm predictions for stress relaxation compare to experimental reality. Table 3 shows an enormous scatter of the experimental data. Thus, a serious check of theoretical predictions is currently severely hampered. The origin of this wide spread of the data is unclear so that even the question arose, whether the dynamical critical behaviour at the gelation transition was indeed universal [107]. Possible explanations for nonuniversal behaviour include the splitting of a static universality class into two dynamical ones [92,17] and, for the case of crosslinking long polymer-chain molecules (vulcanisation), a decrease of the width of the critical region with increasing chain length [108]. The latter may explain the observation of a crossover behaviour to mean-field properties in certain gelation experiments, if measurements were not performed well inside the true critical region. As far as we know, no measurements of the critical behaviour of the first normal stress coefficient have been reported. The Rouse value k ≈ 0.71 for the viscosity and threedimensional percolation statistics agrees well with the experiments of [79,107,80,83] (only [83] was included in Table 3 to demonstrate the broad scatter of the viscosity data). On the other hand, it is not compatible with the possibly oversimplifying albeit attractive proposal [92,17] to interpret the wide variation of the viscosity exponent k as a signature of a splitting of the static universality class of gelation into different dynamic ones. Indeed, Rouse and Zimm dynamics are considered [109,48] to be at the extreme ends of the strength of the hydrodynamic interaction. Since the Zimm model does not even predict a divergence at the transition, the actual value of k should then lie below the Rouse value according to that proposal. Hence, the broad scatter of the experimental data calls for additional relevant interactions beyond those accounted for in the Zimm or Rouse model. This may be due to the preaveraging approximation. In particular, it throws away hydrodynamic interactions among different clusters. But we do not expect this to be the sole relevant simplification of the Zimm model, because linear polymers show a decrease in the viscosity when abandoning the preaveraging approximation [110], and effects of preaveraging for branched molecules are even more § Unfortunately, the value of bη resulting from this scaling argument in the case of Erdős-Rényi random graphs was incorrectly ascribed to d = 6 dimensions in the second last paragraph of [42], leading to the wrong statement bη = 1/2 there. pronounced than those for linear ones [111]. Rather it seems that there are no satisfactory explanations without considering excluded-volume interactions. Indeed, simulations [28] of the bond-fluctuation model deliver higher values k ≈ 1.3 in accordance with the scaling relation k = 2ν−β, which arises from heuristically merging Rouse-type and excluded-volume properties, see above. On the other hand, entanglement effects are neglected, too. These topological interactions are argued to play a vital role in stress relaxation. However, temporary entanglements are expected to play only a minor role [84] for the dynamics close to the gelation transition. This is because the time scale of a temporary entanglement is determined by the smaller clusters, whereas near-critical dynamics is determined by the largest clusters, which contribute the longest time scales. Yet, there remain permanent entanglements due to interlocking loops. They are clearly far beyond the scope of the present and many other theoretical approaches. Closing remarks The list of shortcomings of the Rouse and the Zimm model for crosslinked monomers is long, and it was discussed in Sections 3.3 and 4. 3. Yet, one should not underestimate the importance of these models for our understanding of gelling liquids. First, the success of Theoretical Physics and, in particular, Statistical Physics has always relied on capturing the essence of observable phenomena in simple mathematical models. Models that isolate certain physical mechanisms and, at the same time, sacrifice many details of the observed reality. It is safe to say that, at least for linear polymers, the Rouse and the Zimm model have proven to be among this class [47,48]. Second, simple exactly solvable models always represent cornerstones against which more elaborate theories, approximation methods and numerical simulations can be tested. Moreover, in the absence of an ultimate theoretical picture, the predictions of such minimal models also serve as a standard reference for experimental data. Indeed, this has been common practice in experimental investigations on gelling liquids over the years, see e.g. the review articles [112,44]. All the more it is important to have reliable and mathematically firmly based predictions of these model. Fig. 1 1in the large n-regime for the different values of κ. The horizontal dashed lines in Figs. 2(a) and (b) correspond to the exponent value 1/d (G) f of the respective Kirkwood diffusion constant. The bigger exponent values that occur for small values of κ still show residual Rouse behaviour for the largest system sizes we treated. For bigger values of κ the crossover can hardly be felt any more in the largest systems, and the extracted exponent value b D corresponds to Zimm dynamics. This value is very close to the scaling exponent in(29)for the Kirkwood diffusion constant, and, in fact, we conjecture that Figure 1 .Figure 2 . 12(a) Dn at the gel point for mean field percolation and different hydrodynamic interaction strengths. (b) Same for three-dimensional bond percolation. (a) Critical exponents b D , corresponding to a power-law fit Dn ∼ n −b D in ) are plotted for different strengths of the hydrodynamic interaction parameter κ as a function of the cluster size n on a double logarithmic scale. Power-law fits to the data yield the exponents bη and b Ψ as a function of κ (bottom). respectively, as a function of n on a doublelogarithmic scale for different values of κ. The exponents b η and b Ψ , extracted by fitting the curves in Figs. 3(d) and (e) to a power law for large n, are shown in Fig. 3(f). The numerical values for b η are nearly identical to those obtained for Erdős-Rényi random graphs. Again, one observes a decrease from b η = 0.21 for κ = 0.05 to b η = 0.11 for κ = 0.3. The exponent b Ψ of the normal stress coefficient ranges from b Ψ = 1.1 for κ = 0.05 to b Ψ = 0.78 for κ = 0.25. The corresponding exact Rouse values b η = (2/d s ) − 1 ≈ 1/2 and b Ψ = (4/d s ) − 1 ≈ 2 for κ = 0 follow from Table 1 . 1Numerical values for the critical exponents of the cluster-size distribution (9) and the two fractal dimensions of Gaussian phantom clusters in(27). The values are listed for cluster statistics according to three-dimensional bond percolation (3D) and Erdős-Rényi random graphs (ER).τ σ ds d (G) f 3D 2.18 0.45 1.33 3.97 ER 5/2 1/2 4/3 4 Table 2 . 2Summary of critical exponents for cluster diffusion constants and the incoherent intermediate scattering function (see Eqs.(30),(20) and(34) for their definitions). The numerical values for Rouse and Zimm dynamics-listed for cluster statistics from threedimensional bond percolation (3D) and Erdős-Rényi random graphs (ER)-are compared to experimental findings. cluster are screened by smaller clusters in the reaction bath so that the Rouse rather than the Zimm model should apply. Our analysis supports this conclusion in so far as the exponents of the Rouse model are closer to the experimental values. So the more striking failure of the Zimm model can be traced back to a too slow decay of D n with n.Zimm Rouse Exponent 3D ER 3D ER [68] [77] [78] b D 0.25 1/4 1 1 x 0.80 4/5 1/2 1/2 0.66 0.3 -0.8 0.64 y 0.71 2 0.18 1/2 0.27 0.2 -0.3 0.34 z 0.56 1/2 2.22 2 2.5 a 0.16 ( * ) 1.82 1 1.9 0.5 -1 1.9 ( * ) no divergence models. Excluded-volume interactions could cause a swelling of the clusters, which results in a different fractal Hausdorff dimension. (ii) We chose cluster statistics according to three-dimensional bond percolation. 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[ "The two-body random spin ensemble and a new type of quantum phase transition The two-body random spin ensemble and a new type of quantum phase transition 2", "The two-body random spin ensemble and a new type of quantum phase transition The two-body random spin ensemble and a new type of quantum phase transition 2" ]	[ "Iztok Pižorn \nDepartment of Physics\nFMF\nUniversity of Ljubljana\nJadranska 19SI-1000LjubljanaSlovenia\n", "Tomaž Prosen \nDepartment of Physics\nFMF\nUniversity of Ljubljana\nJadranska 19SI-1000LjubljanaSlovenia\n", "Stefan Mossmann \nInstituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\nC.P. 62132Cuernavaca, MorelosMexico\n", "Thomas H Seligman \nInstituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\nC.P. 62132Cuernavaca, MorelosMexico\n\nCentro Internacional de Ciencias\nApartado postal 6-101C.P.62132Cuernavaca, MorelosMexico\n" ]	[ "Department of Physics\nFMF\nUniversity of Ljubljana\nJadranska 19SI-1000LjubljanaSlovenia", "Department of Physics\nFMF\nUniversity of Ljubljana\nJadranska 19SI-1000LjubljanaSlovenia", "Instituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\nC.P. 62132Cuernavaca, MorelosMexico", "Instituto de Ciencias Físicas\nUniversidad Nacional Autónoma de México\nC.P. 62132Cuernavaca, MorelosMexico", "Centro Internacional de Ciencias\nApartado postal 6-101C.P.62132Cuernavaca, MorelosMexico" ]	[]	We study the properties of a two-body random matrix ensemble for distinguishable spins. We require the ensemble to be invariant under the group of local transformations and analyze a parametrization in terms of the group parameters and the remaining parameters associated with the "entangling" part of the interaction. We then specialize to a spin chain with nearest neighbour interactions and numerically find a new type of quantum phase transition related to the strength of a random external field i.e. the time reversal breaking one body interaction term.	10.1088/1367-2630/10/2/023020	[ "https://arxiv.org/pdf/0711.1218v1.pdf" ]	17,563,567	0711.1218	6759445491d09a3829a2cf3901e1029f88168368	The two-body random spin ensemble and a new type of quantum phase transition The two-body random spin ensemble and a new type of quantum phase transition 2 8 Nov 2007 Iztok Pižorn Department of Physics FMF University of Ljubljana Jadranska 19SI-1000LjubljanaSlovenia Tomaž Prosen Department of Physics FMF University of Ljubljana Jadranska 19SI-1000LjubljanaSlovenia Stefan Mossmann Instituto de Ciencias Físicas Universidad Nacional Autónoma de México C.P. 62132Cuernavaca, MorelosMexico Thomas H Seligman Instituto de Ciencias Físicas Universidad Nacional Autónoma de México C.P. 62132Cuernavaca, MorelosMexico Centro Internacional de Ciencias Apartado postal 6-101C.P.62132Cuernavaca, MorelosMexico The two-body random spin ensemble and a new type of quantum phase transition The two-body random spin ensemble and a new type of quantum phase transition 2 8 Nov 2007numbers: 0367-a0570Fh7510Pq We study the properties of a two-body random matrix ensemble for distinguishable spins. We require the ensemble to be invariant under the group of local transformations and analyze a parametrization in terms of the group parameters and the remaining parameters associated with the "entangling" part of the interaction. We then specialize to a spin chain with nearest neighbour interactions and numerically find a new type of quantum phase transition related to the strength of a random external field i.e. the time reversal breaking one body interaction term. Introduction Eugene Wigner introduced random matrix models about fifty years ago into nuclear physics [1]. The scope of applications has increased over the years [2,3] including fields such as molecular and atomic physics, mesoscopics and field theory. More recently random matrix theory has started to be used in quantum information theory [4,5,6,7,8]. For an introduction to such applications see [9]. There the concept of individual qubits and their interactions becomes important. This implies that we enter the field of two-body random ensembles (TBRE) [10,11], i.e. ensembles of Hamiltonians of n-body systems interacting by two-body forces. While such ensembles have received considerable attention, it was first focussed on fermions and later also included bosons. Yet in quantum information theory the qubits are taken to be distinguishable, and indeed the same holds for spintronics. Interest in both fields has sharply increased recently [12,13]. It is thus very pertinent to formulate and investigate TBRE's for distinguishable qubits. As random matrix ensembles are mainly determined by their symmetry properties this ensemble will be very different from other TBRE's. In particular, as the particles are distinguishable, their interaction can vary from particle pair to particle pair and can indeed be randomly distributed, thus introducing an entirely new aspect. This has the consequence that the topology according to which spins or qubits are distributed or interact will be important, Thus chains, trees and crystals of particles with nearest, second nearest and up to kth order interaction can be represented. As mentioned above, random matrix ensembles are usually basically defined by the invariance group of their measure and, if that is not enough, some minimal information conditions [14,9] or independence condition [15]. Note that we deal with a symmetry of the ensemble, rather than with a symmetry of individual systems. The two concepts are to some degree complementary, and the former has also been called structural invariance. We propose an adequate definition for such ensembles in a very general framework in terms of independent Gaussian distributed variables. We then give an alternate representation in terms of the invariance group and variables that determine the orbits of the Hamiltonian on the ensemble under the action of the group. In order to show the relevance of the new ensemble we address the simplest possible topology, namely the chain with nearest neighbour interactions. For this system we focus on the ensemble averaged structure of the ground state and demonstrate the existence of an unusual quantum phase transition [16], which is triggered by breaking of time-reversal invariance (TRI). Entanglement, a key resource of quantum many-body systems in terms of quantum information, is to large extent related to quantum correlations, localization properties and quantum chaos. Entanglement has also been used as a property, alternative to long-range order in spatial correlation functions, to describe systems undergoing a quantum phase transition [17]. In one-dimensional systems such as quantum spin chains, it was shown [18] that the entanglement entropy of the ground state typically saturates or diverges logarithmically with size when approaching the thermodynamic limit. Furthermore, it has been shown that logarithmic divergence implies quantum criticality. Interesting results emerge when a spatially homogeneous spin model is replaced by its disordered counterpart, where the spin interactions are taken at random. In this case there is often no physical justification why random interactions should still obey specific restricted forms such as Ising or Heisenberg interactions. In this context, we argue, it is more natural to use two-spin random ensembles (TSRE) for distinguishable particles, specifically choosing quantum spins 1/2, though these ensembles can readily be generalized to arbitrary spin. By construction these ensembles, as given in section 2, are invariant with respect to arbitrary local rotations, which we may view as gauge transformations. Another physical motivation for the definition of such ensembles is the coupling among arbitrary and perhaps mutually independent two level quantum systems which may come from completely different physical contexts such as e.g. twolevel atoms, Josephson junctions and photons. In section 3 we concentrate on one-dimensional systems or spin chains and present results of numerical calculations, mainly based on density matrix renormalization group (DMRG) [19], in which we investigate entanglement and correlation properties of the ground state, averaged over an ensemble, and the average spectral gap to the first excited state as well as its fluctuations. If we include the interaction with an external random magnetic field, and hence TRI is broken, we find fast decay of correlations, saturation of entanglement entropy, and power law decay g ≈ N −0.4 of the spectral gap g with the system size N while its distribution displays Wigner-type level repulsion. When the strength of external field goes to zero, and time-reversal invariance is restored, we find long range order, logarithmically divergent entanglement entropy, and exponential decay of the spectral gap, while the level repulsion disappears. We argue that this quantum phase transition is non-conventional from the point of view of established models, since in what we shall call non-critical case we still find slow power law closing of the spectral gap. The embedded ensemble of spin Hamiltonians with random two-body interactions In this section we define the two-spin random ensembles of Hamiltonians for systems with N distinguishable spins or qubits with at most two body interactions and describe its basic (invariance) properties. If we do not allow all spins to interact, we have to define which ones do. The simplest case will be a chain with nearest neighbour interactions, but in general we need a graph, whose vertices correspond to spins and whose edges correspond to two-body interactions. We proceed to formalize this. Let G = (V, E) be an undirected graph with a finite set of N vertices V and a set of M edges E ⊂ V × V. In addition, let λ : V → R + and µ : E → R + be two nonnegative functions defined on the sets of vertices and edges, respectively. To such a graph we assign a 2 N dimensional Hilbert space of N spins or qubits, placed at its vertices H G = ⊗ j∈V C 2 ≡ C 2 N , and a set of N Pauli operators σ α j : H G → H G , α ∈ {1, 2, 3}, j ∈ V satisfying SO(3) commutation relations [σ α j , σ β k ] = iε αβγ σ γ j δ j,k . We also use the notation σ j = (σ 1 j , σ 2 j , σ 3 j ). Let A (j,k) ∈ R 3×3 , (j, k) ∈ E, be a set of M random real 3×3 matrices, and b (j) ∈ R 3 , j ∈ V, be a set of N random 3 dimensional real vectors. The TSRE then consists of the random Hamiltonians H = (j,k)∈E µ(j, k) σ j · A (j,k) σ k + j∈V λ(j) b (j) · σ j .(1) The above defined functions on the edges and vertices of the graph are used to determine the average strength of the corresponding terms in the Hamiltonian. The distribution of random two-body interaction matrices (for short also bond matrices) A (j,k) and the random external field vectors b (j) shall be uniquely determined by requiring the following two conditions: maximum local invariance and maximum independence expressed formally as: (i) An ensemble of Hamiltonians (1) should be invariant with respect to an arbitrary local SO(3) transformation, namely σ ′ j = O j σ j ,(2) where O j ∈ SO(3) j , j ∈ V, meaning that the choice of local coordinate system is arbitrary for each spin/qubit. Obviously, (2) preserves the canonical commutation relations for the Pauli operators. Then it follows immediately that the joint probability distributions of {A (j,k) , b (j) } should be invariant with respect to transformations A (j,k) ′ = OA (j,k) O ′ , b (j) ′ = O b (j) ,(3) where O, O ′ are arbitrary independent SO(3) rotations for each j, k. (ii) The matrix elements of the tensors A (j,k) α,β and of the vectors b (j) α should be independent random variables. Following arguments similar to those presented in [15] it is straightforward to show that, in order to satisfy conditions (i) and (ii) above for pre-determined but general strengths of bonds µ(j, k) and external fields λ(j), A (j,k) α,β and b (j) α should be Gaussian independent random variables of zero mean and equal variance, which are uniquely specified in terms of the correlators A (j,k) α,β A (j ′ ,k ′ ) α ′ ,β ′ = δ jj ′ δ kk ′ δ αα ′ δ ββ ′ , b (j) α b (j ′ ) α ′ = δ jj ′ δ αα ′ , A (j,k) α,β b (j ′ ) α ′ = 0, where • denotes an ensemble average. We abbreviate the ensemble defined in this way by TSRE(G, µ, λ) noting that it depends on the graph and the strength functions µ and λ. Let us now describe some other elementary properties of TSRE. We have seen that each member H of TSRE(G, µ, λ) can be parametrized (1) by 9M + 3N independent random parameters. However, we know for the classical ensembles, that a parametrization in terms of the structural invariance group, i.e. the invariance group of the ensemble and the remaining parameters is very useful. Similarly in the present case for an arbitrary set of local SO(3) rotations O j , the transformation of the parameters A (j,k) ′ = O T j A (j,k) O k , b (j) ′ = O T j b (j) ,(4) preserves the spectrum and all entanglement properties of H. In fact, the transformation (4) can be considered as a gauge transformation since, composed with the local canonical transformation (2), it preserves the Hamiltonian H (1) exactly. Two Hamiltonians, specified by {A (j,k) , b (j) } and {A (j,k) ′ , b (j) ′ },O j+1 = R j O j , where R j := V (j,j+1) [U (j,j+1) ] T(5) and where A (j,j+1) =: U (j,j+1) D (j,j+1) [V (j,j+1) ] T ,(6) is a standard canonical singular value decomposition (SVD) of the original bond matrix (1,2) . This symmetrization is unique provided that singular values of all SVD's (6) are non-degenerate which is the case for a generic member H. Thus the number of parameters specifying the bond matrices is 3 + 6(M − 1) = 6M − 3 and in addition to 3N external field parameters this gives K = 6M − 3 + 3N independent parameters. We recover the original set of parameters if we add the 3N parameters of the group of local rotations. Second, we consider the case of a ring graph with N vertices and M = N bonds, which is obtained from the previous case by specifying the periodic boundary condition N + 1 ≡ N. We see that a general H as given in (1) can now be symmetrized only if the additional topological condition R := R N · · · R 2 R 1 = ½ is satisfied. Now all but one bond matrix can be symmetrized, for example the last one may in addition be multiplied by a topological rotation A (N,1) ′ = A (N,1) ′ symmetric R leading to three additional parameters. Therefore in the case of a ring graph we have K = 6M + 3N independent parameters; again adding the 3N = 3M parameters of the local rotations we obtain the full set of parameters. A (j,j+1) , with U (j,j+1) , V (j,j+1) ∈ SO(3) and D (j,j+1) diagonal matrices of singular values. Since the initial transformation O 1 is still free, we can choose it such O 1 := U (1,2) that the first bond matrix is even diagonalized, A (1,2) ′ = D We can now consider the case of a general (connected) graph. From the previous examples it is evident that the only crucial additional parameter is the number L of primitive cycles, i.e. such cycles which cannot be decomposed into other primitive cycles. It is clear that each primitive cycle adds 3 additional topological parameters (or one SO(3) topological R matrix) to the 6M − 3 + 3N parameters which we would have for the case of a tree graph. Therefore we have K = 6M + 3N + 3L − 3.(7) Counting the cycle-contributions and taking the primitivity criterion into account again we obtain the the total number of parameters by adding those of the local rotations. The above considerations hold for the case of general bond and vertex strength functions µ and λ. If, however, these functions are degenerate, or even constant, i.e. the average interaction strength and field strength do not depend on edges/vertices of the graph, then the structural invariance group of the TSRE may be even larger. In the latter case this group is obtained as a semi-direct product of a discrete symmetry group of the graph G and the gauge group of local rotations, the latter being the normal subgroup. We hope that the considerable invariance properties of the TSRE will prove useful in a future analytical treatment of its properties. Properties of ground states of the TSRE on a 1D chain In this section we shall only consider the simplest case of a TSRE on a 1D chain of N vertices. In addition, we consider the most symmetric case of constant strength functions, say µ(j, j + 1) ≡ 1 and λ(j) ≡ λ = const. Such an ensemble of random spin chain Hamiltonians H = N −1 j=1 σ j · A (j,j+1) σ j+1 + λ N j=1 b (j) · σ j(8) shall be designated as TSRE (N, λ) where we explicitly assume open boundary conditions; we shall, however, also consider a ring graph with periodic boundary conditions in which case we shall stress this separately. In particular we shall be interested in the zero temperature (ground state) properties of TSRE(N, λ). We note that due to the large co-dimension of bond strength space the standard perturbative renormalization group of decimating the strongest bonds [20] would not work and we have at this point to rely on a brute numerical investigation. Still, it turns out that most zero temperature properties of TSRE(N, λ) can be efficiently simulated using White's density matrix renormalization group (DMRG) finitesize algorithm [19] by which spin chains of sizes up to N = 80 could at present be achieved. One should not forget that all numerical estimates of ensemble average or expectation value of some physical quantity A, which will be designated as A , require averaging over many, say N r , realizations from TSRE(N, λ), such that the statistical error estimated as σ A ∼ ( A 2 − A 2 )/N r is sufficiently small. Due to the lack of translational invariance the implementation of the DMRG is non-trivial and was only done for a chain with open boundaries. We note that for λ = 0, any H as defined in (8), or even more generally in (1), commutes with the following anti-unitary time-reversal operation T : σ j → − σ j , H\| λ=0T =T H\| λ=0 ,(9) so, for odd N, all eigenvalues of H have to be doubly degenerate (Kramer's degeneracy [15]). However, as this represents more a technical than conceptual problem for the ground state (or ground plane) properties of TSRE(N, λ) we shall in the following restrict ourselves to the case of even N. Distribution of the spectral gap Let \|0 , \|1 , represent the ground state, and the first excited state, of H, with eigenenergies E 0 , and E 1 , respectively. It is well known that the crucial quantity which determines the rate of relaxation of zero-temperature quantum dynamics is the spectral gap g = E 1 − E 0 . To clarify the difference with respect to TRI, we plot the normalized gap distribution dP/dg whereg = g/ g for both cases, together with the theoretical level spacing distributions for the Gaussian unitary ensemble (GUE) of random matrices [21] and for an uncorrelated Poissonian spectrum (Fig. 1). For the non-TRI case we choose λ = 1 and observe, to our accuracy, good agreement with the GUE case for two chain sizes N = 10 and N = 16. Our results suggest that the GUE-like gap distribution, exhibiting level repulsion, also holds in the thermodynamic limit N → ∞; numerical results for odd N give the same results. In the case of λ = 0 the level repulsion between \|0 and \|1 gradually vanishes as we approach the thermodynamic limit, although no conclusive statement can be made about the limiting distribution. For odd N, the ground state is degenerate for λ = 0 and the present analysis does not apply. Size scaling of the spectral gap Being interested in the thermodynamic limit, it is an important issue to understand how g(N) scales with N. The theory of quantum criticality [16] states that g remains finite in the thermodynamic limit for non-critical systems, and rapidly converges to zero, as N → ∞ for critical systems. In Fig.2 we plot the ensemble averaged spectral gap g versus N for different values of the field strength λ. We find a clear indication that in the non-TRI case the spectral gap exhibits universal asymptotic power law scaling whereas in the TRI case, λ = 0, the asymptotic decay of the gap is faster than a power law, perhaps exponential g ∼ exp(−ξN), with ξ ≈ 0.07 ± 0.02. According to the standard theory [16] both cases, λ = 0 and λ = 0, should be classified as quantum critical, however as we shall see later, the case of slow power-law decaying average gap (10) has many-features of non-critical systems, such as finite correlation length and finite (saturated) entanglement entropy. Therefore we shall, at least for the purposes of the present paper, name the case λ = 0 as random non-critical (RNC) and the case λ = 0 as random critical (RC). We note that the results for odd and even N are in agreement in the RNC case. Also results for the case with periodic boundary conditions up to N = 20 show no significant difference from the results in Fig. 2 for any λ. which measures the entanglement in the ground state between two equal halves of the chain. It has been suggested in non-random systems [18] that for critical cases S ∝ log 2 N whereas in non-critical cases S saturates in the thermodynamic limit. Indeed, as shown in Fig.3, we find for the TSRE(N, λ) that S saturates to a constant finite S ∞ (λ) = lim N →∞ S(N, λ) for the RNC case λ = 0, while in the RC case it grows logarithmically g ∼ N −η , with η ≈ 0.39 ± 0.01,(10)S(N, λ = 0) ≈ c log 2 N + c ′ , with c ≈ 0.17 ± 0.02(12) We also note an interesting even-odd-N/2 effect which slowly diminishes as we approach the thermodynamic limit. As pointed out in Ref. [22] such an effect is induced by open boundary conditions. For periodic boundary conditions the entanglement entropy for RNC case is twice as large as in the TSRE (N, λ) with open boundaries. This further confirms the conjecture that only short-range correlations around the boundary between the two halves contribute to the entanglement. We note that our result is essentially different from results for other models, which can be obtained by perturbative real space renormalization group [20], for example for the disordered critical Heisenberg chain [23], where c = (ln 2)/3 and is in general model dependent [24]. The fact that the entanglement is reduced in the RNC case with local disorder can be explained by chaotic behavior [25] signalized by the level repulsion in the gap distribution. A similar effect can be observed in localization properties where hopping of excitations induced by inter-particle interactions is diminished by introducing local disorder [26,27]. This effect of increased localization is useful for successful quantum computing and has important consequences for transport properties such as conductivity [28]. Correlation functions The most direct probe of criticality is perhaps to investigate of long-range order and (space) correlation functions. In order to do this we compute the ensemble averaged fluctuation of the spin-spin correlation function between two vertices C(j, k) = \| 0\|σ α j σ β k \|0 − 0\|σ α j \|0 0\|σ β k \|0 \| 2 .(13) Note that we have to consider average fluctuations of the spin-spin correlation function instead of the correlation function itself, since the latter have to vanish due to the local gauge invariance properties of the TSRE. Because of the local invariance of the TSRE it is enough to consider a single type of correlation function, as the RHS of (13) does not depend on indices α, β if j = k; in fact in numerical computations we average over α, β in order to improve statistics. We consider the Hamiltonian (8) with periodic boundary conditions. This allows us to average the fluctuation of the correlation over the chain and hence C(r) = 1 N i C(i, i + r). We expect that the results would be qualitatively the same for the model with open boundaries and sufficiently large N, but we obtain better statistics in this way. Fig. 4 shows the averaged correlation function fluctuation C(r) for a few choices of the control parameter λ and of the chain lengths N = 16, 20, 24. The results for chains of different lengths coincide for small distances r whereas for larger r finite-size effects are noticeable. For sufficiently large chains it can be conjectured that the fluctuations (or effective correlations) asymptotically decay in the RNC case as C(r) ≍ C 0 2 −r/ξ with a finite correlation length ξ whereas the decay for the RC case is slower than exponential, perhaps a power-law, which indicates long-range order ξ = ∞. Correlation length and the entanglement entropy saturation value In Fig. 4 we observe that the degree of localization depends on the control parameter λ and the correlation function C(r) decays on larger scales as we approach the critical point λ = 0 which results in a larger correlation length ξ. Eventually, the correlation length becomes infinite at the critical point λ = 0. In Fig. 5 ξ Figure 5. The correlation length ξ as a function of the control parameter λ. Open and full symbols designate correlation length obtained from the best of C(r) with c exp(−r/ξ) on sets r ∈ {4, . . . , 8} and r ∈ {5, . . . , 7}, respectively, all for N = 24. The inset demonstrates ∝ − log 2 λ scaling. Please note that last point at λ = 0.1 is rather inaccurate due to insufficiently large system size N , so it is not used in a linear fit (full line) in the inset, for which other points with λ ≤ 1 have been used. of the correlation length ξ on the control parameter λ as obtained by exponential fit of C(r) for a finite size N = 24. Unlike conventional phase transition as in e.g. [17], the correlation length seems to diverge logarithmically as ξ(λ) ∼ −ξ 0 log 2 λ + const with ξ 0 = 0.26 (indicated in the inset of fig.5), even though algebraic scaling cannot be entirely excluded with the numerical data that are available at present. Large correlation length has strong effect on entanglement. Long range correlations demand longer chains for the entanglement entropy to saturate whereas the saturation value itself also grows when the critical point is approached. In Fig. 6 we plot the entanglement entropy saturation value for various values of parameter λ and observe similar behavior as for the correlation length. However, unlike the correlation length the quantity which diverges logarithmically when λ → 0 is not the entanglement entropy S but its exponential. In fact, numerical data for small λ show good agreement with 2 kS ∝ − log 2 λ + const, where k = 4.0 (indicated in the inset of fig.6). Note that 2 S is to Figure 6. Saturated entanglement entropy S ∞ as a function of the control parameter λ. The inset demonstrates ∝ log 2 log 2 (λ ⋆ /λ) scaling, where λ * = 4, which is indicated by a straight line obtained from a best fit to points with λ < 1. a good approximation proportional to the effective rank, or the Schmidt number of the ground state χ ǫ [18,29], which denotes the number of eigenvalues of the reduced density matrix needed to describe the state of the system up to an error ǫ. In fact, the effective rank χ ǫ , rather than the entanglement entropy, is the decisive indicator of simulability by the DMRG method [30] and, we believe, also a relevant quantity in the description of a quantum phase transition. Conclusions In the present paper we have defined a two-body random matrix ensemble of independent spin Hamiltonians which are invariant under local SO(3) transformations and described them in a framework of undirected graphs. As the simplest example, we have studied a chain with nearest-neighbour interactions in a random external field and observed a non-conventional phase transition when the external field is switched off. The system is always critical in conventional terminology as it has a vanishing gap in the thermodynamic limit in all cases studied. Yet we have shown that, in the presence of a random external field breaking time-reversal invariance, the locally disordered system has many properties of non-critical systems such as finite correlation length and finite bipartite entanglement entropy in the thermodynamic limit, whereas the gap decay obeys a universal power law dependence. The transition towards the critical point with vanishing of the external field exhibits logarithmic divergence for the correlation length and the effective rank of the ground state. We have no explanation for the logarithmic behavior in the quantum phase transition. The model proposed is much richer than the example discussed. Thus we expect, that higher connectivity of the graph will yield very different results, but even an exploration of high temperature behaviour for the chain seems very worthwhile. In view of the large structural invariance group of the ensemble in the case of site independent average coupling and external fields we hope, that some analytic results can be obtained for this ensemble. Figure 1 . 1Distribution of normalized gapg = g/ g for a few choices of even N . In the non-T -invariant case an agreement with GUE level spacing distribution is obtained whereas in theT -invariant case, λ = 0, the level repulsion gradually vanishes in the thermodynamic limit. 3. 3 .Figure 2 . 32Size scaling of the ground state entanglement entropy The second characteristics of quantum phase transitions we choose to investigate in the TSRE(N, λ), is the entanglement entropy of a symmetric bi-partition of the chain S(N, λ) = tr {N/2+1,...N } (tr {1,...,N/2} \|0 0\|) log 2 (tr {1,...,N/2} \|0 0\|) , Spectral gap scaling with the system size for different values of parameter λ (see legend). Note an asymptotic scaling ∝ N −0.39 , unless λ = 0 where faster than power law decay of a gap is observed, perhaps asymptotically exponential (see inset for a semi-log scale). Figure 3 . 3Entanglement entropy versus chain length N for different values of parameter λ (see legend). Logarithmic divergence with an estimated asymptotic slope S ∼ 0.17 log 2 N for the critical case λ = 0 is indicated with a dashed line. Figure 4 . 4Ensemble fluctuations of the ground state spin-spin correlation function C(r) versus distance r, for different values of the parameter λ (indicated in the legend). Open symbols indicate results for chain length L = 16 and L = 20 whereas closed symbols stand for L = 24. can thus be considered equivalent, and the gauge transformation (4) defines a natural equivalence relation in TSRE(G, µ, λ). Therefore, it may be of interest to consider a simplest parametrization of the set of equivalence classes, or in other words the orbits of a Hamiltonian on the ensemble under the structural invariance transformations. We thus ask, what is a general canonical form to which each element H can be brought by gauge transformations (4) and how can it be parametrized? Let the integer K denote the number of such parameters. The equivalent question for the classical ensembles leads to the eigenvalues as canonical parameters, since there the structural invariance group is much bigger.Let us first consider the simplest connected graph, namely an open one dimensional (1D) chain of N vertices {1, . . . N} and M = N − 1 bonds. 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[ "NEW MAXIMAL SURFACES IN MINKOWSKI 3-SPACE WITH ARBITRARY GENUS AND THEIR COUSINS IN DE SITTER 3-SPACE", "NEW MAXIMAL SURFACES IN MINKOWSKI 3-SPACE WITH ARBITRARY GENUS AND THEIR COUSINS IN DE SITTER 3-SPACE" ]	[ "Shoichi Fujimori ", "Wayne Rossman ", "Masaaki Umehara ", "ANDKotaro Yamada ", "Seong-Deog Yang " ]	[]	[]	Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain "cousins" of those maximal surfaces.Cone-like singular points on maximal surfaces are very important, although they are not stable under perturbations of maximal surfaces. It is interesting to ask if cone-like singular points can appear on a maximal surface having other kinds of singularities. Until now, no such examples were known. We also construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges.	10.1007/s00025-009-0443-4	[ "https://arxiv.org/pdf/0910.2768v1.pdf" ]	115,162,141	0910.2768	43630b7e4059687cdab817e2334abf8f4c0e837c	NEW MAXIMAL SURFACES IN MINKOWSKI 3-SPACE WITH ARBITRARY GENUS AND THEIR COUSINS IN DE SITTER 3-SPACE 15 Oct 2009 Shoichi Fujimori Wayne Rossman Masaaki Umehara ANDKotaro Yamada Seong-Deog Yang NEW MAXIMAL SURFACES IN MINKOWSKI 3-SPACE WITH ARBITRARY GENUS AND THEIR COUSINS IN DE SITTER 3-SPACE 15 Oct 2009Dedicated to the memory of Professor Katsumi Nomizu Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain "cousins" of those maximal surfaces.Cone-like singular points on maximal surfaces are very important, although they are not stable under perturbations of maximal surfaces. It is interesting to ask if cone-like singular points can appear on a maximal surface having other kinds of singularities. Until now, no such examples were known. We also construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges. Introduction Maximal surfaces in the Minkowski 3-space R 3 1 arise as solutions of the variational problem of locally maximizing the area among spacelike surfaces. By definition, they have everywhere vanishing mean curvature. Like the case of minimal surfaces in Euclidean 3-space, maximal surfaces possess a Weierstrass-type representation formula [18]. The most significant difference between minimal and maximal surfaces is the fact that the only complete spacelike maximal surfaces are planes [2,3], which is probably the main reason why people have not paid much attention to maximal surfaces. If we allow some sorts of singular points for maximal surfaces, however, the situation changes. Osamu Kobayshi [19] investigated cone-like singular points on maximal surfaces. After that, many interesting examples with cone-like singular points have been found and studied by F. J. López, R. López, and Souam [24], Fernández and F. J. López [6], and Fernández, F. J. López and Souam [7], Fernández [5] and others. the Lorentzian catenoid, the Lorentzian helicoid, (G, η) = (z, dz/z 2 ) on C \ {0} (G, η) = (z, i dz/z 2 ) on the universal cover of C \ {0} On the other hand, for the study of more general singularities, Estudillo and Romero [4] initially defined a class of maximal surfaces with singular points of more general type, and investigated criteria for such surfaces to be planes. Recently, Imaizumi [15] studied the asymptotic behavior of maximal surfaces, and Imaizumi-Kato [16] gave a classification of maximal surfaces of genus zero with at most three embedded ends. In [32], the third and forth authors showed that if admissible singular points are included, then there is an interesting class of objects called maxfaces. In fact, the three surface classes • non-branched generalized maximal surfaces in the sense of [4], • non-branched generalized maximal maps in the sense of [16], and • maxfaces in the sense of [32] are all the same class of maximal surfaces. So in this paper, we shall call such a class of surfaces maxfaces. Maxfaces are spacelike at their regular points, but the limiting tangent plane (that is, the Lorentzian orthogonal complement of the normal vector) at each singular point contains a lightlike direction. For the global study of maximal surfaces, the following terminology given in [32] is useful: Figure 3. A weakly complete triply-periodic maxface with conelike singular points corresponding to the Schwarz-P surface; see [13] for details. Figure 4. A weakly complete triply-periodic maxfaces with fold singular points corresponding to the Schwarz-D surface; see [13] for details. Definition I. A maxface (or more generally, a generalized maximal surface) f : M → R 3 1 is called complete if there exists a symmetric 2-tensor T which vanishes outside a compact set in M , such that ds 2 + T is a complete Riemannian metric on M , where ds 2 is the induced metric by f . If f is complete, the set of singular points is compact in M . On the other hand, a maxface is called weakly complete (in the sense of [32]), if its null holomorphic lift into C 3 (see Section 1) has complete induced metric with respect to the canonical Hermitian metric on C 3 . As shown in [32,Lemma 4.3], completeness implies weak completeness. Conversely, a weakly complete maxface is complete if and only if the singular set is compact and each end is conformally equivalent to a punctured disc (see [33]). A typical well-known complete maxface is the Lorentzian catenoid (see [18], see also [1] in which it is called the Lorentzian elliptic catenoid ; Figure 1, left), which has a cone-like singular point (see Section 2 for the definition). Like minimal surfaces in Euclidean 3-space R 3 , maxfaces have conjugate surfaces. A cone-like singular point on a maxface corresponds to a fold singular point on its conjugate maxface, in general. For the proof of this and the definition of fold singular points, see [23]. The Lorentzian helicoid (see [18], see also [1]; Figure 1, right) is weakly complete (but not complete) and is the conjugate maxface of the Lorentzian catenoid, whose image consists of two surfaces with 'boundary' in R 3 1 . The boundary (that is, the singular set) is a helix, and each interior image point on the surface has two inverse images. Namely, the Lorentzian helicoid can be regarded as a fold along a helix. As shown in [14], generic singular points of maxfaces are cuspidal edges, swallowtails and cuspidal cross caps. Thus, cone-like singular points and folds are non-generic. For example, one can consider an isometric deformation of Lorentzian helicoid corresponding to the family of Weierstrass data (z, e it dz/z 2 ) (t ∈ [0, π/2]). Figure 2 gives the maxface corresponding t = π/4, whose singular points consist only of cuspidal edges. However, they (i.e., cone-like singular points and folds) are important singular points in the theory of maxfaces. For example, fold singular points (i.e., the double surfaces in [16]) appear under a certain situation, see [16,Proposition 7.7 and Page 581]. It should be remarked that a complete maxface automatically has finite total curvature outside of a compact set, and finite topology as well (see [32,Theorem 4.6 and Corollary 4.8]). This is a property that is crucially different from the case of minimal surfaces in R 3 . So, interestingly, there are no complete periodic maxfaces although there exist compact maxfaces in a Lorentzian torus R 3 1 /Γ for a suitable lattice (namely, triply-periodic weakly complete maxface in R 3 1 ). In fact, the same Weierstrass data as for the Schwarz P-surface and the Schwarz D-surface give such examples. See Figures 3 and 4. In [32,Theorem 4.11], it was shown that an Osserman-type inequality G : M −→ S 2 = C ∪ {∞} is the Lorentzian Gauss map and deg G is its degree as a map to the hyperbolic sphere S 2 considered as a compactification of the hyperboloid in R 3 1 , see [32] and [20,Section 5]. Since G is meromorphic at each end of a complete maxface, the left-hand side of (1) is finite (see Fact 1.2). In [12], the authors showed that a similar Osserman-type inequality (1) also holds for the hyperbolic Gauss maps G of complete CMC-1 faces in de Sitter 3-space (for the definition of CMC-1 faces, see Section 4). In contrast to the case of maxfaces, the hyperbolic Gauss map of a CMC-1 face may have an essential singularity at an end. The Lorentzian catenoid satisfies equality in (1). In [32], several examples which attain equality in (1) were given. Recently, complete maxfaces with three embedded ends were classified by Imaizumi and Kato [16]. We mention here two new interesting phenomena on the shape of singular points on maximal surfaces: Example 1. (Trinoids whose graphics seem to show cone-like singular points, although no cone-like singularities exist). We set and consider two Weierstrass data (see Section 1) M = C \ {a, −a} (a > 1/2)(2) G := b − z 2 z , η := z 2 dz (z 2 − a 2 ) 2 b := −a 2 + a 4a 2 − 1 . Substituting these into (1.4), we get a trinoid with three embedded ends at z = ±a, ∞, each of which is asymptotic to a Lorentzian catenoid. The left-hand figure in Figure 5 shows the image with a = 3.67. On the other hand, set (3) M = C \ {1, −1}, G = c z 2 + 3 z 2 − 1 , and η = dz c (c > 0, c = 1). Then we have another trinoid as in Figure 5, right. The ends 1, −1 are asymptotic to the Lorentzian catenoid, and the end ∞ to the plane. The figure shows the image for c = 0.1. Since a maxface is symmetric with respect to a given cone-like singular point (cf. [16] and [23]), neither of these two maxfaces admits any cone-like singular points. However, in Figure 5, we can see two singular sets in each surface which look very much like cone-like singularities. The singular sets do, however, consist of cuspidal edges and swallowtails (see Figure 5). Similarly, there are four cuspidal cross caps on the Enneper maxface (see [32,Example 5.2]). However, it is difficult to recognize the crossing of two sheets on surfaces near these four singular points from computer graphics, since these two sheets are so close to each other. If a maxface admits a cone-like singular point (resp. a swallowtail), its conjugate surface admits fold singular points (resp. a cuspidal cross cap) and vice versa ( [23] and [14,Corollary 2.5]; for the definition of conelike singular points, see Definition 2.1). Thus, in computer graphics, the fact that a union of swallowtails sometimes look like cone-like singular points and the fact that cuspidal cross caps look like fold singular points seems to be a mutually dual phenomena. The authors hope that one could establish a new theory for explaining this. In recent private conversations, Shin Kato [17] said that this phenomenon for trinoids as in Figure 5 seems to occur as a family of trinoids "collapses" to Lorentzian catenoids (which have a cone-like singularity). It is interesting to ask if there exist maxfaces having a cone-like singular point and also having singular points which are not cone-like. We give such an example: Theorem A. There exists a maxface f : M → R 3 1 of genus 0 with two complete ends whose singular set in M consists of cone-like singular points, cuspidal edges, swallowtails, and cuspidal cross caps. Here, the image of the cone-like singular points is a single point. The proof of the first part of this theorem is given in Section 2. In R 3 , examples of complete embedded minimal surfaces of finite total curvature of course are known -for example, the plane, the catenoid, the Costa surface, the Costa-Hoffman-Meeks surface, etc. As a related class of maxfaces, the authors give here the following: Definition II. A complete maxface is called embedded (in the wider sense) if it is embedded outside of some compact set of R 3 1 . From here on out, when we use the word "embedded" for a maxface, we always mean "embedded in the wider sense" as in the above definition. By definition, an embedded maxface attains equality in (1). In a joint work [22] with Kim, the fifth author constructed maximal surfaces of genus k = 1, 2, 3, . . . , which are complete generalized maximal surfaces in the sense of [4] (i.e., they admit branch points when k ≥ 2), and when k = 1 it is a complete embedded maxface, see Figure 6. We shall call this example the Kim-Yang toroidal maxface. We remark that there exist no embedded minimal surface in R 3 with two ends except the catenoid (cf. [29]). Until now, the only known examples of embedded complete maxfaces were • the spacelike plane (which is the only example of a complete maximal surface without singular points), • the Lorentzian catenoid, • the Kim-Yang toroidal maxface. Also, until now the only known complete positive-genus maxfaces were the Lorentzian Chen-Gackstatter surface (given in [32,Example 5.5]) and the Kim-Yang toroidal maxface. In this article, we construct complete maxfaces with two ends and arbitrary genus, which are embedded if the genus is equal to 1. (Theorem B). On the other hand, surfaces of constant mean curvature one (CMC-1 surfaces) in de Sitter 3-space S 3 1 have similar properties to maximal surfaces in R 3 1 (cf. [8] and [12]). Analogous to maxfaces, a notion of CMC-1 faces in S 3 1 , which is CMC-1 surfaces with certain kinds of singular points, was introduced in [8]. Related to this, the second, the third and forth authors introduced in [27] a method to deform minimal surfaces in R 3 to CMC-1 surfaces in hyperbolic 3-space. In this paper, we demonstrate that this method works for maxfaces in R 3 1 and CMC-1 faces in S 3 1 as well (see Section 4 and the appendix). In this article, we shall prove: Theorem B. There exists an family of complete maxfaces f k for k = 1, 2, 3, . . . with two ends, and of genus k if k is odd and genus k/2 if k is even. Moreover, f 1 and f 2 are embedded (in the wider sense). Furthermore, the number of swallowtails and the number of cuspidal cross caps are both equal to 4(k + 1) if k is odd and 2(k + 1) if k is even. In particular, f 1 and f 2 are both of genus one, but are not congruent. (Compare Figures 6 and 9.) Moreover, such maxfaces f k can be deformed to complete CMC-1 faces in S 3 1 , which are embedded (in the wider sense) if k = 1 or 2. We note that f 1 is the Kim-Yang toroidal maxface, but the f k are new examples for all k ≥ 2. The proof of this theorem is given in Sections 3 and 4. The method of construction is somewhat similar to that of [22], but one salient feature of our new family is that it is free of branch points for all k = 1, 2, 3, . . . , whereas the surfaces in the corresponding family of [22] have two branch points whenever k ≥ 2. The first and second authors [11] provided numerical evidence for the existence of CMC-1 faces in S 3 1 of higher genus with two embedded ends. Problem 2. Are there complete maxfaces (resp. complete CMC-1 faces) with more than two ends of positive genus in R 3 1 (or S 3 1 )? The present state of this field is such that it would be very beneficial to have a larger collection of examples, as described in these open problems, for example. A number of types of maximal surfaces can be produced from a canonical correspondence with minimal surfaces in Euclidean 3-space given in Section 5 of [32] (for example, this method applies to the minimal surfaces of arbitrary genus found in [28]), although this construction needs to solve a period problem. (Correspondence of minimal surfaces and maximal surfaces was first introduced in [2] for graphs.) Non-orientable complete maxfaces were recently found in a joint work [10] of the first author with López. (Unfortunately, the ends of these examples are not embedded.) On the other hand, CMC-1 faces in S 3 1 are all orientable (this fact is not trivial, since the surface admits singular points, see [20]). Weakly complete (but not complete) bounded maxfaces (resp. CMC-1 faces) with arbitrary genus were constructed in [25] and [26]. Preliminaries In this section, we review the Weierstrass-type representation formula for maxfaces (see [18,32]), and criteria for singular points (see [32,14]). Throughout this paper, we denote by R 3 1 the Minkowski 3-space with the inner product , of signature (−, +, +). We call the pair (G, η) the Weierstrass data of the maxface f in (1.4), G the Lorentzian Gauss map, and the holomorphic 2-differential Q in (1.5) the Hopf differential, respectively. is compact. In this case, the Weierstrass data (G, η) is well-defined as a pair of a meromorphic function and a meromorphic one form on M , and the compactness of the singular set Σ is equivalent to the condition \|G(p j )\| = 1 for j = 1, . . . , n. As shown in [14], generic singular points of maxfaces (resp. CMC-1 faces) are cuspidal edges, swallowtails and cuspidal cross caps. We recall criteria for generic singular points of maxfaces. For terminology on CMC-1 faces, see Section 4. Re α(p) = 0, α(p) = 0, and Im β(p) = 0. Here, two C ∞ -maps f 1 : (U 1 , p) → N 3 and f 2 : (U 2 , q) → N 3 of domains U j ⊂ R 2 (j = 1, 2) into a 3-manifold N 3 are right-left equivalent at the points p ∈ U 1 and q ∈ U 2 if there exists a local diffeomorphism ϕ of R 2 with ϕ(p) = q and a local diffeomorphism Φ of N 3 with Φ(f 1 (p)) = f 2 (q) such that f 2 = Φ • f 1 • ϕ −1 . Maxfaces with cone-like singular points In this section, we give the proof of Theorem A in the introduction. To do this, we recall a definition and a criterion for cone-like singular points. image f (Σ 0 ) is one point. Moreover, if there is a neighborhood U of Σ 0 and f (U \ Σ 0 ) is embedded, each point of Σ 0 is called a cone-like singular point. Remark 2.2. In [19] and [7], the image f (Σ 0 ) as a single point of the cone-like singular points is called a cone-like singular point. However, we do not use this terminology here, since we treat the singular set not in R 3 1 but in the source manifold. Since dG = 0 on the set Σ 0 of generalized cone-like singular points, Σ 0 is diffeomorphic to the circle S 1 . By [7, Lemma 1], we have the following criterion: (2.1) G = (z − 1)(z 2 + az + 1) (z + 1)(z 2 − az + 1) , η = (z 2 − az + 1) 2 (z − 1) 4 (z + 1) 2 dz. Then the data (G, η) gives no real period in the representation formula (1.4), i.e., it satisfies (1.3). Thus it defines a maxface of genus zero with 2-ends: (2.2) f : M −→ R 3 1 . Since a ∈ R, it holds that \|G\| = 1 on the imaginary axis. When 1 < a < 4 and a = 2, the singular set Σ = {\|G\| = 1} consists of three disjoint (topological) circles on the Riemann sphere, including the imaginary axis of the z-plane (Figure 7, bottom). Set α := dG/(G 2 η) as in Section 1. Then the set {Im α = 0} looks like as in Figure 7, bottom. In particular, Im α = 0 on the imaginary axis, and hence Lemma 2.3 implies that the imaginary axis consists of generalized cone-like singular points. Here, since G is of degree 3 and the singular set consists of three connected components, G is injective on each connected component of the singular set. Since η = 0 on the imaginary axis, Lemma 2.4 implies that the imaginary axis consists of cone-like singular points. On the other hand, other connected components of the singular set contain cuspidal edges, because Im α is not identically zero on the singular set. Since the order of the Hopf differential Q at z = ±1 is −4, both ends are Enneper ends (cf. Example 5.2 in [32]), as shown in Figure 7, top. Proof of The first part of Theorem B In this section, we construct maxfaces f k (k = 1, 2, 3, . . . ) as in the statement of Theorem B. To do it, we proceed as follows for each k: • Take a Riemann surface M k of genus k, which is a compact Riemann surface excluding 2 points (which correspond to the ends), see Section 3.1. • Construct a complete maxfacef k : M k → R 3 1 (k = 1, 2, 3, . . . ), see Sections 3.1-3.4. • When k = 2m is an even number, show that M k is the double cover of a Riemann surface M ′ k of genus m, andf k induces a complete maxface f k : M ′ k → R 3 1 . See Section 3.5. • When k is odd, we set f k =f k . 3.1. The Riemann surface M k . Let (3.1) M k = (z, w) ∈ (C ∪ {∞}) 2 ; w k+1 = z z 2 − 1 k , where k is a positive integer. As a submanifold of (C ∪ {∞}) 2 , M k has singular points at (z, w) = (±1, 0) and (∞, ∞). However, one can define on M k the structure of a Riemann surface using complex coordinates ζ 0 , ζ ∞ , ζ 1 and ζ −1 around (z, w) = (0, 0), (∞, ∞), (1, 0), and (−1, 0), respectively, as follows: (3.2) z = (ζ 0 ) k+1 at (0, 0), z = (ζ ∞ ) −k−1 at (∞, ∞), z = 1 + (ζ 1 ) k+1 at (1, 0), z = −1 + (ζ −1 ) k+1 at (−1, 0). Hence the holomorphic map z : M k → C ∪ {∞} is of degree k + 1 with total branching number 4k. Then by the Riemann-Hurwitz relation, the genus of M k is k. −1 0 1 −1 0 1 o µ 2 (o) γ κ 2 • γ The projection of the loops γ and κ 2 •γ to the z-plane are shown. The left-hand (resp. right-hand) figure shows the z-plane such that arg w = 0 (arg w = 2kλ) when z > 1. We shall construct maxfacesf k : M k → R 3 1 (k = 1, 2, 3, . . . ) with two ends corresponding to (0, 0) and (∞, ∞), where (3.3) M k = M k \ {(0, 0), (∞, ∞)}. Let M k be the universal cover of M k . 3.2. Symmetries and the fundamental group of M k . For simplicity, we set (3.4) λ := π k + 1 . Define reflections (orientation-reversing conformal diffeomorphisms) µ j : M k → M k (j = 1, 2, 3, 4) as (3.5) µ 1 (z, w) = (z,w), µ 2 (z, w) = (z, e 2kiλw ),µ 3 (z, w) = (−z, e −iλw ), µ 4 (z, w) = 1 z , e ikλw z 2 . Using these, we define the following automorphisms of M k : (3.6) κ 1 := µ 2 • µ 1 κ 1 (z, w) = (z, e 2kiλ w) , κ 2 := µ 3 • µ 1 κ 2 (z, w) = (−z, e −iλ w) . Choose a base point o ∈ M k such that (3.7) o ∈ {(t, w) ; 1 < t < ∞, arg w = 0} ⊂ M k , and take a loop γ on M k starting at o as in Figure 8. Then we have the following: Lemma 3.1. The fundamental group π 1 (M k ) of M k is generated by [(κ 1 ) j • γ] and [(κ 1 ) j • κ 2 • γ] (j = 0, . . . , k). 3.3. The Weierstrass data. We now take the Weierstrass data (3.8) G = c w z , η = dz w (c ∈ R + ) on M k , where c is a positive real constant to be determined in (3.12). Define a holomorphic 2-differential Q (3.9) Q := η dG = ck k + 1 Table 1. Orders of G, η, Gη, G 2 η and Q = η dG. z 2 + 1 z 2 (z 2 − 1) dz 2 . (z, w) (0, 0) (∞, ∞) (1, 0) (−1, 0) (±i, * ) Ord G −k −k k k 0 Ord η k − 1 k − 1 0 0 0 Ord Gη −1 −1 k k 0 Ord G 2 η −(k + 1) −(k + 1) 2k 2k 0 Ord Q −2 −2 k − 1 k − 1 1 We call Q the Hopf differential because it will be the Hopf differential of the maxface when the construction is completed. The orders of G, η, Gη, G 2 η and Q are listed as in Table 1, where Ord ω = m (resp. −m) for a positive integer m if ω has a zero (resp. a pole) of order m. Then one can conclude that deg G = 2k for any k ≥ 1. Let (3.10)f k,c := Re Φ : M k −→ R 3 1 , where Φ is the C 3 -valued 1-form as in (1.2) obtained by (G, η) in (3.8). Lemma 3.2. Supposef k,c as in (3.10) is well-defined on M k . Then it is a complete maxface of genus k with 2 ends. Moreover, each end is asymptotic to the k-fold cover of the Lorentzian catenoid. Proof. Observe that η and G 2 η are holomorphic on M k and have no common zeros, and G 2 η has poles of order k+1 at (z, w) = (0, 0) and (∞, ∞). Thus the metric as in (1.1) gives a complete Riemannian metric on M k . Hencef k,c is a weakly complete maxface. Moreover, since G has poles at (z, w) = (0, 0) and (∞, ∞), \|G\| = 1 at the ends. Thenf k,c is complete because of Fact 1.2. At each end, Q has a pole of order 2 and the ramification order of G is k. Then the Weierstrass representation (1.4) yields that the end is asymptotic to the k-fold cover of an end of the Lorentzian catenoid. Remark 3.3. The Hopf differential Q(z) is real if z ∈ R or z ∈ iR. Also, Q(z) is pure imaginary if \|z\| = 1. Then the image of the real and imaginary axes on the z-plane are planar geodesics with respect to the first fundamental form of the surface, and the image of the unit circle consists of line segments joining singular points. 3.4. The period problem. In this section, we shall solve the period problem (1.3). The following lemma can be obtained by straightforward calculations: Lemma 3.4. Let Φ be as in (1.2) for the data (3.8). Then for automorphisms κ j in (3.6) (j = 1, 2), it holds that κ * 1 Φ T =   1 0 0 0 cos 2kλ − sin 2kλ 0 sin 2kλ cos 2kλ   Φ T , κ * 2 Φ T =   1 0 0 0 cos kλ sin kλ 0 − sin kλ cos kλ   Φ T , where T stands for transposition. Because of the matrices in Lemma 3.4, Lemma 3.1 implies that Lemma 3.5. The mapf k,c defined in (3.10) is single-valued on M k if and only if the period condition (1.3) holds for the single loop γ in Figure 8. Now, we determine the value of c ∈ R + in (3.8) such that the condition in Lemma 3.5 holds. Since Gη = (c/z)dz, it holds that Re γ Gη = Re γ c d log z = Re (2πic) = 0. So the condition (1.3) for γ is equivalent to (3.11) γ η + γ G 2 η = 0. To calculate the integrals of the left-hand side, we take two paths on M k as γ 1 := (z, w) = t, e kiλ k+1 t(1 − t 2 ) k t : 1 → 0 , γ 2 := (z, w) = t, e −kiλ k+1 t(1 − t 2 ) k t : 0 → 1 , where we consider k+1 t(1 − t 2 ) k as a positive real number. Roughly speaking, a closed loop γ 1 * γ 2 (the definition of γ 1 * γ 2 is given in the appendix) is homotopic to γ on M k . Here, γ 1 and γ 2 are parametrized regular curves on M k from t = 1 to t = 0 and t = 0 to t = 1, respectively. Adding an exact form to G 2 η, we have G 2 η + c 2 k + 1 k d w z = −2c 2 w 1 − z 2 dz. Since the right-hand side does not have poles at (z, w) = (0, 0) and (1, 0), we have γ G 2 η = −2c 2 γ w 1 − z 2 dz = −2c 2 γ1 w 1 − z 2 dz + γ2 w 1 − z 2 dz = −c 2 · 4i sin kλ · A k A k := 1 0 k+1 t 1 − t 2 dt . On the other hand, since η does not have poles at (z, w) = (0, 0) and (1, 0), we have γ η = γ1 η + γ2 η = 2i sin kλ · B k B k := 1 0 dt k+1 t(1 − t 2 ) k . Therefore, if we set (3.12) c = c k := B k 2A k > 0, then (3.11) holds for γ and hence Now, we prove thatf 1 is embedded (in the wider sense, as in Definition II). When k = 1, each end is asymptotic to a Lorentzian catenoid, that is, each end has no self-intersection. Moreover, the x 0 -component off 1 is calculated as ∞)). This means that the two ends have no self-intersection outside a compact set in M 1 . Hencef 1 is embedded in the wider sense. x 0 := −2 Re p o Gη = −2c 1 log \|z\| + constant, where p = (z, w) ∈ M 1 . Hence x 0 → +∞ (resp. −∞) when (z, w) → (0, 0) (resp. (∞, 3.5. Reduction for the even genus case. When k is even, the Riemann surface M k of genus k = 2m is reduced to the Riemann surface M M ′ k = (Z, W ) ∈ (C ∪ {∞}) 2 ; W 2m+1 = Z m+1 (Z − 1)′ k = M ′ k \ {(0, 0), (∞, ∞)}. Define a map ̟ k from M k of genus k = 2m into M ′ k of genus m as (3.15) ̟ k : M k ∋ (z, w) −→ (Z, W ) = (z 2 , zw) ∈ M ′ k . Then ̟ k (z, w) = ̟ k (−z, −w) for any (z, w) ∈ M k and hence ̟ k is a double cover. Let (G 1 , η 1 ) be the Weierstrass data on M ′ k given by G 1 = c W Z , η 1 = dZ 2W , which satisfy G = ̟ * k G 1 = G 1 • ̟ k , η = ̟ * k η 1 . The data (G 1 , η 1 ) for c = c k as in (3.12) gives a maxface f k : M ′ k → R 3 1 such that f k = f k • ̟ k . Since deg G 1 = m,c 1 > 1 and c k > k 1 2(k+1) √ 2 k k − 1 k−1 2(k+1) (k ≥ 2). In particular, for each k = 1, 2, 3, . . . , it holds that (3.16) 0 < ρ k < 2 ρ k := c −2(k+1) k k . Proof. First, we consider the case k ≥ 2. Set V (s, t) := t(1 − t 2 ) k 1 t(1 − t 2 ) k − s k+1 t 1 − t 2 = 1 − s k+1 t 2 (1 − t 2 ) k−1 for s ∈ R and t ∈ [0, 1]. Then for each fixed value s, V (s, t) attains a minimum at t = 1/ √ k, and V s, 1 √ k = 1 − s k+1 k k − 1 k k−1 . Hence if we set (3.17) s k := k 1 k+1 k k − 1 k−1 k+1 , then V (s k , t) ≥ 0 (t ∈ [0, 1]). Hence k+1 t 1 − t 2 − s k 1 k+1 t(1 − t 2 ) k > 0 holds on t ∈ [0, 1], and then, we have c k = A k 2B k > s k 2 = 1 √ 2 k 1 2(k+1) k k − 1 k−1 2(k+1) . This implies 0 < ρ k < 2 k+1 k k − 1 k k k − 1 1−k k = 2 2 k 1 k k − 1 k k−1 k < 2. Next, consider the case k = 1: A 1 − 2B 1 = 1 0 1 t(1 − t 2 ) − 2 t 1 − t 2 dt = 1 0 1 − 2t t(1 − t 2 ) dt = √ 2 2 −1 u du (1 − u 2 )(3 − u) = √ 2 1 0 u du (1 − u 2 )(3 − u) + 1 0 v dv (1 − v 2 )(3 + v) = √ 2 1 0 u √ 1 − u 2 1 3 − u − 1 3 + u du > 0, where we put u = 1 − 2t and v = −u. Hence c 1 = A 1 /(2B 1 ) > 1, and then ρ 1 < 1 < 2. z − 1 z 2 = r 2 + 1 r 2 − 2 cos 2θ = ρ k ρ k = c k −2(k+1) k , where z = re iθ . Since ρ k ∈ (0, 2) by Lemma 3.6, one can set Γ k := arcsin √ ρ k 2 ∈ 0, π 4 . Thus, the projection of the singular set onto the z-plane consists of two simple closed curves in the z-plane, contained in two angular domains: ∆ + := {−Γ k < arg z < Γ k }, and ∆ − := {π − Γ k < arg z < π + Γ k }. We denote two subsets of the singular set Σ as Σ + := {p ∈ Σ ; z(p) ∈ ∆ + }, and Σ − := {p ∈ Σ ; z(p) ∈ ∆ − }, see Figure 11, where z : M k ∋ (z, w) → z ∈ C is the projection. The loop z(Σ + ) on the z-plane is surrounding the branch point z = 1 of the projection z. Then, the inverse image Σ + of this loop consists of k + 1 copies of z(Σ + ) which forms a single loop in M k , that is, Σ + is a loop in M k . Similarly so is Σ − . Proof. Let α and β be as in (1.6): Finally, since Im β = r 2 + r −2 sin 2θ, Im β = 0 if and only if z ∈ R ∪ iR. Then by Fact 1.3, there are two swallowtails and two cuspidal cross caps on Σ + , on the z-plane. Since M k is a (k + 1)-fold branched cover of the z-plane, we have the conclusion. α := dG G 2 η = Q (Gη) 2 = z 2 + 1 z 2 − 1 = 1 − 2 z 2 − 1 , β := G dG dα = z 2 − 1 z 2 . When k = 2m is even, bothf m and f k give maxfaces of genus m with 2 ends. Moreover, each end is asymptotic to the m-fold cover of the Lorentzian (elliptic) catenoid. Proof. The number of swallowtails on the image off m is 4m + 4. On the other hand, the number of swallowtails on the image of f k (k = 2m) is 4m + 2, which proves the assertion. Proof of the second part of Theorem B In this section, we shall deform the maxfaces given in the previous section to CMC-1 faces in de Sitter space. The technique we use here is similar to that in [27]. However, we need much more technical arguments because the maxfaces f k in the previous section do not have non-degenerate period problem as in [27,Section 5]. So, we accomplish the deformation by computing the derivative of the period matrices. 4.1. Preliminaries. First, we recall some fundamental facts about CMC-1 faces in de Sitter space. For detailed expressions, see [8,9], or [12]. Let R 4 1 be the Minkowski 4-space with the metric , of signature (−, +, +, +). Then de Sitter 3-space is expressed as S 3 1 = {X ∈ R 4 1 ; X, X = 1} with metric induced from R 4 1 , which is a simply-connected Lorentzian 3-manifold with constant sectional curvature 1. We identify R 4 1 with the set of 2 × 2 Hermitian matrices Herm(2) by (4.1) R 4 1 ∋ (x 0 , x 1 , x 2 , x 3 ) ↔ x 0 + x 3 x 1 + ix 2 x 1 − ix 2 x 0 − x 3 ∈ Herm(2). Then de Sitter 3-space is represented as S 3 1 = {X ∈ Herm(2) ; det X = −1} = {F e 3 F * ; F ∈ SL 2 C} = SL 2 C/ SU 1,1 e 3 := 1 0 0 −1 . The first author [8] introduced the notion of CMC-1 faces in S 3 1 , which corresponds to maxfaces in R 3 1 . To state the Weierstrass-type representation formula, we prepare some notions: (4.3) dF F −1 = Ψ Ψ := G −G 2 1 −G Q dG . Then F is a null holomorphic immersion, that is, F is a holomorphic immersion such that det(dF/dz) vanishes identically for each local complex coordinate z on M . And The induced metric ds 2 and the second fundamental form II are expressed as (4.6) ds 2 = (1 − \|g\| 2 ) 2 Q dg 2 , II = Q + Q + ds 2 , respectively. Conversely, any CMC-1 face is obtained in this manner. Remark 4.3. (1) The equation (4.3) should be regarded as an equation on the universal cover M (see (A.6) in the appendix). However, for simplicity, we use the notation here. (2) The condition that \|g\| is not identical to 1 is necessary to avoid the example all of whose points are singular points. Such an example is unique up to isometry, whose image is a lightlike line in S 3 1 (see [ The meromorphic function G, the holomorphic 2-differential Q and the (multivalued) meromorphic function g in Proposition 4.2 are called the hyperbolic Gauss map, the Hopf differential, and the secondary Gauss map, respectively. We call F the holomorphic null lift of the CMC-1 face f . These holomorphic data are related by (4.7) S(g) − S(G) = 2Q, S(h) := h ′′ h ′ ′ − 1 2 h ′′ h ′ 2 dz 2 , ′ = d dz , where z is a local complex coordinate on M and S(·) is the Schwarzian derivative. For an admissible pair (G, Q) on M , there exists a representation ρ F : π 1 (M ) → SL 2 C associated with the solution F of (4.3) as in Proposition A.4 in the appendix: (4.8) F • τ = F ρ F (τ ) −1 τ ∈ π 1 (M ) , where τ ∈ π 1 (M ) is considered as a covering transformation of the universal cover M , as in (A.2). To deform maxfaces to CMC-1 faces, the following facts, which are proved in [27] for CMC-1 surfaces in H 3 , play important roles: 4.9) dF F −1 = tΨ 0 , Ψ 0 = G −G 2 1 −G Q dG , with the initial condition F t (o) = ι(t), where ι(t) is a smooth SL 2 C-valued function in t with ι(0) = e 0 , where (4.10) e 0 := 1 0 0 1 is the identity matrix and o ∈ M is a base point. Let ρ t = ρ Ft (τ ) for τ ∈ π 1 (M ), where ρ Ft is a representation as in (4.8). Then it holds that (4.11) d dt t=0 (ρ t ) −1 = γ Ψ 0 , and γ is a loop in M which represents τ . Proof. By (4.9), F 0 is a constant map. Differentiating (4.9), we have dḞ = Ψ 0 ι(0), whereḞ = (∂/∂t)\| t=0 F t . Hence we have d dt t=0 γ dF t = γ Ψ 0 . Here, the left-hand side is computed as d dt t=0 F t (o)(ρ t ) −1 − F t (o) = d dt t=0 ι(t)(ρ t ) −1 − ι(t) = d dt t=0 (ρ t ) −1 , because ι(0) = e 0 . Hence we have the conclusion. Similar to Definition I in the introduction, we define completeness and weak completeness of CMC-1 faces: 4.2. The holomorphic data. Let k be a positive integer, and take the Riemann surface M k as in (3.3). Take a meromorphic function G, a holomorphic 1-form η and a holomorphic 2-differential Q t (t ∈ R) on M k as (4.12) G := c k w z , Q t := t c k Q = tk k + 1 z 2 + 1 z 2 (z 2 − 1) dz 2 , where (G, η) are the Weierstrass data as in (3.8), Q = ηdG is as in (3.9), and c = c k is as in (3.12). We set (4.13) Ψ := tΨ 0 , Ψ 0 := 1 c k G −G 2 1 −G Q dG . Then one can easily show that (G, Q) (and then (G, Q t ) for t = 0) is an admissible pair on M k , and the metric (4.2) is complete. Then the second part of Theorem B is a conclusion of the following Proposition 4.7. For each positive integer k = 1, 2, 3 . . . , there exists a positive number ε = ε(k) such that for each real number t with 0 < \|t\| < ε, there exists a complete CMC-1 facef k,t : M k → S 3 1 whose hyperbolic Gauss map G and Hopf differential Q t are given as in (4.12). Figure 12. Paths P µ1 , P µ2 and P µ3 (P µ1 is the constant path at o). −1 −1 0 1 0 1 µ 2 (o) o µ 3 (o) P µ 2 P µ 3 Remark 4.8. Considerf k,t as a map into the Minkowski space R 4 1 , and let f k,t = 1 tf k,t : M k −→ S 3 1 (t 2 ) = X ∈ R 4 1 ; X, X = 1 t 2 ⊂ R 4 1 , where S 3 1 (t 2 ) is de Sitter 3-space of constant sectional curvature t 2 . Thenf k,t is a surface of mean curvature t in S 3 1 (t 2 ). Then taking the limit as t → 0 in a similar way as in [30],f k,t converges to the maxfacef k as in the previous section. In this sense, the method provided here is considered as a "deformation". Representation of reflections. The proof is done by a (refined version of) the reflection method, which was introduced in [27] for CMC-1 surfaces in the hyperbolic space. To do this, we first take the reflections on the universal cover M k of M k . Let µ j (j = 1, 2, 3) be the reflections on M k as in (3.5). For a matrix a = (a ij ) i,j=1,2 ∈ SL 2 C and a holomorphic function h on a Riemann surface M , we set (4.14) a ⋆ h := a 11 h + a 12 a 21 h + a 22 . The following lemma is a direct conclusion of (3.5): Lemma 4.9. The pair (G, Q t ) as in (4.12) satisfies G • µ j = σ j ⋆ G, Q t • µ j = Q t (j = 1, 2, 3), where (4.15) σ 1 = 1 0 0 1 (= e 0 ), σ 2 = ψ −2 0 0 ψ 2 , σ 3 = ψ −1 0 0 ψ , and (4.16) ψ := e kiλ/2 = exp iπk 2(k + 1) λ = π k + 1 . Namely, (G, Q t ) is µ j -invariant (j = 1, 2, 3) in the sense of (A.9) in the appendix. We remark that we do not use µ 4 here, because (G, Q) is not µ 4 -invariant in the sense of (A.9) as in the appendix. In fact, Q • µ 4 = −Q holds. We fix the base point o ∈ M k as in (3.7) and take paths P µj on M k associated to µ j as in the appendix which join o and µ j (o) as in Figure 12 for each j = 1, 2, 3. (In Figure 12, the left-hand z-plane is the sheet containing o and the right-hand z-plane is the sheet containing µ 2 (o). These two sheets are connected along two intervals (0, 1) and (−∞, −1) on each real axis.) Here P µ1 is the constant path at o. Then one can take the liftμ j (j = 1, 2, 3) (as orientation-reversing involutions on M k ) of µ j : M k → M k with respect to the path P µj , see (A.4) in the appendix. Then by Proposition A.2, we have Lemma 4.10. Let γ be the loop as in Figure 8, and let κ j (j = 1, 2) be the automorphisms of M k as in (3.6). Then For each t ∈ R and b ∈ SL 2 C, denote by [γ] =μ 3 •μ 2 •μ 3 •μ 1 , [κ 2 • γ] =μ 3 •μ 1 •μ 3 •μ 2 , [(κ 1 ) j • γ] = (μ 2 •μ 1 ) j • [γ] • (μ 1 •μ 2 ) j , [(κ 1 ) j • κ 2 • γ] = (μ 2 •μ 1 ) j • [κ 2 • γ] • (μ 1 •μ 2 ) j hold,Then τ 0 = (μ 3 •μ 2 ) 2(k+1) , τ ∞ = (μ 1 •μ 3 ) 2(k+1) hold. Namely, {μ 1 ,μ 2 ,μ 3 } isF = F t,b : M k −→ SL 2 C the unique solution of the differential equation (4.17) dF F −1 = Ψ = tΨ 0 , F (o) = b, where Ψ 0 is as in (4.13). Then by Lemma 4.9 and Theorem A.6 in the appendix, there existsρ j,t,b ∈ SL 2 C such that (4.18) F t,b •μ j = σ j F t,b (ρ j,t,b ) −1 (j = 1, 2, 3), where σ j (j = 1, 2, 3) are the matrices given in (4.15). By Proposition A.9, the group generated by {ρ j,t,b } j=1,2,3 contains the subgroup ρ F (π 1 (M k )) given in (4.8). Then we have the following by Lemma 4.10 and Theorem A.10, which is the key to our construction: Proposition 4.11. Ifρ j,t,b ∈ SU 1,1 for j = 1, 2, 3, then the CMC-1 facê f t,b := (F t,b )e 3 (F t,b ) * corresponding to F t,b as in (4.4) is well-defined on M k . Here, we summarize properties of the matricesρ j,t,b : Proposition 4.12. Suppose a, b ∈ SL 2 C. Then the following hold: (1)ρ j,t,ba =ā −1 (ρ j,t,b )a. (2)ρ j,0,b =b −1 (σ j )b. In particular,ρ j,0,e0 = σ j . Proof. Since F t,ba = (F t,b )a, (1) holds. When t = 0, F 0,b = b. Then b = F 0,b = F 0,b •μ j = σ j F 0,b (ρ j ) −1 = σ j b(ρ j ) −1 , where ρ j =ρ j,0,b , which implies (2). 4.4. Existence off k . The existence part of Proposition 4.7 is a straightforward conclusion of the following proposition, because of Proposition 4.11: Proposition 4.13. For a sufficiently small positive number ε, there exists a real analytic family ι(t) of matrices in SL 2 C such thatρ j,t,ι(t) ∈ SU 1,1 (j = 1, 2, 3) holds if 0 < \|t\| < ε. The proof is divided into three steps (Claims 1-3): Claim 1. For t ∈ R and for any real matrix b ∈ SL 2 R, it holds thatρ 1,t,b = e 0 . In fact, substituting o into (4.18) for j = 1, we have Claim 1. Claim 2. For sufficiently small ε > 0, there exists a real analytic family {ι(t) ; \|t\| < ε} of matrices in SL 2 R such that ι(0) = e 0 andρ 2,t,ι(t) = σ 2 . Proof. Since (µ 2 •µ 1 ) k+1 = id M k holds, where id M k is the identity map on M k , then by Proposition A.2 in the appendix, (μ 2 •μ 1 ) k+1 is a covering transformation on M k . In fact, such a covering transformation corresponds to the counterclockwise simple closed loop on M k surrounding (z, w) = (1, 0), i.e., (μ 2 •μ 1 ) k+1 = id f M k . Then by Proposition A.9, e 0 = ρ F (id f M k ) = σ 2 σ 1 k+1 ρ 2 ρ 1 k+1 = (−1) k ρ 2 ρ 1 k+1 , where ρ j =ρ j,t,e0 . Hence we have (4.19) (ρ 2 ) k+1 = (−1) k e 0 . Sinceρ 2,t,e0 tends toρ 2,0,e0 = σ 2 as t → 0 (see (2) of Proposition 4.12), the equality (4.19) implies that the eigenvalues ofρ 2,t,e0 are {ψ ±2 }, where ψ is given in (4.16). Hence traceρ 2,t,e0 = 2 cos(kλ) λ = π k + 1 . Then by (A.12), one can writẽ ρ 2,t,e0 = cos(kλ) − iu(t) is 1 (t) is 2 (t) cos(kλ) + iu(t) (cos kλ) 2 + u(t) 2 + s 1 (t)s 2 (t) = 1) , where u = u(t), s j = s j (t) (j = 1, 2) are real analytic functions in t. Sincẽ ρ 2,0,e0 = σ 2 , we have that u(t) + sin(kλ) s 1 (t) −s 2 (t) u(t) + sin(kλ) . By (4.20), (sin kλ) 2 + u(t) sin(kλ) > 0 for sufficient small t, and hence ι(t) is a real matrix. It can be easily checked that ι(t) −1ρ 2,t,e0 ι(t) = σ 2 . Then (1) of Proposition 4.12 yields the assertion. Claim 3. For ι(t) in Claim 2, it holds that (4.21)ρ 3,t,ι(t) = q(t) ir 1 (t) ir 2 (t) q(t) (\|q(t)\| 2 + r 1 (t)r 2 (t) = 1), where q(t) is a complex-valued real-analytic function and r j (t) (j = 1, 2) are realvalued real-analytic functions in t such that (4.22) q(0) = ψ −1 , r 1 (0) = r 2 (0) = 0. Moreover, for t = 0 with sufficiently small absolute value, (4.23) r 1 (t)r 2 (t) < 0 or equivalently, \|q(t)\| > 1 holds. The inequality (4.23) corresponds to the argument in [27, Lemma 6.10]. If (4.23) holds, we can prove the existence of the desired deformation F t,ι1(t) by modifying ι(t) by ι 1 (t), as we shall see later. If r 1 (0)r 2 (0) had been negative, Claim 3 would be obvious. However, in our case, (4.22) implies r 1 (0)r 2 (0) = 0, although all of the examples in [27] satisfy r 1 (0)r 2 (0) = 0. We show (4.23) by examining the derivative of the monodromy matrix. Set (4.24) F t := F t,ι(t) , (4.25) ρ j =ρ j,t,ι(t) (j = 1, 2, 3). Then by Claims 1 and 2, we have (4.26) ρ 1 = e 0 , ρ 2 = σ 2 . Moreover, by Theorem A.6, ρ 3 is written as in (4.21), and by (2) in Proposition 4.12 and the fact that ι(0) = e 0 , we have ρ 3 → σ 3 as t → 0. Hence q, r 1 and r 2 in (4.21) satisfy (4.22). Let τ 0 and τ ∞ be the covering transformations on M k given in Lemma 4.10. Since the initial value ι(t) = F t (o) is real analytic in t, so are ρ Ft (τ 0 ) and ρ Ft (τ ∞ ). Lemma 4.14. F t := F t,ι(t) satisfies (4.27) trace ρ Ft (τ 0 ) = (−1) k 2 cos(πν 0 ), trace ρ Ft (τ ∞ ) = (−1) k 2 cos(πν ∞ ), where ν 0 = ν 0 (t) := k 1 + 4t k + 1 k , (4.28) ν ∞ = ν ∞ (t) := k 1 − 4t k + 1 k . (4.29) Proof. Let g be the secondary Gauss map of F t . Setting z = ζ k+1 , we can take a complex coordinate ζ around (0, 0). Since G is a meromorphic function and Q t has a pole of order 2 at (0, 0), (4.7) implies that the Schwarzian derivative S(g) has a pole of order 2 at (0, 0). Hence there exist a ∈ SL 2 C and a constant ν 0 such that (4.30) a ⋆ g = ζ ν0 1 + o(1) , where o(·) denotes a higher order term. Here, since G has a pole of order k at (0, 0), and Q t = tk k + 1 z 2 + 1 z 2 (z 2 − 1) dz 2 = − tk(k + 1) ζ 2 1 + o(1) dζ 2 (4.7) implies that (see [31, page 233]) S(g) = S(G) + 2Q t = 1 2ζ 2 (1 − k 2 ) − 4tk(k + 1) + o(1) dζ 2 . Similarly, (4.30) implies that S(g) = 1 2ζ 2 (1 − ν 2 0 )dζ 2 . Thus, ν 0 coincides with (4.28). Note that ν 0 is a real number for t with sufficiently small absolute value. Comparing the relation g • τ 0 = ρ Ft (τ 0 ) ⋆ g with (4.30), we can conclude that the eigenvalues of ρ Ft (τ 0 ) are equal to those of the monodromy matrix of the function ζ → ζ ν0 up to sign. Then the eigenvalues of ρ Ft (τ 0 ) are (4.31) e iπν0 , e −iπν0 or −e iπν0 , −e −iπν0 . Similarly, setting 1/z = ζ k+1 , we take a complex coordinate ζ around (∞, ∞). Then we have (4.29), and the eigenvalues of ρ Ft (τ ∞ ) are (4.32) e iπν∞ , e −iπν∞ or −e iπν∞ , −e −iπν∞ . On the other hand, by Lemma 4.10 and Proposition A.9 in the appendix, we have ρ Ft (τ 0 ) = σ 3 σ 2 2(k+1) ρ 3 ρ 2 2(k+1) = (−1) k ρ 3 ρ 2 2(k+1) , (4.33) ρ Ft (τ ∞ ) = (−1) k ρ 1 ρ 3 2(k+1) . (4.34) Here, by (2) in Proposition 4.12 and Claim 2, ρ j → σ j (t → 0) holds for j = 1, 2, 3. Since σ j (j = 1, 2, 3) are given explicitly in (4.15), we have that, as t → 0, (4.35) ρ Ft (τ 0 ) → (−1) k σ 2 σ 3 2(k+1) = (−1) 2k e 0 = e 0 , and ρ Ft (τ ∞ ) → e 0 . Then the eigenvalues of ρ Ft (τ 0 ) and ρ Ft (τ ∞ ) tend to 1 as t → 0. Hence by real analyticity, the right-hand possibilities for eigenvalues in (4.31) and (4.32) never occur, which implies the conclusion. Lemma 4.15. F t := F t,ι(t) satisfies d dt t=0 ρ Ft (τ 0 ) −1 = 2(k + 1)πi 1 0 0 −1 , d dt t=0 ρ Ft (τ ∞ ) −1 = −2(k + 1)πi 1 0 0 −1 . Proof. As in the proof of the previous lemma, we can take the complex coordinate ζ around (0, 0) such that z = ζ k+1 . Then G and Q are expressed in terms of ζ, and by (4.11) and (4.13), we have d dt t=0 ρ Ft (τ 0 ) −1 = γ0 1 c k G −G 2 1 −G Q dG = 2πi Res ζ=0 1 c k G −G 2 1 −G Q dG = 2πi k + 1 0 0 −(k + 1) , where γ 0 is the loop surrounding (0, 0) given in Lemma 4.10. Hence we have the conclusion for ρ Ft (τ 0 ). The derivative of ρ Ft (τ ∞ ) is obtained in a similar way. Now, we shall prove Claim 3: Proof of Claim 3. We have already shown (4.21) and (4.22). Then it is sufficient to show that \|q(t)\| > 1 for t = 0 with sufficiently small \|t\|. We set a 0 (t) = ρ 3 ρ 2 , then (4.33) can be rewritten as (4.36) ρ Ft (τ 0 ) = (−1) k (a 0 (t)) 2(k+1) . Set A 0 (t) := 1 2 trace(a 0 (t)). By Claim 2 and (4.21), we have that (4.37) A 0 (t) = 1 2 (ψ −2 q(t) + ψ 2 q(t)). Letting t → 0, we have A 0 (0) = cos{kπ/(2(k + 1))} by (4.22). Then there exists a real analytic function θ 0 = θ 0 (t) such that (4.38) cos θ 0 (t) = A 0 (t) = 1 2 trace(a 0 (t)), θ 0 (0) = kπ 2(k + 1) . Then the Cayley-Hamilton identity yields that a 0 (t) 2 = 2a 0 (t) cos θ 0 (t) − e 0 . Then by induction, one can prove the identity (purely algebraically) (a 0 ) m = sin(mθ 0 ) sin θ 0 a 0 − sin((m − 1)θ 0 ) sin θ 0 e 0 (m = 1, 2, 3, . . . ). By (4.36), we have ρ Ft (τ 0 ) = (−1) k sin(2k + 2)θ 0 sin θ 0 a 0 − sin(2k + 1)θ 0 sin θ 0 e 0 . Taking the trace of this, Lemma 4.14 yields cos πν 0 = sin(2k + 2)θ 0 sin θ 0 cos θ 0 − sin(2k + 1)θ 0 sin θ 0 = cos(2k + 2)θ 0 . Hence πν 0 (t) ≡ ±2(k + 1)θ 0 (t) (mod 2π). Comparing both sides of this equation at t = 0, (4.38) implies (4.39) πν 0 (t) = 2(k + 1)θ 0 (t) or πν 0 (t) = 2(k + 1)(π − θ 0 (t)), by real analyticity. On the other hand, by Lemma 4.15, it holds that −2(k + 1)πi 1 0 0 −1 = − dρ Ft (τ 0 ) −1 dt t=0 = dρ Ft (τ 0 ) dt t=0 = (−1) k d dt t=0 sin(2k + 2)θ 0 sin θ 0 a 0 − sin(2k + 1)θ 0 sin θ 0 e 0 , where we used the fact that ρ F0 (τ 0 ) = e 0 . Since θ 0 (0) = kπ 2k + 2 , sin((2k + 2)θ 0 (0)) = 0, cos((2k + 2)θ 0 (0)) = (−1) k , by settingθ 0 (0) = dθ 0 /dt\| t=0 , we have that −2(k + 1)πi " 1 0 0 −1 « = (−1) k d dt˛t =0 sin(2k + 2)θ0 sin θ0 a0 − sin`(2k + 2)θ0 − θ0ś in θ0 e0 ! = (−1) k d dt˛t =0 " sin(2k + 2)θ0 sin θ0 a0 − " sin(2k + 2)θ0 sin θ0 cos θ0 − cos(2k + 2)θ0 « e0 « = 2k + 2 sin θ0(0) " a0(0) − cos kπ 2k + 2 e0 «θ 0(0) = 2k + 2 sin kπ 2k+2 " σ3σ2 − cos kπ 2k + 2 e0 «θ 0(0) = −2(k + 1)i " 1 0 0 −1 «θ 0(0). Thus, we have (4.40)θ 0 (0) = e 0 . Then by (4.39), we have (4.41) θ 0 (t) = πν 0 (t) 2k + 2 . Similarly, if we set a ∞ = ρ 1 ρ 3 , then (4.34) can be rewritten as (4.42) ρ Ft (τ ∞ ) = (−1) k (a ∞ (t)) 2(k+1) . Set A ∞ (t) := 1 2 trace(a ∞ (t)). By Claim 2 and (4.21), we have that (4.43) A ∞ (t) = 1 2 (q(t) + q(t)). Letting t → 0, we have A ∞ (0) = cos{kπ/(2k + 2)} by (4.22). Then there exists a real analytic function θ ∞ = θ ∞ (t) such that (4.44) cos θ ∞ (t) = A ∞ (t) = 1 2 trace(a ∞ (t)), θ ∞ (0) = kπ 2k + 2 . Like as in the computation of θ 0 (t), we have (4.45)θ ∞ (0) = −π, θ ∞ (t) = πν ∞ (t) 2k + 2 . By (4.37), (4.38), (4.43) and (4.44), we have that ψ 2 q + ψ −2q = 2 cos θ 0 , q +q = 2 cos θ ∞ . Hence q(t) = −i sin kπ k+1 cos θ 0 (t) − ψ −2 cos θ ∞ (t) and qq = 1 sin 2 kπ k+1 (cos θ 0 ) 2 + (cos θ ∞ ) 2 − 2 cos kπ k + 1 cos θ 0 cos θ ∞ . Differentiating this twice using (4.40), (4.41) and (4.45), we have qq\| t=0 = 1, d dt t=0 qq = 0, d 2 dt 2 t=0 qq = 4(k + 1)π k tan πk 2k + 2 > 0. Thus, (\|q\| 2 =)qq > 1 for t with sufficiently small absolute value, and by (4.21), r 1 r 2 < 0 holds. Proof of Proposition 4.13. Take ι(t) as in Claim 2. We set ι 1 (t) = ι(t) s(t) 0 0 1/s(t) , s(t) = 4 −r 1 (t)/r 2 (t). Then by (1) of Proposition 4.12 and Claims 1 and 2, we haveρ 1,t,ι1(t) = e 0 ∈ SU 1,1 , ρ 2,t,ι1(t) = σ 2 ∈ SU 1,1 and ρ 3,t,ι1(t) = q(t) εi −r 1 (t)r 2 (t) −εi −r 1 (t)r 2 (t) q(t) ∈ SU 1,1 , where ε = 1 (resp. −1) if r 1 (t) > 0 (resp. r 1 (t) < 0). By replacing ι(t) by ι 1 (t), we have the conclusion. Thus, we obtain a one parameter family of CMC-1 faces {f k,t } defined on M k . When k is even, it induces f k : M ′ k → S 3 1 , where M ′ k is the Riemann surface of genus k/2 as in Section 3.5, because {μ 1 ,μ 2 ,μ 3 } generates the fundamental group of M ′ k . 4.5. Completeness and embeddedness. Now, we have weakly complete CMC-1 faces f k,t for positive integers k > 0 and for t = 0 with sufficiently small absolute value. In this subsection, we shall prove completeness of f k,t and embeddedness of f 1,t and f 2,t for sufficiently small t, which shows the second part of Theorem B. Proposition 4.16 (Completeness). For each positive integer k and for t = 0 with sufficiently small absolute value, the CMC-1 face f k,t : M k → S 3 1 is complete, and each end is a regular elliptic end in the sense of [12]. Proof. Since G is meromorphic at the ends, they are regular ends. Moreover, by (4.30) and (4.28), the end (0, 0) is g-regular non-integral elliptic in the sense of [12, Definition 3.3] because ν 0 ∈ πZ. Then by Lemma E1 in [12], the singular set does not accumulate at the end, and hence the end (0, 0) is complete. Similarly, the end (∞, ∞) is also complete. To show embeddedness, we shall look at the asymptotic behavior of the two ends of f k,t . The ideal boundary of S 3 1 consists of two connected components: ∂ + S 3 1 = LC + /R + and ∂ − S 3 1 = LC − /R + , where LC + (resp. LC − ) denotes the positive (resp. negative) light cone in R 4 1 : LC ± = {(v 0 , v 1 , v 2 , v 3 ) ∈ R 4 1 ; ±v 0 > 0}, see [12,Section 4]. Proof. Noticing Q t → 0 as t → 0, (4.7) for (G, Q t ) implies that S(g) → S(G) as t → 0. Since G(0, 0) = G(∞, ∞) = ∞ (see Table 1 in Section 3), this implies that \|g\| − 1 has the same sign at (0, 0) and (∞, ∞). In particular, one can choose g such that \|g\| > 1 on neighborhoods of (0, 0) and (∞, ∞). Then by Proposition 4.2 in [12], f k,t converges to ∂ − S 3 1 at the two ends. Moreover, since G(0, 0) = G(∞, ∞), the proof of [12,Proposition 4.2] implies that both of the ends converge to the same point of ∂ − S 3 1 . Proof. By Table 1 in Section 3, G has poles of order k at (0, 0) and (∞, ∞) in M k . Then there exists a complex coordinate ζ of M k around (0, 0) such that G = ζ −k . On the other hand, by (4.30), g is represented as g = ζ −ν0 α + o(1) α ∈ R \ {0}, where ν 0 > 0 is as in (4.28), since we can replace the secondary Gauss map g by 1/g. Then by Small's formula [12,Equation (1.10)] (see also [21]), the holomorphic lift F is expressed as F = 1 2 √ kν 0    ζ −k+ν 0 2 (k + ν 0 ) −1 √ α + o(1) ζ −k−ν 0 2 (k − ν 0 ) 1 √ α + o(1) ζ k+ν 0 2 (k − ν 0 ) ( √ α + o(1)) ζ k−ν 0 2 (k + ν 0 ) (− √ α + o(1))    , where o(·) denotes a higher order term. Hence the coordinate functions off k,t = F e 3 F * as in (4.1) are expressed as x 0 = −r −(k+ν0) α 0 + o(1) , x 3 = −r −(k+ν0) α 0 + o(1) , x 1 + ix 2 = r −ν0 e ikθ α 1 1 + o(1) , where α 0 , α 1 are positive real numbers and ζ = re iθ . Hence x 0 → −∞ as r → 0 and x 1 + ix 2 = C 1 e ikθ x ν 0 k+ν 0 0 1 + o(1) , x 3 = x 0 1 + o(1) , where C 1 is a non-zero constant. Similarly, the end (∞, ∞) is expressed as x 1 + ix 2 = C 2 e ikθ x ν∞ k+ν∞ 0 1 + o(1) , x 3 = x 0 1 + o(1) , where ν ∞ is as in (4.29) and C 2 is a non-zero constant. Since ν 0 = ν ∞ , these two ends do not intersect each other in sufficiently small neighborhood of the ends. Moreover, each end has no self intersection if and only if k = 1, or k = 2 (notice that M 2 is a double cover of M ′ 2 ). Appendix A. Reflections and fundamental groups In this appendix, we review the properties of the fundamental group and reflections on Riemann surfaces, which were used in Section 4. When two paths γ j : [0, 1] → M (j = 1, 2) satisfy γ 1 (1) = γ 2 (0), we denote by γ 1 * γ 2 the path obtained by joining γ 1 and γ 2 as follows: (A.1) γ 1 * γ 2 (u) := γ 1 (2u) if u ∈ [0, 1/2], γ 2 (2u − 1) if u ∈ [1/2, 1]. We denote the set of loops at o by L o (M ) := {γ ∈ C o (M ) ; γ(1) = o}. The fundamental group π 1 (M ) is the set of homotopy classes of L o (M ) with group multiplication induced from (A.1), which acts on M as covering transformations, as follows: (A.2) τ : M ∋ [γ] −→ [γ 1 * γ] ∈ M τ = [γ 1 ] ∈ π 1 (M ) , where γ 1 ∈ L o (M ) and γ ∈ C o (M ). An orientation-reversing conformal involution µ : M → M is called a reflection of M . In this section, we want to define a reflection of M as a lift of µ. Let P µ : [0, 1] → M be a path starting from o and ending at µ(o). We suppose that P µ is µ-invariant, that is, (A.3) µ • P µ (u) = P µ (1 − u) = P µ −1 (u) (u ∈ [0, 1]). Now we define a mapμ : M −→ M by (A.4)μ([γ]) := [P µ * (µ • γ)], which is called the lift of µ with respect to P µ . Then the following assertion holds: Lemma A.1. The liftμ is an involution of M . Proof. By (A.3), P (µ) −1 * P (µ) is homotopic to the constant map [0, 1] ∋ u → o ∈ M . Thenμ •μ([γ]) =μ([P µ * (µ • γ)]) = [P −1 µ * P µ * (µ • µ • γ)] = [γ]. We now prove the following: Proposition A.2. Let µ 1 ,. . . , µ 2r be a sequence of reflections on M , and take the liftμ j of µ j with respect to the curve P µj for each j = 1, . . . , 2r. Suppose that µ 1 • · · · • µ 2r is the identity map id M on M . Theñ µ 1 • · · · •μ 2r : M −→ M gives a covering transformation on the universal covering M of M which corresponds to the loop P µ1 * (µ 1 • P µ2 ) * (µ 1 • µ 2 • P µ3 ) * · · · * (µ 1 • µ 2 • · · · • µ 2r−1 • P µ2r ). We callμ 1 • · · · •μ 2r the covering transformation associated with µ 1 ,. . . , µ 2r . Proof of Proposition A.2. For the sake of simplicity, we write P j := P µj . Then for each γ ∈ C o (M ), it holds that µ 1 • · · · •μ r ([γ]) = [P 1 * (µ 1 • P 2 ) * · · · * (µ 1 • · · · • µ 2r−1 • P 2r ) * (µ 1 • · · · • µ 2r • γ)]. Since µ 1 • · · · • µ 2r = id M , we have the conclusion. . For such an immersion F , there exists a representation ρ F : π 1 (M ) → SL 2 C such that F • τ = F ρ F (τ ) −1 τ ∈ π 1 (M ) , where τ is considered as a covering transformation. Moreover, we set a holomorphic function g on M as g = −dF 12 /dF 11 = −dF 22 /dF 21 (that is, g is the secondary Gauss map) where F = (F ij ). Then it satisfies g • τ = ρ F (τ ) ⋆ g, where ⋆ denotes the Möbius transformation: (A.8) a ⋆ g := a 11 g + a 12 a 21 g + a 22 a = (a ij ) ∈ SL 2 C . We call ρ F the representation associated to F . Proof of Proposition A.4. By admissibility, Ψ is an sl 2 C-valued holomorphic oneform. Then the existence and uniqueness of F follows. Since Ψ is τ -invariant, F • τ is also a solution of (A.6). Hence the existence of ρ F follows. The final assertion can be shown directly. Let µ be a reflection on M . Then an admissible pair (G, Q) is said to be µ- where ⋆ denotes the Möbius transformation in (A.8). Moreover, such a matrix σ(µ) is unique up to ±-ambiguity, and is written in the following form: σ(µ) = q µ is µ is µ q µ , q µ ∈ C, s µ ∈ R, \|q µ \| 2 + (s µ ) 2 = 1 . In particular σ(µ)σ(µ) = e 0 holds, where e 0 is the identity matrix. Proof. By (A.9), the pull-back ds 2 F S := 4\|dG\| 2 /(1 + \|G\| 2 ) 2 of the Fubini-Study metric of C ∪ {∞} by G is µ-invariant. Hence G • µ is an orientation-preserving developing map of ds 2 F S , as well as G. Then there exists σ ∈ SU 2 such that G • µ = σ ⋆ G. Since G = G • µ • µ = σσ ⋆ G,σσ = ±e 0 holds. Ifσσ = −e 0 holds, σ = ± 0 −1 1 0 because σ ∈ SU 2 . In this case, for a fixed point z of µ, G(z) = G • µ(z) = σ ⋆ G(z) = − 1 G(z) , that is, \|G(z)\| 2 = −1 holds, which is impossible. Then σσ = e 0 , and by a direct calculation we have the conclusion. where σ(µ) is as in Lemma A.5. Moreover, ρ(μ) is written as (A.12) ρ(μ) = q is 1 is 2 q q ∈ C, s j ∈ R (j = 1, 2), \|q\| 2 + s 1 s 2 = 1 . Proof. By (A.9) and Lemma A.5, Ψ • µ = σΨσ −1 holds, where σ = σ(µ). Then d(σF )(σF ) −1 = σΨσ −1 = Ψ • µ = d(F •μ)(F •μ) −1 , which implies that σ −1 F and F •μ satisfy the same equation. Thus, there exists ρ = ρ(μ) ∈ SL 2 C such that F •μ = σF ρ −1 . Sinceμ is an involution and σσ = e 0 , ρρ = e 0 holds. Noticing ρ ∈ SL 2 C, we have (A.12). Finally, we write a representation ρ F as in Proposition A.4 of the fundamental group in terms of reflections. Definition A.7. Let µ 1 ,. . . , µ N be mutually distinct reflections on M and take the liftμ j of µ j for j = 1, . . . , N as in (A.4). If each covering transformation τ ∈ π 1 (M ) has an expression (A. 13) τ =μ i1 • · · · •μ i2r , then {μ 1 , . . . ,μ N } is called a generator of π 1 (M ). We now fix a generator {μ 1 , . . . ,μ N } and take an admissible pair (G, Q) on M which is µ j -invariant for each j = 1, . . . , N . Choose σ(µ j ) as in Lemma A.5 for each j = 1, . . . , N . Lemma A.8. If a covering transformation τ ∈ π 1 (M ) is written as in (A.13) in terms of a generator, σ(µ i1 )σ(µ i2 ) . . . σ(µ i2r−1 )σ(µ i2r ) is equal to e 0 or −e 0 . Proof. Since µ i1 • · · · • µ i2r = id M , we have G = G • µ i1 • · · · • µ i2r = σ(µ i1 )σ(µ i2 ) . . . σ(µ i2r−1 )σ(µ i2r ) ⋆ G. Thus, we have the conclusion. Then, by the definition of the representation ρ F in Proposition A.4, we have Proposition A.9. Under the situations above, we have ρ F (τ ) = σ(µ i1 )σ(µ i2 ) . . . σ(µ i2r−1 )σ(µ i2r ) ρ(μ i1 )ρ(μ i2 ) . . . ρ(μ i2r−1 )ρ(μ i2r ) . Note that the ±-ambiguity of σ(µ j ) does not affect this expression, because if one were to choose −σ(µ j ) instead of σ(µ j ), ρ(μ j ) changes to −ρ(μ j ). Hence we have Theorem A. 10. Let M be a Riemann surface with reflections {µ 1 , . . . , µ N }, and assume its lift {μ 1 , . . . ,μ N } is a generator of π 1 (M ). Take an admissible pair (G, Q) on M which is µ j -invariant for each j = 1, . . . , N , and let F be a solution of (A.6), ρ F a representation as in Proposition A.4, and ρ(μ j ) (j = 1, . . . , N ) as in (A.11). Then, if ρ(μ j ) ∈ SU 1,1 holds for all j = 1, . . . , N , the image ρ F (π 1 (M )) lies in SU 1,1 . Figure 1 . 1The duality between cone-like singular points and fold singular points. The pair (G, η) denotes the Weierstrass data, see Section 1. Figure 2 . 2An associated surface of the Lorentzian helicoid, with Weierstrass data (G, η) = (z, e πi/4 z −2 dz). G ≥ −χ(M ) + (number of ends) holds for the degree of the Gauss map of complete maxfaces f : M → R 3 1 , and equality holds if and only if all ends are properly embedded. Here of (2) for a = 3.67 Singular set of (3) for c = 0.1 Both trinoids have eight swallowtails and cuspidal edges for singular points. No cuspidal cross caps appear. For the notations, see Section 1. Figure 5 . 5The trinoids in Example 1. However, there is still a scarcity of examples of complete maxfaces, especially embedded examples. In light of the relationships between maxfaces and CMC-1 faces, we give a pair of open problems about finding new surfaces: Problem 1. Are there complete maxfaces (resp. complete CMC-1 faces) with embedded ends of genus greater than one in R 3 1 (or S 3 1 )? Furthermore, could such examples actually be embedded (in the wider sense)? Figure 6 . 6The example for k = 1 (Kim-Yang toroidal maxface) and half of it. Fact 1 . 1 11([32, Theorem 2.6]). Let M be a Riemann surface with a base point o ∈ M , and (G, η) a pair of a meromorphic function and a holomorphic 1-form on M such that (1.1) (1 + \|G\| 2 ) 2 \|η\| 2 gives a (positive definite) Riemannian metric on M , and \|G\| is not identically 1. . . . , N ), for loops {γ j } N j=1 such that the set {[γ j ]} of homotopy classes generates the fundamental group π 1 (M ) of M . Then , 1 + G 2 , i(1 − G 2 ) η is well-defined on M and gives a maxface in R 3 1 . Moreover, any maxfaces are obtained in this manner. The induced metric ds 2 and the second fundamental form II are expressed as (1.5) ds 2 = 1 − \|G\| 2 2 \|η\| 2 and II = Q + Q, (Q = η dG), respectively. Weak completeness as in Definition I in the introduction is equivalent to completeness of the metric (1.1). The point p ∈ M is a singular point of the maxface (1.4) if and only if \|G(p)\| = 1. Fact 1 . 2 12([33]). Let M be a Riemann surface, and f : M → R 3 1 a weakly complete maxface. Then f is complete if and only if there exists a compact Riemann surface M and a finite number of points p 1 , . . . , p n ∈ M such that M is conformally equivalent to M \ {p 1 , . . . , p n }, and the set of singular points Σ = {p ∈ M ; \|G(p)\| = 1} Fact 1 . 3 13([32, Theorem 3.1], [14, Theorem 2.4]). Let U be a domain of the complex plane (C, z) and f : M → R 3 1 a maxface, (G, η) its Weierstrass data. Set (1) A point p ∈ M is a singular point of f if and only if \|G(p)\| = 1. (2) f is right-left equivalent to a cuspidal edge at p if and only if Im α(p) = 0, (3) f is right-left equivalent to a swallowtail at p if and only if Im α(p) = 0 and Re β(p) = 0, (4) and f is right-left equivalent to a cuspidal cross cap at p if and only if 2. 1 . 1Cone-like singular points. Definition 2 . 1 ( 21Cone-like singular points). Let Σ 0 be a connected component of the set of singular points of the maxface f as in (1.4) which consists of non-degenerate singular points in the sense of [32, Section 3] and [14]. Then each point of Σ 0 is called a generalized cone-like singular point if Σ 0 is compact and the Lemma 2. 3 . 3A connected component Σ 0 of the set of singular points of the maxface (1.4) consists of generalized cone-like singular points if and only if it is compact, and α = 0 and Im α = 0 holds, where α is a function on M as in (1.6). Proof. Take a complex coordinate z around p ∈ Σ 0 and identify the tangent plane of M with C. The point p is non-degenerate if and only if α = 0, because of [32, Lemma 3.3]. The singular direction (the tangential direction of the singular set) and the null direction (the direction of the kernel of df ) are represented as i(G ′ /G) and i/(Gη), where ′ = d/dz and η =η dz (see [32, Proof of Theorem 3.1]). Here, the image f (Σ 0 ) consists of one point if and only if these two directions are linearly dependent. Thus we have the conclusion. Lemma 2. 4 . 4Assume a connected component Σ 0 of the singular set of a maxface consists of generalized cone-like singular points. Then it consists of cone-like singular points if and only if G\| Σ0 : Σ 0 → S 1 ⊂ C is injective and η does not vanish on Σ 0 , where (G, η) is the Weierstrass data. The image of the surface with (2.1) and a = 2.5. cone-like singular setThe singular set in the z-plane. The imaginary axis corresponds to the set of cone-like singular points, and the other connected components each consist of cuspidal edges, four swallowtails (the intersection points of the singular set with the curve Im α = 0, the curves shown in thin lines), and four cuspidal cross caps (the intersection points of the singular set with the curve Re α = 0, the dotted curves). Figure 7 . 7A maxface with a cone-like singularity and other singularities 2.2. Proof of Theorem A. Let M := C ∪ {∞} \ {−1, 1} and let a be a real constant such that 1 < a < 4 and a = 2. Set Figure 8 . 8The loops γ and κ 2 • γ. ( 3 . 313)f k :=f k,c k is single-valued on M k . By Lemma 3.2,f k is a complete maxface of genus k with 2 ends. See Figures 6, 9 and 10. Figure 9 . 9The example for k = 2 and half of it. Figure 10 . 10The example for k = 3 and half of it. 2m of genus m, where m is a positive integer. As a submanifold of (C ∪ {∞}) 2 , M ′ k has singular points at z = 0, 1 and z = ∞. However, in a similar way to the case of M k (see (3.2)), the structure of a Riemann surface can be introduced to M ′ k . The map π : M ′ k ∋ (Z, W ) −→ Z ∈ C ∪ {∞} is a meromorphic function of degree 2m + 1 with total branching number 6m. Then, by the Riemann-Hurwitz relation, the genus of M ′ k is equal to m. Let (3.14) all ends are embedded if and only if m = 1 (cf. [32, Theorem 4.11]). Moreover, if this is the case, embeddedness of f 2 can be shown in a similar way to the case off 1 . Compare Figures 6 (for k = 1) and 9 for the case k = 2.3.6. Swallowtails and cuspidal cross caps of f k . In this section, we investigate the properties of the singular points. Lemma 3. 6 . 6The number c k in (3.12) satisfies Lemma 3. 7 ( 7The singular curve). The set of singular points off k consists of 2 simple closed curves on M k . The projection of the singular set onto the z-plane is shown inFigure 11.Proof. The set of singular points is represented as Σ := {p ∈ M k ; \|G(p)\| = 1} (cf. Fact 1.1). Here, by (3.8), the condition \|G\| = 1 is equivalent to \|c k w/z\| = 1, and then, by (3.1), it is equivalent to Figure 11 . 11The singular set off k .Lemma 3.8. On each connected component of the singular set in Lemma 3.7, there are 2(k + 1) swallowtails and 2(k + 1) cuspidal cross caps. Singular points other than these points are cuspidal edges. α = 0 if and only if z ∈ R ∪ iR. On the other hand, since Re β = r 2 − r −2 cos 2θ (z = re iθ ), Corollary 3 . 9 . 39When k = 2m,f m and f k are not congruent. Definition 4.1. A pair (G, Q) of a meromorphic function G and a holomorphic 2-differential Q on M is said to be admissible if (positive definite) Riemannian metric on M . Proposition 4. 2 . 2Let M be a Riemann surface and (G, Q) an admissible pair on M . Let F = (F ij ) : M → SL 2 C be a holomorphic map of the universal cover M of M such that F e 3 F 3* : M −→ S 3 1 is a CMC-1 face if \|g\| is not identically 1, where g is a meromorphic function on M defined by ( 4 . 45) g := − dF 12 dF 11 = − dF 22 dF 21 . Lemma 4. 4 4(cf.[27, Lemma 4.8]). Take an admissible pair (G, Q) on a Riemann surface M . Then (G, Q t = tQ) is also an admissible pair for all t ∈ R \ {0}. Let F := F t : M → SL 2 C be a solution of ( Definition 4.5 ([12, Definitions 1.2 and 1.3]). A CMC-1 face f : M → S 3 1 is called complete if there exists a symmetric 2-tensor T which vanishes outside a compact set in M , such that ds 2 + T is a complete Riemannian metric on M , where ds 2 is the induced metric by f as in (4.6). On the other hand, f is called weakly complete if the metric ds 2 # in (4.2) is complete. Like as in the case of maxfaces, we have Fact 4 . 6 46([33]). Let M be a Riemann surface, and f : M → S 3 1 a weakly complete CMC-1 face. Then f is complete if and only if there exists a compact Riemann surface M and a finite number of points p 1 , . . . , p n ∈ M such that M is conformally equivalent to M \ {p 1 , . . . , p n }, and the set of singular points Σ = {p ∈ M ; \|g(p)\| = 1} is compact. where [ ] denotes the homotopy class and the meaning of the above equalities is explained in Remark A.3. On the other hand, let τ 0 and τ ∞ be covering transformations corresponding to counterclockwise simple closed loops γ 0 and γ ∞ in M k around (z, w) = (0, 0) and (∞, ∞), respectively. (The projections of γ 0 and γ ∞ to the z-plane are both (k + 1)-fold coverings of simple closed loops in C.) a generator of the fundamental group of M k in the sense of Definition A.7 in the appendix. sin kλ) 2 + u(t) sin(kλ) Lemma 4. 17 . 17For t = 0 with sufficiently small absolute value, the two ends of f k,t are asymptotic to the same point of the same connected component of the ideal boundary. Proposition 4 . 18 . 418For t = 0 with sufficiently small absolute value, the CMC-1 face f k,t is embedded if and only if k = 1 or 2. A. 1 . 1Properties of reflections. Let M be a connected Riemann surface and fix a base point o ∈ M . We denote the set of continuous paths starting at o by C o (M ) := {γ : [0, 1] → M ; γ is continuous and γ(0) = o}. We denote by [γ] the homotopy class containing γ ∈ C o (M ). Then the universal covering space M can be canonically identified with the quotient space {[γ] ; γ ∈ C o (M )}, and the covering projection is given by π : M ∋ [γ] −→ γ(1) ∈ M. Remark A. 3 .F 3Proposition A.2 gives a method to explicitly write down the isomorphism between the covering transformation group and the fundamental group π 1 (M ). A.2. A certain analytic property of reflections. Let G be a meromorphic function and Q a holomorphic 2-differential on the Riemann surface M . Such a pair (G, Q) is called admissible if (positive definite) Riemannian metric on M . Let π : M → M be the universal covering as in Appendix A.1, and letõ ∈ M be the point corresponding to the constant path at o ∈ M (then π(õ) = o holds). For an admissible pair (G, Q) on M , we define G := G • π, Q := Q • π. Then ( G, Q) is an admissible pair on M which is invariant under the covering transformations. Consider the following ordinary differential equation (õ) = a ∈ SL 2 C. Proposition A.4. 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[ "PREVALENCE ESTIMATION FROM RANDOM SAMPLES AND CENSUS DATA WITH PARTICIPATION BIAS", "PREVALENCE ESTIMATION FROM RANDOM SAMPLES AND CENSUS DATA WITH PARTICIPATION BIAS" ]	[ "Stéphane Guerrier [email protected] \nGeneva School of Economics and Management & Faculty of Science\nDepartment of Economics\nUniversity of Geneva\nSwitzerland\n", "Christoph Kuzmics [email protected] \nGeneva School of Economics and Management\nUniversity of Graz\nAustria\n", "Maria-Pia Victoria-Feser [email protected] \nUniversity of Geneva\nSwitzerland\n" ]	[ "Geneva School of Economics and Management & Faculty of Science\nDepartment of Economics\nUniversity of Geneva\nSwitzerland", "Geneva School of Economics and Management\nUniversity of Graz\nAustria", "University of Geneva\nSwitzerland" ]	[]	Countries officially record the number of COVID-19 cases based on medical tests of a subset of the population with unknown participation bias. For prevalence estimation, the official information is typically discarded and, instead, random survey samples are taken. An exception is the surveys recorded by the Statistics Austria Federal Institute, were the sample contains information about the number of positive COVID-19 tests in the sample as well as the participants with a positive COVID-19 test measured through the official procedure in the population during the same period. We derive (maximum likelihood and method of moment) prevalence estimators, with possible measurement errors, based on a survey sample, that additionally utilize the official information. We show that they are substantially more accurate than the simple survey sample proportion of positive cases. Put differently, using the proposed estimators, the same level of precision can be obtained with substantially smaller survey sample sizes. Moreover, the proposed estimators are less sensitive to measurement errors due to the sensitivity and specificity of the medical testing procedure. The proposed estimators and associated confidence intervals are implemented in the companion open source R package cape.	null	[ "https://arxiv.org/pdf/2012.10745v2.pdf" ]	229,340,122	2012.10745	abd17d54760e688730d161b27e810e25b5a88b6f	PREVALENCE ESTIMATION FROM RANDOM SAMPLES AND CENSUS DATA WITH PARTICIPATION BIAS December 25, 2020 Stéphane Guerrier [email protected] Geneva School of Economics and Management & Faculty of Science Department of Economics University of Geneva Switzerland Christoph Kuzmics [email protected] Geneva School of Economics and Management University of Graz Austria Maria-Pia Victoria-Feser [email protected] University of Geneva Switzerland PREVALENCE ESTIMATION FROM RANDOM SAMPLES AND CENSUS DATA WITH PARTICIPATION BIAS December 25, 2020maximum likelihood estimation · (generalized) method of moments · sample proportion · infectious disease · Clopper-Pearson confidence interval · measurement error Countries officially record the number of COVID-19 cases based on medical tests of a subset of the population with unknown participation bias. For prevalence estimation, the official information is typically discarded and, instead, random survey samples are taken. An exception is the surveys recorded by the Statistics Austria Federal Institute, were the sample contains information about the number of positive COVID-19 tests in the sample as well as the participants with a positive COVID-19 test measured through the official procedure in the population during the same period. We derive (maximum likelihood and method of moment) prevalence estimators, with possible measurement errors, based on a survey sample, that additionally utilize the official information. We show that they are substantially more accurate than the simple survey sample proportion of positive cases. Put differently, using the proposed estimators, the same level of precision can be obtained with substantially smaller survey sample sizes. Moreover, the proposed estimators are less sensitive to measurement errors due to the sensitivity and specificity of the medical testing procedure. The proposed estimators and associated confidence intervals are implemented in the companion open source R package cape. Introduction In the ongoing COVID-19 pandemic, governments face a trade-off between reducing the wealth or the health of citizens when choosing the degree of economic slowdown in their policy measures. The key to assess this trade-off is an understanding of the number or proportion of cases in the population and their evolution. Acquiring this understanding, in turn, depends on reliable estimates of the number of cases (at different points in time). The officially recorded number of positive cases can probably only be seen as a lower bound of the actual number of cases. The selection of participants to be medically tested is typically not complete and, importantly, also not random, but instead suffers from an unknown participation bias. The whole official procedure can, in fact, be understood as a complete census with (a possibly large) participation bias. It is typically unclear how many undetected positive cases there are in the population. Acknowledging this problem, for the case of COVID-19, some studies have proposed estimates for the prevalence among asymptomatic patients (see e.g. Nishiura et al., 2020, Mizumoto et al., 2020, or arXiv:2012.10745v2 [stat.ME] 23 Dec 2020 have attempted to infer from the prevalence obtained through the official procedure to the population one (see e.g. Manski and Molinari, 2020). In this paper, we instead propose to combine the information available in the data obtained through the official procedure that, as argued, suffers from participation bias, with data collected using a random sample of participants from the population all of which are medically tested. From this random sample, an unbiased estimator of the population proportion of positive cases, ignoring the information available from the official procedure, is then simply the proportion of positive cases in the sample; see e.g. Bendavid et al. (2020); SORA (2020); Stringhini et al. (2020) for the analysis of COVID-19 prevalence. More precisely, we demonstrate that the information gathered through the official procedure, while not useful in its own, can be used to improve the accuracy of the best estimators derived from random samples. All what is needed, is to also record, for each participant in the random sample, whether they are already part of the official statistics, i.e., whether they have been already declared positive through the official procedure. Appropriate estimators can then be derived whose key input is the number of new cases found in the sample. We show that these estimators are substantially more accurate than the standard proportion of cases in the random sample. Or put differently, appropriately utilizing the information obtained through the official procedure, means that the sample sizes for the survey can be substantially smaller and yet achieve the same statistical accuracy, thus, substantially reducing the costs and/or time for data acquisition. Alternatively, from the same survey sample, finer analysis at sub-population levels (e.g. regions) can reasonably be done even if the number of participants in these levels is rather small. We also provide several standard approaches to building confidence interval bounds for the proportion of positive cases, and compare, in a simulation study, their (finite sample) coverage properties. We also take into account possible misclassification errors of the (medical) testing devices used to collect the data see e.g., Kobokovich et al. (2020) and Surkova et al. (2020). The associated misclassification errors are actually induced by their sensitivity, i.e., the complement to the False Positive (FP) rate, and by their specificity, i.e., the complement to the False Negative (FN) rate, and adjusting for these errors avoids biased estimates (see e.g. Diggle, 2011;Lewis and Torgerson, 2012, and the references therein). Using a sensitivity analysis with the Austrian survey data, we actually find that the proposed estimators are much less influenced by the value of the FN rate, than the survey sample proportion, allowing, in practice, to limit the impact of the choice for the medical test specificity when estimating the proportion of positive cases. Such misclassification adjustments are also necessary with binary outcomes in logistic regression; see e.g. Ni et al. (2019) and Meyer and Mittag (2017), and the references therein. In this paper, we consider the case of estimating the proportion of positive cases, but the framework could easily be extended to the case of logistic regression. Moreover, while the data from the November 2020 survey collected by Statistics Austria (2020) is suitable for prevalence estimation, i.e. the population proportion of Austrians infected by the COVID-19 in November 2020, the same approach can be used to estimate other proportions such as the incidence of the COVID-19 (see e.g. Woodward, 2014). For the sensitivity and specificity, we use cutoff values, hence without the need to specify a (prior) distribution for these quantities (see e.g. McDonald and Hodgson, 2018;Bouman et al., 2020, and the references therein). Finally, the data from the Austrian survey (Statistics Austria, 2020) is performed using the cape R package which includes the new methods developed in this paper (see Section 7 for more details). The paper is organised as follows. We first present the formal setup in Section 2. In Section 3 we derive associated estimators and inference procedures, also treating the case of possible (partially) missing information. In Section 4 we present a simulation study that confirms and quantifies the theoretical results we develop in the previous sections. In Section 5 we apply the methodology to the case of the COVID-19 prevalence estimation and associated confidence bounds in Austria. The Model Consider taking a (random) survey sample of n participants in some population in order to estimate the population proportion π of, for example, a given infectious disease. Our framework also supposes that prior to the collection of the survey sample, a known proportion of individuals in the population have been declared positive through an official procedure based on an incomplete census or a census with participation bias. The official procedure has two steps. First, participants are selected based on some unknown criteria. Second, selected participants are medically tested for the disease. For each participant i = 1, . . . , n in the survey sample, there are three random variables of interest. X i := 1 if participant i is positive, 0 otherwise; Y i := 1 if participant i is tested positive in the survey sample, 0 otherwise; Z i := 1 if participant i was declared positive with the official procedure, 0 otherwise. (1) We assume that, for each participant i = 1, . . . , n in the survey sample, we observe Y i and Z i , but not X i . The objective is to provide an estimator for the unknown population proportion π := P (X i = 1) . We allow for the possibility that the outcome of (medical) tests can be subject to misclassification error. Let π 0 := P(Z i = 1), α 0 := P(Z i = 1\|X i = 0), α := P(Y i = 1\|X i = 0), β 0 := P(Z i = 0\|X i = 1), β := P(Y i = 0\|X i = 1). The probabilities α and β, are the (assumed known) FP rates (α = 1 − specificity) and FN rates (β = 1 − sensitivity) of the particular medical test employed in the survey. The probabilities α 0 and β 0 , are respectively the (assumed known) FP and FN rates of the official procedure. REMARK A: Note that α 0 is not the FP rate of the medical test administered in the official procedure. It is the probability that a participant has been incorrectly declared positive through the official procedure and, therefore, the product of two probabilities: the probability that a negative individual is selected to be tested in the official procedure multiplied with the probability that the medical test is positive conditional on this individual being (selected and) negative. In many applications α 0 will, therefore, be, sometimes substantially, smaller than the FP rate of the medical test. The FN rate β 0 of the official procedure is not known (otherwise we would know the population proportion π) and depends on π 0 , π and α 0 as follows: π 0 = P(Z i = 1) = P(X i = 0)P(Z i = 1\|X i = 0) + P(X i = 1)P(Z i = 1\|X i = 1) = (1 − π)α 0 + π(1 − β 0 ), Thus, β 0 = 1 − π 0 − α 0 (1 − π) π . It is useful to make three small assumptions. ASSUMPTION A: α + β < 1. ASSUMPTION B: α 0 + β 0 < 1. ASSUMPTION C: The survey sample is collected completely at random, without replacement. Its size n is small compared to the population size. With Assumption A, we rule out the uninteresting case α + β = 1. Indeed, if α + β = 1, Y i would be completely uninformative about the random variable of interest X i , as P(X i = 1\|Y i = 1) = P(X i = 1\|Y i = 0) = π. Otherwise Assumption A is without loss of generality in the following sense. If α + β > 1, we could just use Y i = 1 − Y i instead of Y i , which would have FP and FN rates of α = 1 − α and β = 1 − β, with α + β < 1. Assumption B is similarly without loss of generality. It implies that α 0 ≤ π 0 . To see this suppose that α 0 > π 0 = (1 − π)α 0 + π(1 − β 0 ). This is equivalent to 0 > −πα 0 + π(1 − β 0 ), which in turn, is equivalent to 0 > 1 − α 0 − β 0 , a contradiction. Assumption C specifies the type of sampling method assumed in this paper. Extensions to weighted sampling methods, with non random weights, would require a relatively straightforward adjustment of the proposed estimators, that we omit for clarity of exposition. Moreover, assuming that the sample size is relatively small compared to the population size, allows one to consider distributional properties of the variables that can be easily defined, in that binomial distributions can be used to approximate hypergeometric distributions. REMARK B: The unknown population proportion of positive cases π is bounded from below by π := π0−α0 1−α0 . To see this, recall that the equality π 0 = (1 − π)α 0 + π(1 − β 0 ) must hold (with both π and β 0 unknown parameters). The lowest admissible value for π is achieved when β 0 = 0, in which case we get the lower bound π0−α0 1−α0 . If α 0 = 0 then π = π 0 . Note that, given the assumptions, 0 ≤ π0−α0 1−α0 ≤ 1. From these variables we construct the following random variables that will be used to formulate the models: R 11 := n i=1 Y i Z i , R 10 := n i=1 (1 − Y i )Z i ,(2)R 01 := n i=1 Y i (1 − Z i ), R 00 := n i=1 (1 − Y i )(1 − Z i ) = n − R 11 − R 10 + R 01 . In words, R 11 is the number of participants in the survey sample that are tested positive and have also been declared positive through the official procedure; R 10 is the number of participants in the survey sample that are tested negative but have been declared positive through the official procedure; R 01 is the number of participants in the survey sample that are tested positive but have been declared negative through the official procedure; R 00 is the number of participants in the survey sample that are tested negative and have been declared negative through the official procedure. We also make use of R * 1 = n i=1 Y i = R 11 + R 01 , the number of participants that are tested positive in the survey sample. The success probabilities (see Supplementary Material A for their derivation), denoted by τ ij (π) associated to each R ij , i, j ∈ {0, 1} in (2) are given by τ 11 (π) := P(Z i = 1, Y i = 1) = π∆α 0 + (π 0 − α 0 )(1 − β) + αα 0 , τ 10 (π) := P(Z i = 1, Y i = 0) = −π∆α 0 + (π 0 − α 0 )β + (1 − α)α 0 , τ 01 (π) := P(Z i = 0, Y i = 1) = π∆(1 − α 0 ) − (π 0 − α 0 )(1 − β) + α(1 − α 0 ), τ 00 (π) := P(Z i = 0, Y i = 0) = −π∆(1 − α 0 ) − (π 0 − α 0 )β + (1 − α)(1 − α 0 ),(3) where ∆ := 1 − (α + β). Without misclassification error, we would have τ 11 (π) = π 0 , τ 10 (π) = 0, τ 01 (π) = π − π 0 , τ 00 (π) = 1 − π. Moreover, it is easy to verify that given our Assumptions, we have that the τ 's are non-negative and sum up to 1. Estimation and Inference In this section we derive Maximum Likelihood Estimators (MLE), a marginal MLE when some data is missing, and some Generalized Method of Moment (GMM) estimators. We also provide (exact) fiducial confidence intervals when possible, such as for a Method of Moment Estimator (MME) estimator under the assumption that the FP rates are zero. We also provide confidence intervals based on the estimators' asymptotic distribution. We compare the accuracy of the proposed estimators (that utilize the information from the official procedure) with the survey MLE that is the sample proportion of positive cases in the survey sample (that ignores the information from the official procedure). Estimators Survey MLE The benchmark estimator which is based only on R * 1 (= R 11 + R 01 ), the number of positive cases in the survey sample, is given byπ = R * 1 /n − α ∆ ,(4) which reduces toπ = R * 1 /n, when α = β = 0. It is actually the MLE of π based only on the survey sample. Its variance is given by var(π) = (τ 11 (π) + τ 01 (π))(1 − τ 11 (π) − τ 01 (π)) n∆ 2 = (π∆ + α)(1 − π∆ − α) n∆ 2 .(5) Conditional MLE Under Assumption C, the likelihood function for π can be obtained from the multinomial distribution with categories provided by R 11 , R 10 , R 01 , R 00 and their associated success probabilities τ 11 (π), τ 10 (π), τ 01 (π), τ 00 (π). The log-likelihood function is, therefore, given by (π\|R 11 , R 10 , R 01 , R 00 ) = C + 1 i=1 1 j=0 R ij ln(τ ij (π)),(6) where C is a quantity independent of π. The conditional MLE, i.e., the one based on the log-likelihood given in (6), which is hence conditional on the information provided by the official procedure, is defined bŷ π := argmax π∈[π,1] (π\|R 11 , R 10 , R 01 , R 00 ),(7) with π given in Remark B. The conditional MLEπ, generally, has no closed-form solution but can be computed numerically. However, in the case when α 0 = 0, we obtain a closed-form solution given bŷ π = π 0 R 00 + R 01 ∆ (R 01 + R 00 ) − π 0 β ∆ − α ∆ .(8) When α 0 = α = β = 0, this further reduces tô π = π 0 n − R * 1 n − R 11 + R 01 (n − R 11 ) .(9) REMARK C: The closed form expression in (8) is the conditional MLE only if the estimate is within the interval [π, 1]. There are, however, possible (but unlikely in practice) combinations of parameter values and sample realisations for which the likelihood function is maximized at the boundaries, i.e. either at π or at 1. In the case of no misclassification errors (α 0 = α = β = 0) the estimate given in 9 is automatically within [π, 1]. REMARK D: When α 0 = α = β = 0, in (1), Y i = X i and Z i ≤ X i , so that, in (2), R 10 = 0, R 01 = n i=1 Z i , and R * 1 = n i=1 X i with R 01 ≤ R * 1 . Under Assumption C, we have that R * 1 ∼ B(n, π) and, conditionally on R * 1 , we obtain the conditional model R 01 \|R * 1 ∼ B(R * 1 , π0 π ). The associated (conditional) likelihood function is, therefore, given by L(π\|R 01 , R * 1 ) = n R * 1 (π) R * 1 (1 − π) (n−R * 1) r R 01 π 0 π R01 1 − π 0 π (R * 1−R01) , with associated conditional MLE given in (9). In Proposition 1 below, we show the consistency and asymptotic normality of the conditional MLE defined in (7). PROPOSITION 1: The conditional MLEπ defined in (7) is consistent for π. Moreover, if π ∈ (π, 1), we have √ n (π − π) D − −−− → n→∞ N 0, 1 I(π) , where I(π) =                dτ11(π) dπ 2 τ 11 (π) + 1 j=0 dτ0j (π) dπ 2 τ 0j (π) if α 0 = β = 0 1 i=0 1 j=0 dτij (π) dπ 2 τ ij (π) otherwise. The proof of Proposition 1 is provided in Supplementary Material B. GMM Estimators Alternatively, we can consider an estimator from the class GMM estimators (Hansen, 1982) based on the random variable R := [R 11 /n, R 10 /n, R 01 /n, R 00 /n] with expectation E[R] := τ (π) = [τ 11 (π), τ 10 (π), τ 01 (π), τ 00 (π)]. A GMM estimatorπ is given byπ := argmin π∈[π,1] Q(π\|R), with Q(π\|R) := (R − τ (π)) T Ω (R − τ (π)) , where Ω is a fixed 4 by 4 positive definite matrix with entries ω ij , i, j = 1, ..., 4. Since τ (π) is a linear combination of π, we can write τ (π) := aπ + b, with a = [a l ] l=1,...,4 , b = [b l ] l=1,...,4 two vectors derived from (3). Then, assuming an interior solution exists (a remark similar to Remark C applies),π is the root of d dπ Q(π\|R) = −2 (R − τ (π)) T Ωa. Therefore, we obtainπ = (R − b) T Ωa a T Ωa ,(10) and it follows that E[π] = π. For a general matrix Ω,π is a linear combination of the elements of R, and it would be useful to choose Ω such that the distribution ofπ is known (for all n), for the construction of exact confidence bounds. One such case is obtained when ω ij = 1 for i = j = 3 and 0 otherwise, i.e. the GMM is reduced to a MME based on R 01 (with expectation τ 01 (π)), which, again assuming an interior solution exists, is given byπ ∈ [π, 1] that solves τ 01 (π) = R 01 n . This yieldsπ = 1 ∆(1 − α 0 ) R 01 n + π 0 − βπ 0 − α 0 ∆ − α .(11) When α 0 = α = β = 0, this reduces toπ = π 0 + R 01 n .(12) REMARK E: Interestingly, in the case of no misclassification errors (α 0 = α = β = 0),π can also be seen as an approximation to the MLE (in 9) for small values of π 0 and π, i.e., by simplifying (n − R * 1 )/(n − R 11 ) ≈ 1 and π 0 (n − R 11 ) ≈ π 0 n. Moreover, we have E[π] = π, i.e., the moment estimator is unbiased, and the variance is easily determined to be var (π) = 1 ∆ 2 (1 − α 0 ) 2 var R 01 n = τ 01 (π)(1 − τ 01 (π)) n∆ 2 (1 − α 0 ) 2 .(13) The possible advantage of the MMEπ in (11) is that is has a known finite sample distribution, based on R 01 ∼ B(n, τ 01 (π)), so that exact confidence bounds can be computed using, for example, the Clopper-Pearson method, see below. Actually, using (10) and setting ω ij = 1 for i = j = l and 0 otherwise, l = 1, . . . , 4, we can obtain all the MME corresponding to the different variables in R, asπ (l) = R l /n − b l a l ,(14) with E[π (l) ] = π for all l = 1, . . . , 4, and also known finite sample distribution. In Supplementary Material C we propose an alternative and more efficient moment estimator based on a (variance minimizing) linear combination of thẽ π (l) , but unfortunately without known finite sample distribution. However, when α 0 tends to zero (recall Remark A for the interpretation of α 0 ), this minimum variance GMM estimator is in fact the MME in (11). Missing information In some cases it might be that the information in R 10 (and R 00 ) in (2) is not easily available, for example, when additional data is collected using follow-up procedures. In that case, one can proceed with the marginalization of the likelihood function in (6) on the unknown quantities, leading to * (π\|R 11 , R 01 ) = C + R 11 ln(τ 11 (π)) + R 01 ln(τ 01 (π))+ + E [R 10 ] ln(τ 10 (π)) + (n − R − E [R 10 ]) ln(τ 00 (π)) = C + R 11 ln(τ 11 (π)) + R 01 ln(τ 01 (π))+ + nτ 10 (π) ln(τ 10 (π)) + (n − R − nτ 10 (π)) ln(τ 00 (π)), where C is a quantity independent of π. The marginal MLE is given by π := argmax π∈[π,1] * (π\|R 11 , R 01 ),(15) and, generally, has no closed form. It can however be easily computed using a numerical optimisation method. As for the conditional MLE, we show the consistency and asymptotic normality of the marginal MLE in (15) in Proposition 2 below. The proof is omitted as it follows closely the one of Proposition 1. Also, the exact expression of the asymptotic variance denoted by I * (π) −1 , is not explicitly provided here but implemented in the cape R package (see Section 7). PROPOSITION 2: The marginal MLEπ in (15) is consistent for π. Moreover, if π ∈ (π, 1), we have √ n (π − π) D − −−− → n→∞ N 0, 1 I * (π) . Efficiency In this section, we compare the variance of the various estimators to assess their efficiency relative to the Cramer-Rao lower bound variance (that the conditional MLE achieves asymptotically) of all unbiased estimators. The closed form expressions for the variance are given in (5) for the survey MLE and in (13) for the MME. No closed form expressions of the finite sample variance of the conditional MLE and the marginal MLE are easily obtained, not even for the case of no misclassification errors. The Cramer-Rao lower bound, which is also the asymptotic variance of the conditional MLE, is given by the reciprocal of the Fisher information, that is I(π) −1 = −E ∂ 2 ∂π 2 (π\|R) −1 .(16) One can provide a lengthy closed form expression for I(π) −1 , see Proposition 1. In practice, based on simulations (not presented here), the sample variance appears indistinguishable from the asymptotic variance, from sample sizes of n ≥ 500. In Section 4, we perform a simulation study, with parameter values loosely inspired by what one might expect for estimating the COVID-19 prevalence using PCR tests, to empirically assess the efficiency of the various estimators by considering the ratio of the Cramer Rao lower bound and the variance of the estimator. In this section we formally compute efficiency ratio, in the case of no misclassification errors, in order to highlight the increased precision that we get by considering the information from the official procedure. To do so, let α 0 = α = β = 0 and consider the ratio of the variance ofπ (in 5) relative toπ (in 13): var (π) var (π) = π(1 − π) (π − π 0 )(1 + π 0 − π) = π(1 − π) π(1 − π) − π 0 (1 + π 0 − 2π) . Therefore, when 2π > 1 + π 0 we have var(π) < var(π), while when 2π < 1 + π 0 we have var(π) > var(π). A sufficient condition for the variance of the MME to be lower than the variance of the survey MLE is, therefore, that the true population proportion π is below one half. On the other hand, the efficiency of the survey MLE relative to the (asymptotic) conditional MLE, in this case, is given by e(π) = I(π) −1 var(π) = π − π 0 π(1 − π 0 ) < 1, since π 0 ≤ π < 1. Moreover, since the variance of the conditional MLE is also the Cramer-Rao lower bound for the variance of any unbiased estimator of π, the MME, being unbiased, must have a higher variance. Indeed, the relative efficiency ofπ versus the conditional MLE (for sufficiently large n) is given by e(π) = I(π) −1 var(π) = (1 − (π − π 0 ))(1 − π 0 ) 1 − π ≤ 1, since π ≥ π 0 . The efficiency loss ofπ relative toπ can also be expressed in terms of the increase in sample size needed when usingπ rather thanπ. Let n * denote the sample size that is needed to obtain a variance for the survey MLEπ that is equal to the one of MMEπ using a sample size of n. We obtain n * n = 1 − π 0 1 − π0 π , which, for small π 0 , is approximately equal to 1 1−π0/π . If, for instance, π = 2π 0 then n * n ≈ 2. The added value in using the additional information provided in R 11 , therefore, is equivalent to using the survey MLE with a sample with twice the size. Confidence bounds Although the MMEπ has a (typically small) efficiency loss relative to the conditional MLE, it has the advantage of having a known distribution through R 01 ∼ B(n, τ 01 (π)). This allows one to construct (exact, but possibly conservative) confidence intervals even in finite samples without appealing to the estimator's asymptotic normal distribution, using the (fiducial) approach put forward in Clopper and Pearson (1934) (see also e.g. Fisher, 1935;Brown et al., 2001). A Clopper-Pearson (CP) (1 − γ) confidence interval based on the survey MLE, i.e., based on R * 1 , is given by I γ 2 (R * 1 ) − α ∆ < π < I 1− γ 2 (R * 1 ) − α ∆ , where, generally, I γ 2 (r) = B γ 2 ; r, n − r + 1 , I 1− γ 2 (r) = B 1 − γ 2 ; r + 1, n − r , and where B(p; v, w), 0 ≤ p ≤ 1, is the cumulative distribution function of a beta distribution with shape parameters v and w. A CP (1 − γ) confidence interval can be constructed based on the moment estimator (11), i.e., based on the information provided by R 01 . Given that E (3)), a (1 − γ) confidence interval for π, is given by [R 01 ] = π∆(1 − α 0 ) − (π 0 − α 0 )(1 − β) + α(1 − α 0 ) (seeI γ 2 (R 01 ) + (π 0 − α 0 )(1 − β) − α(1 − α 0 ) ∆(1 − α 0 ) < π < I 1− γ 2 (R 01 ) + (π 0 − α 0 )(1 − β) − α(1 − α 0 ) ∆(1 − α 0 ) . Using the conditional and marginal MLEs we can also provide confidence intervals based on their asymptotic normal distribution. All these confidence intervals are compared in our COVID-19 inspired simulation study in Section 4 and in our case study using actual COVID-19 data from an Austrian survey sample in Section 5. Simulation study In this section, we present the efficiencies, coverage and confidence interval lengths of the different methods, in finite samples. This section is parameterized in such a way that it is loosely compatible with the case of COVID-19 prevalence estimation using PCR tests. In particular, The FP and FN rates have been chosen so that they correspond to sensitivity and specificity commonly encountered in COVID-19 medical tests, as for example reported by the Center for Health Security of the John Hopkins University (Kobokovich et al., 2020), see also (Surkova et al., 2020). Throughout we choose α 0 = 0, the FP rate of the official procedure. We do so because, as pointed out in Remark A, α 0 is the product of two probabilities, here the probability of a COVID-19 negative person being selected to be tested in the official procedure and the FP rate of the PCR test employed in the official procedure. Given the relative low official prevalence of COVID-19, at least at the moment of writing this article, this product must be fairly close to zero. If, for instance, 1% of the member of a population have been found positive through the official procedure and if the FP rate of the PCR test is another 1%, we get an α 0 = (0.01) 2 = 0.01%. We consider three settings. Setting I is without misclassification error, i.e. with α 0 = α = β = 0. Setting II has only a FN rate, i.e. α 0 = α = 0, β = 2%. Setting III, finally, has both types of misclassification errors, i.e., α 0 = 0, α = 1%, β = 2%. We consider a sample size of n = 2, 000 which leads to the same conclusions (not presented here) as a somewhat smaller sample size (e.g. n = 1, 500). For π, we consider three rather different values, i.e. 5%, 20% and 75% in order to cover a wide range of possible prevalence rates. For π 0 , we consider, for each value of π, 30 equally spaced values between 1.025 min(α 0 , π) and 0.975π, so that, conditionally on the information brought in by Z i , one can appreciate the efficiency and accuracy gain of the approach based on the conditional model. As estimators, we consider the survey MLEπ in (4), the conditional MLEπ in (7), the MMEπ in (11) as well as the marginal MLEπ in (15) for the plausible cases when the information on R 10 and R 00 in (2) is not available. is a substantial efficiency loss for the survey MLEπ that increases drastically as π 0 approaches π, with or without misclassification errors. This is in line with the fact that the information brought in by considering Z i (1), is more important as π 0 is near π, and ignoring it, lowers the efficiency. Second, for the marginal MLE, the efficiency loss is negligible throughout the different settings, so there is little gain in considering R 10 and R 00 in (2), especially when this information is difficult/costly to obtain. Third, for the MME, the efficiency loss is negligible for π = 5% and π = 20% when π 0 is not too near to π, while the efficiency loss is rather important for small values of π 0 (relative to π), compared to the one of the survey MLE when π = 75%. Figure 2 presents the coverage (at the 95% level), computed using simulations, for the CP method based on R * 1 in (2), which is associated to the survey MLEπ, the CP method based on R 01 in (2), which is associated to the MMEπ, and the asymptotic method based on the conditional MLEπ. The coverage for the asymptotic method based on the marginal MLEπ are not presented as they are the same as the ones for the asymptotic method based on the conditional MLE. Overall, as expected, the CP method provides slightly conservative coverage across settings, while the asymptotic method based on the survey MLE is slightly liberal, especially for π = 5%. Moreover, for both the CP method based on R 01 and the asymptotic method based on the conditional MLE, for π = 5% and π = 20%, the coverage worsens (even if they remain quite accurate) as π 0 approaches π. For the asymptotic method, this can be explained by the fact Figure 2: Empirical coverage (at the 95% level) for the CP method based on R * 1 in (2), the CP method based on R 01 in (2) and the asymptotic method based on the conditional MLEπ. Top panels: α 0 = α = β = 0. Middle panels: α 0 = α = 0, β = 2%. Bottom panels: α 0 = 0, α = 1%, β = 2%. The sample size is 2, 000 and the number of Monte Carlo simulations is 50, 000. that confidence intervals might have bounds falling outside the domain of π (e.g. below π 0 ), especially when π is near π 0 and in settings such as Setting II. Given that the coverage is reasonable across methods, it is worth comparing the confidence interval lengths. Figure 3 presents the relative confidence interval (at the 95% level) lengths, computed using simulations, for the CP method based on R * 1 in (2) (associated to the survey MLEπ) and the CP method based on R 01 in (2) (associated to the MMẼ π), relative to the confidence interval (at the 95% level) lengths for the asymptotic method based on the conditional MLEπ. One can observe, as expected, that the (mean) confidence interval lengths can be a lot larger when ignoring the information provided by Z i in (1), especially as the information increases, i.e. as π 0 approaches π. An interesting feature appears, however, for a small population proportion (π = 5%) when π 0 approaches π, in that the mean confidence interval length for the CP based on R 01 (associated to the MME) is smaller than the one of the asymptotic method based on the conditional MLE. However, for a large population proportion (π = 75%), the mean confidence interval length for the CP based on R * 1 are relatively smaller than the ones based on R 01 , while remaining larger than the mean confidence interval length for the asymptotic method based on the conditional MLE. This is especially the case for small values of π 0 relative to π, and is in line with the study of the efficiencies provided in Figure 1. Figure 3: Relative empirical confidence interval (at the 95% level) mean lengths for the CP method based on R * 1 in (2) and the CP method based on R 01 in (2), relative to the empirical confidence interval (at the 95% level) mean lengths for the asymptotic method based on the conditional MLEπ. Top panels: α 0 = α = β = 0. Middle panels: α 0 = α = 0, β = 2%. Bottom panels: α 0 = 0, α = 1%, β = 2%. The sample size is 2, 000 and the number of Monte Carlo simulations is 50, 000. Case study: Application to Austrian COVID-19 survey We use the methodology developed in this paper for the case of the COVID-19 prevalence estimation using the results of a survey done in November 2020 by Statistics Austria (2020). We also compare the different approaches, in order to illustrate, in practice, the impact of choosing one method rather than another one. In November 2020, a survey sample of n = 2287 was collected to test for COVID-19 using PCR-tests. Seventy-one participants (R * 1 = 71) were tested positive, and among these ones, thirty-two (R 11 = 32) had declared to have been tested positive with the official procedure, during the same month. In November, there were 93914 declared cases among the official (approximately) 7166167 inhabitants in Austria (above 16 years old), so that π 0 ≈ 1.3105%. The sensitivity (1 − α) and the specificity (1 − β) are not known with precision, so that we present estimates of the prevalence without misclassification error as well as for values for the FP and FN rates, that are plausible given the data and according to the sensitivity and specificity reported in Kobokovich et al. (2020) or Surkova et al. (2020). Table 1 provides various estimates of π, the COVID-19 prevalence in Austria in November 2020, for the case of no misclassification error and for the case of misclassification errors with α = 1%, β = 10%, and α 0 = 0. Recall Remark A for the choice of α 0 = 0. We also chose a small α (FP rate for the medical test in the survey sample), because Table 1: Prevalence estimation for the Austrian data (November 2020) with associated 95% confidence intervals, using the conditional MLE (CMLE) with asymptotic confidence intervals, the moment estimator (MME) with Clopper-Pearson intervals and the survey MLE (SMLE) with asymptotic and Clopper-Pearson confidence intervals. For the later, two additional estimation are provided with n * = kn and R * * 1 = kR * 1 , with k = 1.5 (SMLE-CP * ) and k = 2 (SMLE-CP * * ). The original data are π 0 ≈ 1.3105%, n = 2287, R * 1 = 71, R 11 = 32. The CI are illustrated as horizontal bars with lengths associated to respectively 80%, 95% and 99% confidence levels, with a dot representing the estimate. The first three columns are under the assumption of no misclassification errors. The second three columns assume α = 1%, β = 10%, and α 0 = 0. α0 = α = β = 0 α0 = 0, α = 1%, β = 10% we only observe 71 positive cases out 2287 participants. If α were larger, say α = 5%, we would also expect a larger number of (misclassified) positive cases, i.e. 114 positive cases just because of false positives. From the first three lines of Table 1, one can derive a series of insights. First, we note that without misclassification errors, the estimates are very similar across methods. Second, as expected, the confidence intervals for the SMLE are wider than the ones associated to the conditional MLE (CMLE) or the MME. These two statements are true for both the case of no misclassification errors and the case of some misclassification errors. Third, in the case of misclassification errors, the estimates differ more substantially between the sample MLE and the conditional MLE or MME, with a difference of 10% in the estimate. Since the FP rate α has a limited number of possible values, given the data, we present in Figure 4 a sensitivity analysis of the prevalence estimation by the survey MLE and the MME, when the FN β varies from 0% to 30%. What is striking is that the sample MLE is much more influenced by the value of the FP β compared to the MME, which shows a far better stability. To understand this feature, from (11), we get, for the MME, under the sensitivity analysis conditions, π = 1 ∆ R01 n + (1 − β)π 0 . With increasing values for β, ∆ = 1 − (α + β) decreases, but at the same time, the quantity (1 − β)π 0 also decreases. On the other hand, with the survey MLE given in (4), an increase in the FP β directly induces an increased value for the estimator. Finally, in order to illustrate the accuracy gain of using a conditional MLE or MME, in Table 1, last two lines, we provide the prevalence estimate using the sample MLE with associated CI with 1.5 and 2 times as many sample data. In other words, the (hypothetical) data are built up by choosing n * = kn and R * * 1 = kR * 1 , with k = 1.5, 2. The aim of this exercise it to see if with more data, the sample MLE can provide an estimator that is as accurate as the conditional MLE or MME. One can see that, roughly, one would need twice as much survey sample data, in order to achieve the same level of accuracy provided by the MME or the conditional MLE. This is in line with the theoretical results provided in Section 3.2. Conclusions While we have cast this paper in the language of disease prevalence estimation, the method we propose has a more general range of applications. We actually propose a method to estimate the proportion of some characteristic in a population using information both from a random sample and from an incomplete census or census with participation bias. In other words, we are interested in the prevalence (or proportion) of population members having characteristic A, conditional on another characteristic B, such that having characteristic B implies having characteristic A, but not necessarily vice versa. We study this problem with and without the possibility of misclassification errors for A as well as for B. The approach that we propose for such settings is that when a random survey sample is drawn to not only record for each participant whether they have characteristic A or not, but also whether they have characteristic B or not. The key idea, to improve the accuracy of the estimate of the prevalence of characteristic A in the population, is to base the estimate appropriately on the number of participants in the sample that have characteristic A and not B. We propose MLE as well as MME derived from this idea. We show that our approach provides estimates that are substantially more accurate than the simple sample proportion (of participants with characteristic A), the maximum likelihood estimate that ignores the information available for characteristic B. As an important consequence, our approach can provide a given level of desired accuracy, with a substantially smaller sample size. This is useful when data collection is costly or, as for our COVID-19 example, medical tests (or lab spaces to evaluate test) are in limited supply. It would be straightforward to adapt the estimators to the case of weighted sampling, with non random weights, as well as to include explanatory variables in our model in the same vein as in generalized linear models by postulating a relationship of the proportion parameter of interest and an array of additional observable characteristics. Finally, there is some similarity of our approach and that of capture-recapture models (see e.g. Chao et al., 2001, and the references therein) used to estimate the size of a population. In capture-recapture models several samples are drawn randomly from a population with unknown size. Estimates of the size then, as in our approach, rely on the possibility of participants in a first sample showing up again in a second sample. To see the difference between the two approaches, we can place our framework in the language of capture-recapture models as follows. In our case, the first capture is taken as an incomplete census or a census with participation bias from a population of known size, and not a random sample from a population of unknown size. Software All computations presented in this paper were done using the ContionAl Prevalence Estimation, or cape R package that can be downloaded from https://github.com/stephaneguerrier/cape. Installation instructions as well as a user guide (vignette) of the package are provided in https://stephaneguerrier.github.io/cape/. All simulation results (as well as additional ones), can be reproduced and the simulation script is available on GitHub. SUPPLEMENTARY MATERIAL A Success probabilities The success probabilities τ j (π) for R j , j = 1, 2, 3, 4, in (2), can be deduced from the following table. There are two fundamental cases X i = 0 and X i = 1, and conditionally on each one of these cases, errors are independently and identically distributed. X i prob P(Y i = 1) P(Z i = 1) 1 π 1 − β 1 − β 0 0 1 − π α α 0 , where β 0 = 1 − π0−α0(1−π) π . We, thus, have τ 11 = P(X i = 1)P(Y i = 1\|X i = 1)P(Z i = 1\|X i = 1) + P(X i = 0)P(Y i = 1\|X i = 0)P(Z i = 1\|X i = 0) = π(1 − β)(1 − β 0 ) + (1 − π)αα 0 . Plugging in β 0 = 1 − π0−α0(1−π) π and using ∆ := 1 − (α + β) we obtain τ 11 (π) = π∆α 0 + (π 0 − α 0 )(1 − β) + αα 0 . The remaining probabilities τ 10 , τ 01 and τ 00 can be similarly obtained. B Proof of Proposition 1 PROOF: The identifiability of the model is straightforward from (3) and by the extreme value theorem we have E[\| ln p(R\|π)\|] < ∞, where p(R\|π) denotes the probability mass function of a multinomial distribution with event probabilities τ i , i = 1, 2, 3, 4 as defined in (3). Therefore, by applying the information inequality (see e.g. Lemma 2.2 of Newey and McFadden, 1994), we can verify the identification ofπ. By combining the compactness of Π, the (uniform) law of large numbers and/or Theorem 2.1. of Newey and McFadden (1994),π is a consistent estimator for p 0 . Then, if p 0 ∈ (π, 1), standard techniques can be used to show that √ n (π − p 0 ) D − −−− → n→∞ N 0, 1 I(p 0 ) , where I(π) = ∆ 2 α 2 0 τ 11 (π) [αα 0 + (1 − β)(π 0 − α 0 ) + πα 0 ∆] 2 + α 2 0 τ 10 (π) [β(π 0 − α 0 ) + α 0 (1 − π∆ − α)] 2 + (1 − α 0 ) 2 τ 01 (π) [(1 − β)(π 0 − α 0 ) − π (1 − α 0 ) ∆ − α(1 − α 0 )] 2 + (1 − α 0 ) 2 τ 00 (π) [(1 − α 0 ) (1 − π∆ − α) − β(π 0 − α 0 )] 2 . Finally, we verify that Assumption A guarantee that I(π) exists and is finite. Indeed, none of the equations: αα 0 + (1 − β)(π 0 − α 0 ) + πα 0 ∆ = 0 β(π 0 − α 0 ) + α 0 (1 − π∆ − α) = 0 (1 − β)(π 0 − α 0 ) − π (1 − α 0 ) ∆ − α(1 − α 0 ) = 0 (1 − α 0 ) (1 − π∆ − α) − β(π 0 − α 0 ) = 0, have a solution in (π, 1), which concludes the proof. C Alternative GMM estimators for the conditional model A possibly more efficient and closed form estimator can be obtained by choosing a weighted sum of theπ (l) , with weights summing to one to obtain an unbiased estimator with a smaller variance. Indeed, let for example ω ll = γ l , l = 1, 2, 3 and 0 otherwise, such thatπ and γ = [γ l ] l=1,2,3 , with 3 l=1 γ l = 1, we can choose γ such that min γ var(π(γ)). The fourth term l = 4 is omitted as it does not provide additional information, since we have that 1 i=0 1 j=0 R ij = n. As is shown below, we have that γ 1 = λα 0 2 1 − α 0 τ 00 + α 0 τ 11 γ 2 = λα 0 2 α 0 τ 10 − 1 − α 0 τ 00(19)γ 3 = λ(1 − α 0 ) 2 1 − α 0 τ 00 + 1 − α 0 τ 01 . One can see that the weight γ 3 is the most important, as α 0 is usually very small, see Remark A. Unfortunately, the weights γ depend on π, so that one needs to plug in a value. This could be chosen as being the one provided byπ in (11), which is a consistent estimator of π. Nevertheless, the finite sample distribution ofπ(γ) in (18) is unknown, so that one would need to resort to asymptotic theory, and this would not bring any advantage, in terms of inference, compared to the MLE. To obtain (19), we first develop (14) using (3) to obtaiñ π 1 = 1 ∆α 0 R 11 n − (π 0 − α 0 )(1 − β) − αα 0 π 2 = 1 ∆α 0 (π 0 − α 0 )β + (1 − α)α 0 − R 10 n π 3 = 1 ∆(1 − α 0 ) R 01 n + π 0 − βπ 0 − α 0 ∆ − α . Letting τ j := τ j (π), j = 1, . . . , 4, the variance of the GMMπ in (18), using the properties of the multinomial distribution, is given by var(π) = γ 2 1 n∆ 2 α 2 0 τ 11 (1 − τ 11 ) + γ 2 2 n∆ 2 α 2 0 τ 10 (1 − τ 10 ) + γ 2 3 n∆ 2 (1 − α 0 ) 2 τ 01 (1 − τ 01 ) +2 γ 1 γ 2 n∆ 2 α 2 0 τ 11 τ 10 − 2 γ 1 γ 3 n∆ 2 α 0 (1 − α 0 ) τ 11 τ 01 + 2 γ 2 γ 3 n∆ 2 α 0 (1 − α 0 ) τ 10 τ 01 . Minimizing the variance subject to j γ j = 1 is then equivalent to minimizing H(γ) = γ 2 1 α 2 0 τ 11 (1 − τ 11 ) + γ 2 2 α 2 0 τ 10 (1 − τ 10 ) + γ 2 3 (1 − α 0 ) 2 τ 01 (1 − τ 01 ) +2 γ 1 γ 2 α 2 0 τ 11 τ 10 − 2 γ 1 γ 3 α 0 (1 − α 0 ) τ 11 τ 01 + 2 γ 2 γ 3 α 0 (1 − α 0 ) τ 10 τ 01 −λ(γ 1 + γ 2 + γ 3 − 1). The first order conditions for minimality are then given by ∂H ∂γ 1 = 2γ 1 α 2 0 τ 11 (1 − τ 11 ) + 2γ 2 α 2 0 τ 11 τ 10 − 2γ 3 α 0 (1 − α 0 ) τ 11 τ 01 − λ = 0 ∂H ∂γ 2 = 2γ 2 α 2 0 τ 10 (1 − τ 10 ) + 2γ 1 α 2 0 τ 11 τ 10 + 2γ 3 α 0 (1 − α 0 ) τ 10 τ 01 − λ = 0 ∂H ∂γ 3 = 2γ 3 (1 − α 0 ) 2 τ 01 (1 − τ 01 ) − 2γ 1 α 0 (1 − α 0 ) τ 11 τ 01 + 2γ 2 α 0 (1 − α 0 ) τ 10 τ 01 − λ = 0 which can be simplified as 2γ 1 α 0 (1 − τ 11 ) + 2γ 2 α 0 τ 10 − 2γ 3 1 − α 0 τ 01 = λα 0 2τ 11 (20) 2γ 2 α 0 (1 − τ 10 ) + 2γ 1 α 0 τ 11 + 2γ 3 1 − α 0 τ 01 = λα 0 2τ 10 2γ 3 1 − α 0 (1 − τ 01 ) − 2γ 1 α 0 τ 11 + 2γ 2 α 0 τ 10 = λ(1 − α 0 ) 2τ 01 Using (20) in (21) to simplify for γ 3 yields γ 2 α 0 + γ 1 α 0 = λα 0 2 1 τ 11 + 1 τ 10 .(23) Similarly using (20) in (22) leads to γ 1 α 0 (1 − τ 11 − τ 01 ) + γ 2 α 0 τ 10 = λ 2 1 − α 0 + (1 − τ 01 )α 0 τ 11 .(24) Then, from (23) and (24), knowing that 1 i=0 1 j=0 τ ij = 1, we obtain γ 1 α 0 τ 00 = λ 2 1 + α 0 τ 11 (τ 00 − τ 11 ) , which leads to γ 1 in (19). Using γ 1 in e.g. (23), we obtain γ 2 in (19), and finally γ 3 is deduced as in (19). Figure 1 Figure 1 : 11presents the relative efficiencies, as measured by the relative empirical RMSE, for the MMEπ, the survey MLEπ, and the marginal MLEπ relative to the conditional MLEπ. The main messages are the following. First, there Relative efficiencies, as measured by the relative empirical RMSE, for the MMEπ (green lines), the survey MLEπ (red lines) and the marginal MLE (blue lines) relative to the conditional MLEπ. First raw with no misclassification error, middle row with FN positive rates (α 0 = α = 0, β = 2%), bottom row with both types of misclassification errors (α 0 = 0, α = 1%, β = 2%). The sample size is n = 2, 000 and the number of Monte Carlo simulations is 50, 000. Figure 4 : 4Sensitivity analysis for the prevalence estimation using the moment estimator and the survey MLE, according the the FP rate β. The confidence bounds are computed using the CP-method for both estimators. α 0 = 0, α = 1%, π 0 = 1.3105%, n = 2287, R * 1 = 71 and R 11 = 32. AcknowledgmentsStéphane Guerrier is partially supported by Swiss National Science Foundation grant #176843 and Innosuisse-Boomerang Grant 37308.1 IP-ENG. Maria-Pia Victoria-Feser is partially supported by a Swiss National Science Foundation grant #182684. We are grateful to Michael Greinecker, Helmut Kuzmics, Hans Manner, Michael Richter, Michael Scholz and Dominique-Laurent Couturier for helpful comments and suggestions. . E Bendavid, B Mulaney, N Sood, S Shah, E Ling, R Bromley-Dulfano, C Lai, Z Weissberg, R Saavedra-Walker, J Tedrow, D Tversky, A Bogan, T Kupiec, D Eichner, R Gupta, J Ioannidis, J Bhattacharya, Bendavid, E., B. Mulaney, N. Sood, S. Shah, E. Ling, R. Bromley-Dulfano, C. Lai, Z. Weissberg, R. Saavedra- Walker, J. Tedrow, D. Tversky, A. Bogan, T. Kupiec, D. Eichner, R. Gupta, J. Ioannidis, and J. Bhattacharya (2020). 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[ "A POLYHEDRAL APPROACH TO COMPUTING BORDER BASES", "A POLYHEDRAL APPROACH TO COMPUTING BORDER BASES" ]	[ "Gábor Braun ", "Sebastian Pokutta " ]	[]	[]	Border bases can be considered to be the natural extension of Gröbner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gröbner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is independent from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with preference on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree.The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces.	null	[ "https://arxiv.org/pdf/0911.0859v3.pdf" ]	115,174,075	0911.0859	c1a04db19e8946ed374674379ff4355b72f8a728	A POLYHEDRAL APPROACH TO COMPUTING BORDER BASES Gábor Braun Sebastian Pokutta A POLYHEDRAL APPROACH TO COMPUTING BORDER BASES Border bases can be considered to be the natural extension of Gröbner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gröbner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is independent from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with preference on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree.The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces. INTRODUCTION Gröbner bases are fundamental tools in commutative algebra to model and perform important operations on ideals such as intersection, membership test, elimination, projection, and many more. More precisely, the Gröbner bases framework makes these operations computationally accessible allowing to perform actual computations on ideals. Comparing border bases and Gröbner bases. Unfortunately, Gröbner bases are not always well suited to perform theses operations, in particular when the actual ideals under consideration are inferred from measured data. In fact, Gröbner bases do not react smoothly to small error in the input data. Border bases are a natural generalization of Gröbner bases that are believed to deform smoothly in the input (cf. [26]) and that have been used for computations with numerical data (cf., e.g., [17,1]). In [17] a numerically stable version of the Buchberger-Möller algorithm has been derived which heavily relies on the usage of border bases in order to ensure numerical stability. An alternative algorithm based on border bases with similar stability behavior was investigated in [1]. Both algorithms calculate bases of ideals of points which can be used to derive polynomial models from empirical datasets. It is well known that every Gröbner basis with respect to a degree-compatible term ordering can be extended to a border basis (see [20, p. 281ff]) but not every border basis is an extension of a Gröbner basis. Moreover, not every order ideal supports a border basis even if it has the right cardinality. An example illustrating these two cases is presented in [20,Example 6]. While the border basis algorithm in [20], which is a specification of Mourrain's generic algorithm [28], allows for computing border bases of zero-dimensional ideals for order ideals supported by a degree-compatible term ordering, it falls short to provide a border basis for more general order ideals: The computed border basis is supported by a reduced Gröbner basis. The alternative algorithm presented in [20,Proposition 5] which can potentially compute arbitrary border bases requires the a priori knowledge of the order ideal that might support a border basis so that the order ideal has to be guessed in advance. As we cannot expect this prior knowledge, the algorithm does not solve the problem of characterizing those order ideals for which a border basis does exist. Further, as pointed out in [20, p. 284], the basis transformation approach of this algorithm is unsatisfactory as it significantly relies on Gröbner basis computations. It can be favorable though to be able to compute border bases for order ideals that do not necessarily stem from a degree-compatible term ordering. It has been an open question to characterize those order ideals of a given zero-dimensional ideal I that support a border basis. We will answer this question for degree-compatible order ideals by resorting to polyhedral combinatorics, mentioned in Subsection 1.2. The restriction to degree-compatible order ideals is due to the design of the algorithm to proceed degree-wise and the authors believe that this restriction can be overcome as well. The preference of border bases over Gröbner bases partly arises from the iterative generation of linear syzygies, inherent in the border basis algorithm, which allows for successively approximating the basis degree-by-degree. Moreover, the border basis algorithm is a linear algebra algorithm with a tiny exception, the final step, which does not contribute to the inherent complexity though as its running time is polynomial in the input size. Polynomial method. In discrete mathematics and combinatorial optimization, polynomial systems have been used to formulate combinatorial problems such as the graph coloring problem, the stable set problem, and the matching problem (see e.g., [13] for an extensive list of references). This well-known method, which Alon referred to as the polynomial method (cf. [3,4]) recently regained strong interest. This emphasizes the alternative view of the border basis algorithm as a proof system which successively uncovers hidden information by making it explicit. In [14, Section 2.3] and [13,12] infeasibility of certain combinatorial problems, e.g., 3-colorability of graphs is established using Hilbert's (complex) Nullstellensatz and the authors provide an algorithm NulLA to establish infeasibility by using a linear relaxation. The core of the algorithm is identical to the L-stable span procedure used in the border basis algorithm, which intimately links both procedures. The difference is of a technical nature: whereas NulLA establishes infeasibility, the classical border basis algorithm as presented in [20] computes the actual border bases of the ideal. Another recent link between border bases and the Sherali-Adams closure (introduced in [33]), a method for convexification, was investigated in [31]. The authors show that the Sherali-Adams procedure can be understood as a weaker version of the L-stable span procedure, which is an essential part of the border basis algorithm. As a consequence a new tighter relaxation than the Sherali-Adams closure is derived by exploiting the stronger L-stable span procedure. Applications of border bases. Surprisingly, it turns out that there are deep connections to other mathematical disciplines, and border bases represent the combinatorial structure of the ideal under consideration in a canonical way. Although the use of border basis as a concise framework is quite recent, introduced in [19,20,21] and [25,Section 6.4], the concept of border basis is rather old and appeared in different fields of mathematics including computer algebra, discrete optimization, logic, and cryptography (independently) under different names. Originally, border bases were introduced in computer algebra as a generalization of Gröbner bases to address numerical instabilities. Border bases have been successfully used since for solving zero-dimensional systems of polynomial equations (see, e.g., [5,27,28]), which in particular include those with solutions in 0/1 and thus a large variety of combinatorial problems. Border bases have been also used to solve sparse quadratic systems of equations thus giving rise to applications in cryptography in a natural way. Such systems arise from crypto systems (such as AES, BES, HFE, DES, CTC variants, etc.) when rewriting the S-boxes as polynomial equations. The celebrated XL, XSL, MutantXL attacks, which are based on relinearization methods, are essentially equivalent to the reformulation-linearization-technique (RLT) of Sherali and Adams [33] and use a version of the Nullstellensatz to break ciphers. In fact the XL algorithm (see e.g., [22,8]) in its classical form is actually identical to a level d Sherali-Adams closure of the associated system and therefore it is a border bases computation at its core. Motivated by the success of the aforementioned methods, border bases have also been used in crypto analysis and coding theory, see [6]. Another application of border bases is the modeling of dynamic systems from measured data (see e.g., [17,23,1]) where the better numerical stability can be advantageous. Our contribution. A perennial problem in various applications is that the classical computation of a border basis depends on a degree-compatible term ordering and hence finds only the border bases supported by an order ideal induced by the ordering. These border bases do represent only a rather small fraction of all possible border bases (see Example 2.6). Further, in this case the border basis contains a reduced Gröbner basis, and for example the theoretically nice numerical/symmetry properties are often lost, as the vector space basis of K[ ]/I (where I is a zero-dimensional ideal) might not be optimally suited for numerical computation. Moreover, for example when solving large systems of polynomial equations, it is desirable to guide the solution process by having a vector space basis that does actually have a nice interpretation attached. These bases often cannot be obtained by a degree-compatible term ordering. Techniques from commutative algebra, and in particular border bases (and alike) have been of great value to discrete mathematics and combinatorial optimization. This time we will proceed the other way around. We will apply the combinatorial optimization toolbox to the combinatorial structure of zerodimensional ideals and we solve the aforementioned problems by freeing the computation of a border basis from the term ordering. Our contribution is the following: Polyhedral characterization of all border bases: We provide a complete, polyhedral characterization of all degree-compatible border bases of any zero-dimensional ideal I: we associate an order ideal polytope P to I that characterizes all its degree-compatible border bases. The integral 0/1 points in the order ideal polytope are in one-to-one correspondence with degreecompatible order ideals supporting a border basis of I. (An order ideal is degree-compatible if it is a degree-wise complement of I, see Definition 2.3.) This explicitly establishes the link between the combinatorial structure of the basis of the factor space and the structure of the ideal. Whether an order ideal supports a border basis is solely determined by the combinatorial structure of the order ideal polytope. A related result for Gröbner bases of a vanishing ideal of generic points was established in [29]. Computation of border bases not coming from a term ordering: We outline an algorithm based on the classical border basis algorithm as defined in [20] and show how to compute border bases for arbitrary degree-compatible order ideals without relying on a specific degreecompatible term ordering. Recall that not every order ideal supports a border basis and guessing the order ideal in advance is permissive. It is also not advised to search for an admissible order ideal with brute force as the combinatorial structure of the ideal can be so complicated that it is not even possible to easily guess a (non-term ordering induced) feasible order ideal, not even to mention to find one that has a preferred structure (see also the next point). Finding preferred order ideals: Our approach is also able to compute order ideals maximizing a prespecified preference function (and compute their border bases). We will show that computing a preference-optimal order ideal supporting a border basis is NP-hard in general, which is surprising, as choosing the order ideal is merely a basis transformation. The NP-hardness does not come from the hardness of computing the L-stabilized span, as the problem remains NPhard in cases where the L-stabilized span is small enough to be determined efficiently. Computational feasibility: We provide computational tests that demonstrate the feasibility of our method. Applications: The structure of ideals and counting of border bases: Having the order ideal polytope available for a zero-dimensional ideal I, it is possible to examine the structure of the ideal based on its border bases. One straight-forward application is counting the number of degreecompatible border bases for a zero-dimensional ideal I. Order ideals and determining those with maximum score do appear in a very natural way in combinatorial optimization as the so called maximum weight closure problem (cf. e.g., [30]) and they have a variety of applications, e.g., in open-pit mining where any feasible production plan is indeed an order ideal; clearly we can only mine the lower levels after having mined the upper ones. A good survey as well as an introduction to the problem can be found in [18]. Other, more involved applications might arise, e.g., in computational biology where the structure of boolean network is inferred from the Gröbner fan. Due to their higher numerical stability, border bases might be a better choice here and the order ideal polytope allows for a detailed study of the underlying combinatorial structure of the networks. Outline. We start with the necessary preliminaries in Section 2 and recall the classical border basis algorithm in Subsection 2.1. In Section 3 we introduce the order ideal polytope and establish the oneto-one correspondence between the 0/1 points of this polytope and degree-compatible border bases. We also derive an equivalent characterization that is better suited for actual computations. In Section 4 we then show how the results from Section 3 can be used to compute admissible degree-compatible border bases using a preference function for the structure of the order ideal. We also establish the NP-hardness of the optimization variant of this problem. As an immediate application, we also outline how to count all admissible border bases of a zero-dimensional ideal. We conclude with computational results in Section 6 and a few final remarks in Section 7. ]} and similarly, for a set of polynomials P ⊆ K[ ] we define the support of P to be supp(P) := p∈P supp(p). Given a term ordering σ, the leading term LT σ (p) of a polynomial p is LT σ (p) := t with t ∈ supp(p) such that for all t ∈ supp(P) \ {t} we have t > σ t ; the leading coefficient LC σ (p) of p is the coefficient of LT σ (p). We drop the index σ if the ordering is clear from the context. Recall that the degree of a polynomial p ∈ K[ ] is deg(p) := max x m ∈supp(p) m 1 . The leading form LF(p) of a polynomial p = l i=1 a i x m i ∈ K[ ] is defined to be LF(p) = l i=1, m i =d a i x m i where d = deg(p), i.e., we single out the part with maximum degree. Both LF and LT generalize to sets in the obvious way, i.e., for a set of polynomials P we define LF(P) := {LF(p) \| p ∈ P} and LT(P) := {LT(p) \| p ∈ P}. In the following we will frequently switch between considering polynomials M = {p 1 , . . . , p s }, the generated ideal, and the generated vector space whose coordinates are indexed by the monomials in the support of M . We denote the ideal generated by M as 〈M 〉 K[ ] and the vector space generated by M as 〈M 〉 K . For n ∈ we define [n] := {1, . . . , n}. All other notation is standard as to be found in [9,24]; we have chosen the border basis specific notation to be similar to the one in [20] where the border basis algorithm was first introduced in its current form. Central to our discussion will be the notion of an order ideal: Definition 2.1. Let be a finite subset of n . If for all t ∈ and t ∈ n such that t \| t we have t ∈ , i.e., is closed under factors, then we call an order ideal. Furthermore, the border ∂ of a non-empty order ideal is the set of terms ∂ := {x j t \| j ∈ [n], t ∈ } \ . As an exception, we set ∂ := {1} for the empty order ideal. (1) the polynomials in have the form g j = b j − µ i=1 α i j t i for j ∈ [ν] and α i j ∈ K; (2) 〈 〉 K[ ] = I; (3) K[ ] = I ⊕ 〈 〉 K as vector spaces. If there exists an -border basis of I then the order ideal supports a border basis of I. Note that the condition 〈 〉 K[ ] = I is a consequence of ⊆ I, the particular form of the elements in , and K[ ] = I ⊕ 〈 〉 K . More precisely, 〈 〉 K[ ] + 〈 〉 K is closed under multiplication by the x i , and hence it is an ideal. As it contains 1, it must be the whole ring, and hence 〈 〉 K[ ] = I by the modular law. See [21,Proposition 4.4.2] for another proof. In particular, an order ideal supports an -border basis of I if and only if K[ ] = I ⊕ 〈 〉 K . Moreover, for any given order ideal and ideal I the -border basis of I is unique as b j has a unique representation in K[ ] = I ⊕ 〈 〉 K for all j ∈ [ν]. Furthermore, as K[ ] = I ⊕ 〈 〉 K it follows that \| \| = dim 〈 〉 K is invariant for all choices of . The requirement for I being zero-dimensional is a consequence of this condition as well, as ∂ (and in consequence ) as part of the output of our computation should be finite. Clearly, as a vector space, I has a degree filtration, i.e., I = i∈ I ≤i where I ≤i := {p ∈ I \| deg(p) ≤ i}. For a set of monomials we define =i := {m ∈ \| deg(m) = i} (and similarly for ≤ instead of =). In the following we consider special types of order ideals, i.e., those that essentially preserve the filtration: Definition 2.3. Let I ⊆ K[ ] be a zero-dimensional ideal and let ⊆ n be an order ideal. We say that is degree-compatible (to I) if dim =i K = dim n =i K − dim I ≤i I ≤i−1 for all i ∈ . Thus, the -border basis of a zero-dimensional ideal I with respect to any degree-compatible order ideal has a pre-determined size for each degree i ∈ . Intuitively, the degree-compatible order ideals are precisely those that correspond to degree-compatible orderings on the monomials. The important difference is that the orderings do not have to be term orderings. The definition above only requires local compatibility with multiplication as is an order ideal and thus downwardly closed and degreecompatible, i.e., if p, q are polynomials and deg(p) < deg(q) then p ≤ q. The following example shows that the requirements of Definition 2.3 are not automatically satisfied by all order ideals . We also give an example of a degree-compatible ideal not coming from a term ordering. Recall that the order ideal associated to a Gröbner basis of an ideal consists of all monomials not divisible by any leading term in the Gröbner basis. Example 2.4 (A non-degree-compatible order ideal). With the degree lexicographic ordering on two variables, the set of polynomials x 3 1 + x 1 x 2 , x 2 1 x 2 , x 1 x 2 2 , x 3 2 is a Gröbner basis of an ideal with associated order ideal consisting of all monomials of degree at most 2: {1, x 1 , x 2 , x 2 1 , x 1 x 2 , x 2 2 }. The element x 3 1 + x 1 x 2 of the ideal enables us to replace x 1 x 2 with x 3 1 and thus obtain another order ideal of the ideal, which is not degree-compatible: {1, x 1 , x 2 , x 2 1 , x 3 1 , x 2 2 }. Example 2.5 (A degree-compatible order ideal not coming from a term ordering). The homogeneous ideal having a Gröbner basis in the degree lexicographic term ordering x 2 2 + x 1 x 2 + x 2 1 , x 1 x 2 2 , x 4 2 has the associated order ideal {1, x 1 , x 2 , x 1 x 2 , x 2 2 , x 3 2 }. Via a basis change, we obtain another order ideal {1, x 1 , x 2 , x 2 1 , x 2 2 , x 3 2 } , which cannot come from a term ordering. The reason is that every Gröbner basis of the ideal must contain a polynomial with degree-two leading term, which therefore must be a multiple of the first generator x 2 2 + x 1 x 2 + x 2 1 modulo higher-degree terms. The degree-two leading term cannot be x 1 x 2 because if e.g., x 1 < x 2 then the term x 2 2 is larger then x 1 x 2 . So the leading term must be either x 2 1 or x 2 2 , which excludes the order ideal. Example 2.6 (Generic ideal). Let k and n be positive integers and let {a i j } i∈[n], j∈[k] be algebraically independent real numbers over . Let I be the ideal of polynomials in the variables x 1 , . . . , x n which are zero on the points (a 1 j , . . . , a n j ) for j ∈ [k]. Thus, the ideal is zero-dimensional, and K[ ]/I has dimension k. Every k distinct monomials form a complementary basis of I, since they are linearly independent on the k points (a 1 j , . . . , a n j ). An equivalent formulation of linear independence is that the determinant of the matrix formed by the values of the monomials on these points is non-zero. The determinant is indeed non-zero, as it is a non-trivial polynomial of the algebraic independent a i j with integer coefficients. In particular, every order ideal of size k is an order ideal of I. The degree-compatible order ideals are the ones where the monomials have the least possible degree, i.e., consisting of all monomials of degree less than l and k − n+l−1 l−1 monomials of degree l, where l is the smallest non-negative integer satisfying k ≤ n+l l , i.e., there are at least k monomials of degree at most l. 2.1. Computing border bases for term ordering induced . Without proofs, we recall the classical border basis algorithm introduced in [20] as it will serve as a basis for our algorithm. The interested reader is referred to [21,19] for a general introduction to border bases and to [20] in particular for an introduction of the border basis algorithm. The classical border basis algorithm calculates border bases of zero-dimensional ideals with respect to an order ideal which is induced by a degree-compatible term ordering σ. First, the border basis algorithm heavily relies on the following neighborhood extension: Definition 2.7. (cf. [20, Definition 7.1 and paragraph preceding Proposition 13]) Let V be a vector space. We define the neighborhood extension of V to be V + := V + V x 1 + · · · + V x n . For a finite set W of polynomials, its neighborhood extension is W + = W ∪ W x 1 ∪ · · · ∪ W x n . Note that for a given set of polynomials W such that 〈W 〉 K = V we have W + K = 〈W 〉 + K = V + as multiplication with x i is a K-vector space homomorphism. It thus suffices to perform the neighborhood extension on a set of generators W of V . Let F be a finite set of polynomials and let L ⊆ n be an order ideal, then F ∩ 〈L〉 K = { f ∈ F \| supp( f ) ⊆ L}, i.e., F ∩ 〈L〉 K contains only those polynomials that lie in the vector space generated by L. Clearly, for L = n ≤d we have 〈F 〉 K ∩ 〈L〉 K = 〈F 〉 ≤d K . Using the neighborhood extension we can define: Definition 2.8. (cf. [20, Definition 10]) Let L be an order ideal and let F be a finite set of polynomials such that supp(F ) ⊆ L. The set F is L-stabilized if F + K ∩ 〈L〉 K = 〈F 〉 K . The L-stable span F L of F is the smallest vector space G containing F satisfying G + ∩ 〈L〉 K = G. A straightforward construction of the L-stable span of F is to inductively define the following increasing sequence of vector spaces: F 0 := 〈F 〉 K and F k+1 := F + k ∩ 〈L〉 K for k > 0. The union k≥0 F k is the L-stable span F L of F . The set L represents our computational universe and we will be in particular concerned with finite sets L ⊆ n . Note that L-stability is a property of the vector space and does not depend on the basis: Remark 2.9. The L-stable span of a finite set F depends only on the generated vector space 〈F 〉 K , as F + K = 〈F 〉 + K . In the following we will explain how the L-stable span can be computed explicitly for L = n ≤d . We will use a modified version of Gaussian elimination as a tool, which allows us to extend a given basis V with a set W as described in the following: (4) If f = 0 or i > r + η then go to step (7). Lemma 2.10. [20, Lemma 12] Let V = {v 1 , . . . , v r } ⊆ K[ ] \ {0} be a finite set of polynomials such that LT(v i ) = LT(v j ) whenever i, j ∈ [r] with i = j and LC(v i ) = 1 for all i ∈ [r].(5) If LT( f ) = LT(v i ) then replace f with f − LC( f ) · v i . Set i := 1 and go to step (4). (6) Set i := i + 1. Go to step (4). (7) If f = 0 then put η := η + 1 and let v r+η := f / LC( f ). Go to step (2). We can now compute the L-stable span using the Gaussian elimination algorithm 2.11: (1) V := GaussEl( , F ). (2). (2) W := GaussEl(V, V + \ V ). (3) W := {w ∈ W \| supp(w) ⊆ L} = {w ∈ W \| deg(w) ≤ d}. (4) If \|W \| > 0 set V := V ∪ W and go to step(5) Return V . The last ingredient that we need in order to formulate the border basis algorithm is the final reduction algorithm. This algorithm basically transforms the output of the border basis algorithm to bring it into the desired form of a border basis by applying linear algebra steps only. It interreduces the elements so that they only have support in the leading term and . It is easy to see that 〈L〉 K = F L ⊕ 〈 〉 K , so the algorithm will do a form of Gaussian elimination on the basis V to obtain a new basis V R with all non-leading terms supported on . Of course, this new basis will contain an -border basis. Algorithm 2.15 (Final Reduction Algorithm-FinalRed). Input: V , as in Lemma 2.14. Output: as in Lemma 2.14. (1) Let V R := . (2) If V = then go to step (8). (3) Let v ∈ V such that v has minimal leading term. Put V := V \ {v}. (4) Let H := supp(v) \ (LT(v) ∪ ). (5) If H = then append v/ LC(v) to V R and go to step (2). (6) For each h ∈ H choose w h ∈ V R and c h ∈ K such that LT(w h ) = h and h / ∈ supp(v − c h w h ). (7) Set v := v − h∈H c h w h , append v/ LC(v) to V R , and go to step (2). (8) Return := {v ∈ V R \| LT(v) ∈ ∂ }. We will now formulate the border basis algorithm. (1) Let d := max f ∈F deg( f ). (2) V = {v 1 , . . . , v r } := LStabSpan(F, n ≤d ). (3) Let := n ≤d \ {LT(v 1 ), . . . , LT(v r )}. (4) If ∂ n ≤d then set d := d + 1 and go to step (2). (5) Return := FinalRed(V, ). It is worthwhile to note that step (4) in Algorithm 2.17 is essentially for testing if the L-stable span is large enough to support an -border basis. The rationale for searching such a large span is due to the following proposition that serves as a stopping criterion. It is obvious from its Corollary 2.19 that the span is indeed minimal in the sense that it is the smallest span that contains a degree-compatible order ideal that supports a border basis (and thus all such degree-compatible order ideals). We obtain the following corollary: Corollary 2.19. LetĨ be an n ≤d -stabilized vector space satisfyingĨ + n ≤d−1 K = n ≤d K . Then Ĩ K[ ] ∩ n ≤d K =Ĩ. Proof. We apply Proposition 2.18 with the choice L := n ≤d , I := Ĩ K[ ] and := n ≤d \LT(Ĩ) where the leading terms are with respect to any degree-compatible term ordering. Clearly, n ≤d K =Ĩ ⊕ 〈 〉 K . The conditionĨ + n ≤d−1 K = n ≤d K ensures that consists of monomials of degree less than d, so ∂ ⊆ n ≤d . Hence the proposition applies, and we obtain K[ ] = I ⊕ 〈 〉 K . Together with n ≤d K =Ĩ ⊕ 〈 〉 K this gives I ∩ n ≤d K =Ĩ. The border basis algorithm decomposes into two components. The first one is the calculation of the L-stable span for L = n ≤d for sufficiently large d (this is the main 'work') and the second component is the extraction of a border basis via the final reduction algorithm. Effectively, after using any degreecompatible term ordering in order to compute the L-stable span one can choose a different ordering with respect to which the basis can be transformed. This approach is very much in spirit of the FGLM algorithm, Steinitz's exchange lemma, and related basis transformation procedures. In Remark 2.9 it is shown that L-stability does not depend on the basis and thus any sensible basis transformation that results in an admissible order ideal is allowed. In the following we will characterize all admissible order ideals. THE ORDER IDEAL POLYTOPE We will now introduce the order ideal polytope P (a 0/1 polytope) that characterizes all order ideals that support a border basis (for a given zero-dimensional ideal I) in an abstract fashion independent of particular vector space bases for the stable span approximation. Its role will be crucial for the later computation (Algorithm 4.1) of border bases for general degree-compatible order ideals. In the first subsection, we introduce the polytope in an abstract, invariant way highlighting its main property that its integral points are in bijection with degree-compatible order ideals supporting a border basis (Theorem 3.2). In the second subsection, we give a more direct reformulation targeted to actual computations. Given d ∈ in advance for which Algorithm 2.17 stops, LStabSpan computes the actual border basis (and some unused extra polynomials). The computation of the border basis is performed by Gaussian elimination applied to the matrices obtained within LStabSpan. The order ideal for which we effectively compute the border basis is solely determined by the pivoting rule when doing the elimination step. Degree-compatible term orderings ensure that we obtain an order ideal for which a border basis exists (given the correct d). If we would now pivot arbitrarily, which effectively means permuting columns, it is not clear that, first, the resulting ideal is an order ideal and, second, that it does actually support a border basis. If we remain in the setting of degree-compatible order ideals (which means that we only permute columns of monomials with same degree) then we face the combinatorial problem that when permuting a column such that it will end up being an element of we also have to ensure that all its divisors will also end up in . On the other hand, due to the degree-compatibility constraint (see Definition 2.3) we must choose an exactly determined amount of elements for each degree that will end up in . We will show now how to transform this combinatorial problem of faithful pivoting into a polyhedral setting. We obtain a 0/1 polytope P, the order ideal polytope that characterizes all admissible degree-compatible order ideals that support a border basis for the ideal at hand. Theoretical point of view. (3.1) We are ready to relate the order ideal polytope with order ideals. From now on, let Λ(I) denote the set of degree-compatible order ideals of a zero-dimensional ideal I. (3.2) (3.3) (3.4) z m 1 ≥ z m 2 ∀m 1 , m 2 ∈ n : m 1 \| m 2 m∈ n =i z m = dim n =i K − dim I ≤i I ≤i−1 ∀i m∈U z m ≤ dim U ∪ I ≤i I ≤i−1 K − dim I ≤i I ≤i−1 ∀i, U ⊆ n =i : \|U\| = dim n =i K − dim I ≤i I ≤i−1 z m ∈ [0, 1] ∀m ∈ n Theorem 3.2. Let I be a zero-dimensional ideal. There is a bijection between the set Λ(I) of its degreecompatible order ideals and the set of integral points of the order ideal polytope of I. The bijection is given by ξ: z ∈ P(I) ∩ n → (z) := {m ∈ n \| z m = 1}. Proof. In fact, we will see that the order ideal polytope is defined exactly to this end. First, the integral solutions z of Condition 3.4 are exactly the 0/1 points, i.e., the characteristic vectors of sets of terms (z) := {m ∈ n \| z m = 1}. Second, it is easy to see that Condition 3.6 means that (z) is indeed an order ideal, as whenever m 1 \| m 2 and m 2 ∈ (z), i.e., z m 2 = 1, then it follows that z m 1 = 1, i.e., m 1 ∈ (z) as well. In the third step we will provide an algebraic characterization of Conditions 3.2 and 3.3. Clearly, Condition 3.2 can be rewritten to \| (z) =i \| = dim n =i K − dim I ≤i I ≤i−1 . We will now show that Condition 3.3 is equivalent to (3.5) (z) =i K ∩ I ≤i I ≤i−1 = {0}, i.e., the image of (z) =i is linearly independent in the factor n =i K I ≤i I ≤i−1 . Similarly as above, Condition 3.3 can be rewritten to U ∩ (z) =i ≤ dim U ∪ I ≤i I ≤i−1 K − dim I ≤i I ≤i−1 , i.e., the size of U ∩ (z) =i is at most the dimension of the vector space generated by the image of U in the factor n =i K I ≤i I ≤i−1 . This is obviously necessary for the image of (z) =i to be linearly independent in the factor. (Here the size of U does not matter.) For sufficiency choose U := (z) =i . Then the dimension of the vector space generated by the image of (z) =i is at least \| (z) =i \|, so the image of (z) =i is independent. So far we have proved that the integral points of the order ideal polytope correspond bijectively to order ideals with the properties \| (z) =i \| = dim n =i K − dim I ≤i I ≤i−1 and (z) =i K ∩ I ≤i I ≤i−1 = {0} for all i. Lastly, we will show now that this is equivalent to \| (z) =i \| = dim n =i K − dim I ≤i I ≤i−1 and I ⊕ 〈 (z)〉 K = 〈 n 〉 K and thus the assertion follows. By dimensionality it follows, \| (z) =i \| = dim n =i K − dim I ≤i I ≤i−1 and (z) =i K ∩ I ≤i I ≤i−1 = {0} for all i together are equivalent to I ≤i I ≤i−1 ⊕ (z) =i K = n =i K for all i. For brevity, we will omit the phrase 'for all i'. Using the filtration argument, the latter is equivalent to I ≤i ⊕ (z) ≤i K = n ≤i K . Via a dimension argument on embeddings, this is further equivalent to I ⊕ 〈 (z)〉 K = 〈 n 〉 K and dim I ≤i + dim (z) ≤i K = dim n ≤i K . Finally, filtrating the dimension by degree shows that the latter is equivalent to I ⊕ 〈 (z)〉 K = 〈 n 〉 K and dim I ≤i I ≤i−1 + dim (z) =i K = dim n =i K . Computational point of view. Throughout this subsection we assume that M = i∈ M i ⊆ K[ ] is a finite set of polynomials with degree-filtration {M i \| i ∈ } that generates a zero-dimensional ideal 〈M 〉 K[ ] ⊆ K[ ] such that all p ∈ M i have degree i. Furthermore, for each i ∈ we have an enumeration M i = {p i j \| j ∈ [k i ]} with k i ∈ . As seen in Subsection 2.1, an important component of the border basis algorithm is the computation of L-stable spans with respect to some computational universe L ⊆ n . Computing a border basis with respect to a different order ideal is merely a basis transformation of the vector space obtained from the LStabSpan procedure. In the following we assume that L is of the form L = n ≤d where d ∈ is such that Algorithm 2.17 stops. In view of Remark 2.9 and Definition 2.3 we have that for a given computational universe L either all border bases (supported by degree-compatible order ideals) are contained in L or none. Furthermore we assume that M is L-stabilized and has in particular a convenient form. Effectively one might want to think of M being the output of the LStabSpan procedure, which is then brought into the following reduced form: Here we can freely choose leading terms of the polynomials with the only constraint that they have to be maximal-degree terms. We give a visual interpretation of the definition. The coefficient matrix A ∈ K M × n ≤ of M is the matrix where the rows are the elements of M , the columns are all the monomials of degree at most , and the entries are the coefficients of the terms in the elements of M . We use the convention that for terms t 1 , t 2 with deg(t 1 ) > deg(t 2 ) we put column t 1 to the left of column t 2 . Similarly, we put leading terms of a polynomial to the left of the other terms. Now M is in canonical form if the matrix A has the structure as depicted in Figure 3.2, i.e., it consists of degree blocks and each degree block is maximally interreduced. The degree blocks correspond to the leading forms of the polynomials in M . Any finite set can be brought into canonical form by applying Gaussian elimination and column permutations of terms with same degree if necessary. In particular, the output of the LStabSpan procedure can be easily brought into this form. The following lemma summarizes the basic properties of a set M in canonical form: (1) 〈M 〉 ≤i K[ ] 〈M 〉 ≤i−1 K[ ] ∼ = LF(M i ) K A =                   0 0 0 1                    FIGURE 3.2. canonical form (2) 〈M 〉 ≤i K[ ] = j≤i M j K (3) M i <i K = 0 and thus M i <i K ⊆ 0≤ j≤i−1 M j K Proof. We first show that M i <i K = 0 for all i ∈ [d]. Let i ∈ [d] be arbitrary and observe that each nonzero element p ∈ M i has degree i. As M is in canonical form, the polynomials in M i are interreduced (see the matrix in Figure 3.2 for Definition 3.3) and thus we also obtain each nonzero element p ∈ M i K has degree i. By Corollary 2.19, 〈M 〉 K[ ] ∩ 〈L〉 K = 〈M 〉 K . Hence 〈M 〉 ≤i K[ ] = 〈M 〉 ≤i K for i ∈ [d] . Now the statements of the lemma are obvious consequences of M being in canonical form. The following lemma provides us a practical way to compute the sizes of the degree components of degree-compatible order ideals, which are the same for all order ideals of a given ideal. \| =i \| = dim L =i K − dim LF(M i ) K for every i ∈ [d]. Proof. In view of Definition 2.3 it suffices to observe that I ≤i I ≤i−1 ∼ = LF(M i ) K by Lemma 3.4 (1) where I := 〈M 〉 K[ ] . We are ready to provide a reformulation of the definition of order ideal polytopes, which is better suited for actual computations, partly as it only involves direct matrix operations via replacing dimensions with ranks of subsets: Second, we replace all occurrences of I ≤i I ≤i−1 with LF(M i ) K (or simply LF(M i )). This almost results in the inequality system of Figure 3.3, with the only difference that instead of Condition 3.8 we have z m 1 ≥ z m 2 ∀m 1 , m 2 ∈ L : m 1 \| m 2 m∈L =i z m = dim L =i K − dim LF(M i ) K ∀i ∈ [d − 1] m∈U z m ≥ \|U\| − rk(Ũ) ∀i ∈ [d − 1], U ⊆ L =i : \|U\| = dim LF(M i ) K z m ∈ [0, 1] ∀m ∈ L(3.9) m∈U z m ≤ dim U ∪ LF(M i ) K − dim LF(M i ) K ∀i ∈ [d − 1], U ⊆ L =i : \|U \| = dim L =i K − dim LF(M i ) K , where we have deliberately replaced U with U . We will show that the difference of Conditions 3.7 and 3.8 is equal to Condition 3.9 with the choice U := L =i \ U, which has size \|U \| = dim L =i K − dim LF(M i ) K . This will finish the proof. Let U ⊆ L =i as above and compute the difference of Condition 3.7 and Condition 3.8. We obtain m∈L =i \U z m ≤ dim L =i K − dim LF(M i ) K − \|U\| + rk(Ũ). It is easy to see that \|U\| = dim 〈L =i 〉 K 〈L =i \U〉 K . We claim that it suffices to show that rk(Ũ) = dim LF(M i ) ∪ (L =i \ U) K L =i \ U K . Indeed, using this we can rewrite the inequality as m∈L =i \U z m ≤ dim L =i K − dim LF(M i ) K − dim L =i K L =i \ U K + dim LF(M i ) ∪ (L =i \ U) K L =i \ U K = dim LF(M i ) ∪ (L =i \ U) K − dim LF(M i ) K , which is (3.9) for U := L =i \ U as claimed. We will show now that rk (Ũ) = dim LF(M i ) ∪ (L =i \ U) K − dim L =i \ U K . Let B denoterk(Ũ) = dim LF(M i ) ∪ U K 〈U 〉 K = dim LF(M i ) ∪ (L =i \ U) K L =i \ U K and thus the result follows. COMPUTING BORDER BASES USING THE ORDER IDEAL POLYTOPE In the following we explain how Theorem 3.2 can be used to actually compute border bases for general degree-compatible order ideals. We cannot expect to be able to compute a border basis for any degree-compatible order ideal, simply as such a basis does not necessarily exist. Having the order ideal polytope at hand something slightly more subtle can be done: By choosing a linear objective function c ∈ n and optimizing it over the order ideal polytope, we can actually search for an order ideal with preferred monomials in its support (see Subsection 4.2). Having the order ideal polytope available we can also count the number of degree-compatible border bases that exist for a specific ideal. Before we can address this application though, we will first show how to obtain an -border basis for ∈ Λ(I) where I ⊆ K[ ] is a zero-dimensional ideal. Computing border bases for ∈ Λ(I). As the computation of the L-stable span of a set of generators M is independent of the actual chosen vector space basis (see Remark 2.9), we can adapt the classical border basis algorithm (Algorithm 2.17) to compute border bases for general degreecompatible order ideal. We first determine the right computational universe L = n ≤d for some d ∈ such that the associated L-stable span M contains all border bases. In a second step we optimize over the order ideal polytope P(M , L) and then perform the corresponding basis transformation. We will first formulate the generalized border basis algorithm by adding two steps after (3) in Algorithm 2.17 to the classical border basis algorithm, formulate the missing parts, and then prove its correctness: Note that step (3) is a convenient way to quickly check whether V is already L-stabilized and if it contains all degree-compatible order ideals. In this augmented algorithm we added the steps (4) and (5). The first step will be extensively discussed in Subsection 4.2 as there are various ways to determine ∈ Λ(〈V 〉 K[ ] ) and this is precisely one of the main features, i.e., to choose the order ideal more freely. Note that by Theorem 3.2 we already know that does support an -border basis of 〈F 〉 K[ ] (as 〈F 〉 K[ ] = 〈V 〉 K[ ] ) and our task is now to actually extract this basis from V . This extraction is performed in step (5). Let A be a matrix representing a set of polynomials M where the columns correspond to the monomials in some fixed ordering. Let the head (short: Head(a)) of a row a of A be the left-most monomial in the matrix representation whose coefficient is non-zero. Note that the notion of head replaces the notion of leading term of a polynomial as we do not (necessarily) have a term ordering anymore. The main idea is to reorder the columns of V and then to bring V into a reduced row echelon form such that no m ∈ is head of a row of the resulting matrix -a classical basis transformation: Proof. First, the algorithm finds from M . As ∈ Λ(〈V 〉 K[ ] ) we have that supports a border basis and in particular we have K[ ] = 〈V 〉 K[ ] ⊕ 〈 〉 K , and hence 〈L〉 K = 〈V 〉 K ⊕ 〈 〉 K by the modular law. We will now show that Condition (1) of Definition 2.2 is satisfied. This in turn follows from the fact that the algorithm creates every element g j of to have the form g j = b j − µ i=1 α i j t i with α i j ∈ K. By construction b j ∈ ∂ and thus is an -border basis of 〈V 〉 K[ ] . Note that Algorithm 4.3 works for any order ideal that supports a border basis of 〈V 〉 K[ ] , i.e., also those that are not necessarily degree-compatible. When the order ideal is known to be degreecompatible, it is enough to do the permutations in each degree block in the first step, and then use Algorithm 2.11 in the second step. We will show now that Algorithm 4.1 computes an -border basis for ∈ Λ(I). Proof. Whenever we reach step (4) in Algorithm 4.1, we have that V is L-stabilized for some L = ⊆ L via 〈V 〉 ≤d K /〈V 〉 ≤d−1 K ∼ = n =d K = L =d K . An improved version of the border basis algorithm has been also considered in [20]. Basically, the improvement can be traced back to considering more restricted computational universes L that arise from choosing L to be the smallest order ideal that contains the support of the initial system and then successively extending it using the + operation. This improvement due to the restriction of the computational universe L cannot work in our setting anymore: Suppose that V is L-stabilized with respect to some computational universe L = n ≤d for all d ∈ (i.e., L is not obtained by bounding the total degree of the monomials in ) and contains the order ideal that is induced by the chosen degreecompatible term ordering in the classical border basis algorithm. Then the associated polytope P(V, L) would only contain a subset of all possible degree-compatible order ideals, as we might be lacking monomials that we need to represent certain alternative choices of . If a subset of all admissible degree-compatible order ideals is sufficient, then the same optimizations can be applied though. Computing preferred border bases. Let V and L be as obtained after step (3) in Algorithm 4.1. As shown in Theorem 3.2 and Lemma 3.6, the order ideal polytope P(V, L) characterizes all degreecompatible order ideals that support a border basis of 〈V 〉 K[ ] . Every z ∈ P(V, L) ∩ L induces an order ideal (z) which supports an (z)-border basis of 〈V 〉 K[ ] . This also shows that we cannot expect that every order ideal supports an -border basis of 〈V 〉 K[ ] as the characterization is one-to-one. The natural question is therefore how to specify which order ideal should be computed, i.e., which z ∈ P(V, L) ∩ L to choose. As we cannot always get what we would like to have, it suggests itself to specify a preference, i.e., which monomials we would like to be contained in (z) and which ones we would rather not. As the coordinates of z are in direct correspondence with the monomials in L we can define: Definition 4.5. A preference is a vector c ∈ L which assigns a weight to each monomial m ∈ L. If z ∈ P(V, L) ∩ L , then cz is the score or weight of z. As P(V, L) ⊆ [0, 1] n is a polytope we can optimize over P(V, L) ∩ L and compute an element z 0 ∈ P(V, L) ∩ L that has maximal score, i.e., we can compute z 0 ∈ P(V, L) ∩ L such that cz 0 = max{cz \| z ∈ P(V, L) ∩ L }. In this sense a preference is an indirect way of specifying an order ideal. For certain choices of c ∈ L though it can be hard to compute an order ideal that maximizes the score as we will show now. COMPLEXITY OF FINDING PREFERRED ORDER IDEALS In this section, we show that finding a weight optimal, border basis supporting order ideal of a zerodimensional ideal given by generators is NP-hard (Theorem 5.3). Note that this also translates to large classes of other choice functions as P(V, L) ∩ L is a 0/1 polytope and thus its extremal points are given by an inequality description. So any choice function that implicitly asks for a k-clique (which we will use in our reductions) can be replaced by the corresponding linear function and hardness also follows in this case. The hardness for computing a weight optimal order ideal is unexpected in the sense that we merely ask for a basis transformation. On the other hand it highlights the crucial role of order ideals in describing the combinatorial structure of the ideal. As an immediate consequence it follows that it is rather unlikely that we can obtain a good characterization of the integral hull conv(P(V, L) ∩ L ) and we will not be able to compute degree-compatible order ideals that support a border basis and have maximum score efficiently unless P = NP. This shows that not only computing the necessary liftings of the initial set of polynomials via the LStabSpan procedure is hard but also actually determining an optimal choice of an order ideal once an L-stable span has been computed. As mentioned before, this is in some sense surprising as the actual interference has been already performed at that point and we are only concerned with choosing a nice basis. From a practical point of view this is not too problematic as, although NP-hard, computing a weight optimal order ideal is no harder than actually computing the LStabSpan in general. For bounds on the degree d ∈ needed to compute border bases see, e.g., [13,Lemma 2.4]; the border basis algorithm generates the Nullstellensatz certificates and is therefore subject to the same bounds. Further, state-of-the-art mixed integer programming solvers that can solve the optimization problem such as scip ([2]), cplex ([10]), or gurobi ( [16]) can handle instance sizes far beyond the point for which the actual border bases can be computed. Very good solutions can also be generated using simple local search schemes starting from a feasible order ideal derived from a degree-compatible term ordering. Fast without constraint. Determining an order ideal of maximum score in a computational universe L without having any additional constraints can be done in time polynomial in \|L\|, as was shown in [30]. One simply transforms it into a minimum cut problem in graphs (see, e.g., [32]): Let c ∈ L be a preference vector. Define a graph Γ := (V, E) with V := L ∪ {s, t} andẼ := {(u, v) \| u, v ∈ L and v \| u}, i.e., whenever v \| u we add an arc from u to v. In fact, it is enough to have an arc when u = v x for some variable x, i.e., to consider the transitive reduction ofẼ. Define E :=Ẽ ∪ {(s, u) \| u ∈ L, c u > 0} ∪ {(u, t) \| u ∈ L, c u < 0}. Further we set all the capacities of the arcs with both vertices in L to ∞, for any arc (s, u) with u ∈ L we set the capacity to c u , and for any arc (u, t) with u ∈ L we set the capacity to \|c u \|; let κ(u, v) denote the capacity of the arc (u, v) ∈ E. An example is depicted in Figure 5 The last line asks for a minimum weight cut in the graph Γ. Note that we can indeed drop the condition that has to be an order ideal as it is guaranteed implicitly by all finite weight cuts as explained above. It is well-known that the computation of a minimum cut can be performed in time polynomial in the number of vertices and arcs (see [34]). Thus we can indeed compute an order ideal of maximum score efficiently in this case. NP-hard with constraints. So far we did not include the additional requirements as specified by the order ideal polytope (see Figure 3.3), in order to obtain degree-compatible order ideals that do actually support a border basis of the ideal I under consideration. Unfortunately, when including these additional requirements, the problem of computing an order ideal of maximum score is NP-hard as we will show in the following. We will show NP-hardness by a reduction from the MAX-CLIQUE problem, which is well known to be NP-complete (see, e.g., [15] or [11,GT22]). Given a graph Γ = (V, E), recall that a clique C is a subset of V such that for all distinct u, v ∈ C we have (u, v) ∈ E. MAX-CLIQUE: Let Γ = (V, E) be a graph. Determine the maximum size of a clique C contained in Γ. We will in particular use the following variant: k-CLIQUE: Let Γ = (V, E) be a graph. Determine whether Γ contains a clique C of size k. Note that if Γ contains a clique of size k for some k ∈ , then so does it for any 1 ≤ l ≤ k. Further the maximum size of a clique is bounded by \|V \|. It follows that already testing whether Γ contains a clique C of size k for some given k ∈ has to be NP-complete, as otherwise, we could solve the MAX-CLIQUE problem with O(log 2 \|V \|) calls of the algorithm that solves the test variant via binary search. In [18, Discussion after Definition 3.2] it was indicated that determining a maximum weight order ideal of a pre-defined size is NP-hard by a reduction from MAX-CLIQUE. While already indicating hardness due to one additional cardinality constraint, this is a slightly different problem than the one that we are facing here: in addition to the size constraint, the order ideal has to be degree-compatible. Thus we have a cardinality/degree-constraint for each degree of the monomials (Constraints (3.7) and (3.9) in Figure 3.3). Further, the degree constraints are not completely independent of each other. So it could be a priori perfectly possible that this particular restriction of the problem can be actually solved efficiently, which is not the case as we will see soon. We consider the following optimization problem: MAX-BOUNDED ORDER IDEAL: Let L = n ≤d for some d ∈ and c ∈ L be a preference. c m = 1, if m = x u x v and (u, v) ∈ E; 0, otherwise. A vertex v ∈ V is represented by the degree-one monomial x v and each edge (u, v) ∈ E is represented by the degree-two monomial x u x v . In some sense we extended the graph (V, E) to the complete graph K \|V \| that has \|V \|(\|V \| − 1)/2 edges as we consider all possible degree-two monomials that correspond to the edges; those edges that do not belong to E have weight 0 though. The order ideals satisfying the size constraints are those consisting of the monomials 1, x v i , x v i x v j for 1 ≤ i, j ≤ k for some distinct vertices v 1 ,. . . ,v k . The score of the order ideal is the number of edges in the subgraph spanned by the x v i . Thus when maximizing c we ask for a k-vertex subgraph with maximal number of edges. The subgraph is given by the degree-one monomials contained in the order ideal, which we denote by , i.e., C := {v ∈ V \| x v ∈ }. So the maximum score is equal to k(k − 1)/2 if and only if Γ contains a clique of size k. Thus we can solve k-CLIQUE efficiently if we can solve MAX-BOUNDED ORDER IDEAL efficiently and so the latter has to be NP-hard. Finally, it remains to show that for every graph Γ = (V, E) and k ∈ [\|V \|] there exists a system of polynomials F \|V \|,k ⊆ K[x v \| v ∈ V ] spanning a zero-dimensional ideal such that solving the MAX-ORDER IDEAL OF IDEAL problem for F \|V \|,k solves the k-CLIQUE problem for Γ. For this, we construct an ideal encoding all k-cliques of the complete graph on n vertices: Let n ∈ and k ∈ [n] and define v j := i∈ [n] i j x i , F n,k := {v j \| j ∈ [n − k]} ∪ n =3 . We consider the ideal generated by F n,k . We show that its order ideals are in one-to-one correspondence with the k-element subsets of the set of n variables x 1 , . . . , x n as stated in the following lemma. Lemma 5.2. Let n ∈ and k ∈ [n]. Then F n,k generates a zero-dimensional ideal such that ∈ Λ F n,k K[ ] if and only if =1 ⊆ n =1 with \| =1 \| = k, =2 = {x y \| x, y ∈ =1 }, and = = for all ≥ 3. Proof. We will first characterize the vector space K[x 1 , . . . , x n ] F n,k K[ ] . Observe that the coefficient matrix A := (v j ) j∈[n−k] is actually a Vandermonde matrix and in particular every square submatrix of A is invertible. Furthermore, the polynomials v j with j ∈ [n − k] are homogeneous of degree one. Thus, for any k variables of {x 1 , . . . , x n }, without loss of generality say, x 1 , . . . , x k , we have that {x 1 , . . . , x k , v 1 , . . . , v n−k } is a basis for the homogeneous polynomials of degree one. This is just another way of saying that removing the columns belonging to x 1 , . . . , x k from A, the resulting square submatrix is invertible. It follows 〈 〉 K ∼ = K[x 1 , . . . , x n ] F n,k K[ ] ∼ = K[x 1 , . . . , x k , v 1 , . . . , v n−k ] F n,k K[ ] ∼ = K[x 1 , . . . , x k ] k =3 K[ ] . As the substitution preserves degrees, homogeneity, etc., it follows, that any degree-compatible order ideal has to have \| =1 \| = k, =2 = {x y \| x, y ∈ =1 }, and = = for all ≥ 3. The other direction follows immediately as each order ideal with \| =1 \| = k, =2 = {x y \| x, y ∈ =1 }, and = = for all ≥ 3 is actually a degree-compatible order ideal such that K[x 1 , . . . , x n ] = F n,k K[ ] ⊕ 〈 〉 K as vector spaces, by the argumentation above. Note that the order ideals of F n,k indeed correspond to the k-cliques of the complete graph on n vertices: If ∈ Λ(F n,k ), then =1 = {x i 1 , . . . , x i k } and x i j x i l ∈ =2 if and only if x i j , x i l ∈ =1 . If we now remove all elements of the form x 2 i j with x i j ∈ =1 , and there are k of those, then \| =2 \ {x 2 i j \| x i j ∈ =1 }\| = k(k − 1) 2 , the size of a k-clique. We are ready to state the main result of this section. Consider the following problem: MAX-ORDER IDEAL OF IDEAL: Let M ⊆ K[ ] be a system of polynomials generating a zero-dimensional ideal and let c ∈ n be a preference on the monomials. Compute an order ideal supporting an -border basis of 〈M 〉 K[ ] with maximum score with respect to c. Theorem 5.3. MAX-ORDER IDEAL OF IDEAL is NP-hard. Proof. The proof is by a reduction from the NP-hard k-CLIQUE along the lines of the proof of Theorem 5.1. Let Γ = (V, E) be a graph with n := \|V \| and k ∈ [n] be an instance of k-CLIQUE. We consider M := F n,k and define c ∈ n ≤3 via c m = 1, if m = x u x v and (u, v) ∈ E; 0, otherwise. By Lemma 5.2, we have that the degree-compatible order ideals of 〈M 〉 K[ ] are in one-to-one correspondence with the k-cliques of the complete graph on n vertices. Similarly to the proof of Theorem 5.1, the score of an order ideal is just the number of edges between the corresponding k vertices of the graph, so the maximum score is k(k − 1)/2 if and only if the graph contains a k-clique. Thus, we obtain a test whether Γ contains a clique of size k. COMPUTATIONAL RESULTS We performed a few computational tests to verify the practical feasibility of our method. All computations were performed with CoCoA 4.7.5 ([7]) and scip 1.1.0 ([2]) on a 2 GHz Dual Core Intel machine with 2 GB of main memory. We performed computations on various sets of systems of polynomial equations. The employed methodology was as follows. We first computed a border basis using the classical border basis algorithm. From the last run of the algorithm we extracted the L-stabilized span and brought it into canonical form. We generated the constraints (3.7) from the order ideal that we obtained; from the L-stabilized span in matrix from, we generated the constraints (3.8). As we needed to get access to the L-stable span computed in the last round of the border basis algorithm, we had to use an implementation of the border basis algorithm in CoCoA-L which is slower in terms of speed compared to a C or C++ implementation. We then transcripted these constraints into the CPLEX LP format which served as input for scip. For the optimization we chose various preference functions. One was a random function, and the other one was chosen with the intent to make the optimization particularly hard by giving monomials deep in the order ideal negative weights and assigning positive weights for the outer elements. In all cases the optimization, i.e., the computation of the weight optimal order ideal was performed in less than a second, whereas the actual calculation of the initial border bases was significantly more time consuming. As indicated before, this is not unexpected as the computation of the L-stable span is significantly more involved than computing a weight optimal order ideal (in the worst case double exponential vs. single exponential). When computationally feasible we also counted all feasible order ideals with scip, which basically means enumerating all feasible solutions. This in fact is equivalent to optimizing all potential preference functions simultaneously and thus emphasizes the computational feasibility. We considered 7 systems of polynomials of various complexity in terms of number of variables and order ideal degree. We would have liked to test significantly larger instances but we were not able to compute the initial border basis (or more precisely the L-stable span), neither with our implementation nor with the C/C++ implementation of CoCoA 5 (BBasis5). This shows once more that the limiting factor is the actual computation of the L-stable span. In Table 1 we report our results. CONCLUDING REMARKS We provided a way to characterize all degree-compatible order ideals that support a border basis for a given zero-dimensional ideal I by borrowing from combinatorial optimization and in particular polyhedral theory. We established a one-to-one correspondence of the integral points of a certain polytope, the order ideal polytope, and those degree-compatible order ideals that support a border polynomial system order ideal signature optimization [s] counting [s] # order ideals x 3 , x y 2 + y 3 (1, 3, 1, 1, 1) < 0.01 0.02 3 vanishing ideal of the points (0, 0, 0, 1), (1, 0, 0, 2), (3, 0, 0, 2), (5, 0, 0, 3), (−1, 0, 0, 4), (4,4,4,5), (0, 0, 7, 6)). (1, 4, 2) < 0.01 0.02 45 x + y + z − u − v, x 2 − x, y 2 − y, z 2 − z, u 2 − u, v 2 − v (1, 4, 5) < 0.01 0.35 1,260 x + y + z − u − v, x 3 − x, y 3 − y, z 2 − z, u 2 − u, v 2 − v (1, 4, 7, 6) 0.02 51.50 106,820 (1,4,8,9) 0.02 53.00 108,900 (1,4,9,12,9) 0.08 300.00* > 1,349,154 Computational results. The first column contains the considered polynomial system. The second column contains the degree vector of the order ideal, i.e., (dim I ≤i I ≤i−1 ) i starting with i = 1 and I ≤0 := 0. The third column contains the average time (in seconds) needed to optimize a random preference over the order ideal polytope (we performed 20 runs for each system). The fourth column contains the time (in seconds) needed to count all admissible degree-compatible order ideals and the last column contains the actual number of admissible degree-compatible order ideals. The '*' indicates that the counting had been stopped after 300 seconds. The number of order ideals reported in this case is the number that have been counted up to that point in time. x + y + z − u − v, x 3 − x, y 3 − y, z 3 − z, u 2 − u, v 2 − vx + y + z − u − v, x 3 − x, y 3 − y, z 3 − z, u 3 − u, v 2 − vx + y + z − u − v + a, x 2 − x, y 2 − y, z 2 − z, u 2 − u, v 2 − v, a 2 − basis of I. This connection in particular links the ideals to their combinatorial structure of the factor spaces. Using this polytope we adapted the classical border basis algorithm in order to be able to compute border basis for general degree-compatible order ideal based on a preference ordering on terms contained therein. Effectively, the algorithm can be used for any integral point contained in the order ideal polytope. We also showed that computing a border basis for a preference on the monomials one might want to have included in the order ideal is NP-hard and thus we cannot expect to be able to efficiently compute preferred order ideals in general although it is merely a basis transformation. On the other hand, this is not restricting the applicability of our method in any practical application because the preceding computation of the L-stable span dominates in terms of computational complexity. We finally presented a few computational results showing the applicability of our method for actual computations. a polynomial ring K[ ] over the field K with indeterminates = {x 1 , . . . , x n }. For convenience we define x m := j∈[n] x m j j for m ∈ n and let n := {x m \| m ∈ n } be the monoid of terms. For any d ∈ we let n ≤d := {x m ∈ n \| m 1 ≤ d} be the set of monomials of total degree at most d. For a polynomial p ∈ K[ ] with p = l i=1 a i x m i we define the support of p to be supp(p) := {x m i \| i ∈ [l Recall that an ideal I ⊆ K[ ] is zero-dimensional, if and only if K[ ]/I is finite dimensional. The -border basis of a zero-dimensional ideal I is a special set of polynomials: Definition 2.2. Let = {t 1 , . . . , t µ } be an order ideal with border ∂ = {b 1 , . . . , b ν }. Further let = {g 1 , . . . , g ν } ⊆ K[ ] be a (finite) set of polynomials and let I ⊆ K[ ] be a zero-dimensional ideal. Then the set is an -border basis of I if: Further let G = {g 1 , . . . , g s } be a finite set of polynomials. Then Algorithm 2.11 computes a finite set of polynomials W ⊆ K[ ] with LC(w) = 1 for all w ∈ W , LT(u 1 ) = LT(u 2 ) for any distinct u 1 , u 2 ∈ V ∪ W , and 〈V ∪ W 〉 K = 〈V ∪ G〉 K . (V , W may be empty.) Algorithm 2.11 (Gaussian Elimination for polynomials-GaussEl). Input: V , G as in Lemma 2.10. Output: W ⊆ K[ ] as in Lemma 2.10. (1) Let H := G and η := 0. (2) If H = then return W := {v r+1 , . . . , v r+η } and stop. (3) Choose f ∈ H and remove it from H. Let i := 1. Lemma 2.12. [20, Proposition 13] Let L = n ≤d and F ⊆ K[ ] be a finite set of polynomials supported on L. Then Algorithm 2.13 computes a vector space basis V of F L with pairwise different leading terms. Algorithm 2.13 (L-stable span computation-LStabSpan).Input: F , L as in Lemma 2.12.Output: V as inLemma 2.12. Lemma 2.14. [20, Proposition 17] Let F be a system of generators of a zero-dimensional ideal I, let L be an order ideal, and let V be a vector space basis of F L with pairwise different leading terms and := L \ LT(V ) such that ∂ ⊆ L. Then Algorithm 2.15 computes the -border basis of I. Proposition 2.18. [20, Proposition 16] Let L be an order ideal. Further letĨ be an L-stabilized generating vector subspace of a zero-dimensional ideal I ⊆ K[ ], i.e.,Ĩ + ∩ 〈L〉 K =Ĩ and Ĩ K[ ] = I. If is an order ideal such that 〈L〉 K =Ĩ ⊕ 〈 〉 K and ∂ ⊆ L then supports a border basis of I. FIGURE 3 . 1 . 31Order ideal polytope P(I) Definition 3.1. Let I be a zero-dimensional ideal. Its order ideal polytope P(I) is given by the system of inequalities in Figure 3.1. The order ideal polytope is actually a finite dimensional polytope as all the z m are 0 when the degree of m is large enough. Indeed, for large i, we have =i K ∼ = I ≤i I ≤i−1 and hence Condition 3.7 gives z m = 0 for every m of degree i. Definition 3. 3 . 3Let M be a finite set of polynomials of degree at most for some ∈ . Then M is in canonical form if the leading term of any element of M does not occur in the other elements. Lemma 3. 4 . 4Let M be in canonical form and L-stabilized with L = n ≤d . Let n =d ⊆ 〈M 〉 K + n ≤d−1 . Then the following hold for all i ∈ [d]: Lemma 3. 5 . 5Let M be in canonical form and L-stabilized with L = n ≤d and d = max m∈∂ deg(m). Let n =d ⊆ 〈M 〉 K + n ≤d−1 . Further let be an order ideal of 〈M 〉 K[ ] . Then is degree-compatible if and only if Lemma 3. 6 . 6Let M be L-stabilized and in canonical form with L = n ≤d and 〈M 〉 ≤d K Then the order ideal polytope P(M , L) ⊆ [0, 1] L of 〈M 〉 K[ ] is given by the system of inequalities in Figure 3.3. Proof. Let I := 〈M 〉 K[ ] . We successively transform the defining inequalities of the order ideal polytope in Figure 3.1 into the desired form of Figure 3.3. The reformulation is mostly based on I ≤i I ≤i−1 ∼ = LF(M i ) K from Lemma 3.4(1). First, as I ≤d I ≤d−1 ∼ = n =d K , we can remove the variables z m with m degree at least d together with the inequalities involving them. These variables are always zero. FIGURE 3 . 3 . 33Order ideal polytope P(M , L). In Condition 3.8, the matrixŨ is the induced sub-matrix of LF(M i ) with column monomials only in U. the matrix obtained when writing the elements in LF(M i ) as rows and let L =i index the columns. Clearly, L =i K = ∈L =i Ke as a vector space and U K = ∈U Ke as a sub vector space with U := L =i \U. We obtain L =i K U K ∼ = ∈L =i \U Ke . Note that LF(M i ) K ⊆ L =i K and thus we can consider LF(M i ) ∪ U K U K ⊆ L =i K U K . Now dim LF(M i ) ∪ U K U K = rk(Ũ) whereŨ is obtained from B by removing the columns in U . We obtain Algorithm 4. 1 (( 1 )( 3 ) 113Generalized border basis algorithm-BBasis). Input: F a finite generating set of a zero-dimensional ideal. Output: a border basis of the ideal. Let d := max f ∈F {deg( f )} and put L := n ≤d . (2) V = {v 1 , . . . , v r } := LStabSpan(F, L). If n =d LT(V ) then set d := d + 1 and put L := n ≤d and go to step (2). (4) Choose ∈ Λ(〈V 〉 K[ ] ) (⇔ z ∈ P(V, L) ∩ L and = (z)). ( 5 ) 5Let := BasisTransformation(V, ). Lemma 4. 2 . 2Let L = n ≤ with ∈ , let V be a finite set of polynomials satisfying 〈V 〉 K = 〈V 〉 K[ ] ∩ 〈L〉 K and let = {t 1 , . . . , t µ } be an order ideal with ∂ ⊆ L and ∈ Λ(〈V 〉 K[ ] ). Then Algorithm 4.3 returns the -border basis of 〈V 〉 K[ ] . Algorithm 4. 3 ( 3Basis transformation algorithm-BasisTransformation). Input: V, as in Lemma 4.2. Output: as in Lemma 4.2. (1) Set := max m∈M deg(m). (2) Permute the columns of V such that t is right of m in the matrix representation of V for all m ∈ n ≤ and t ∈ . ( 3 ) 3Reduce V using Gaussian elmination: like Algorithm 2.11, but use Head instead of LT and the coefficient of the head instead of LC. Let be the result. (4) Let := {g ∈ : Head(g) ∈ ∂ }. ( 5 ) 5Return . Proposition 4. 4 . 4Let F = { f 1 , . . . , f s } ⊆ K[ ] be a finite set of polynomials that generates a zerodimensional ideal I = 〈F 〉 K[ ] . Then Algorithm 4.1 computes the -border basis of I for any (chosen) ∈ Λ(I). ≤d with d ∈ and it contains all degree-compatible order ideals supporting a border basis, i.e., all ∈ Λ(I). Observe that I = 〈F 〉 K[ ] = 〈V 〉 K[ ] and thus, by Lemma 4.2, it follows that is indeed an -border basis of 〈F 〉 K[ ] . Note that step (3) ensures ∂ FIGURE 5 . 1 . 51Order ideal computation as minimum cut problem. Capacities denoted next to the arc and weight c u denoted next to the respective node u. The dashed arcs have capacity ∞. . 1 . 1For U, W ⊆ V , we define C(U, W ) := (u,w)∈U×W κ(u, w). A (s,t)-cut (S,S) is a partition S · ∪S = V of the vertices of V with s ∈ S and t ∈S and the weight of the cut is C(S,S).We would like to compute an order ideal contained in L with maximum score:max u∈ c u ⊆ L order ideal .Observe that, if (S,S) is a cut in Γ of finite weight, there exist no arc (u, v) ∈Ẽ with u ∈ S and v ∈S. Therefore, (S,S) is a cut in Γ of finite weight if and only if S \ {s} is an order ideal. We can therefore rewrite the optimization problem as follows:max u∈ c u ⊆ L order ideal = max{C({s}, ) − C( , {t}) \| ⊆ L order ideal} = max{C({s}, L) − C({s}, L \ ) − C( , {t}) \| ⊆ L order ideal} = C({s}, L) − min{C({s}, L \ ) + C( , {t}) \| ⊆ L order ideal} = C({s}, L) − min{C({s} ∪ , (L \ ) ∪ {t}) \| ⊆ L}. Further let d i ∈ for i ∈ [d − 1]. Determine an order ideal 0 ⊆ L with \| =i 0 \| = d i for all i ∈ [d − 1] and maximum score, i.e., c(ξ −1 ( 0 )) = max{c(ξ −1 ( )) \| ⊆ L order ideal and \| =i \| = d i for all i ∈ [d − 1]}.By a reduction from k-CLIQUE we obtain:Theorem 5.1. MAX-BOUNDED ORDER IDEAL is NP-hard. Proof. Let Γ = (V, E) be an arbitrary simple graph and choose k ∈ [\|V \|]. We consider the polynomial ring K[x v \| v ∈ V ]. Let L = \|V \| ≤2 . Further choose d 1 = k and d 2 = k(k + 1)/2. Define Proposition 2.16. [20, Proposition 18] Let F ⊆ K[ ] be a finite set of polynomials that generates a zero-dimensional ideal I = 〈F 〉 K[ ] . Then Algorithm 2.17 computes the -border basis of I. Algorithm 2.17 (Border basis algorithm-BBasis). Input: F as in Proposition 2.16. Output: as in Proposition 2.16. 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[ "An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem", "An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem" ]	[ "A Karim Abu-Affash " ]	[]	[]	Given two sets of points in the plane, P of n terminals and S of m Steiner points, a Steiner tree of P is a tree spanning all points of P and some (or none or all) points of S. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, P and S, and a positive integer k ≤ m, find a bottleneck Steiner tree of P with at most k Steiner points. The problem has application in the design of wireless communication networks.We first show that the problem is NP-hard and cannot be approximated within factor √ 2, unless P = N P . Then, we present a polynomial-time approximation algorithm with performance ratio 2.	null	[ "https://arxiv.org/pdf/1012.1502v1.pdf" ]	6,745,380	1012.1502	e0efa72b8e9c31850dad3295285f14eadd44ce97	An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem A Karim Abu-Affash An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem Given two sets of points in the plane, P of n terminals and S of m Steiner points, a Steiner tree of P is a tree spanning all points of P and some (or none or all) points of S. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, P and S, and a positive integer k ≤ m, find a bottleneck Steiner tree of P with at most k Steiner points. The problem has application in the design of wireless communication networks.We first show that the problem is NP-hard and cannot be approximated within factor √ 2, unless P = N P . Then, we present a polynomial-time approximation algorithm with performance ratio 2. Introduction Consider a wireless communication network with n stations, each station has a limited power so that it can only communicate with stations within a limited range, and suppose that, in order to make the network connected and due to budget limits, we are only allowed to put at most k new stations in given potential locations in this network. Clearly, we would like to select locations such that distance between stations as small as possible. This application motivates the following problem: The Bottleneck Steiner Tree (k-BST) problem. Given two sets in the plane, P of terminal points and S of Steiner points, and a positive integer k, one is asked to find Steiner tree T of P with at most k Steiner points, such that the bottleneck (i.e., length of the longest edge) of T is minimized. In the classical Steiner tree (ST) problem, the goal is to find a Steiner tree T such the total length of edges of T is minimized. This problem has been shown to be NP-complete [7] and many approximation algorithms have been proposed [2,3,8]. A general version of the k-BST, where k = \|S\|, has been studied by Sarrafzadeh and Wong [12]. They showed that this version can be solved in polynomial time. Another version, where S is the whole plane R 2 , has been studied extensively in the last decade. In [13], this version was shown to be NP-hard to approximate within ratio √ 2. The best known upper bound on approximation ratio is 1.866 [14]. Bae et al. [1] presented an O(n log n) time algorithm to the problem for k = 1 and an O(n 2 ) time algorithm for k = 2. Li et al. [11] presented a ( √ 2 + )-approximation algorithm with inapproximability within √ 2 for a special case of the problem where there should be no edge connecting any two Steiner points in the optimal solution. These versions have many important applications in VLSI design, network communication and computational biology [4,6,9,10]. We are not aware of any previous work studying our version. However, in this paper, we show that the k-BST problem is NP-hard and we present a polynomial-time algorithm with constant factor approximation ratio for the problem. Hardness Result Given a set P of n terminals in the plane, a set S of m Steiner points and an integer k ≤ m, the goal in the k-BST problem is to find a Steiner tree with at most k Steiner points from S and bottleneck as small as possible. In this section we prove hardness of the problem. 2-Approximation Algorithm In this section, we develop a polynomial-time approximation algorithm for computing a Steiner tree with at most k Steiner points (k-ST for short) such that its bottleneck is at most 2 times the bottleneck of an optimal (minimum-bottleneck) k-ST. Let G = (V, E) be the complete graph over V = P ∪ S. We assume, without loss of generality, that E = {e 1 , e 2 , . . . , e l } such that \|e 1 \| ≤ \|e 2 \| ≤ . . . ≤ \|e l \|. It is not hard to see that the bottleneck of an optimal k-ST is a length of an edge from E. For an edge e i ∈ E, let G i = (V, E i ) be the graph with E i = {e j ∈ E : \|e j \| ≤ \|e i \|}. The idea behind our algorithm is to devise a procedure that, for a given edge e i ∈ E, does one of the following: (i) It constructs a k-ST of P in G with bottleneck at most 2 times \|e i \|. (ii) It returns the information that G i does not contain any k-ST of P . For two points p, q ∈ P , let δ i (p, q) be a shortest Steiner path between p and q in G i , i.e., a path connecting p and q with minimum number of Steiner points in G i . Let G P = (P, E P ) be the complete graph over P . For each edge (p, q) in E P , we assign a weight w(p, q) equal to the number of Steiner points in δ i (p, q). Let T be a minimum spanning tree of G P under w. We define the normalized weight of T as C(T ) = e∈T w(e)/2 . Proof: Let T * be a k-ST of P in G i . A Steiner tree is full if all terminals are leaves. We decompose T * into a union of full trees. For each full tree T * j of T * , we will construct a spanning tree T j of the terminals of T * j in G P , such that the union of these tree is a spanning tree T of P in G P with C(T ) ≤ k. We arbitrary select a Steiner point as the root of T * j ; see Figure 1(a). The construction of T j is bottom-up by an iterative process. In each iteration, we select the deepest leaf p in the rooted tree, which is a terminal, and we connect it to its nearest terminal q by an edge of weight equal to the number of Steiner points between them. Let s be the first common parent of p and q. We then remove the Steiner points between p and s (in the last iteration, we may remove all of the remaining points). In the example in Figure 1(b), we first select the terminal a, which is the deepest one, we connect it to the terminal b by an edge of weight 3 and we remove the points s 1 and s 2 . Next, we select the terminal d, we connect it to the terminal c by an edge of weight 2 and we remove the point s 3 . In the last iteration, we select the terminal b, we connect it to the terminal c by an edge of weight 3 and we remove all of the remaining points. Notice that, since, in each iteration, we select the deepest terminal, we add an edge (p, q), of weight w(p, q), and we remove at least w(p, q)/2 Steiner points from T * j . This implies that C(T j ) = e∈T j w(e)/2 ≤ k j , where k j is the number of Steiner points in T * j . Moreover, the union T of the trees T j is a spanning tree of G and has C(T ) ≤ k. Thus, since T is a minimum spanning tree of G , we have C(T ) ≤ C(T ) ≤ k. We now describe our approximation algorithm. We traverse the edges of E in the sorted order and, for each edge e i ∈ E, we construct a minimum spanning tree T of G P = (P, E P ) and check whether C(T ) ≤ k. If so, we construct a k-ST of P , otherwise, we move to the next edge e i+1 . Algorithm 1 EBST (G = (V, E), P, k) 1: C(T ) ← ∞ 2: G P = (P, E P ) ← the complete graph over P 3: i ← 0 4: while C(T ) > k do 5: i ← i + 1 6: construct the graph G i 7: for each edge (p, q) ∈ E P do 8: w(p, q) ← the number of Steiner points in δ i (p, q) 9: construct a minimum spanning tree T of G P under w 10: C(T ) ← e∈T w(e)/2 11: Construct-k-ST (T, G i ) The construction of a k-ST is done as follows. For each edge e = (p, q) ∈ T , we select w(e)/2 Steiner points on any shortest Steiner path between p and q in G i , such that, the path from p to q that passes through these points has a bottleneck at most 2\|e i \|, and we connect these points to form a path; see Figure 2. Clearly, the obtained Steiner tree contains at most k Steiner points and its bottleneck is at most 2\|e i \|. Proof: Let e i be the first edge satisfying the condition C(T ) ≤ k. Thus, by Lemma 3.1, the bottleneck of any k-ST in G is at least \|e i \|, and, therefore, the constructed k-ST has a bottleneck at most 2 times the bottleneck of an optimal k-ST. Proof: G i can be constructed in O((n + m) 2 ) time. In order to construct the graph G P , we can compute in O((n + m) 3 ) time the shortest Steiner paths between each pair of points in P [5]. Once G P is constructed, computing a minimum spanning tree of G P can be done in O(n 2 ) time, and selecting the relevant Steiner points can be done in O(k(n + m)) time. By combining Lemma 3.2 and Lemma 3.3, we get the following theorem. Theorem 3.4. There exists a polynomial-time approximation algorithm with performance ratio 2 for the k-BST problem. Conclusion In this paper, we studied the problem of finding bottleneck Steiner trees in the Euclidean plane. We proved that the k-BST problem in the plane does not admit any approximation algorithm with performance ratio less than √ 2, unless P = N P , and that there exists a polynomial-time approximation algorithm with performance ratio 2. It would be interesting to find better approximation algorithm for the k-BST problem. Another interesting question is how efficient can one solve the k-BST problem for a constant k > 0? Theorem 2 . 1 . 21The k-BST problem cannot be approximated within √ 2 in polynomial time, unless P = N P . The proof directly follows by a slight modification of the proof of Theorem 1 in[13]. Lemma 3. 1 . 1If G i contains a k-ST of P , then C(T ) ≤ k. Figure 1 : 1(a) The rooted tree, and (b) the construction of T j . Figure 2 : 2The constructed k-ST consists of the black circles and the dotted lines. Lemma 3. 2 . 2The algorithm above constructs a k-ST of P with bottleneck at most 2 times the bottleneck of an optimal k-ST. Lemma 3 . 3 . 33The algorithm above has a polynomial running time. On exact solutions to the euclidean bottleneck Steiner tree problem. S W Bae, C Lee, S Choi, Information Processing Letters. 110S.W. Bae, C. Lee, and S. Choi. On exact solutions to the euclidean bottleneck Steiner tree problem. Information Processing Letters, 110:672-678, 2010. Improved approximation for the Steiner tree problem. P Berman, V Ramaiyer, Journal of Algorithms. 17P. Berman and V. Ramaiyer. Improved approximation for the Steiner tree problem. Journal of Algorithms, 17:381-408, 1994. The k-Steiner ratio in graphs. A Borchers, D Z Du, SIAM Journal on Computing. 26A. Borchers and D.Z. Du. The k-Steiner ratio in graphs. SIAM Journal on Computing, 26:857-869, 1997. Steiner Tree in Industry. X Cheng, D Z Du, Kluwer Academic PublishersDordrecht, NetherlandsX. Cheng and D.Z. Du. Steiner Tree in Industry. Kluwer Academic Publishers, Dordrecht, Netherlands, 2001. Introduction to Algorithms. T H Cormen, C E Leiserson, R L Rivest, C Stein, MIT Press2nd editionT.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction to Algorithms, 2nd edition. MIT Press, 2001. Advances in Steiner Tree. D Z Du, J M Smith, J H Rubinstein, Kluwer Academic PublishersDordrecht, NetherlandsD.Z. Du, J.M. Smith, and J.H. Rubinstein. Advances in Steiner Tree. Kluwer Aca- demic Publishers, Dordrecht, Netherlands, 2000. The complexity of computing Steiner minimal trees. M R Garey, R L Graham, D S Johnson, SIAM Journal of Applied Mathematics. 324M.R. Garey, R.L. Graham, and D.S. Johnson. The complexity of computing Steiner minimal trees. SIAM Journal of Applied Mathematics, 32(4):835-859, 1977. A 1.598 approximation algorithm for the Steiner problem in graphs. S Hougardy, H J Prommel, Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00). the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00)S. Hougardy and H.J. Prommel. A 1.598 approximation algorithm for the Steiner problem in graphs. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00), pages 448-453, 1999. The Steiner Tree Problem. F K Hwang, D S Richards, P Winter, Annuals of Discrete Mathematics. F.K. Hwang, D.S. Richards, and P. Winter. The Steiner Tree Problem. Annuals of Discrete Mathematics, Amsterdam, 1992. On Optimal Interconnection for VLSI. A B Kahng, G Robins, Kluwer Academic PublishersDordrecht, NetherlandsA.B. Kahng and G. Robins. On Optimal Interconnection for VLSI. Kluwer Academic Publishers, Dordrecht, Netherlands, 1995. Approximation algorithm for bottleneck Steiner tree problem in the Euclidean plane. Z.-M Li, D.-M Zhu, S.-H Ma, Journal of Computer Science and Technology. 196Z.-M. Li, D.-M. Zhu, and S.-H. Ma. Approximation algorithm for bottleneck Steiner tree problem in the Euclidean plane. Journal of Computer Science and Technology, 19(6):791-794, 2004. Bottleneck Steiner trees in the plane. M Sarrafzadeh, C K Wong, IEEE Transactions on Computers. 413M. Sarrafzadeh and C.K. Wong. Bottleneck Steiner trees in the plane. IEEE Trans- actions on Computers, 41(3):370-374, 1992. Approximations for a bottleneck Steiner tree problem. Algorithmica. L Wang, D.-Z Du, 32L. Wang and D.-Z. Du. Approximations for a bottleneck Steiner tree problem. Algo- rithmica, 32:554-561, 2002. Approximation algorithm for a bottleneck k-Steiner tree problem in the Euclidean plane. L Wang, Z.-M Li, Information Processing Letters. 81L. Wang and Z.-M. Li. Approximation algorithm for a bottleneck k-Steiner tree prob- lem in the Euclidean plane. Information Processing Letters, 81:151-156, 2002.	[]
[ "BranchGAN: Branched Generative Adversarial Networks for Scale-Disentangled Learning and Synthesis of Images", "BranchGAN: Branched Generative Adversarial Networks for Scale-Disentangled Learning and Synthesis of Images" ]	[ "Zili Yi \nHuawei Technologies Co. Ltd\n\n\nSimon Fraser University\nCanada\n\nMemorial University of Newfoundland\nCanada\n", "Zhiqin Chen \nSimon Fraser University\nCanada\n", "Hao Cai \nMemorial University of Newfoundland\nCanada\n", "Xin Huang \nMemorial University of Newfoundland\nCanada\n", "Minglun Gong \nMemorial University of Newfoundland\nCanada\n", "Hao Zhang \nSimon Fraser University\nCanada\n" ]	[ "Huawei Technologies Co. Ltd\n", "Simon Fraser University\nCanada", "Memorial University of Newfoundland\nCanada", "Simon Fraser University\nCanada", "Memorial University of Newfoundland\nCanada", "Memorial University of Newfoundland\nCanada", "Memorial University of Newfoundland\nCanada", "Simon Fraser University\nCanada" ]	[]	We introduce BranchGAN , a novel training method that enables unconditioned generative adversarial networks (GANs) to learn image manifolds at multiple scales. The key novel feature of Branch-GAN is that it is trained in multiple branches, progressively covering both the breadth and depth of the network, as resolutions of the training images increase to reveal finer-scale features. Specifically, each noise vector, as input to the generator network, is explicitly split into several sub-vectors, each corresponding to, and is trained to learn, image representations at a particular scale. During training, we progressively "defreeze" the sub-vectors, one at a time, as a new set of higher-resolution images is employed for training and more network layers are added. A consequence of such an explicit sub-vector designation is that we can directly manipulate and even combine latent (sub-vector) codes which model different feature scales. Experiments demonstrate the effectiveness of our training method in scale-disentangled learning of image manifolds and synthesis, without any extra labels and without compromising quality of the synthesized high-resolution images. We further demonstrate three applications enabled or improved by BranchGAN.	null	[ "https://arxiv.org/pdf/1803.08467v2.pdf" ]	53,960,495	1803.08467	90f3977184cee09ba52267f8471a05b39867c7c6	BranchGAN: Branched Generative Adversarial Networks for Scale-Disentangled Learning and Synthesis of Images Zili Yi Huawei Technologies Co. Ltd Simon Fraser University Canada Memorial University of Newfoundland Canada Zhiqin Chen Simon Fraser University Canada Hao Cai Memorial University of Newfoundland Canada Xin Huang Memorial University of Newfoundland Canada Minglun Gong Memorial University of Newfoundland Canada Hao Zhang Simon Fraser University Canada BranchGAN: Branched Generative Adversarial Networks for Scale-Disentangled Learning and Synthesis of Images Generative Adversarial NetworkImage SynthesisImage Representation LearningMulti-ScaleDisentanglement We introduce BranchGAN , a novel training method that enables unconditioned generative adversarial networks (GANs) to learn image manifolds at multiple scales. The key novel feature of Branch-GAN is that it is trained in multiple branches, progressively covering both the breadth and depth of the network, as resolutions of the training images increase to reveal finer-scale features. Specifically, each noise vector, as input to the generator network, is explicitly split into several sub-vectors, each corresponding to, and is trained to learn, image representations at a particular scale. During training, we progressively "defreeze" the sub-vectors, one at a time, as a new set of higher-resolution images is employed for training and more network layers are added. A consequence of such an explicit sub-vector designation is that we can directly manipulate and even combine latent (sub-vector) codes which model different feature scales. Experiments demonstrate the effectiveness of our training method in scale-disentangled learning of image manifolds and synthesis, without any extra labels and without compromising quality of the synthesized high-resolution images. We further demonstrate three applications enabled or improved by BranchGAN. Introduction Unconditioned GANs [1] have been intensively studied as a means for unsupervised learning and data synthesis. Compared to their conditional counterparts [2,3,4,5,6,7,8,9], unconditioned GANs place less burden on the training data but are less steerable at the same time. In an unconditional GAN, a well-trained generator could synthesize novel data by sampling a random noise vector from the learned manifold as input and altering values "parameterizing" the dimensions of the manifold. However, this synthesis process is typically uncontrollable and counterintuitive, since we have little understanding how each manifold dimension impacts the synthesized output. For manifold learning of images or other visual forms, the notion of feature scales is of great importance. An ability to learn multi-scale or scale-invariant features often leads to a deeper and richer understanding of representations and distributions of images. In the last few years, scale-aware unconditioned GANs have been developed, ıe.g., StackGAN [6], LPGAN [10] and PGGAN [11], where correlated GANs are trained in a coarse-to-fine manner, using lower-and then higher-resolution images, with the goal of improving the quality of the final full-resolution images. However, factors which impact image features at various scales remain entangled in PGGAN networks. In StackGAN and LPGAN, factors (dropout layer or noisy perturbation) were added at various scales to learn scale-independent image features, though they are neither explicit nor controllable. Under the setting of conditional GANs, several recent works [12,3], including DNA-GAN [13] and InfoGAN [14], aim to disentangle the latent codes which correspond to different image attributes. Namely, the image structures are disentangled by attributes rather than scales. In this paper, we introduce a novel training method that enables unconditioned GANs to learn image manifolds in a "scale-disentangled" manner, aiming to improve the controllability of image synthesis and editing. The key novel feature of our learning paradigm is that each noise vector, as input to the generator, is explicitly split into a prescribed number of sub-vectors, ıe.g., 5 for learning 256 2 images and 6 for 512 2 images, where each sub-vector corresponds to, and is trained to learn, image representations at a particular scale. A consequence of such a sub-vector designation is that we can directly manipulate and even com- Training pipeline of BranchGAN. We start with both the generator (G) and discriminator (D) having a low spatial resolution. During the first training period, we feed z 0 with random vectors of uniform distribution and z t ( t > 0) with zero vectors 0, and make those linear layers corresponding to z t ( t > 0) untrainable. As the training advances, we incrementally add layers to G and D, thus increasing the spatial resolution of the generated images. Meanwhile, we "de-freeze" more z vectors for training by feeding them with non-zero uniform-random vectors. This process is repeated until the target resolution is reached. During training, "branch suppression" (see Section 3) happens as z 0 has well encoded large-scale structures and will maintain its dominance in coarse-level encoding. When z 1 is de-frozen, it is suppressed in terms of coarse-level encoding but has the chance to encode finer-scale features. bine latent (sub-vector) codes modeling different feature scales, leading to novel applications of unconditional adversarial learning that were not possible before. Figure 1 shows an example of cross-scale image fusion, where we intentionally synthesize an image by integrating the coarse-scale features of one image with finer-scale features of another. At the high level, our learning method employs the standard GAN framework, which comes with an unconditioned generator and a discriminator, and follows the standard GAN training paradigm as described in [1,15]. To achieve scale-disentangled learning, our network is trained progressively, bearing some similarity to Karras et al. [11]. However, instead of progressing only on network depth (adding network layers as the resolutions of training images increase), our network training also progresses over network width by progressively activating sub-vectors that correspond to different feature scales; see Figure 2 and more details in Section 3. As a result, we explicitly designate dimensions of the image manifold to different image scales, leading to scale-specific "training branches". Hence, we refer to our network as BranchGAN . By disentangling the image scales, BranchGAN enables multi-scale learning of image manifolds and more controllable image editing and synthesis, without requiring extra labels. We tested our novel training method on several highquality image datasets to verify its effectiveness in learning scale-disentangled image representations, compared with alternative GAN training schemes. We show that BranchGAN enables new applications of GANs to coarse-to-fine image synthesis and scale-aware image fusion. We further demonstrate improved performance of iGAN [16], an interactive image generation method, when the generator is trained by BranchGAN in place of DCGAN. Related work Multi-scale image representation An inherent property of visual objects is that they only exist as meaningful entities over certain ranges of scales in an image. How to describe image structures at multiple scales remains an essential and challenging problem in image analysis, processing, compression, as well as image synthesis. Early methods for multi-scale image representing such as Discrete Fourier/Cosine Transforms (DFT/DCT) [17] and Discrete Wavelet Transformation (DWT) [18] are widely used in decomposing small-scale details and large-scale structures. In our paper, DFT is employed as a metric for scale disentanglement. Another scale-independent representation of images is the layer activations of a well-trained Convolutional Neural Network (CNN) [19,20,21]. In a CNN, top activation layers roughly represent large-scale image structures such as objects and scenes, while bottom activations represent small-scale details such as edges, colors, or textures. Other than CNNs, stacked models such as Deep Belief Network (DBN) [22], Stacked AutoEncoders (SAE) [23,24] or multi-scale sparse Coding [25] can also be utilized to retrieve multi-scale representations of images, though the effectiveness could be limited. Coarse-to-fine image synthesis Scale-aware image synthesis has been explored in StackGAN [6], LPGAN [10], and PGGAN [11], as we discussed in Section 1. The goal of these methods is to synthesize higher-quality images, rather than to learn multi-scale image manifolds. We extend the idea of progressive training from progressively adding layers to progressively growing both layers and branches. In addition, instead of training multiple GANs [6,10], our method trains only one GAN. Controllability in image synthesis One of the most frequently adopted approaches to improving the controllability of image generation is conditional modeling. For example, conditional GANs or semi-conditional GANs can condition the image synthesis task on image attributes [3,12,13,14], classes [4], input texts [5,6], or images [7,8,9]. These methods either require extra labels or paired training images, or need strict inherent relations between the priors and the outputs. Our method, as a type of Fig. 3. Training sub-procedure at one scale level. After a new layer is added to the generator, we first only train the last layer of the generator while holding other layers untrainable (Stage I). After the last layer is well-trained, we then de-freeze all pretrained layers (the branches in green) plus the newly-added branch for training (Stage II). To avoid "sudden shock" to the well-trained layers, we feed the newly de-frozen z sub-vector with a random vector following uniform distribution U (−α, α), where α increases smoothly from 0.0 to 1.0 throughout Stage II. unconditional GAN, conditions image generation on random noise of uniform distribution and does not require any extra labels or priors. For unconditional GANs, the method known as iGAN [16] provides a way for users to synthesize or manipulate realistic images in a more controllable way. In iGAN, users could add certain constraints on the appearance of desired images (ıe.g., draw edges, add color strokes or set an exemplar image) and the latent code is then optimized to satisfy these constraints. Nonetheless, a sophisticated optimization method is required, as gradient descent is particularly vulnerable to local minima. We demonstrate through experiments that the scale-disentangled latent spaces learned with our method can help iGAN avoid falling into local minima and hence boost its performance. BranchGAN training As shown in Figure 2, we start with training only the sub-vector corresponding to the coarsest level features, ıi.e., using the lowest-resolution images, while keeping the other sub-vectors "frozen". Then we progressively "de-freeze" the sub-vectors, one at a time, as a set of higher-resolution images is employed for training and more network layers are added. When training for a finer scale, the network weights learned from the previous coarser scales are used for initialization. Note that after training, these weights are often changed to adapt to the new training data. What had motivated our branched GAN idea and what made it work effectively is a phenomenon that we observed during our experiments with multibranch data generators and coined as "branch suppression". Roughly speaking, we found that when multiple noise vectors, with their respective training branches are at play, GAN training typically results in one dominant branch while the other branches are either fully or partially suppressed. In other words, the already-trained weights (branches) will have priority in maintaining their role in encoding the image structures that are already encoded and suppress the other branches. Branch suppression does happen to the proposed branched training method. When de-freezing one sub-vector during progressive training, branch suppression helps inhibit the ability of the newly defrozen branch in the network to encode coarser-scale structures, thus "encouraging" it to encode the finer-scale structures in the new set of higher-resolution training images. Note that the inhibition or suppression is not absolute; the network weights in previously trained branches are still altered. BranchGAN progressively adds the depth (or layers) and breadth (or z sub-vectors) to the generator, and then start training with images of higherresolution. During the process, branch suppression helps encourage the newlyadded sub-vector to encode the finer-scale structures, as shown in Figure 2. At each scale, a two-staged sub-procedure is used to avoid sudden shock to already well-trained, smaller-resolution layers: see Figure 3. Note that all already-added layers of the discriminator are trainable throughout the sub-procedure. More details about hyper-parameters are available in the supplementary material. The architecture of the generator is the same as shown in Figure 2. For 256 × 256 image generation, we use 5 z sub-vectors. The number of z subvectors is subject to change according to the resolution of output images. We use generator and discriminator networks that are mirror images of each other and always grow in synchrony, and use the standard non-saturated loss as in DCGAN [15] for training. Results, evaluation, and applications We have tested and evaluated BranchGAN on three datasets: church_outdoor from LSUN [26], celeba_hq [11], and car. The original car dataset has 800 × 600 pixel resolution. To speed up the training, we used downsampled versions of celeba_hq (256 × 256) and car (400 × 300). We trained our models on a GTX TITAN XP GPU, which took roughly 20 ∼ 40 hours per (full) dataset. Evaluation of scale disentanglement Qualitative evaluation. We wish to show how each designated sub-vector affects images generated via BranchGAN training. To this end, we vary the values of each sub-vector while holding the other sub-vectors fixed, as shown in Figures 4, 5, and 6. From visual examination and as detailed in the figure captions, we can observe that z 0 affects the output images most significantly as it mainly controls large-scale structures, which contrasts the effects of z 4 or z 5 as they are tied more to smaller-scale details. This suggests that scale disentanglement by splitting the training to progressively activated sub-vectors has been achieved. The output setting is the same as in Figure 4. The first row basically reflects the property of the car dataset: it contains left-views and right-views, but no front-views. GAN was not able to learn a smooth interpolation between the two views, resulting in a messy image in the middle of the first row. However, once z 0 is fixated with a view, the other sub-vectors can generate smooth interpolations. z 1 appears to alter the view angles slightly, z 2 impacts the front parts of the car, while changing the other sub-vectors influences more minor details. That being said, it is also clear that we do not have direct control for feature localization or image semantics. Quantitative evaluation. To examine how each dimension of the latent manifold space impacts appearance of the output images, we designed a metric to evaluate the variance of the output images when the latent vector is manipulated. The metric, which we refer to as variance by scale or VBS, denoted by V, measures the variation of output images with respect to any sub-vector z of z, at a specific scale, as reflected by a frequence interval [f 1 , f 2 ]. That is, z can correspond to a single dimension of z or to one of the designated sub-vectors z t , t = 0, . . . , 4. Figure 4. Similarly, as more clearly reflected in the variance images, the sub-vectors z t , with t from 0 to 4, appear to control higher-level to finer image details. Specifically, V f 2 f 1 (z ) = h,w,d E c∼U (−1,1) σ z ∼U (−1,1),z ←c , DFT f 2 f 1 (G(z)) , and V f 2 f 1 (z ) = V f 2 f 1 (z )/ E z ⊆z V f 2 f 1 (z ),(1) where z is the set of dimensions of z excluding z . impact of a manifold dimension (or a subset of manifold dimensions) on the output. To examine the distributions of VBS at specific scales, we split the frequency domain into five ranges: (0, 1/16), (1/16, 1/8), (1/8, 1/4), (1/4, 1/2), and (1/2, 1), which roughly correspond to increasingly fine image scales. We then visualize the VBS distributions for various GANs using histogram plots, as shown in Figure 8. To produce the histrograms, we randomly sampled 10 z vectors to generate 10 images using the trained model for the celeba_hq dataset. For each z vector, which is of dimension 150, and for each dimension, we vary it while keeping all the other dimensions fixed. This results in a set of varied images, from which we compute the VBS value for the selected dimension. Overall, we collect [15], PGGAN [11] and InfoGAN [14] predominantly falls into the interval (0.5, 1.5), whereas the VBS of BranchGAN spans over a wider range of (0.1, 2.5). Peak VBS values indicate maximal impact. For example, sub-vector z 0 exhibits higher impact over the lowest frequency range, which corresponds to larger image scales. for each frequency range. As can be observed from Figure 8, the VBS values of BranchGAN exhibit a much greater variance (ıi.e., wider histogram) than those of traditional GANs, such as DCGAN [15], PGGAN [11], and InfoGAN [14]. Specifically, the VBS values of these traditional GANs at all scale levels mainly fall into the range of [0.5, 1.5], implying that the corresponding image representations are more scale-entangled, in comparison to BranchGAN, whose VBS values vary over a larger interval [0.1, 2.5]. Finally, to examine whether the image manifolds learned by BranchGAN are disentangled by the designated sub-vectors z t , t = 0, . . . , 4, we show how the VBS of each z t varies against frequencies in Figure 9. These VBS values were obtained in the same way as for the histogram plots in Figure 8. As we can observe, V(z 0 ) sees its peak value in the (0, 1/16) range, implying that z 0 mainly controls larger-scale structures of the generated images. The remaining sub-vectors z 1 , z 2 ...z 4 show their peaks at frequency intervals which reflect their respective controls over increasingly finer image features or structures. New applications Coarse-to-fine image synthesis. We show that the scale disentanglement afforded by BranchGAN facilitates coarse-to-fine image synthesis. To this end, we developed a new interactive application; see Figure 10 (a). A user can select bestmatching faces from randomly-generated ones displayed on the right panel. At the coarsest scale, the images are mapped from different z 0 values with other sub-vectors set to zero. If the user is satisfied with a coarse-level image, then he/she can select it and move on to the next scale. Then, the value of z 0 will be fixed and images mapped from different z 1 values will be displayed for selection. As a result, the user can progressively improve the appearance of a synthesized face, as shown in Figure 10 (b). Cross-scale image fusion. Scale disentanglement facilitates another new application: cross-scale image fusion, where latent codes representing different scales are joined to create hybrid images. Figures 1 and 7 show some examples of such image fusion, which are synthesized by integrating coarse-scale features of one image with fine-scale features of another. Through swapping sub-vectors representing different scales, our approach can achieve coarse-level fusion, such as face swap, as well as fine-level fusion, such as expression and face shape transfer. Improving interactive image editing (iGAN) We show how BranchGAN can improve the performance of iGAN [16], an interactive image editing tool. In the original paper, DCGAN [15] was adopted. We now compare different choices of GANs as replacement for DCGAN, including PGGAN [11], InfoGAN [14], and BranchGAN. In the iGAN framework, a user makes interactive edits (ıe.g., scribbles, warping) to an existing image. The edits may be unprofessional and lead to various artifacts. The tool can automatically adjust the output image to keep all user edits as natural as possible. We conduct our experiment using a modified iGAN workflow, as shown in Figure 11. Specifically, we abandoned "edit transfer" [16] to improve performance for images with cluttered backgrounds, as is the case for the celeba_hq dataset. On the other hand, we enriched the available operations in the GUI, as shown in Figure 12. We also removed the smoothness term from the original optimization objective to ensure more faithfulness to user inputs. Aside from these changes, the modified iGAN workflow is the same as the original iGAN implementation [16] and has the following objective consisting of a color term and an edge term: z * (C, M, E) = arg min z \|C − G(z)\| · M/\|M\|+ α\|HOG(G(z)) − HOG(E)\|,(2) where C is the color map (edited image) and M is the mask. M(i, j) = 1 if pixel (i, j) has color assigned and M(i, j) = 0 if (i, j) has color erased. \|M\| denotes the sum of elements in M. E is the edge map drawn with the edge tool and HOG(·) is the differentiable HOG operator [27] which maps an image to a HOG descriptor. The parameter α is used to balance the two terms in the objective function. We set α = 10 throughout our experiment. Note that if the user does not provide an edge map, the edge term is disabled. The modified iGAN framework can be integrated with the generator of any unconditional GAN. To examine which GAN manifold serves iGAN better, we used the minimum optimization value of the objective in Eq. (2) as the metric. The minimum optimization loss indicates to what extent the output image fits user inputs. Smaller objective value represents greater effectiveness of the GAN manifold in avoiding local-minima traps. We compared the minimum optimization loss of DCGAN [15], PGGAN [11], InfoGAN [14], and our BranchGAN, given the same set of user inputs. To simplify the comparison, we used 60 images from the training dataset as the user inputs. Table 1 shows that BranchGAN has the lowest minimum objective value than traditional GANs, implying that the scale-disentangled manifolds learned by BranchGAN can better fit user inputs. Figure 13 shows a few image editing examples using different GAN manifolds as the latent space of iGAN. The visual comparison between BranchGAN and other GANs indicates that BranchGAN generally performs better in fitting both coarse-level structures and fine-scale features. Conclusion, limitation, and future work We have introduced BranchGAN, a novel, progressive training procedure for unconditional GANs which enables multi-scale image manifold learning and manipulation. The key idea is to not only progressively increase network depth by adding layers, but also increase the network width by creating multiple, progressively activated training branches triggered by different sub-vectors of the network input. Each sub-vector corresponds to, and is trained to learn, image representations at a particular scale, leading to a scale-disentangled learning scheme. Experimental results on several well-known high-quality image datasets verify the effectiveness of our method in disentangling image manifolds by scales. We also demonstrated new and improved applications by GANs via BranchGAN training. BranchGAN is scale-aware, but not feature-aware. This is a major limitation to progressive training using images at a selected set of resolutions. One reason is that not all image features are well represented in this selected set of training images. Another reason is that similar or repeated features in an image may not always be in the same scale, ıe.g., due to perspective projection. While such features are often manipulated as a collection during editing, they are difficult to learn using the current BranchGAN. In addition, the scale disentanglement afforded by our current approach is only a partial one, since adding a new training layer, which corresponds to a newly activated sub-vector, can still impact weights learned for the preceding sub-vectors. As a result, all learned weights may be correlated with image features across multiple scales. Overall, while the controllability enabled by BranchGAN training for image manipulation has been improved, it is still inherently limited. In future work, we would like to extend BranchGAN to feature-or semanticaware progressive training, where sub-vector designation can be based on more meaningful or more visually apparent image features; this would add more meaning to sub-vector manipulation for image editing and synthesis. We believe that the progressive training paradigm introduced by BranchGAN is a generic approach and can be tuned for different forms of disentanglement by adjusting the training targets. In addition, we shall explore potential values of scaledisentangled image manifolds in tasks such as image compression, filtering, and denoising. Finally, it is a curious question whether branch suppression exists in other "multi-branch" neural networks, such as ResNet [28], DenseNet [29], and capsule networks [30]. We are interested in whether this phenomenon may offer insights to the training of other convolutional and/or generative networks. Appendix Networks architecture and hyperparameters Code for the models is available will be made available. The detailed information about the networks architecture of generators and discriminators are presented in Table 2, 3, 4, 5. The non-architecture hyper-parameters are listed in Table 6. Please note that lrelu is leaky relu layer. activation size filter size input [30], [30], NA [30], [30], [ 6.2 Initialization of neural weights, "freeze" and "defreeze" For the untrained linear or deconv/conv/linear layers, the filter weights are initialized with normally random numbers N (µ, σ) and biases are initialized with 0. For instance normalization layer, we initialize the scale with 1.0 and assign the center with 0.0. Table 6. Non-architecture training hyperparameters. We "freeze" certain branches (or weights) by feeding the corresponding z vector with 0. Here we intend to explain the reason why feeding zero vector makes the corresponding weights untrainable. Therefore, the activations of linear layer when fed with 0 are given by f (0) ≡ θ0 + 0 ≡ 0, where θ are the linear weights. The activations of conv/deconv layer are given by f (0) ≡ θ 0 + 0 ≡ 0 (or θ 0 ≡ 0 after concatenation), where θ are filter weights. So we have the gradients ∇f θ (0) ≡ 0. For instance normalization layer and leaky relu layer, g(0) ≡ lrelu((0 − 0) · 1.0 + 0.0) ≡ 0. We have the gradients g β (0) ≡ 0, where β is the scale. In this way, the branches (or weights) could be "frozen" when fed with 0. To "defreeze" these branches (or weights), simply feed them with non-zero vectors. Branch suppression We observed "branch suppression" in all kinds of multi-branch generators as shown in Figure 14, among which some are fully suppressed, some are partially suppressed. In "branch suppression", the already-trained weights (branches) will have priority in maintaining their role in encoding the image structures that are already well encoded and suppress the other branches. To explain it in more details, we present a few examples of branch suppression in Figure 14. Consistency of V BS with human perception We are not aware of universally accepted metrics to assess variation of outputs "by scale", as Frechet Inception Distance and Inception Score do not. We proposed VBS as objective metrics for variation-by-scale. To assess consistency of the metric with human perception, we conducted a user study. We selected 48 pairs of result images (18 for each dataset) that were produced by controlling the value of different latent vectors of BranchGAN. We hired 20 Turkers to rate each pair in terms of level of variation. In the test, three options with elaborate explanations were shown to the Turkers: (a) large-scale variation; (b) median-scale variation; or (c) small-scale variation. The label with the most votes is treated as ground-truth. Then we compare the human-labeled results with those estimated by VBS (the scale level with highest score is used) and compute the percentages of agreements is 85,4%. The agreement rates of VBS and human perception are quite high from this preliminary study (significantly better than random), serving as an initial validation. Further explorations are certainly warranted. Experiments with LAPGAN [10] LAPGAN [10] is very similar to BranchGAN in terms of coarse-to-fine image synthesis and adding noise at multiple scale level, though the noise is added though dropout layer, which is neither controllable nor explicit. We did re-implement LAPGAN and attempt to add noise vectors explicitly as inputs to generators at each step. The results showed that the noise vectors are not responsible for any variation of the residual images. The reason could be that by using strictly paired training data, the upscaled conditioning image would deterministically generate the residual image. 11. Overview of modified iGAN workflow. As in [16], the encoder that projects an image onto a manifold needs to be trained in advance. Once the user makes an edit to the image, the edited image is mapped to a latent code in the manifold space, which is assigned as the initial value (z0) of the latent vector. Then the latent vector z is optimized to minimize the objective in Eq. (2). In these examples, we employ the training loss and discriminator of dcgan [15]. Here we change the architecture of the generator a bit by conditioning image generation on split z vectors (z t , t ∈ {1, 2, 3}). In the upper row, the left branch is already well trained for image generation, and the middle and right branches are initialized randomly (see more details about the initialization in the supplementary material). Then we train the GAN by following the standard GAN training procedure as in [15]. After the training converges, the left branch dominates the output while the other are fully suppressed, as seen from the variance image on the right (see Fig. 4 for the meaning of variance image). In the lower row, the generator architecture is the same as traditional GAN except that the z vector is split. We train the left branch till converging, then de-freeze the middle branch for training till converging, and finally the right branch. Note that the number of training steps for each stage are equal and the pre-trained weights (or branches) are not frozen even after new branches are de-frozen. As a result, the middle branch is slightly suppressed and the right branch is severely suppressed as seen from the rightmost variance images. Figure 4 except that there is one more sub-vector z 5 and each row is generated independently. From top to bottom, changing z t (t ∈ {0, 1, 2, 3, 4, 5}) leads to smaller and smaller image variations, as reflected by intensity drop in the variance images. Similar to Figure 4, sub-vector z 0 dominates the overall color, z 1 controls some facial features and hair features, while the rest bring minor changes near ear, mouth, and hair. Fig. 1 . 1Cross-scale image fusion by directly combining coarse-scale features in one image with finer-scale features from another. Please note that x 0 (x ∈ {a, b}) encodes image-wide structures and x t (t ∈ {1, 2, 3, 4}) encodes increasingly fine-scale features. Given a pair of images, we compose new images by cross-combining coarse-scale structures and fine-scale features of the two, accomplishing expression transfer (a) and face swap (b). Fig. 2 . 2Fig. 2. Training pipeline of BranchGAN. We start with both the generator (G) and discriminator (D) having a low spatial resolution. During the first training period, we feed z 0 with random vectors of uniform distribution and z t ( t > 0) with zero vectors 0, and make those linear layers corresponding to z t ( t > 0) untrainable. As the training advances, we incrementally add layers to G and D, thus increasing the spatial resolution of the generated images. Meanwhile, we "de-freeze" more z vectors for training by feeding them with non-zero uniform-random vectors. This process is repeated until the target resolution is reached. During training, "branch suppression" (see Section 3) happens as z 0 has well encoded large-scale structures and will maintain its dominance in coarse-level encoding. When z 1 is de-frozen, it is suppressed in terms of coarse-level encoding but has the chance to encode finer-scale features. Fig. 4 . 4Effects on generated images for celeba_hq dataset by varying individual subvectors. We first initialize z randomly, and then replace one of the sub-vectors z t , t = 0, . . . , 4, by pI, where p = −0.8, −0.4, 0, 0.4, or 0.8 and I is the all-one vector of length \|z t \|, while holding all the other sub-vectors fixed. Columns 1 to 5 show images generated by BranchGAN and the last column shows a variance image for the five generated images on the left, where lightness reflects pixel variance. From top to bottom, changing z t leads to smaller and smaller image variations, as reflected by intensity drop in the variance images. Sub-vector z 0 dominates the overall color, z 1 controls some facial features, while the rest bring minor changes near ear, mouth, and hair. Fig. 5 . 5Effects on generated images for car dataset by varying individual sub-vectors. Fig. 6 . 6Effects on generated images for church_outdoor dataset by varying individual sub-vectors. The output settings is the same as in σ z ∼U (−1.0,1.0),z ←c f (z) refers to the deviation of the value of f (z) when z follows the uniform distribution U (−1.0, 1.0) and z is fixed as a constant vector c. G(z) is the output image of the generator G given z. h, w, d are the height, width, and depth of images (or layer activations). E(·) is the expectation operator. In Eq. (1), DFT f2 f1 (·) refers to the discrete Fourier transform of an image, and (f 1 , f 2 ) is a frequency range. In order to avoid the impact of image size, VBS is further normalized by division over their expected values. Intuitively, a larger value of VBS implies a greater Fig. 7 . 7Cross-scale image fusion. The notations and synthesis setup are the same as inFigure 1. From the first three columns, we can see that the sub-vector x 1 mainly controls the facial features while x 2 controls the face shape. By swapping x 1 and x 2 , a face swap may be achieved. x 3 and x 4 have less significant impacts, such as lighting, shading, and minor changes in hair and ear. More such examples can be found in the supplementary material. Fig. 8 . 8Distributions of VBS over specific frequency ranges or image scales. The VBS of DCGAN Fig. 9 . 9Plot of VBS values for sub-vectors z 0 , z 1 , . . . , z 4 against frequencies. 6. 4 4More image editing results with BranchGAN and iGANFigure 16 shows the outputs of BranchGAN on celeba_hq 512 × 512 dataset. Figures 15 shows more results of cross-scale image fusion. Figures 17, 19, 18, 20 and 21 show more results of iGAN using our multi-scale image manifold as the latent codes. Fig. 10 . 10Coarse-to-fine image synthesis. (a): GUI of the application. (b): Sequences of images selected in a coarse-to-fine manner. Bounding boxes with the same color highlight the changes made by the user at each step: yellow box → thinner face, red box → less dimple, and blue box → red lip. Fig. Fig. 11. Overview of modified iGAN workflow. As in [16], the encoder that projects an image onto a manifold needs to be trained in advance. Once the user makes an edit to the image, the edited image is mapped to a latent code in the manifold space, which is assigned as the initial value (z0) of the latent vector. Then the latent vector z is optimized to minimize the objective in Eq. (2). Fig. 12 . 12GUI of the modified iGAN tool. The main window includes an edit zone (left) and a display zone (right). The edit zone provides various tools and a canvas to help edit the color map and mask or produce the edge map. The display zone shows the result generated by iGAN based on the edits. See video in the supplemental material for more details. Fig. 13 . 13Comparison of iGAN results when using different GAN manifolds as the latent space. Original images were edited by users in different ways: (a) no edit, (b) face lightened, (c) hair erased, (d) adding edge map. Notably, none of the results perfectly fit the edited images, as patterns do exist in images generated by the same GAN model. Other than this, we observe that in (a) and (b), the head poses rendered by BranchGAN and PGGAN fit the inputs better. In terms of smaller-scale image features, BranchGAN generally performs better than other GANs, which could be observed in regions highlighted in red bounding boxes. Fig. 14 . 14Two examples of branch suppression in GANs. Fig. 15 . 15Results of Cross-scale image fusion. The notations and synthesis setup are the same as inFigure 7. Fig. 16 . 16Effects on generated images for celeba_hq 512 × 512 dataset by varying individual sub-vectors. The output setting is similar as in Fig. 17 . 17The edge maps and color maps drawn by users and the corresponding image generation results with improved iGAN. Fig. 18 . 18Edits by users and the corresponding results generated by improved iGAN. (a) face erased. (b) face slimmed. (c) mouth replaced with a patch from another image. (d) hair darkened. (e) mouth closed. (f) hair turned brown. (g) hair turned brown, eye shadowed, and lips reddened. (h) face whitened and lips reddened. Fig. 19 . 19Face image (512x512) generation and editing results with improved iGAN. (a)-(b), results based on edge maps. (c)-(d), results based on masked color maps. (e)-(h), image editing results. Fig. 20 . 20Car image generation and editing results with improved iGAN: (a-b) results based on edge maps; (c-d) results based on masked color maps; (e-h) manipulation of existing images, including erasing license plate (e), changing body color (f & h), and adding extra edge map (g). Fig. 21 . 21Church image generation and editing results with improved iGAN: (a-b) results based on edge maps; (c-d) results based on masked color maps; and (e-h) image editing results. Average minimum optimization loss of iGAN using different GANs. DCGAN was used by the original iGAN.Dataset celeba_hq car church_outdoor Image size 256 2 512 2 400 × 300 256 2 DCGAN 0.24 0.26 0.29 0.27 PGGAN 0.18 0.17 0.19 0.22 InfoGAN 0.23 0.25 0.29 0.24 Ours 0.15 0.14 0.17 0.18 Table 1. Network architecture of the generator for 256 × 256 image synthesis.30] concat [150] NA linear [32768] [32768,150] reshape [8,8,512] NA deconv+instanceNorm+lrelu [16,16,256] [5,5,512,256] deconv+instanceNorm+lrelu [32,32,128] [5,5,256,128] deconv+instanceNorm+lrelu [64,64,64] [5,5,128,64] deconv+instanceNorm+lrelu [128,128,64] [5,5,64,64] deconv+sigmoid (output) [256,256,3] [5,5,64,3] Table 2. Network architecture of the discriminator for 256 × 256 image synthesis. Network architecture of the generator for 400 × 300 image synthesis. Network architecture of the discriminator for 400 × 300 image synthesis.activation size filter size input [256,256,3] NA deconv+instanceNorm+lrelu [128,128,64] [5,5,3,64] deconv+instanceNorm+lrelu [64,64,64] [5,5,64,64] deconv+instanceNorm+lrelu [32,32,128] [5,5,64,128] deconv+instanceNorm+lrelu [16,16,256] [5,5,128,256] deconv+instanceNorm+lrelu [8,8,512] [5,5,256,512] reshape [32768] NA linear [1] [32768,1] Table 3. activation size filter size input [30], [30], NA [30], [30], [30] concat [150] NA linear [17920] [17920,150] reshape [5,7,512] NA deconv+instanceNorm+lrelu [10,13,256] [5,5,512,256] deconv+instanceNorm+lrelu [19,25,128] [5,5,256,128] deconv+instanceNorm+lrelu [37,50,64] [5,5,128,64] deconv+instanceNorm+lrelu [75,100,64] [5,5,64,64] deconv+instanceNorm+lrelu [150,200,64] [5,5,64,64] deconv+sigmoid (output) [300,400,3] [5,5,64,3] Table 4. activation size filter size input [300,400,3] NA deconv+instanceNorm+lrelu [150,200,64] [5,5,3,64] deconv+instanceNorm+lrelu [75,100,64] [5,5,64,64] deconv+instanceNorm+lrelu [38,50,64] [5,5,64,64] deconv+instanceNorm+lrelu [19,25,128] [5,5,64,128] deconv+instanceNorm+lrelu [10,13,256] [5,5,128,256] deconv+instanceNorm+lrelu [5,7,512] [5,5,256,512] deconv+instanceNorm+lrelu [17920] NA linear [1] [17920,1] Table 5. name value Optimizer AdamOptimizer learning rate 0.0002 beta1 0.5 beta2 0.999 #sub-vector 5 for 256 × 256 images 6 for 512 × 512 or 400 × 300 images #epoch/scale 20 for 256 × 256 images 12 for 512 × 512 or 400 × 300 images #batch/epoch subject to dataset size and batch size we use full dataset for each epoch batch size 20 for 256 × 256 images 12 for 512 × 512 or 400 × 300 images I Goodfellow, J Pouget-Abadie, M Mirza, B Xu, D Warde-Farley, S Ozair, A Courville, Y Bengio, Generative adversarial nets. 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[ "Shtukas adiques, modifications et applications", "Shtukas adiques, modifications et applications" ]	[ "Kieu Nguyen ", "Hieu " ]	[]	[]	Dans cet article, via l'étude des modifications de fibrés vectoriels sur la courbe de Fargues-Fontaine, on prouve une formule géométrique reliant les tours de Lubin-Tate avec les espaces de Rapoport-Zink non ramifiés simples basiques de type EL de signature (1, n − 1), (p 1 , q 1 ), · · · , (p k , q k ) où p i q i = 0. En particulier, on en déduit le calcul des groupes de cohomologie de ces derniers.Abstract. In this paper, via the study of the modifications of vector bundles on the Fargues-Fontaine curve, we prove a geometric formula relating the Lubin-Tate towers with the simple basic unramified Rapoport-Zink spaces of EL type of signature (1, n−1), (p 1 , q 1 ), · · · , (p k , q k ) where p i q i = 0. In particular, we deduce the computation of the cohomology groups of the latter.	10.24033/bsmf.2819	[ "https://arxiv.org/pdf/1904.12381v1.pdf" ]	139,101,081	1904.12381	bf542d2f993a3fb129f99bdd800f1f834b134599	Shtukas adiques, modifications et applications 28 Apr 2019 April 30, 2019 Kieu Nguyen Hieu Shtukas adiques, modifications et applications 28 Apr 2019 April 30, 2019arXiv:1904.12381v1 [math.NT] Dans cet article, via l'étude des modifications de fibrés vectoriels sur la courbe de Fargues-Fontaine, on prouve une formule géométrique reliant les tours de Lubin-Tate avec les espaces de Rapoport-Zink non ramifiés simples basiques de type EL de signature (1, n − 1), (p 1 , q 1 ), · · · , (p k , q k ) où p i q i = 0. En particulier, on en déduit le calcul des groupes de cohomologie de ces derniers.Abstract. In this paper, via the study of the modifications of vector bundles on the Fargues-Fontaine curve, we prove a geometric formula relating the Lubin-Tate towers with the simple basic unramified Rapoport-Zink spaces of EL type of signature (1, n−1), (p 1 , q 1 ), · · · , (p k , q k ) where p i q i = 0. In particular, we deduce the computation of the cohomology groups of the latter. Introduction Le programme de Langlands prédit une bijection de nature arithmétique entre d'un côté les représentations galoisiennes et de l'autre les représentations automorphes. Pour construire des réalisations, on cherche des objets géométriques dont la cohomologie ℓ-adique s'exprimera en termes de ces correspondances. Les premiers exemples sont donnés par les espaces de Rapoport-Zink. Par ailleurs, plus récemment, Scholze a construit des variétés de Shimura locales qui se décrivent comme des espaces de modules de Shtukas dont la partie supercuspidale de la cohomologie est décrite par la conjecture de Kottwitz. Commençons par un triplet (G, µ, b) où G est un groupe réductif sur Q p , b ∈ G(Q p ) et µ ∈ X + * (T ) avec b ∈ B(G, µ) (c.f. section 3) auquel on associe un G-fibré E b sur la courbe de Fargues-Fontaine. De manière informelle, l'espace de module de Shtukas Sht(G, µ, b) est un faisceau sur Perf Fp classifiant les modifications de type µ entre E b et E 1 . L'espace H • c (V ⊗Q p C p , Z/ℓ n Z) où V parcourt les ouverts relativement compacts deM µ Kp et où ℓ = p est un nombre premier. On note également H • c (M µ Kp , Q ℓ ) := H • c (M µ Kp , Z ℓ )⊗Q ℓ lesquels sont des Q ℓ -représentations de G(Q p ) × D × × W Ep où Jb(Qp) = D × est le groupe des inversibles de l'algèbre de division d'invariant 1 n et où W Ep est le groupe de Weil du corps de définition E p de µ. Les groupes de cohomologie de la tour de Lubin-Tate ont été entièrement calculés dans [Boyer99], [HT01] et [Boyer09]; par dualité [Fal] [FGL] [SW17], on obtient aussi le cas de Drinfeld. Pour un espace de type EL non ramifié général, la partie supercuspidale est traitée dans [Far04], [Shin]. Sous l'hypothèse précédente, on obtient alors la description complète de groupes de cohomologie suivante avec les notations de la section 7. Théorème 1.2. Pour tout diviseur g de n = gs et toute représentation irréductible cuspidale π de GL g (F ), on a des isomorphismes G(Q p ) × W F -équivariants lim − → K H n−1−i c (M µ K )[π[s] D ] =    LT π (s, i) ⊗ L(π)\| · \| − s(g+1)−2(i+1) 2 · τ ∈J ω i • ( τ rec −1 F ) 0 ≤ i < s 0 i < 0 où π[s] D désignera la représentation JL −1 (St s (π)) ∨ de D × , ω i est le caractère central de LT π (s, i) et rec −1 F est le morphisme de réciprocité d'Artin et où τ rec −1 F := τ · rec −1 F ·τ −1 . La preuve repose sur le théorème 1.1 qui donne une relation entre la tour (M µ Kp ) Kp et la tour de Lubin-Tate ainsi que les résultats dans [Boyer09] décrivant la cohomologie de la dernière. Remarque 1.3. En utilisant les résultats de [Boyer14], on peut prouver que les groupes de cohomologie lim − → K H • c (M µ Kp , Z ℓ ) sont Z ℓ -libres. Remerciements. Je remercie profondément Pascal Boyer et Laurent Fargues tant pour leur aide mathématique déterminante que pour leurs constants encouragements. 2 La courbe de Fargues-Fontaine d'après [Far-Fon] Soit E/Q p une extension finie de corps résiduel F q d'uniformisante π. Soit F/F q un corps parfait muni d'une valuation non-triviale v : F −→ R ∪ {∞}. On pose E = W O E (F ) 1 π où W O E (F ) désigne l'anneau des vecteurs de Witt ramifiés à coefficients dans F . Plus précisément, on a E = n≫−∞ [x n ]π n \| x n ∈ F . Le corps E est une extension non ramifiée complète de E dont le corps résiduel est F . Il y a un relèvement de Teichmüler F −→ O E , x −→ [x]. On introduit alors le sous-anneau de E B bd = n≫−∞ [x n ]π n ∈ E \| ∃C, ∀n \|x n \| < C . On définit ensuite des normes de Gauss sur B bd . Pour r ∈]0, ∞[ et ρ = q −r ∈]0, 1[ on pose pour x = n [x n ]π n ∈ B bd v r (x) = inf n∈Z v(x n ) + nr et \|x\| ρ = sup n∈Z \|x n \|ρ n . Il s'agit de normes multiplicatives, autrement dit v r est une valuation sur B bd . Soient I ⊂ I ′ ⊂]0, 1[ alors Y I est un ouvert rationnel de Y I ′ . Si I = [\|a\|, \|b\|] avec a, b ∈ F × alors Y I = Y I ′ [a] π , π [b] . On pose Y = lim − →I Y I où I parcourt les intervalles compacts de ]0, 1[ d'extrémités dans \|F × \|. C'est un espace adique tel que Γ(Y, O Y ) = B. On peut donc voir les éléments de B I comme les fonctions holomorphes de la variable π sur la couronne de rayons définis par I. Le Frobenius ϕ de B bd induit un isomorphisme ϕ : B I ∼ − → B ϕ(I) et on a donc un isomorphisme ϕ : Y ϕ(I) ∼ − → Y I . En prenant la limite sur les intervalles I on obtient un automorphisme ϕ : Y ∼ − → Y. Ce morphisme ϕ agit de manière proprement discontinue sur Y et on définit alors la courbe de Fargues-Fontaine adique de la manière suivante. Définition 2.4. La courbe de Fargues-Fontaine adique est X ad = Y /ϕ Z . On rappelle enfin la version relative de la courbe de Fargues-Fontaine sur un espace affinoïde perfectoïde S. Intuitivement, on peut y penser comme une famille de courbes (X k(s) ) s∈S . Soit S = Spa(R, R + ) un espace affinoïde perfectoïde sur Spa(F q ). Posons A S = W O E (R + ) = n≥0 [x n ]π n \| x n ∈ R + , où W O E désigne les vecteurs de Witt ramifiés. Notons B bd S = A S 1 π bd = n≫−∞ [x n ]π n \| x n ∈ R, sup n \|x n \| < +∞ . Pour ρ ∈]0, 1[, il y a une norme de Gauss sur B bd S définie par n≫−∞ [x n ]π n ρ = sup n \|x n \|ρ n . De même, pour I ⊂]0, 1[ un intervalle compact, on note B I,S le complété de B bd S par rapport aux normes ( · ρ ) ρ∈I . On note B S = lim ← −I B I,S le complété de B bd S par rapport aux normes ( · ρ ) ρ∈]0,1[ . Comme auparavant, Y I,S = Spa B I,S , (B I,S ) o ainsi que Y S = lim − →I Y I,S sont des espaces adiques. Finalement, on définit la courbe de Fargues-Fontaine relative comme suit X ad S := Y S /ϕ Zoù P = d≥0 H 0 (X ad S , O X ad S (d)) et O X ad S (1) = (B S ) ϕ=π . Débasculements et diviseurs de Cartier sur la courbe Soit R une F q -algèbre perfectoïde. Notons R o ⊂ R le sous anneau des éléments de puissances bornées ainsi que R oo ⊂ R o l'ensemble des éléments topologiquement nilpotents. Un élément f = n≥0 [a n ]π n ∈ A R est dit primitif de degré 1 si a 0 ∈ R oo ∩ R × et a 1 ∈ R o× . Proposition 2.5. (prop. 1.18 de [Far16], [Fon]) • Soit R ♯ un débasculement de R sur E. Alors, le noyau du morphisme de Fontaine θ : W O E (R o ) ։ R ♯,o est engendré par un élément primitif de degré 1. • Inversement, si f ∈ A R est un élément primitif de degré 1 alors W O E (R o ) 1 π /f est une E-algèbre perfectoïde qui est un débasculement de R. On obtient ainsi une bijection entre l'ensemble des débasculements de R sur E et l'ensemble des immersions fermées T ֒→ Y R définies localement par un élément primitif de degré 1. Autrement dit, si S ∈ Perf Fq alors à chaque débasculement S ♯ correspond un diviseur de Cartier D : S ♯ ֒→ Y S et de plus ce diviseur de Cartier induit un diviseur de Cartier ϕ-invariant k∈Z ϕ k * D et on obtient donc un diviseur de Cartier sur X ad S . Supposons que F q ⊂ F . Pour tout h ∈ N * , il existe une unique extension non ramifiée E h de degré h de E. Notons X E h la courbe adique associé à E h , alors π h : X E h = X E ⊗ E E h −→ X E est un revêtement étale fini de degré h. Pour tout λ = d h ∈ Q où (d, h) = 1 et h > 0, on définit un fibré vectoriel O X E (λ) par la formule O X E (λ) = (π h ) * O X E h (d). On peut définir les fonctions rang et degré sur les classes d'isomorphisme des fibrés vectoriels et puis appliquer le formalisme des filtrations de Harder-Narasimhan sur la catégorie Définition 2.6. Un fibré vectoriel non nul X est semi stable si pour tout sous fibré vectoriel strict non nul X ′ de X, on a µ(X ′ ) ≤ µ(X). Théorème 2.7. ( [Far-Fon] théo 8.5.1 ) Supposons F algébriquement clos. • Les fibrés semi-stables de pente λ sur X, à isomorphisme près, sont les O X (λ) m . • La filtration de Harder-Narasimhan d'un fibré vectoriel sur X est scindée. • Tout fibré vectoriel sur X est isomorphe à un fibré de la forme n i=1 O X (λ i ) où λ 1 ≥ · · · ≥ λ n est une suite décroissante dans Q. Notons ϕ-Mod B la catégorie des ϕ-modules libres sur B. On a une équivalence de catégories ϕ − Mod B ∼ − → F ib X (M, ϕ) −→ d≥0 M ϕ=π d . D'autre part le foncteur sections globales implique une équivalence entre la catégorie des fibrés ϕ-équivariants sur Y et celle des ϕ-modules sur B. Théorème 2.8. ([Far14] théo. 3.5) Supposons F algébriquement clos. Il y a une équivalence de catégories entre faisceaux cohérents sur X et sur X ad . Notons ϕ-ModȆ la catégorie des isocristaux surȆ oùȆ := E nr . Il y a un foncteur naturel de ϕ-ModȆ dans F ib X . Soit (D, ϕ) un isocristal, on pose E(D, ϕ) = Y × ϕ Z D −→ Y /ϕ Z = X ad qui est un fibré vectoriel sur X ad . Via GAGA cela correspond au fibré associé au P - module gradué d≥0 D ⊗ O(Y ) ϕ⊗ϕ=π d . Le théorème 2.7 implique que le foncteur E(−) : ϕ − ModȆ −→ F ib X est essentiellement surjectif. Définition 2.9. Soit G un groupe réductif sur E. Notons Bun X la catégorie des fibrés vectoriels sur X. Un G-fibré sur X (ou X ad ) est un foncteur tensoriel exact Rep E G −→ Bun X . On considère l'ensemble de Kottwitz B(G) = G(Ȇ)/σ-conj des classes d'isomorphismes de G-isocristaux. Si b ∈ G(Ȇ) on peut lui associer un G-fibré sur X, que on note E b , par composition Rep(G) −→ ϕ − ModȆ E(−) − −− → Bun X (V, ρ) −→ (VȆ, ρ(b)σ). Théorème 2.10. ([Far16] théorème 2.13) On suppose F algébriquement clos. Il y a une bijection entre B(G) et l'ensemble des G-fibrés. B(G) ∼ − → H 1 et (X, G) [b] −→ [E b ]. Cette bijection généralise le théorème 2.7. Si S −→ Spa(F q ) un espace affinoïde perfectoïde et b ∈ G(Ȇ), on définit comme aupar- avant un G-fibré E b sur X S , Rep(G) −→ ϕ − ModȆ E S (−) − −−− → Bun X S (V, ρ) −→ (VȆ p , ρ(b)σ). Par abus de langage, on le note encore E b . De plus E 1 est le G-fibré trivial. 3 La B dR -Grassmanienne affine Soient S = Spa(R, R + ) affinoïde perfectoïde de caractéristique p et (S ♯ , ι) un débasculement de S sur Q p et donc un diviseur de Cartier D S ♯ ֒→ X S . On a une application surjective θ : W (R 0 ) −→ (R ♯ ) 0 dont le noyau est engendré par ξ ∈ W (R 0 ) qui n'est pas un diviseur de zéro. Alors B + dR (R ♯ ) est défini comme le complété ξ-adique de W (R 0 ) 1 p et B dR (R ♯ ) = B + dR (R ♯ )[ξ −1 ]. Le complété de X S au-dessus de D S ♯ est alors Spf(B + dR (R ♯ )). Notons B e (R ♯ ) = H 0 (X S \ D S ♯ , O X S ). Etant donné un G-fibré E, posons E e sa restriction sur X S \ D S ♯ = Spec B e (R ♯ ) ainsi que E + dR son complété au-dessus de D S ♯ . La proposition suivante implique que E est déterminé par E e et E + dR . Proposition 3.1. [BL95] (Recollement de Beauville-Laszlo) La catégorie des G-fibrés sur X S est équivalent à la catégorie des triplets E e , E + B dR , ι où E e est un G-fibré sur B e (R ♯ ), E + B dR est un G-fibré sur B + dR (R ♯ ) et ι : E e ⊗ Be(R ♯ ) B dR (R ♯ ) ∼ − → E + B dR ⊗ B + dR (R ♯ ) B dR (R ♯ ) est un isomorphisme. Définition 3.2. Soit E un G-fibré sur X S . Une modification de E au-dessus de D S ♯ est un G-fibré E ′ avec un isomorphisme α : E\| X S \D S ♯ ∼ − → E ′ \| X S \D S ♯ . En vertu du recollement de Beauville-Laszlo, une modification entre E = E e , E + B dR , ι et E ′ = E ′ e , (E ′ ) + B dR , ι ′ est donnée par un isomorphisme α : E e ∼ − → E ′ e . Supposons maintenant R ♯ = C est un corps complet algébriquement clos. On a X C b = Proj d≥0 (B C b ) ϕ=π d . et le diviseur D C ֒→ X C b correspond à une injection O X C b ֒→ O X C b (1) des fibrés en droites sur X C b et donc un élément t ∈ H 0 (X C b , O X C b (1)) = (B X C b ) ϕ=π . Maintenant comme B e (C) = H 0 (X C b \D C , O X S ) on voit que B e (C) = Spec (B C b [t −1 ]) ϕ=Id . Soit (D, ϕ) un isocristal, on y a associé un fibré vectoriel E(D, ϕ). En terme du recolle- ment de Beauville-Laszlo, E(D, ϕ) est donné par (B C b [t −1 ] ⊗Ȇ D) ϕ⊗ϕ=Id , (B + dR (C)) n , ι où ι est le morphisme trivial (B dR (C)) n ∼ − → (B + dR (C)) n ⊗ B dR (C). On suppose désormais que G est un groupe réductif connexe défini sur Q p . Définition 3.3. [SW17] La B dR -Grassmanienne affine Gr B dR G est la faisceautisation étale du foncteur qui à un espace affinoïde perfectoïde S = Spa(R, R + ) avec un morphisme S −→ SpdQ p correspondant à un débasculement S ♯ = Spa(R ♯ , (R ♯ ) + ), associe l'ensemble des classes d'isomorphismes de G-torseurs sur Spec B + dR (R ♯ ) trivialisés sur Spec B dR (R ♯ ). Comme tout G-torseur sur Spa(R ♯ , (R ♯ ) + ) devient trivial sur un recouvrement étale de Spa(R ♯ , (R ♯ ) + ), on en déduit que Gr B dR G est le faisceau étale associé au préfaisceau (R ♯ , (R ♯ ) + ) −→ G(B dR (R ♯ ))/G(B + dR (R ♯ )). Notons G * le groupe réductif forme quasi-déployée de G (i.e G Q p ≃ G * Q p ) ainsi que T ⊂ B ⊂ G * un tore maximal contenu dans un sous groupe de Borel de G * . Posons X * (G) := Hom(G m,Q p , G Q p ). Le groupe de Galois Γ = Gal(Q p /Q p ) ainsi que G(Q p ) agissent sur X * (G) et on a Γ\ X * (G * )/G * (Q p ) = Γ\ X * (G)/G(Q p ) = Γ\X * (T ) + . Pour S = Spa(C, C + ) avec C/Q p un corps perfectoïde algébriquement clos alors B + dR (C) est un anneau de valuation discrète et on voit que Gr B dR G (C, C + ) = G(B dR (C))/G(B + dR (C)). On a la décomposition de Cartan G(B dR (C)) = µ∈X * (T ) + G(B + dR (C))µ(ξ) −1 G(B + dR (C)). Notation 3.4. Notons ≤ l'ordre de Bruhat sur X * (T ) + , i.e µ ′ ≤ µ si et seulement si µ − µ ′ est une somme des co-racines positives avec des coefficients positifs rationnels. [SW17], il suffit de considérer le morphisme Grass(d, n) ⋄ −→ Spa(Q) ⋄ . Dans ce cas, le résultat découle de [Hub96]. B dR G (C(x)) = µ∈X * (T ) + G(B + dR (C))µ(ξ) −1 G(B + dR (C)) appartiennent à G(B + dR (C))µ(ξ) −1 G(B + dR (C)), resp. à µ ′ ≤µ G(B + dR (C))µ ′ (ξ) −1 G(B + dR (C) L'espace de modules de Shtukas Supposons maintenant que le G-fibré E est trivialisé sur B + dR (R ♯ ). En terme du recollement de Beauville-Laszlo, E correspond à un triplet E e , E + 1,B dR , ι où E + 1,B dR est le B + dR -réseau trivial dans E 1,B dR (R ♯ ) et où ι est un isomorphisme ι : E e ⊗ Be(R) B dR (R ♯ ) ∼ − → E + 1,B dR ⊗ B + dR (R ♯ ) B dR (R ♯ ) ≃ E 1,B dR . (4.1) Autrement dit, un fibré E trivialisé sur B + dR (R ♯ ) est déterminé par un couple E e , ι avec ι comme décrit dans (4.1). Soit R ♯ = C un corps perfectoïde algébriquement clos ainsi que R = C b . Considérons une modification α : E b 1 \| X C b \D C ∼ − → E b 2 \| X C b \D C où b 1 , b 2 ∈ G(Q p ). Puisque E b 1 et E b 2 possèdent une trivialisation naturelle sur B + dR (C), la modification α induit un automorphisme g = ι b 2 • α ⊗ Id B dR (C) • ι −1 b 1 de E 1,B dR . D'après la décomposition de Cartan, il existe un unique µ ∈ X * (T ) + de sorte que g ∈ G(B + dR (C))µ(t) −1 G(B + dR (C)) . On dit alors que la modification α est de type µ. Revenons à la situation générale où R est une algèbre affinoïde perfectoïde et R ♯ un débasculement de R. Une modification α : E\| X S \D S ♯ ∼ − → E ′ \| X S \D S ♯ . est dite de type µ si et seulement si pour tout point géométrique x C b : C b −→ Spa(R, R + ), la modification x * C b α est de type µ. Définition 4.1. Supposons nous donné un triplet (G, µ, b) où G est un groupe réductif sur Q p , b ∈ G(Q p ) et µ ∈ X + * (T ) avec b ∈ B(G, µ) . Notons E le corps de définition de µ. L'espace de modules de Shtukas Sht(G, µ, b) −→ Spa(Ȇ) ⋄ est le préfaisceau sur Perf Fp qui associe à chaqueQ p -espace perfectoïde S ♯ , avec un débasculement (S ♯ ) b = S, l'ensemble des modifications α : E b \| X S \D S ♯ ∼ − → E 1 \| X S \D S ♯ . qui est de type µ ′ plus petit que µ. Le foncteur Sht(G, µ, b) admet une action de G(Q p ), respectivement de J b (Q p ) via α −→ g • α pour g ∈ G(Q p ), respectivement α −→ α • h −1 pour h ∈ J b (Q p ). Puisque b σ = b −1 · b · b σ alors [b] = [b σ ] dans B(G). Il y a donc un isomorphisme canonique σ : E b σ ∼ − → E b . On définit alors la donnée de descente de Sht(G, µ, b) comme l'isomorphisme Fr : Sht(G, µ, b) ∼ − → Sht(G, b σ , µ) α −→ α • σ. Les actions de G(Q p ) et J b (Q p ) commutent avec la donnée de descente. La proposition suivante est bien connue des experts. Proposition 4.3. Soit G un groupe réductif connexe défini sur Q p et b ∈ G(Q p ). Soient S = Spa(R, R + ) un espace affinoïde perfectoïde sur Spa(F p ) ainsi que Spa(R ♯ , (R ♯ ) + ) un débasculement. Si α : E b\|X S \D S ♯ ∼ − → E b\|X S \D S ♯ est une modification de G-fibrés de type µ = 0 alors α s'étend en un isomorphisme de G-fibrés. En particulier on a une identification Sht(G, Id, 0) ≃ G(Q p ). Proof. D'après le formalisme tanakien, il suffit de démontrer le résultat pour G = GL n . Supposons tout d'abord que Spa(R, R 0 ) = Spa(C b , O C b ) où C est un corps complet algébriquement clos. En utilisant le recollement de Beauville-Laszlo, on peut exprimer E b sous forme (B C b [t −1 ] ⊗Q pQ n p ) ϕ⊗Id bσ , B + dR (C) n . Comme la modification α est de type 0, on en déduit que g := α ⊗ Id B dR (C) est donné par un élément dans GL n (B + dR (C)). En particulier le couple (α, g) est un isomorphisme de E b . Le cas particulier où R = C b couplé avec le lemme 3.4.6 de [Car-Sch] implique le cas général où Spa(R, R + ) est un espace affinoïde perfectoïde. Enfin, on voit que Sht(G, Id, 0) classifie les automorphismes du G-fibré trivial E 1 . D'après l'exemple 2.21 de [Far16], on a une identification de diamants Sht(G, Id, 0) = G(Q p ). Torsions des modifications des G-fibrés Soient X un schéma et G un X-schéma en groupes. Un G-torseur est un Q p -morphisme fidèlement plat T −→ X muni d'une action G × T −→ T sur l'action triviale de X de sorte que, fppf-localement sur X, on a un isomorphisme G-équivariant T ≃ G × X. Lemme 5.1. (lemme 4.6.1 de [KW]) Soit G −→ X un schéma en groupes réductifs. La catégorie des G-fibrés sur X est équivalente à celle des G-torseurs sur X. Soit S = Spa(R, R + ) un espace affinoïde perfectoïde sur Spa(F q ). Considérons maintenant la courbe de Fargues-Fontaine X S . Pour chaque b ∈ G(Q p ), on a un G-fibré E b sur X S . Notons T b le G-torseur correspondant à E b via le lemme 5.1. On peut décrire ce G-torseur par la formule T b = Proj d≥0 H 0 (Y S , O Y S ) ⊗Q pQ p [G] ϕ S ⊗bσ=π d . (5.1) oùQ p [G] est l'algèbre de définition de GQ p . L'action de G sur T b est donnée par celle de G sur lui même par translation à droite. Désormais, notons λ g et ρ g respectivement la multiplication à gauche par g et à droite par g −1 . Étant donné un groupe réductif connexe G défini sur Q p et ι : Z 0 G ֒→ G la composante connexe neutre du centre Z G . On va construire un morphisme naturel Bun G × Bun Z 0 G −→ Bun G . D'après le lemme 5.1, it suffit de définir le morphisme au niveau des torseurs. Considérons T un G-torseur ainsi que T ′ un Z 0 G -torseur sur X. Puisque Z 0 G est contenu dans le centre de G, on peut munir T d'une action de Z 0 G telle que l'action de G et de Z 0 G commutent. Plus précisément, un élément g ∈ Z 0 G agit par action de g sur T . Le produit contracté T × Z 0 G T ′ est alors un G-torseur. Proposition 5.2. Pour b ∈ B(G) et h ∈ B(Z 0 G ), l'image de (E b , E h ) via le morphisme ci-dessus est E bh où h = ι(h). Proof. Il suffit de démontrer T b × Z 0 G T h = T bh pour b ∈ B(G) et h ∈ B(Z 0 G ). D'après la formule (5.1) on a T b = Proj d≥0 H 0 (Y S , O Y S ) ⊗Q pQ p [G] ϕ S ⊗bσ=π d . où l'action de G sur T b est donnée par action de G sur lui même par translation à droite et action de Z 0 G est donnée par translation à gauche. On a le diagramme commutatif suivant où µ désigne la multiplication de G. GQ p × (Z 0 G )Q p GQ p GQ p × (Z 0 G )Q p GQ p λ b × ρ h −1 λ bh µ µ On en déduit que la co-multiplication µ : Q p [G] −→Q p [G]⊗Q pQ p [Z 0 G ] induit l'isomorphisme voulu T b × Z 0 G T h = T bh . Pour T un tore défini surQ p , d'après Kottwitz on a une bijection κ : B(T ) basic = B(T ) ∼ − → X * (T ) Γ . On note b λ l'élément correspondant à un λ ∈ X * (T ) Γ via cette bijection. On voit également que B(T, λ) = {b λ }, en particulier pour C un corps algébriquement clos, il existe une modification de type λ α : E b\|X C b \D C ∼ − → E 1\|X C b \D C si et seulement si E b ≃ E b λ . Étant données deux modifications α, α ′ de type µ = 0 ∈ X * (T ), d'après la proposition 5.2, on peut construire une autre modification qui est aussi de type µ = 0 E 1 = E 1 × T E 1 α×α ′ − −− → E 1 × T E 1 = E 1 . En utilisant cette construction, on peut munir l'espace de Shtukas Sht(T, 0) d'une structure de diamant en groupes. En effet, pour S = Spa(R, R + ) un espace affinoïde perfectoïde sur Spa(F p ) ainsi que S ♯ = Spa(R ♯ , (R ♯ ) + ) un débasculement. Étant données deux modifications de type µ = 0 α, β : E 1\|X S \D S ♯ ∼ − → E 1\|X S \D S ♯ , on définit le produit de α et β par la formule α × β : E 1 = E 1 × T E 1\|X S \D S ♯ α×β − −− → E 1 × T E 1\|X S \D S ♯ = E 1 . Alors Sht(T, 0) avec ce produit est un diamant en groupes. De plus, d'après la proposition 4.3, on a une identification de diamants en groupes Sht(T, 0) ≃ T (Q p ). Lemme 5.3. Pour une modification α : E \|X S \D S ♯ −→ E 1\|X S \D S ♯ de G-torseurs et deux modifications β, γ : E \|X S \D S ♯ −→ E 1\|X S \D S ♯ de Z 0 G -torseurs on a α × β × γ = α × β × γ comme modifications de G-torseurs. Proof. Il s'agit d'utiliser le lemme 5.1 pour calculer explicitement les isomorphismes des torseurs. Étant donné un groupe réductif connexe G défini sur Q p et ι : Z 0 G ֒→ G la composante connexe du tore central. Le morphisme ι induit un morphisme θ : X * (Z 0 G ) −→ X * (T ) + où T est le tore maximal de G. Afin d'alléger les notations, pour λ ∈ X * (Z 0 G ), on note encore λ son image par θ dans X * (T ) + . Soient S = Spa(R, R + ) un espace affinoïde perfectoïde sur Spa(F p ) ainsi que S ♯ = Spa(R ♯ , (R ♯ ) + ) un débasculement. Si l'on dispose d'une modification α : E h\|X S \D S ♯ −→ E 1\|X S \D S ♯ de Z 0 G -fibrés de type λ, alors pour tout G-fibré E b et tout isomorphisme f : E b ∼ − → E b on a une modification de G-fibrés α : E bh ≃ E b × Z 0 G E h\|X S \D S ♯ f ×α − −− → E b × Z 0 G E 1\|X S \D S ♯ ≃ E b . De même, pour une modification de G-fibrés β : E b −→ E 1 de type µ, on peut construire une modifications de G-fibrés β : E b = E b × Z 0 G E 1\|X S \D S ♯ β×Id − −− → E 1 × Z 0 G E 1\|X S \D S ♯ = E 1 . Proposition 5.4. Supposons que G est déployé surQ p . (i) La modification α : E bh −→ E b est de type λ. (ii) La modification β est de type µ. Proof. D'après le formalisme tanakien, il suffit de démontrer le résultat pour toutes les représentations ρ : G −→ GL(V ) de G. Puisque l'on est dans le cas de caractéristique 0 et que G est réductif, la catégorie Rep Qp G est semi-simple. Il suffit donc de traiter les représentations irréductibles. Lorsque ρ : G −→ GL(V ) est irréductible, on a ρ(Z 0 G ) ⊂ Z GL(V ) . On obtient également un cocaractère de Z GL(V ) ρ • λ : G m,Q p −→ Z GL(V ),Q p . On peut supposer que G = GL(V ). On a alors Z 0 G = GL 1 et X * (Z 0 G ) ≃ Z, de plus le cocaractère λ correspond à un d ∈ Z. On en déduit que h = [p d ] ∈ B(Z 0 G ) puisque la modification α : E h\|X S \D S ♯ −→ E 1\|X S \D S ♯ est de type λ. Pour calculer le type de α, il suffit de considérer la cas R = C b où C est un corps complet algébriquement clos. D'après le recollement de Beauville-Laszlo, on a la description suivante : E p d = (B C b [t −1 ] ⊗Q pQ p ) ϕ⊗p d σ , B + dR (C) E 1 = (B C b [t −1 ] ⊗Q pQ p ) ϕ⊗Id σ , B + dR (C) . La modification α de type λ est alors donnée par l'isomorphisme (B C b [t −1 ] ⊗Q pQ p ) ϕ⊗p d σ −→ (B C b [t −1 ] ⊗Q pQ p ) ϕ⊗Id σ x −→ t −d C b x. D'après la formule (5.1), la modification α s'écrit en termes de Z 0 G -torseurs T p d \|X C b \D C ≃ B C b [t −1 ] ⊗Q pQ p [Z 0 G ] ϕ⊗p d σ ∼ − → B C b [t −1 ] ⊗Q pQ p [Z 0 G ] ϕ⊗Id σ ≃ T 1\|X C b \D C x ⊗ x ′ −→ t −d x ⊗ x ′ . En utilisant le diagramme 5, la modification α s'écrit en termes de G-torseurs T p d b\|X C b \D C ≃ B C b [t −1 ] ⊗Q pQ p [G] ϕ⊗p d bσ ∼ − → B C b [t −1 ] ⊗Q pQ p [G] ϕ⊗Id σ ≃ T 1\|X C b \D C x ⊗ x ′ −→ t −d C b x ⊗ x ′ . Finalement la modification α s'écrit sous la forme (B C b [t −1 ] ⊗ Qp V ) ϕ⊗p d bσ −→ (B C b [t −1 ] ⊗ Qp V ) ϕ⊗Id σ x −→ t −d x. Il est alors aisé de voir que la modification α est de type (d, · · · , d) ∈ Z dim Qp V = X * (GL(V )), autrement dit α est de type λ, ce qui démontre le point (i). On démontre le point (ii) par le même argument. On peut ainsi utiliser la proposition 5.4 pour définir une action de Sht(Z 0 G , 0) ≃ Z 0 G (Q p ) sur Sht(G, µ, b) et sur Sht(Z 0 G , λ). Soient S = Spa(R, R + ) un espace affinoïde perfectoïde sur Spa(F p ) ainsi que S ♯ = Spa(R ♯ , (R ♯ ) + ) un débasculement. Étant donnée une modification de type µ = 0 de Z 0 G -fibrés α : E 1\|X S \D S ♯ ∼ − → E 1\|X S \D S ♯ et une modification de type µ de G-fibrés α ′ : E b\|X S \D S ♯ ∼ − → E 1\|X S \D S ♯ , on définit l'action de α sur α ′ par la formule E b = E b × Z 0 G E 1\|X S \D S ♯ α ′ ×α − −− → E 1 × Z 0 G E 1\|X S \D S ♯ = E 1 . D'après la proposition 5.4, α ′ × α est une modification de type µ. Le lemme 5.3 implique que cela définit une action de Z 0 G (Q p ) sur Sht(G, µ, b). De même, si l'on a une modification de type λ de Z 0 G -fibrés α ′ : E b λ \|X S \D S ♯ ∼ − → E 1\|X S \D S ♯ , on peut définir une autre modification de type λ par la formule E b λ = E 1 × Z 0 G E b λ \|X S \D S ♯ α −1 ×α ′ − −−−− → E 1 × Z 0 G E 1\|X S \D S ♯ = E 1 . où les actions de Z 0 G sur E 1 × Z 0 G E b λ et E 1 × Z 0 G E 1 sont induites par celle sur la première composante. Remarque 5.5. Étant donné un élément g ∈ Z 0 G (Q p ) correspondant à une modification de type µ = 0 de Z 0 G -fibrés α : E 1\|X S \D S ♯ ∼ − → E 1\|X S \D S ♯ alors action de α est celle de ι(g) où ι : Z 0 G (Q p ) ֒→ G(Q p ) est l'injection canonique. En effet, on a α ′ × α = α ′ • Id E 1 × Id E 1 •α = α ′ × Id E 1 • Id E 1 ×α = α ′ • ι(g). Lemme 5.6. Soient T un tore défini sur Q p et α, β : E λ −→ E 1 deux modifications de T -fibrés de type λ. Alors il existe une modification γ : E 1 ∼ − → E 1 (de type 0) de sorte que β = α × γ. Lemme 6.2. Notons σ bb λ : E (bb λ ) σ ∼ − → E bb λ l'isomorphisme canonique. On a alors σ b × σ b λ = σ bb λ . Proof. D'après la formule (5.1) on a T b = Proj d≥0 H 0 (Y S , O Y S ) ⊗Q pQ p [G] ϕ S ⊗bσ=π d , et de même pour le torseur T b σ . On en déduit que l'isomorphisme canonique σ b : T b σ ≃ T b est donné par Id S ⊗b. D'après la proposition 5.2, on sait que la co-multiplication µ : Q p [G] −→Q p [G] ⊗Q p Q p [Z 0 G ] induit l'isomorphisme T b × Z 0 G T b λ = T bb λ . On en déduit que σ b × σ b λ = σ bb λ . Théorème 6.3. Il y a un isomorphisme G(Q p ) × Jb(Qp)-équivariant de faisceaux pro-étale qui commute avec les données de descentes. Sht(G, µ, b) × Z 0 G (Qp) Sht(Z 0 G , λ) −→ Sht(G, bb λ , µ · λ). Proof. Pour uneQ p -algèbre perfectoïde R ♯ avec (R ♯ ) b = R et un couple (α, β) ∈ Sht(G, µ, b)× Spa(Qp) ⋄ Sht(Z 0 G , λ) R ♯ , R , on peut construire, d'après la proposition 5.2, une modification α × β : E bb λ \|X S \D S ♯ −→ E 1\|X S \D S ♯ D'autre part, en écrivant α = Id E b •α et β = β • Id E 1 , on voit que α × β = α × Id E 1 • Id E b ×β . où Id E b ×β et α × Id E 1 sont des modifications de G-fibrés Id E b ×β : E bb λ −→ E b α × Id E 1 : E b −→ E 1 qui sont respectivement de type λ et µ d'après la proposition 5.4. Afin de calculer le type de α × β, on utilise la décomposition de Cartan G(B dR (C)) = η∈X * (T )/W G(B + dR (C))η(ξ) −1 G(B + dR (C)) où C est un corps perfectoïde algébriquement clos et T un tore maximal de G. Pour g ∈ G(B + dR (C))µ(ξ) −1 G(B + dR (C)) et h ∈ G(B + dR (C))λ(ξ) −1 G(B + dR (C)), on voit que gh ∈ G(B + dR (C))µ · λ(ξ) −1 G(B + dR (C)) car λ(ξ) −1 est central. Autrement dit la modification α × β est de type µ · λ. On a construit un morphisme Φ : Sht(G, µ, b) × Spa(Qp) ⋄ Sht(Z 0 G , λ) −→ Sht(G, bb λ , µ · λ) . Montrons ensuite que ce morphisme passe au quotient par Sht(Z 0 G , 0). Soient (α, β) ∈ Sht(G, µ, b) × Spa(Qp) ⋄ Sht(Z 0 G , λ) et γ ∈ Z 0 G (Q p ), il faut montrer que α × γ × γ −1 × β = α × β. Or d'après le lemme 5.3, cette identité est vérifiée. On va montrer que Φ est en fait un isomorphisme. Il s'agit de montrer que Φ R ♯ ,R est un isomorphisme pour tous les couples (R ♯ , R). On aura besoin de l'ensemble auxiliaire suivant S = (α, β, γ) ∈ Sht(G, µ · λ, bb λ ) × Spa(Qp) ⋄ Sht(Z 0 G , λ −1 ) × Spa(Qp) ⋄ Sht(Z 0 G , λ)(R ♯ , R) / ∼ • b ∈ B(G, µ) est l'unique élément basique. Pour un tel triplet, on a Jb(Qp) = D × n/F , le groupe des inversibles de l'algèbre de division d'invariant 1 n , de centre F . Le corps de définition de µ est F . On note encore J ⊂ I \ {τ 0 } l'ensemble de τ = τ 0 tel que (p τ , q τ ) = (n, 0). Les espaces de Rapoport-Zink associés sont non ramifiés de type EL. Par la suite, on notera Sht(µ) (resp.M µ K ) pour Sht(G, µ, b) (resp. M K (G, µ, b)). Lorsque la signature µ LT = (1, n − 1), (0, n), · · · , (0, n), on retrouve la tour de Lubin-Tate. Pour σ une représentation irréductible de D × n/F , on notera H q c (M µ K )[σ] = Hom D × n/F H q c (M µ K , Q ℓ ), σ . Définition 7.2. [Boyer09] -Soient π 1 et π 2 des représentations de respectivement GL n 1 (F ) et GL n 2 (F ), on note π 1 × π 2 l'induite parabolique Ind GL n 1 +n 2 (F ) P n 1 ,n 1 +n 2 (F ) (π 1 {n 2 /2} ⊗ π 2 {−n 1 /2}). -Soit g un diviseur de n = sg et π une représentation cuspidale irréductible de GL g (F ) : • π{ 1−s 2 } × π{ 3−s 2 } × · · · × π{ s−1 2 } possède un unique quotient (resp. sous espace) irréductible. C'est une représentation de Steinberg (resp. de Speh) généralisée notée habituellement St s (π) (resp. Speh s (π)). • St s−i (π){ −i 2 }× Speh i (π){ s−i 2 } possède un unique sous espace irréductible que l'on note LT π (s, i). En particulier, pour i = 0 (resp. i = s − 1), on retrouve St s (π) (resp. Speh s (π)) -Pour π une représentation irréductible cuspidale de GL g (F ) et t > 0, π[t] D désignera la représentation JL −1 (St t (π)) ∨ de D × n/F . Théorème 7.3. Pour tout diviseur g de n = gs et toute représentation irréductible cuspidale π de GL g (F ), on a alors des isomorphismes G(Q p ) × W F -équivariants où ω i est le caractère central de LT π (s, i) et rec −1 F est le morphisme de réciprocité d'Artin et où τ rec −1 F := τ · rec −1 F ·τ −1 . Remarque 7.4. Tout énoncé valable sur l'espace de Lubin-Tate trouvera son analogue dans la situation plus générale précédente. En particulier, dans [Boyer14] il est annoncé que les lim − → K H • c (M µ LT Kp , Z ℓ ) sont Z ℓ -libres, ce qui impliquerait la même propriété pour lim − → K H • c (M µ Kp , Z ℓ ). Proof. On exploitera les résultats obtenus dans les sections précédentes pour calculer la cohomologie de la tour M µ Kp Kp . Il s'agit de trouver les relations de ce dernier avec la tour de Lubin-Tate. Tout d'abord, d'après [Boyer09], on a des isomorphismes G(Q p ) × W Féquivariants de fibrés vectoriels. Pour λ = d h avec (d, h) = 1 alors O X (λ) est un fibré de degré d et de rang h et donc de pente de Harder-Narasimhan µ(O X (λ)) = λ. Définition 3. 5 . 5. Pour un cocaractère µ ∈ X * (T ) + , notons E son corps de définition etȆ := E ·Q.Considérons les sous-foncteursGr B dR G,µ ⊂ Gr B dR G,≤µ ⊂ Gr B dR Gqui sont définis par les conditions qu'un morphisme S −→ Gr B dR G avec un morphisme S −→ SpdȆ où S ∈ Perf Fp se factorise par Gr B dR G,µ resp. Gr B dR G,≤µ si et seulement si pour tout point géométrique x = Spa(C(x), C(x) + ) −→ S, les éléments correspondant dans Gr Théorème 4. 2 . 2[SW17] Étant donné un triplet (G, b, µ) comme ci-dessus alors Sht(G, µ, b) est un diamant localement spatial. De plus, le morphisme f :Sht(G, µ, b) −→ Spa(Q p ) ⋄ est partiellement propre et dim.trg f < ∞.Proof. On a en effet Sht(G, µ, b) = lim ← − K Sht(G, µ, b) K , le théorème 23.1.3 de[SW17] implique alors que Sht(G, µ, b) est un diamant localement spatial. D'autre part, la proposition 23.2.1 de loc.cit couplée avec 3.6 implique que f est partiellement propre et dim.trg f < ∞. Le groupe G = Res F/Qp GL F (V ) est déployé sur F et G F (F ) ≃ τ ∈I GL n (F ). Le centre de G est Z = Res F/Qp GL 1 et il est aussi déployé sur F : Z F ≃ τ ∈I GL 1 (F ). Identifions I Définition 2.1. Pour I ⊂]0, 1[ un intervalle, on note B I le complété de B bd par rapport aux normes (\|.\| ρ ) ρ∈I . Notons B = B ]0,1[ . On a alors B = lim ← −I B I où I parcourt les intervalles compacts de ]0, 1[. L'anneau E possède un morphisme de Frobenius ϕ tel que Ce morphisme de Frobenius induit alors un automorphisme ϕ de B avec E = B ϕ=Id . On rappelle ensuite la version adique de la courbe de Fargues-Fontaine. Si I = [ρ 1 , ρ 2 ] ⊂ ]0, 1[ avec ρ 1 , ρ 2 ∈ \|F × \| alors B I est une E-algèbre de Banach qui est un anneau principal. On pose alors Y I = Spa(B I , B o I ) comme espace topologique muni d'un préfaisceau d'anneaux. Théorème 2.3. ([Far16] 2.1) L'espace Y I est adique i.e le préfaisceau O Y I est un faisceau.ϕ n [x n ]π n = n [x q n ]π n . On note également ϕ : [0, 1] ∼ − → [0, 1] défini par ϕ(ρ) = ρ q . Le Frobenius ϕ de B bd s'étend en un isomorphisme ϕ : B I ∼ − → B ϕ(I) . Définition 2.2. La courbe schématique de Fargues-Fontaine est X = Proj(P) où P = d≥0 B ϕ=π d , vue comme E-algèbre graduée. On note O X (1) = P [1] le fibré en droites tautologique sur X et pour d ∈ Z, O X (d) = O X (1) ⊗d . On a alors P = d≥0 H 0 (X, O X (d)). ). On dispose alors du théorème suivant. est un sous-foncteur fermé qui est propre sur Spa(Q) ⋄ et Gr B dR G,µ ⊂ Gr B dR G,≤µ est un sous-foncteur ouvert. De plus le morphisme Gr B dR G,≤µ −→ Spa(Q) ⋄ est de dimension de transcendance finie (i.e dim.trg < ∞). Proof. Toutes les assertions, sauf la dernière, sont démontrées dans la proposition 19.2.3 et le théorème 19.2.4 de [SW17]. L'assertion sur la finitude de la dimension de transcendance est démontrée implicitement dans les lemmes 19.3.2 et 19.3.3. D'après le lemme 19.1.5 de loc.cit, toute injection de groupes réductifs G ֒→ GL n induit une immersion fermée Gr B dR G,≤µ ֒→ Gr B dR GLn,≤µ . On peut donc supposer G = GL n . Puisqu'il y a un nombre fini de µ ′ ≤ µ et que chaque point x ∈ \|Gr B dR GLn,≤µ \| appartient à \|Gr B dR GLn,µ ′ \| pour un µ ′ ≤ µ, il suffit de montrer la dernière assertion pour Gr B dR GLn,µ . GLn,µ de Gr B dR GLn,µ (définition 19.3.1 de loc.cit). Nous allons suivre la suite de réductions du lemme 19.3.2. Le lemme 21.3 de [S17] nous permet de supposer que µ est minuscule. Ensuite, la question 21.4 1 de loc.cit nous permet de passer au recouvrement ouvert et comme dans le lemme 19.3.2 deThéorème 3.6. ([SW17]) Pour tout µ ∈ X * (T ) + , Gr B dR G,≤µ est un diamant spatial. Le sous-foncteur Gr B dR G,≤µ ⊂ Gr B dR G En vertu du lemme 19.3.3 de loc.cit., il suffit de considérer la résolution de Demazure Gr Nous utilisons la définition au début du page 120 et le lemme 21.3 ainsi que la question 21.4 sont donc vérifiés. b λ étant un élément central de G(Qp), on a alors Jb(Qp) = J bb λ (Qp). Mais l'isomorphisme (7.2) implique que l'inclusion (7.4) est en fait un isomorphisme de représentations de W F . En prenant la limite projective lorsque le niveau K varie, on en déduit que, pour 0 ≤ i < s, Proof. Soit α ′ : E λ −1 −→ E 1 une modification de T -fibrés de type λ −1 (une telle modification existe). Considérons la modification suivantePuisque E λ × T E λ −1 = E 1 (d'après la proposition 5.2), la modification α × α ′ est en fait une modification de type 0 de E 1 . Posons α ′ × α × α ′ −1 × β le produit contractéEn particulier d'après la proposition 5.2, E λ −1 × T E 1 × T E λ = E 1 et on voit que α ′ × α × α ′ −1 × β est une modification de type 0 de E 1 . Finalement, d'après le lemme 5.3, on a α × α ′ × α × α ′ −1 × β = Id ×β = β ce qui termine la démonstration.Modifications centrales des espaces de ShtukasDans cette section, en utilisant la proposition 5.4, on prouve un énoncé géométrique reliant deux espaces de modules de Shtukas. On donnera également une relation cohomologique de ces espaces.Puisque le produit fibré de diamants existe ([S17],Prop. 11.4), on peut définir le faisceau quotient. Définition 6.1. On définit Sht(G, µ, b)× Z 0 G (Qp) Sht(Z 0 G , λ) comme le faisceau quotient. Plus précisément, c'est la faisceautisation du pré-faisceau sur Perf Fp qui associe à chaqueQ pespace perfectoïde S ♯ , avec (, on peut utiliser cette action pour définir une action de ce groupe sur le faisceau Sht(G, µ, b) × Z 0 G (Qp) Sht(Z 0 G , λ). Un élément g ∈ G(Q p ) correspond à un isomorphisme E 1 ∼ − → E 1 et g agit sur Sht(G, µ, b) par composition avec cet isomorphisme. On définit alors l'action de g sur Sht(G, µ, b) × Z 0Il y a des isomorphismes canoniquesOn définit alors la donnée de descente de Sht(G, µ, b)On vérifie aisément que cela est bien défini.On a donc une application canoniqueOn montre que cela est une bijection.. On a alors f (α, δ × β, γ) = α. Autrement dit f est une surjection. Le même type d'argument couplé avec le lemme 5.6 montre également que f est injective, finalement on en déduit que f est une bijection.Maintenant on définit l'application Φ −1 R ♯ ,R par la formuleReste finalement à voir que Φ R ♯ ,R commute avec toutes les structures. La commutativité avec l'action de G(Q p ) × Jb(Qp) est claire et la commutativité avec les données de descentes résulte du lemme 6.2.On aimerait descendre en niveau fini l'énoncé géométrique de la proposition 6.3. Considérons un sous groupe compactCorollaire 6.4. Il y a un isomorphisme Jb(Qp)-équivariant de faisceaux pro-étale qui commute avec les données de descentes.Proof. D'après la remarque 5.5, l'action de g ∈ Z 0D'après la proposition 6.3, il y a un isomorphisme J(Qp)-équivariant qui commute avec les données de descentesest un isomorphisme (ensembliste). En effet, (x, y) et (z, t) dans Sht(G, µ, b) × Sht(Z 0 G , λ) sont dans la même classe dans l'ensemble à gauche si et seulement sDe même, (x, y) et (z, t) sont dans la même classe dans l'ensemble à droite si et seulement s'il existe g ∈ Z 0 G (Q p ) et k ∈ K de sorte que z = x · k · g et t = g −1 · y. Puisque Z 0 G (Q p ) est contenu dans le centre de G(Q p ), on voit que g · k = k · g. On en déduit que le morphisme (6.1) est un isomorphisme.LD'autre part, l'action de Z 0 G (Q p )/K Z sur Sht(Z 0 G , λ)/K Z étant sans point fixe, on en déduit qu'il est de même pour l'action de Z 0. On remarque que Φ admet une section. En effet, C p étant un corps perfectoïde, on peut choisir une modification α dans Sht(Z 0En particulier, on a un isomorphisme (qui, à priori, ne commute pas avec les données de descentes)Il y a également un isomorphismequi, à priori, ne commute pas avec les données de descentes.Groupes de cohomologie de quelques espaces de Rapoport-ZinkDans ce paragraphe, on va calculer la cohomologie de quelques espaces de Rapoport-Zink non ramifiés en utilisant les résultats des paragraphes précédents et[Boyer09],[Boyer14]. Considérons un triplet (G, µ, b) où G est un groupe réductif (non ramifié) sur Q p , b ∈ G(Q p ) et µ ∈ X + * (T ) minuscule avec b ∈ B(G, µ). On suppose que (G, µ, b) correspond à une donnée de Rapoport-Zink de type EL. À une telle donnée (G, µ, b), on associe un groupe p-divisible X avec structures additionnelles. Notons M(G, µ, b) le foncteur qui classifie les déformations par quasi-isogénies de X avec structures additionnelles. Ce foncteur est représentable par un schéma formel défini sur Spf(OQ p ) que l'on notera encore M(G, µ, b). Considérons ensuite l'espace rigideM(G, µ, b) défini surQ p associé au schéma formel M(G, µ, b). Pour chaque sous groupe compact ouvert K p ⊂ G(Z p ), il existe un espace rigideMKp (G, µ, b)qui est défini comme le revêtement étale deM(G, µ, b) classifiant lesSoit K p ⊂ G(Z p ) un niveau. D'après[Far04], on peut montrer queoù V parcourt les ouverts relativement compacts deM Kp (G, µ, b) et où ℓ = p est un nombre premier. On note égalementExemple 7.1. ([Far04]) Soit Z un tore non ramifié et K Z ⊂ Z(Q p ) un sous groupe compact. Alors, on aoù action de Jb(Qp) = Z(Q p ) se fait par la représentation régulière. L'action du groupe de Weil W E est l'action triviale. D'après la remarque 6.5, il y a un isomorphismeOn considère maintenant les triplets (G, µ, b) tels que• µ est un cocaractère minuscule de G. Un tel µ est déterminé par des couples d'entiers (p τ , q τ ) τ ∈I où I = Hom Qp (F, Q p ). On suppose de plus que• Il existe τ 0 ∈ I tel que (p τ 0 , q τ 0 ) = (1, n − 1),• Pour τ = τ 0 on a (p τ , q τ ) = (0, n) ou (p τ , q τ ) = (n, 0).avec Z/dZ, le groupe de caractère est donné par X * (Z) = (x ij ) i∈Z/dZ,1≤j≤n \| ∀i, j x i,j ∈ Z .Le groupe de Weil associé est W = S d n . Le cocaractère µ s'écrit sous la forme (x i,j ) i∈Z/dZ,1≤j≤n(1, 0, · · · , 0) si i = i 0 (1, 1, · · · , 1) si i ∈ J (0, 0, · · · , 0) sinon.Considérons maintenant le cocaractère λ = (z i,j ) i∈Z/dZ,1≤j≤n où z i,j = 1 si i ∈ J 0 sinon. Il est clair que λ(p) est dans le centre de G(Q p ) et d'autre part, on constate que µ = µ LT · λ. On voit également que b LT · λ(p) est l'unique classe basique de B(G, µ). D'après la remarque 6.5, il y a un isomorphisme G(Q p ) × Jb(Qp)-équivariantPour toute représentation supercuspidale π de GL g (F ), il y a alors des isomorphismes G(Q p )-équivariants= 0 pour i < 0. D'après le corollaire 6.4, il y a un isomorphisme D × n/F -équivariant d'espaces rigides qui commute avec la donnée de descentȇOn en déduit qu'il y a un isomorphisme D × n/F × W F -équivariantD'après la remarque 6.5, on voit que (M µ LT K ) Cp ×M K Z (Z, λ, λ(p)) Cp est un Z(Q p )/K Ztorseur trivial au-dessus de (M µ K ) Cp . Alors, le produit tensoriel dérivé dans (7.3) dégénère en un produit tensoriel. Puisque les groupes de cohomologie supérieure deM K Z (Z, λ, λ(p)) Cp s'annulent, la formule de Kunneth implique qu'il y a un isomorphisme D × n/F ×W F -équivariantOn en déduit qu'il y a un isomorphismeD'après[Chen], il y a des isomorphismes Z(Q p ) × W F -équivariantsOn en déduit que pour 0 ≤ i < s, il y a une inclusion LT π (s, i) K ⊗ L(π)\| · \| − s(g+1)−2(i+1) On a donc une tour d'espaces rigides M Kp (G, µ, b) Kp qui possède à la fois une action de G(Q p ) × J b (Q p ) et une donnée de descente. O F trivialisations modulo K p du module de Tate p-adique de groupe p-divisible universel surM(G, µ, b). Il y a un espace de module de Shtukas Sht(G, µ, b) associé à chaque donnée (G, µ, bO F trivialisations modulo K p du module de Tate p-adique de groupe p-divisible universel surM(G, µ, b). On a donc une tour d'espaces rigides M Kp (G, µ, b) Kp qui possède à la fois une action de G(Q p ) × J b (Q p ) et une donnée de descente. Il y a un espace de module de Shtukas Sht(G, µ, b) associé à chaque donnée (G, µ, b). On a une identification de diamants sur Spa(Q p ) : Sht(G, µ, b) = lim ← − Kp Sht(G, µ, b)/K p. On a une identification de diamants sur Spa(Q p ) : Sht(G, µ, b) = lim ← − Kp Sht(G, µ, b)/K p . . /K P =m Kp, G, µ, b) ⋄Puisque µ est minuscule, d'après le théorème 24.2.5 de [SW17Puisque µ est minuscule, d'après le théorème 24.2.5 de [SW17], on a Sht(G, µ, b)/K p =M Kp (G, µ, b) ⋄ . Un faisceau Z ℓ -adique sur X ét est un système projectif (F n ) n∈N de faisceaux en Z ℓ -module sur X. Soit X Un C P -Espace, ét vérifiant ℓ n F n = 0. D'après [Far04], pour un faisceau Z ℓ -adique (F n ) n∈N , on définit le foncteur sections globalesSoit X un C p -espace analytique tel que \|X\| est séparé. Un faisceau Z ℓ -adique sur X ét est un système projectif (F n ) n∈N de faisceaux en Z ℓ -module sur X ét vérifiant ℓ n F n = 0. 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A relation between two moduli spaces studied by v.g. Drinfeld. In Algebraic number theory and algebraic geometry, volume 300 of Contemp. Math., pages 115?129, 2002. ] L.Fargues -J M Fon, Fontaine, Courbes et Fibrés vectoriels en théorie de Hodge. Fon] L.Fargues -J.M.Fontaine, Courbes et Fibrés vectoriels en théorie de Hodge p- adique. Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales. L Fargues, Astérisque. 291Variétés de Shimura, espaces de Rapoport-Zink de correspondances de Langlands localesL. Fargues, Cohomologie des espaces de modules de groupes p-divisibles et corre- spondances de Langlands locales, In "Variétés de Shimura, espaces de Rapoport-Zink de correspondances de Langlands locales", Astérisque 291 (2004), 1-199. L Fargues, Quelques résultats et conjectures concernant la courbe. 369L.Fargues, Quelques résultats et conjectures concernant la courbe, Asterisque 369. 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Math. 134 (2012), no. 2, 407-452. MR2905002. F-93430NGUYEN Kieu Hieu • Email [email protected], Université Paris 13. Sorbonne Paris-Cité, LAGA, CNRS, UMR; Villetaneuse, FRANCE, PerCoLarTor7539NGUYEN Kieu Hieu • Email [email protected], Université Paris 13, Sorbonne Paris-Cité, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, FRANCE, PerCoLarTor, ANR-14-CE25	[]
[ "Nanophotonic supercontinuum based mid-infrared dual-comb spectroscopy", "Nanophotonic supercontinuum based mid-infrared dual-comb spectroscopy" ]	[ "Hairun Guo \nLaboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nKey Laboratory of Specialty Fiber Optics and Optical Access Networks\nShanghai University\n200343ShanghaiChina\n", "Wenle Weng \nLaboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Junqiu Liu \nLaboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Fan Yang \nGroup for Fiber Optics (GFO)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Wolfgang Hänsel \nMenlo Systems GmbH\n82152MartinsriedGermany\n", "Camille Sophie Brès \nPhotonic Systems Laboratory (PHOSL)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Luc Thévenaz \nGroup for Fiber Optics (GFO)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Ronald Holzwarth \nMenlo Systems GmbH\n82152MartinsriedGermany\n", "Tobias J Kippenberg \nLaboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n" ]	[ "Laboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Key Laboratory of Specialty Fiber Optics and Optical Access Networks\nShanghai University\n200343ShanghaiChina", "Laboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Laboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Group for Fiber Optics (GFO)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Menlo Systems GmbH\n82152MartinsriedGermany", "Photonic Systems Laboratory (PHOSL)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Group for Fiber Optics (GFO)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Menlo Systems GmbH\n82152MartinsriedGermany", "Laboratory of Photonics and Quantum Measurements (LPQM)\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland" ]	[]	High resolution and fast detection of molecular vibrational absorption is important for organic synthesis, pharmaceutical process and environmental monitoring, and is enabled by mid-infrared (mid-IR) laser frequency combs via dual-comb spectroscopy. Here, we demonstrate a novel and highly simplified approach to broadband mid-IR dual-comb spectroscopy via supercontinuum generation, achieved using unprecedented nanophotonic dispersion engineering that allows for flat-envelope, ultra-broadband mid-IR comb spectra. The mid-IR dual-comb has an instantaneous bandwidth covering the functional group region from 2800 − 3600 cm −1 , comprising more than 100,000 comb lines, enabling parallel gas-phase detection with a high sensitivity, spectral resolution, and speed. In addition to the traditional functional groups, their isotopologues are also resolved in the supercontinuum based dual-comb spectroscopy. Our approach combines well established fiber laser combs, digital coherent data averaging, and integrated nonlinear photonics, each in itself a state-of-theart technology, signalling the emergence of mid-IR dual-comb spectroscopy for use outside of the protected laboratory environment.	10.1364/cleo_si.2019.sth1g.7	[ "https://arxiv.org/pdf/1908.00871v1.pdf" ]	165,121,907	1908.00871	a2a2a36a95595020b06224e3c4b84476ec60037b	Nanophotonic supercontinuum based mid-infrared dual-comb spectroscopy Hairun Guo Laboratory of Photonics and Quantum Measurements (LPQM) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Key Laboratory of Specialty Fiber Optics and Optical Access Networks Shanghai University 200343ShanghaiChina Wenle Weng Laboratory of Photonics and Quantum Measurements (LPQM) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Junqiu Liu Laboratory of Photonics and Quantum Measurements (LPQM) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Fan Yang Group for Fiber Optics (GFO) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Wolfgang Hänsel Menlo Systems GmbH 82152MartinsriedGermany Camille Sophie Brès Photonic Systems Laboratory (PHOSL) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Luc Thévenaz Group for Fiber Optics (GFO) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Ronald Holzwarth Menlo Systems GmbH 82152MartinsriedGermany Tobias J Kippenberg Laboratory of Photonics and Quantum Measurements (LPQM) Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Nanophotonic supercontinuum based mid-infrared dual-comb spectroscopy High resolution and fast detection of molecular vibrational absorption is important for organic synthesis, pharmaceutical process and environmental monitoring, and is enabled by mid-infrared (mid-IR) laser frequency combs via dual-comb spectroscopy. Here, we demonstrate a novel and highly simplified approach to broadband mid-IR dual-comb spectroscopy via supercontinuum generation, achieved using unprecedented nanophotonic dispersion engineering that allows for flat-envelope, ultra-broadband mid-IR comb spectra. The mid-IR dual-comb has an instantaneous bandwidth covering the functional group region from 2800 − 3600 cm −1 , comprising more than 100,000 comb lines, enabling parallel gas-phase detection with a high sensitivity, spectral resolution, and speed. In addition to the traditional functional groups, their isotopologues are also resolved in the supercontinuum based dual-comb spectroscopy. Our approach combines well established fiber laser combs, digital coherent data averaging, and integrated nonlinear photonics, each in itself a state-of-theart technology, signalling the emergence of mid-IR dual-comb spectroscopy for use outside of the protected laboratory environment. Mid-infrared (mid-IR) is known as one of the most useful wavelength regions for spectroscopy due to the presence of fundamental vibrational transitions in molecules [1]. Moreover, it has medical potential as human breath contains numerous volatile chemical compounds (VOC), many of which can be associated with diseases [2]. Presently, mid-IR spectroscopy is primarily based on Fourier transform infrared (FTIR) spectrometers [3] that are bulky and have limited resolution and acquisition time. Over the past decade dual-comb spectroscopy (DCS) has emerged as an approach that can alleviate some of these shortcomings [4,5]. This approach, emerged with the invention of optical frequency combs [6,7], enables fast detection, scanning without moving parts, high resolution spectra, and have no limitation on size as the device length is independent on resolution, contrary to FTIR. DCS has today seen significant advances [8], and has also been successfully applied to an increasing portion of the mid-IR spectrum using a number of mid-IR comb sources [9], including quantum cascade lasers (QCLs) [10], microresonator Kerr frequency combs [11], difference frequency generation (DFG) [12,13], cascaded quadratic nonlinear process [14,15], and optical parametric oscillators (OPO) [16]. Yet to date, generating phase locked mid-IR frequency combs that exhibit high brightness, broad bandwidth, and fine resolution remains challenging. The most advanced approaches have used DFG that typically requires the synchronization of two laser beams and features limited instantaneous spectral bandwidth. To reach a larger spectral coverage, either mechanically tuning the phase matching of the nonlinear crystal or implementing chirped quasi-phasematching (QPM) is required [12]. OPO based mid-IR frequency combs have attained some of the broadest spectra to date, allowing for massively parallel sensing of trace molecules [16]. Although capable of a miniature size, this approach is mostly based on solid-state laser cavities that contain discrete bulk optics and components. Here we demonstrate a new approach to mid-IR DCS that offers unprecedented simplicity, based on supercontinuum generation in "coupled" nanophotonic integrated silicon nitride (Si 4 N 4 ) waveguides, and driven by conventional low-noise fiber-laser-based optical frequency combs in the well developed telecommunication band, as shown schematically in Fig. 1. In particular, we propose a coupled waveguide structure that represents an advanced approach to dispersion engineering, under which the supercontinuum process is along with the mode coupling in the waveguide( Fig. 1(b)), and is tailored to have a large spectral bandwidth and high flatness in the mid-IR ( Fig. 1(c)). The dual-comb spectrometer we presented offers state-of-the-art signal to noise, large instantaneous spectral bandwidth in the mid-IR (over 800 cm −1 ). It combines both low-noise fiber lasers and photonic integrated waveguides. Each in itself is a well developed technology and is commercially available, and therefore can contribute to high-performance mid-IR DCS. RESULT Nanophotonic supercontinuum generation with advanced dispersion engineering -Supercontin- 1: Nanophotonic supercontinuum-based mid-IR dual comb spectroscoppy. (a) Schematic setup for mid-IR dual-comb gas-phase spectroscopy, in which two mid-IR frequency combs are generated via coherent supercontinuum process in nanophotonic chip-based Si3N4 waveguides, seeded by a mutually locked dual-frequency-comb source at the telecom-band (i.e. ∼ 1550 nm). HWP: half wave plate; PD: (mid-infrared) photodetector. (b) Microscopic pictures of a photonic integrated chip with coupled Si3N4 waveguides, corresponding to both the input facet where the beginning section of each waveguide contains an inverse taper structure, and the output facet showing dual-core waveguide structures. The false-colored scanning electron microscopic (SEM) picture of the waveguide cross section is also presented. (c) Illustration of the supercontinuum process in a coupled Si3N4 waveguide, where the spatial mode distributions are calculated by means of the finite element method, at different wavelengths. It shows the detailed spectral broadening underlying the supercontinuum process is along with the mode coupling in the waveguide. (d) Principle of enhanced mid-IR continuum generation serving as the frequency comb, which corresponds to a dispersion landscape that is engineered and flattened in the mid-IR, particularly by the anomalous GVD produced in the coupled waveguide. Dispersion landscape is defined as relative phase constant compared with the pump pulse, i.e. β(ω) − βs(ω). In addition, visible dispersive wave is also supported in this type of waveguide due to the phase matching. uum generation is one of the most dramatic nonlin-ear optical processes [17], and has been essential for enabling femtosecond lasers to be self-referenced, making fully stabilized optical frequency combs [18]. Yet it constitutes a way to access ultra-broadband and coherent comb sources potentially imperative to applications. While being extensively studied in the past decade, the supercontinuum process has to date rarely been exploited for mid-IR DCS. Recent advances have been on nanophotonic integrated waveguides, where lithographically tailorable supercontinuum generation is enabled at low pulse energies [19][20][21][22][23], and have been applied for laser self-referencing [24] and offset frequency detection [25], as well as for DFG [26]. In particular, Si 3 N 4 waveguides [27], combining a wide transparency with nanophotonic dispersion engineering, have been demonstrated to support mid-IR frequency combs based on dispersive wave generation from a femtosecond erbium-fiber laser [22]. In this way, they provide access to the high-demand mid-IR range by bridging an efficient and coherent link with well developed fiber laser technology in the near-infrared, amiable for DCS. The spectral extent and efficiency of supercontinuum generation in nanophotonic waveguides critically depends on the dispersion properties. While the geometry (width and height) of the waveguide can be controlled to achieve dispersion engineering, leading to mid-IR dispersive waves [22], this control poses limitations on achieving a high conversion efficiency and ultra-broadband mid-IR continuum. Here, we employ coupled structures consisting of multiple Si 3 N 4 waveguide cores, typically dualcore waveguides (cf. Fig. 1(b)). When two waveguide cores are in close proximity, the optical mode propagating in one core is coupled to the other core, which effectively changes its phase, i.e. the propagation constant of the mode (β(ω), where ω is the angular frequency of the light). In this way, the group velocity dispersion (GVD) is also changed as it corresponds to the frequencydependent phase change induced by the mode coupling (GVD = ∂ 2 β/∂ω 2 ). Physically, mode coupling leads to the hybridization of mode-field distributions, resulting in a pair of supermodes, namely the symmetric and anti-symmetric superpositions of the original uncoupled waveguide modes [28]. The coupling induced dispersion is then reflected by the phase profile of these supermodes, which is curved to bridge that of the uncoupled modes, and feature an avoided crossing between each other(cf. Fig. 2(a)). Deterministically, anomalous GVD is always produced by the anti-symmetric mode, while normal GVD is by the symmetric mode. In principle, such mode coupling (formally termed the mode hybridization) can be engineered at arbitrary wavelength regions, particularly in the mid-IR where anomalous GVD is essential for tailoring a flattened dispersion landscape, but is hardly accessible in conventional single-core waveguides. The dispersion landscape is defined as the relative propagation constant compared with the pump pulses (as-sumed as solitons): ∆β(ω) = β(ω) − β s (ω) (1) = β(ω) − (β(ω s ) + v −1 g (ω − ω s ))(2) where β s (ω) indicates the dispersionless phase profile of the soliton pulse, ω s is the angular frequency of the pump and v g is the soliton group velocity. Figure. 2 illustrates the design of the Si 3 N 4 dual-core waveguide. The cross-section of the two Si 3 N 4 cores are separately selected, in which two modes (one from each core) feature the hybridization in the mid-IR region, by matching their propagation constants, (or equivalently by matching the effective refractive index (n eff ), since β = n eff ω/c, where c indicates the speed of light in vacuum). For a choice of Si 3 N 4 core widths of w 1 = 1.3 µm and w 2 = 3.4 µm, respectively, and for an identical core height of h = 0.85 µm, the fundamental TE 00 mode in the narrow core and the TE 10 mode in the wide core have the same n eff at the wavelength of 3200 nm ( Fig. 2(a)). The propagation constant and the mode-field distribution of supermodes are calculated for a coupling gap of g = 0.8 µm, which exhibit a strongly curved phase profile in the mid-IR compared with the original uncoupled mode (the uncoupled TE 00 mode in the narrow core is selected as the reference in this plot). Therefore, strong and dominant anomalous GVD is produced in the mid-IR (at ∼ 3200 nm, cf. Fig. 2(b)). Counterintuitively, the larger change in dispersion is not achieved by a closer proximity of the two waveguides as this causes the mode hybridization to occur over a larger spectral range such that the frequency-dependency of mode's phase constant is reduced. Moreover, to exploit anomalous GVD in the mid-IR for engineering the supercontinuum, it is imperative to selectively excite the anti-symmetric mode only. This is accomplished by designing the waveguide input section to be a single-core waveguide (with an inverse taper in the beginning [29], which excites the TE 00 mode, followed by the dual-core section (cf. Fig. 1(b)). The length of the input section is chosen such that the pulse propagation enters into the dual-core section before signifiant broadening occurs. In this way, the spectral broadening will extend into the mode hybridization region, and excite the anti-symmetric mode in the mid-IR. The designed dispersion landscape is shown in Fig. 2(c), in which the mid-IR portion is particularly tailored and flattened. This is to compare with a conventional single-core waveguide which shows a similar phase matching (i.e. ∆β(ω) = 0) wavelength for mid-IR dispersive wave [22,30]. We next carried out experiments to investigate the supercontinuum generation in designed Si 3 N 4 coupled waveguides. The waveguides are fabricated using the photonic Damascene process [31], in which thermal annealing steps are critical, in order to reduce the hydrogen content and the related absorption losses [32]. The of symmetric (purple curve) and anti-symmetric (orange curve) modes in a coupled dual-core Si 3 N 4 waveguide, compared with those of original uncoupled modes. The geometry of the waveguide is: w 1 = 1.3 µm, w 2 = 3.4 µm, h = 0.85 µm, and g = 0.8 µm. The mode coupling is between the the fundamental TE 00 mode (marked as n 1 (λ)) in the narrow core (blue curve) and the TE 10 mode in the wide core (green curve). Insets are the electric-field distribution of the supermodes at different wavelengths. (b) Calculated dispersion (orange curves) of the anti-symmetric mode (corresponding to three gap distances g = 0.6, 0.7, 0.8 µm), which produce additional anomalous GVD in the mid-IR compared with the uncoupled mode (blue curve). (c) Calculated dispersion landscape of the anti-symmetric mode (with g = 0.8 µm), compared with that of a selected single-core waveguide (w = 1.8 µm, h = 0.85 µm). Both show a similar phase matching wavelength for the mid-IR dispersive wave (i.e. 3500 nm). (d) Experimentally observed supercontinuum generation in a Si 3 N 4 dual-core Si 3 N 4 waveguide, with measured geometry w 1 = 1.3 µm, w 2 = 3.5 µm, h = 0.90 µm, and g = 0.65 µm. OSA: optical spectral analyzers; FTIR: Fourier-transform infrared spectrometer. (e) Simulations of the supercontinuum generation in both the dual-core waveguide and in the single-core waveguide, corresponding to the dispersion landscape in (c). (f ) Spectral overlapped two mid-IR continua from two separate Si 3 N 4 coupled waveguides, with similar cross-section geometry. waveguide length is 5 mm including the inverse taper section at the beginning. In waveguides similar to the design, supercontinuum generation was observed ( Fig. 2(d)), which is seeded by the amplified femtosecond fiber laser in the telecom-band (pulse duration < 70 fs, max-imum averaged power > 350 mW, pulse energy > 1 nJ, repetition rate ∼ 250 MHz). Significantly, with respect to the pumped wavelength (i.e. 1550 nm) the supercontinuum is mostly extended to the long wavelength side, leading to ultra-broadband mid-IR continuum ranging Parallel gas-phase spectroscopy by mid-IR broadband dual-comb spectrometer. (a) One retrieved midinfrared spectrum from the detected and coherent averaged interferogram trace, after the averaging of 52 s (net data acquisition time, cf. details in SI), which covers a large span from 2800 − 3600 cm −1 . The gas species in the gas cell, i.e. methane and acetylene, as well as water vapor in the circumstance, are featured as sharp absorbance in the spectrum. Insets show the temporal interferogram trace recorded by the photodetector. The blue shading area marks the mode hybridization region which exhibits a big dip in the retrieved spectrum. (b) In this panel, the measured gas absorbance (blue curves), methane (left) and acetylene (right) are compared with the HITRAN database (red curves, inverted for clarity) in a large span of wavenumber. The gas cell has the total pressure of 1 atm, with methane at 136.4 ppm, acetylene at 406.5 ppm, and nitrogen as the buffer gas. (c) In this panel, detailed gas absorbance, including both the intensity and the phase information, are compared with the HITRAN database. Residual data from the fitting is also presented, an offset value is artificially imposed merely for plotting purpose. 1.0 -1.0 0 ln(T/T 0 ) 0 0.3 -0.3 Phase (2π) from 2000 − 3700 nm, while at the short wavelength side, it features a sharp edge (stopping at 1000 nm) followed by a dispersive wave in the visible range (at 600 nm). The spectral envelope exactly reflects the designed dispersion landscape, i.e. the spectrum amplitude is inverse proportional to the relative phase constant. Significantly, filtering out the mid-IR continuum by an edge-filter (cuton at 2500 nm), we measured the net power in the mid-IR to be 1 − 3 mW, depending on the intensity of the pump. Note that such a power level already comprises the insertion loss of the waveguide and the loss in the light collecting component (e.g. a mid-IR collecting lens that has a transparency of ∼ 70%). The conversion efficiency is then estimated 1 − 5%, producing thereby sufficient power to implement dual-comb spectroscopy. Interestingly, we also observed a narrow-band mid-IR wave generation in the opposite polarization direction to the pump wave, at mode coupling wavelength 3200 nm ( Fig. 2(d)). This radiation is understood as the result of mode hybridization such that the polarization of the mode in the coupling region is rotated as well. We also performed numerical simulations of the supercontinuum process on the coupled waveguide and the single-core waveguide, see Fig. 2(e). The simulation is based on the generalized nonlinear Schrödinger equation (cf. SI for equation details), in which the dispersion landscape is of the anti-symmetric mode in the dual-core waveguide. Simulation result confirms that the supercontinuum in the dual-core waveguide does exhibit an enhanced mid-IR generation over a ultra-broadband wavelength span. This spectral structure qualitatively agrees with our experimental results (cf. Fig. 2(d)). It was confirmed that the coupled waveguide, with tailored dispersion landscape, does support broadband mid-IR continuum, while the single-core waveguide supports the narrowband dispersive wave. In addition, the power of the mid-IR wave in the coupled waveguide is also higher than that in the single-core waveguide. From separate Si 3 N 4 waveguide chips, similar mid-IR continuum can be generated with similar dual-core waveguides (Fig. 2(f)). A high level of spectral overlap is implemented, which is essential for building up the mid-IR broadband dual-comb spectrometer. Nanophotonic supercontinuum based mid-IR dual-comb spectrometer -The schematic setup of the nanophotonic supercontinuum based mid-IR dualcomb spectrometer is shown in Fig. 1(a). The pump source consists of two ultra-low noise femtosecond fiber lasers with sub-mHz individual linewidth (FC-1500-ULN from Menlo Systems, wavelength ∼ 1550 nm, repetition rate f rep ∼ 250 MHz). Both lasers have the carrier-offset frequency locked via self-referencing [7,18], and one comb mode optically locked to a shared reference laser (at ∼ 1541 nm, the laser is free-running and a daily shift in frequency is O(10 MHz)). The locked mode index is different by one, which leads to a small difference in the repetition rate, i.e. ∆f rep ≈ 320 Hz, and in principle allows the dual-comb spectrometer to cover a large span in the optical window, i.e. ∼ 100 THz (cf. details in the SI). In principle, the mid-IR frequency comb from the supercontinuum process is viewed as the spectral extension of the frequency comb structure of the original pump source, therefore it inherits the full properties of the source comb. Based on such a configuration, a phase-resolved mid-IR dual-comb spectrometer was built up, with one mid-IR comb passing through a gas cell for gas-phase detection, and the other comb serving as the reference. After detection, the two combs are interfered on a mid-IR photodetector (VIGO PV-4TE, mercury cadmium telluride (HgCdTe) detector). In the presence of a difference in the repetition rate, they generate a radio frequency (RF) comb composed of distinguishable heterodyne beats between pairs of optical comb teeth. In the time domain, it corresponds to a periodic interferogram pattern that can be directly recorded by the detector. The data acquisition was implemented by a field programmable gate array (FPGA). Real-time coherent averaging process is also enabled with a computer for multiple sets of signal (cf. SI for coherent averaging details). The normalized signal-to-noise ratio of our spectrometer has a peak value of 25/ √ s at the region of 3400 cm −1 (where the spectral intensity is strongest). The averaged signal-to-noise ratio is estimated as 10/ √ s. Therefore, we can conclude a figure of merit of 1.0 × 10 6 / √ s for our ultra-broadband mid-IR dual-comb spectrometer, which is mostly limited by the RIN on the mid-IR frequency combs (cf. details in SI). Although this figure of merit does not reach the shotnoise limit, it is comparable to reported results in other works, which mostly for DFG-based dual-comb spectrometers is 1 − 6 × 10 6 / √ s. Mid-IR gas-phase spectroscopic results We next applied the DCS for gas phase spectroscopy. Figure 3(a) shows the result of mid-IR dual-comb gasphase spectroscopy, where the spectral coverage is 2800 − 3600 cm −1 (∼ 25 THz) corresponding to the number of comb teeth of 100, 000. The dynamical range is > 40 dB supported by sufficient power (> 1 mW) in the mid-IR. This spectrum is retrieved from a coherent averaged temporal interferogram pattern which in the Fourier domain corresponds to the RF comb constructed by the interference of the two optical frequency combs. The gas cell (length 104 cm) was constructed with wedged sapphire windows to avoid etalon effects, and filled with a low concentration of methane and acetylene as targeted gas species, and nitrogen as the buffer gas. The overall pressure in the cell is 1 atm. To extract the absorption spectrum, we measure first the spectrum through the sample gas T , then purge the cell, fill it back to the original pressure with pure nitrogen, and measure the reference spectrum T 0 . The spectral absorbance is then determined as −ln(T /T 0 ). The gas concentrations are extracted from the absorption spectrum via a nonlinear least square fitting with the data (line center frequency, line intensity, pressure broadening and shift coefficients) from HITRAN 2016 database. The HI-TRAN phase spectra are calculated from the Kramers-Kronig transformation of the absorption spectra. The retrieved gas absorbance and phase spectra agree very well with HITRAN database (Fig. 3(b,c)). In addition the water vapor in the circumstance was also detected as ∼ 1.8 %. Moreover, the performance of our supercontinuum-based mid-IR dual-comb spectroscopy was also benchmarked by successful detection of natural isotopologues of methane (Fig. 4), i.e. 12 CH 4 , 13 CH 4 and 12 CH 3 D. For such a measurement, the gas cell is operated at low pressure such that the collisional broadening of spectral lines is reduced and those corresponding to isotopologues (i.e. 13 CH 4 , natural abundance 1.11%; 12 CH 3 D, 0.06%) can be resolved as separated from traditional elements ( 12 CH 4 ). In experiments, we set the pressure of the gas cell to be ∼ 0.1 atm and the methane concentration to ∼ 12.5%. At this pressure, the fullwidth-at-half-maximum spectral linewidth of methane is reduced to ∼ 480 MHz, which is both sufficient for separating isotopes, and resolvable by our sub-Doppler resolution (i.e., 250 MHz, determined by the mode spacing) 12 of the mid-IR frequency comb. The capability of identifying natural abundance of isotopes is of high importance as it provides signatures in earth science as well as in cosmology. DISCUSSION We have demonstrated a high-performance mid-IR dual-comb spectrometer based on the supercontinuum process in nanophotonic integrated nonlinear Si 3 N 4 waveguides. The proposed coupled waveguide structure has revealed unprecedented ways of performing dispersion engineering, which can lead to flat-envelope, ultrabroadband mid-IR frequency combs. Such a supercontinuum based dual-comb spectrometer has not only detected traditional chemical species, but can also trace their isotopologues. Our approach combines fiber-laser combs technology with nanophotonic integrated devices, each in itself a well established technology, yet it exhibits a performance competitive to DFG-based dual-comb spectrometers, which is amiable for applications outside the protected laboratory environment. In addition, this approach can benefit from superior laser stabilization methods, e.g. adaptive laser stabilization [33], frequency alignment [34] or feed forward locking [35]. At the moment, the long wavelength edge of our spectrometer is limited to 4.0 µm, which is mostly as a result of the SiO 2 cladding in the fabrication of Si 3 N 4 waveguides. Although Si 3 N 4 shows a much larger transparency window reaching the beginning of the backbone region (5 µm), mode coupling will expose the propagating light mostly to the cladding and therefore feature strong loss in SiO 2 . Such a problem can be solved with air-cladding waveguides or substrates that are mid-IR transparent (e.g. sapphire). Note: During the preparation of this work, DCS in the near IR (< 2500 nm) with two Si 3 N 4 single-core waveguides was reported [36]. In theory, nonlinear wave propagating dynamics in a waveguide can be described by the following wave equation: (where we only consider the spontaneous response in the cubic nonlinearity) ∂Ẽ(ω, r) ∂z = −iβ(ω)Ẽ(ω, r) − i ωχ (3) 2cn F E(t, r) 3 ω (3) where E(t, r) indicates the electric field of the light wave in the time domain (t-axis), and its amplitude spectral density isẼ(ω, r) in the frequency domain (w-axis), namely via the Fourier transform (operator F) there has:Ẽ(ω, r) = dtE(t, r)e −iωt ∆ = F [E(t, r)] ω ; r = {x, y, z} indicates the space frame and the light propagation direction in the waveguide is defined as the zaxis; β(ω) indicates the propagation constant of the light wave in a waveguide, which is frequency dependent reflecting dispersion properties; n is the effective refractive index of the waveguide; χ (3) is the cubic nonlinear susceptibility of the waveguide material; c is the speed of light in vacuum. The electric-field can be further expressed as:Ẽ (ω, r) =B(ω, x, y)Ã(ω, z)(4) withB the normalized mode distribution such that: dxdyB 2 = 1. Thus the propagation dynamics of the light field is enfolded inÃ and the Eq. 3 can be modified to: (if only considering the nonlinear phase modulation effect, i.e. the Kerr nonlinearity) ∂Ã(ω, z) ∂z = −iβ(ω)Ã(ω, z)−i ω c χ (3) 2nA eff F \|A(t, z)\| 2 A(t, z) ω(5) where information of the mode confinement in the waveguide is reflected on the parameter of the effective mode area, A eff . We further consider the case that the light wave consists of both a primary wave packet (seeded by the pump wave and assumed as solitons)Ã s and a nonlocal small waveσ (i.e. the dispersive wave). Therefore, in the frequency range w > 0, we can define: A(ω > 0, z) =Ã s (Ω, z)e −iβsz +σe −iβ(ω d )z(6) where Ω = ω − ω s defines a relative frequency frame with respect to the pumping frequency ω s ; β s (Ω) indicates the phase constant of the soliton, which is dispersionless, i.e.: β s (Ω) = β(ω s ) + Ωβ (1) (ω s ) β (m) (ω) = ∂ m ∂ω m β(ω) indicates the m-th order of dispersion with respect to ω s ; v g = 1/β (1) (ω) is also known as the group velocity of the soliton; ω d indicates the central frequency of the small wave and ω d = ω s . Using Eq. 6 in Eq. 5, we obtain the following equations: m! Ω m . In particular, with anomalous group velocity dispersion (GVD), i.e. β (2) < 0, solitons are supported in the waveguide. Equation 9 is derived at the sideband ofÃ s , i.e. when ω = ω d , and the nonlinear effect is also neglected. ∂Ã s ∂z = −i∆β(ω)Ã s − i ω c 3χ (3) 8nA eff F \|A s \| 2 A s Ω (8) σ ≈Ã s e i∆β(ω d )z(9) Significantly, from Eq. 9, the phase matching condition betweenÃ s andσ is: ∆β(ω d ) = 0. This is also understood as the phase matching between the soliton and the dispersive wave (Note: soliton would have an extra nonlinear induced phase constant (q) enfolded inÃ s , which however is usually small valued and is neglected). Moreover, the conversion efficiency of the dispersive wave depends on the intensity of the soliton sideband. Described by Eq. 8, A s would experience the nonlinear self-phase modulation (second term on the right hand side) that in the frequency domain, results in its spectral broadening (i.e. raising the sideband power). Conventionally, this effect will be counterbalanced by certain dispersion in the system, i.e. ∆β(ω) = 0. Nevertheless, the maximum sideband power comes where there is the lowest dispersion, i.e. : ∆β(ω) → 0. Therefore, apart from the phase matching condition, the overall landscape of ∆β actually determines the conversion efficiency of the dispersive wave, which in the range ω ∈ [ω s , ω d ] can be assimilated to a "spectral barrier" between the soliton and the dispersive wave. The purpose of our design is indeed to implement a flattened and reduced dispersion landscape, in the mid-IR, such that the supercontinuum generation can be broadband with enhanced efficiency. Engineered mid-IR continuum via geometry control Here we present the effect of tuning the waveguide geometry on the mid-IR spectral structure via the supercontinuum process, see Fig. 5. The pumping condition for all spectra is similar (pulse energy ∼ 1 nJ) and the waveguide length is always 5 mm. Generally, we find that the mid-IR continuum is built up by two parts: the long wavelength edge is raised by the mid-IR dispersive wave that is determined by the phase matching; the moderatewavelength part (in between the dispersive wave and the edge of the broadened soliton sideband) that is by the mode hybridization where the dispersion landscape is changed such that the overall phase mismatch is reduced. As a result, it can be observed that by decreasing the gap distance between the two Si 3 N 4 cores, the mid-IR dispersive wave is slightly shifted to longer wavelengths, and in the meantime, the effect of mode hybridization on raising the moderate-wavelength part is reduced. The latter is counter-intuitive as a closer gap usually means stronger mode coupling effect. However, this increased mode coupling would also involve a large wavelength span such that the relative phase-change over wavelength (which is directly linked to the induced group velocity dispersion) is reduced. The fingerprint of the mode hybridization is also revealed by measured orthogonal polarized light generation with respect to the pump. While the pump beam is coupled to the horizontally polarized TE 00 mode in the narrow core, the moderate-wavelength part is supported by a narrow-band vertically polarized beam with its wavelength in accordance to the designed mode coupling region. This orthogonal polarized beam is found almost independent on the change of the geometry, i.e. the gap distance as well as the width of the wide waveguide core. Locking of fiber-laser frequency combs The two comb sources are fully stabilized both in the carrier-envelope offset frequency (f CEO ) and in the repetition rate (f rep ). The latter is implemented by having one comb mode in the C-band optically locked to a stable continuous wave (CW) reference laser (at the frequency f CW ), resulting in a beat signal ν beat . f CEO and ν beat are locked to have the same radio frequency, i.e. f CEO + ν beat = 0. Therefore the repetition rate of the frequency comb can be easily derived as: f CW = f CEO + N f rep + ν beat ,(10)f rep = f CW N(11) where N is the index of the locked comb mode. Sharing the same reference laser, the two combs are mutually locked as well. The index of the locked mode is different by one in between the two combs, namely the second comb has f rep,2 = fCW N +1 . Therefore the difference in the repetition rate is: ∆f rep = f CW N (N + 1)(12) which is usually small valued compared with the repetition rate, such that the dual-comb configuration can cover a large span of optical window. In our work, we have f rep ≈ 250 MHz and ∆f rep ≈ 320 Hz. Therefore, the optical spectral coverage is estimated as: f rep × f rep /2 ∆f rep ≈ 100 THz(13) which is sufficient to cover a broadband mid-infrared spectroscopic window. In addition, the beat signals (ν beat ) for the mutual optical referencing are derived in close proximity compared with the free-space section (i.e. the mode spacing of the comb) in order to keep phase drifts at a minimum. This leads to a coherence time > 1 s as demonstrated in the RF spectra (cf. the following section), and at the same time enables coherent averaging for at least 84 interferograms (cf. the section of coherent averaging). Relative intensity noise spectra of the dualcomb system. Comparison between the measured RIN spectra of the amplified near-IR pump laser and the mid-IR emission from the waveguide shows that the RIN of the mid-IR light is approximately 10 dB higher than that of the pump source at frequency range between 100 Hz and 100 kHz. At relatively low frequencies the RIN measurement of the mid-IR light is limited by the the noise floor of the mid-IR photodetector, which is also presented. Relative intensity noise and DCS signal linewidth We used an FFT analyzer to measure the RIN of the dual-comb system. Fig. 6 shows the measured RIN spectra. The mid-IR RIN spectrum exhibits a level approximately 10 dB higher than that of the pump laser source. A similar deterioration of the RIN performance has been previously observed and investigated for the supercontinuum generation in fiber optics [37]. At frequencies from 100 Hz to 1 kHz we observed multiple spectral peaks, which we attribute to the mechanical and acoustical vibrations in the experimental setting. In future works passive vibration cancellation and active laser intensity control can be applied to improve the RIN of the mid-IR dual-comb system. Next we measured the RF comb spectra of the dualcomb system at both the near-IR pump branch and the mid-IR branch with an electrical spectrum analyzer (ESA). The spectra are displayed in Fig. 7. The RF beat signals show linewidths of sub-hertz level, which is limited by the minimum resolution bandwidth (RBW) of the ESA. The results show that the mutual coherence time of the mid-IR combs is at least of the order of 1 s, which potentially allows us to carry out realtime coherent averaging for a period that is significantly longer than the 0.26 s in this work, which is currently limited by the FPGA function. Averaging of interferograms Triggered by the pulse repetition rate of one of the near-IR pump lasers, the FPGA data acquisition unit can record the output voltage level of the mid-IR photodetector and continuously save up to 84 interferograms. The data is then read out on a computer, which in the meantime co-adds these interferograms and performs averaging to get a single averaged interferogram. We refer to this onboard averaging as realtime coherent averaging. In fact, after each 84 interferograms recored in the FPGA, the data communicating and saving on the computer will introduce a dead time of ∼ 8 s. We then post-process tens to hundreds of such saved and coherent-averaged interferograms with phase calibration, and thus obtain a single interferogram by averaging the phase-corrected interferograms. This post processing is referred to as "offline averaging" in this work. To calculate the time-normalized DCS SNR and the figure of merit (DCS quality factor), for offline averaging only the effective data acquisition time is taken into account (dead time excluded). Signal-to-noise ratio and figure of merit In Fig. 8 (a) we present the dual-comb spectrum without gas sample absorption after the gas cell was purged, as a reference to the spectrum shown in Fig. 3 (a). To calculate the DCS SNR we choose 9 wavelengths that are At each wavelengths a section of data containing 500 to 1000 data points are picked, showing no obvious absorption features. Sum-of-multiple-sine function is applied to fit the data, in order to remove the background etalons. The standard deviations of the normalized fitting residuals (σ) are computed, and the DCS SNR is derived as 1 σ . As an example, σ of different averaging times around 3120nm are displayed in Fig. 8 (b). The av-eraged SNR of both realtime coherent averaging at short averaging times and offline averaging at relatively long averaging times are presented in Fig. 8 (d), showing the effectiveness of the averaging approaches with the similar dependence on averaging time as SNR = 10 × t 1/2 . The DCS figure of merit (DCS quality factor [38]) is computed as the product of the averaged SNR and the number of modes contained in a spectrum. FIG. 2 : 2Design and mid-IR supercontinuum generation in coupled Si 3 N 4 waveguides. (a) Calculated effective refractive indices FIG. 3: Parallel gas-phase spectroscopy by mid-IR broadband dual-comb spectrometer. (a) One retrieved midinfrared spectrum from the detected and coherent averaged interferogram trace, after the averaging of 52 s (net data acquisition time, cf. details in SI), which covers a large span from 2800 − 3600 cm −1 . The gas species in the gas cell, i.e. methane and acetylene, as well as water vapor in the circumstance, are featured as sharp absorbance in the spectrum. Insets show the temporal interferogram trace recorded by the photodetector. The blue shading area marks the mode hybridization region which exhibits a big dip in the retrieved spectrum. (b) In this panel, the measured gas absorbance (blue curves), methane (left) and acetylene (right) are compared with the HITRAN database (red curves, inverted for clarity) in a large span of wavenumber. The gas cell has the total pressure of 1 atm, with methane at 136.4 ppm, acetylene at 406.5 ppm, and nitrogen as the buffer gas. (c) In this panel, detailed gas absorbance, including both the intensity and the phase information, are compared with the HITRAN database. Residual data from the fitting is also presented, an offset value is artificially imposed merely for plotting purpose. FIG. 4 : 4Methane isotopologues resolved by mid-IR dual-comb spectrometer. Measured absorbance of methane (blue curve) is compared with the HITRAN database (red curve, inverted for clarity), in the case of high concentration, i.e. ∼ 12.5 %. The total pressure of the gas cell is ∼ 0.1 atm. Insets show the absorbance of 12 CH4, 13 CH4 and 12 CH3D, at selected regions. Authors acknowledge Miles Anderson for fruitful discussions and suggestions regarding the manuscript. This publication was supported by Contract W31P4Q-16-1-0002 (SCOUT) from the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO), and the Swiss National Science Foundation under grant agreement 163864. W.W. acknowledge funding from the European Unions H2020 research and innovation programme under grant agreement No. 753749 (SOLISYNTH). F.Y. and L.T. acknowledge funding from the Swiss National Science Foundation under grant agreement No.200021 178895. Nanophotonic Si 3 N 4 waveguide chips were fabricated at the Center for MicroNanoTechnology (CMi) at EPFL. whereÃ s = F [A s ] Ω . Equation 8is written in the frequency domain (Ω-axis), and its form in the time domain is the well-known nonlinear Schrödinger equation with full dispersion (i.e. the dispersion landscape), namely ∆β(ω) = β(ω) − β s = m≥2 β (m) FIG. 5 : 5Engineered mid-IR wave generation in hybrid dual-core waveguides. (a) A panel of experimentally acquired spectra in three dual-core Si3N4 nano-photonic waveguides, in which the gap g parameter is tuned while other parameters are fixed as: w1 = 1.3 µm, w2 = 3.4 µm and h = 0.9 µm. (b) A similar panel of spectra in the second set of waveguides where only the parameter w2 = 3.5 µm is different to that in (a). Note: the pumping source is horizontally polarized and the overall spectrum is measured and shown as the red curve, while the blue curve represents the optical spectrum measured after a linear polarizer that is set orthogonal to the pump. FIG. 6: Relative intensity noise spectra of the dualcomb system. Comparison between the measured RIN spectra of the amplified near-IR pump laser and the mid-IR emission from the waveguide shows that the RIN of the mid-IR light is approximately 10 dB higher than that of the pump source at frequency range between 100 Hz and 100 kHz. At relatively low frequencies the RIN measurement of the mid-IR light is limited by the the noise floor of the mid-IR photodetector, which is also presented. FIG. 7 :FIG. 8 : 78Mutual linewidths of the dual-comb system. Measured RF spectra of the dual-comb beat signals of the near-IR pump lasers (left) and the mid-IR emissions (right). The insets show the magnified spectra of individual beat signals, which exhibit linewidths of sub-hertz level that are limited by the 1 Hz resolution bandwidth of the ESA. Calculation of the signal-to-noise ratio and the DCS figure of merit. (a) Mid-IR dual-comb spectrum of the purged gas cell. 9 numbered red dash lines indicate the 9 wavelengths where the DCS SNR is computed. (b) The standard deviations (std) of the normalized fitting residuals around 3120 nm. Both the realtime coherent averaging and the offline averaging show a relation as std = 0.08 × t −1/2 . (c) Three histograms of the normalized fitting residuals at 3120 nm, corresponding to three different averaging times indicated in the figures respectively. The red curves are Gaussian function fittings. (d) The averaged SNR with varied averaging times, exhibiting a dependence of SNR = 10 × t 1/2 . distributed over an obtained spectrum. These 9 wave-lengths are indicated by the vertical dashed lines in the figure. arXiv:1908.00871v1 [physics.optics] 2 Aug 2019HWP Si 3 N 4 waveguides Gas cell Dual-comb source 1550 nm Ref. Filter PD Coherent Averaging DAQ . . . N Wavelength Near-IR Mid-IR Pump: fiber laser comb Relative Phase Const. Optical power Single-core Coupled structure Enhanced mid-IR frequency comb F il t e r Anomalous GVD in mid-IR Engineered dispersion landscape Visible dispersive wave Visible c) b) a) 1 m SiO 2 Si 3 N 4 Si 3 N 4 w 1 w 2 h g Inverse taper section Coupled waveguides Tapered Sec. Coupled waveguide structure d) = 15 50 nm = 20 00 nm = 25 00 nm = 30 00 nm = 35 00 nm FIG. 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[ "The Effective ∆m 2 ee in Matter", "The Effective ∆m 2 ee in Matter", "The Effective ∆m 2 ee in Matter", "The Effective ∆m 2 ee in Matter" ]	[ "Peter B Denton \nNiels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA\n", "Stephen J Parke \nNiels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA\n", "Peter B Denton \nNiels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA\n", "Stephen J Parke \nNiels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA\n" ]	[ "Niels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA", "Niels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA", "Niels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA", "Niels Bohr International Academy\nTheoretical Physics Department\nFermi National Accelerator Laboratory\nUniversity of Copenhagen\nThe Niels Bohr Institute\nBlegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA" ]	[]	In this paper we generalize the concept of an effective ∆m 2 ee for νe/νe disappearance experiments, which has been extensively used by the short baseline reactor experiments, to include the effects of propagation through matter for longer baseline νe/νe disappearance experiments. This generalization is a trivial, linear combination of the neutrino mass squared eigenvalues in matter and thus is not a simple extension of the usually vacuum expression, although, as it must, it reduces to the correct expression in the vacuum limit. We also demonstrated that the effective ∆m 2 ee in matter is very useful conceptually and numerically for understanding the form of the neutrino mass squared eigenstates in matter and hence for calculating the matter oscillation probabilities. Finally we analytically estimate the precision of this two-flavor approach and numerically verify that it is precise at the sub-percent level.	10.1103/physrevd.98.093001	[ "https://arxiv.org/pdf/1808.09453v1.pdf" ]	119,068,508	1808.09453	579bdcd5a2f3cda236b5535591143dd1d8bbc6af	The Effective ∆m 2 ee in Matter Peter B Denton Niels Bohr International Academy Theoretical Physics Department Fermi National Accelerator Laboratory University of Copenhagen The Niels Bohr Institute Blegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA Stephen J Parke Niels Bohr International Academy Theoretical Physics Department Fermi National Accelerator Laboratory University of Copenhagen The Niels Bohr Institute Blegdamsvej 17P. O. Box 500DK-2100, 60510Copenhagen, BataviaILDenmark, USA The Effective ∆m 2 ee in Matter (Dated: August 28, 2018) In this paper we generalize the concept of an effective ∆m 2 ee for νe/νe disappearance experiments, which has been extensively used by the short baseline reactor experiments, to include the effects of propagation through matter for longer baseline νe/νe disappearance experiments. This generalization is a trivial, linear combination of the neutrino mass squared eigenvalues in matter and thus is not a simple extension of the usually vacuum expression, although, as it must, it reduces to the correct expression in the vacuum limit. We also demonstrated that the effective ∆m 2 ee in matter is very useful conceptually and numerically for understanding the form of the neutrino mass squared eigenstates in matter and hence for calculating the matter oscillation probabilities. Finally we analytically estimate the precision of this two-flavor approach and numerically verify that it is precise at the sub-percent level. I. INTRODUCTION Since the discovery that neutrinos oscillate [1,2] tremendous progress has been made in understanding their properties. The oscillation parameters are all either well-measured or will be with the advent of next generation experiments. As the final parameters are measured, precision in the neutrino sector becomes more important than ever. In vacuum, an effective two-flavor oscillation picture was presented in [3] for calculating the ν e → ν e disappearance probability which introduced an effective ∆m 2 , ∆m 2 ee ≡ cos 2 θ 12 ∆m 2 31 + sin 2 θ 12 ∆m 2 32 , which precisely and optimally determines the shape of the disappearance probability around the first oscillation minimum. That is, even in the three favor framework, for ν e disappearance in vacuum (P 0 ), the two-flavor approximation P 0 (ν e → ν e ) : ≈ 1 − sin 2 2θ 13 sin 2 ∆ ee , where ∆ ee ≡ ∆m 2 ee L/(4E) , is an excellent approximation at least over the first oscillation. ∆m 2 ee has been widely used by the short baseline reactor experiments, Daya Bay [4] and RENO [5] in their shape analyses around the first oscillation minimum and will be precisely measured to better than 1% in the medium baseline JUNO [6] experiment. The matter generalization of the three-flavor ν e disappearance probability in matter (P a ) can also be ade- * [email protected]; 0000-0002-5209-872X † [email protected]; 0000-0003-2028-6782 quately approximated by a two-flavor disappearance oscillation probability in matter P a (ν e → ν e ) ≈ 1 − sin 2 2θ 13 ∆m 2 ee ∆ m 2 ee 2 sin 2 ∆ ee ,(3) where ∆ ee ≡ ∆ m 2 ee L/(4E) , and x denotes the exact matter version of a variable and is a function of the Wolfenstein matter potential [7]. This new ∆ m 2 ee would be the dominant frequency, over the first few oscillations, for ν e disappearance at a potential future neutrino factory [8] in the same way that ∆m 2 ee is for short baseline reactor experiments. As we will find in section II, satisfies all of the necessary criteria to describe ν e disappearance in matter in the approximate two-flavor picture of eq. 3 above and trivially reproduces eq. 1 in vacuum. We will also discuss an alternate expression ∆ m 2 EE which numerically behaves quite similarly, but is somewhat less useful analytically. The layout of this paper is as follows. In section II we define the matter version of ∆m 2 ee denoted ∆ m 2 ee . We review the connection between the three-flavor and twoflavor expressions in section III which naturally leads to a slightly different expression dubbed ∆ m 2 EE . In section IV we show how the natural definition of ∆ m 2 ee matches the expression given from a perturbative description of oscillation probabilities. We analytically and numerically show that both expressions are very close in section V. We perform the numerical and analytical calculations to show the precision of this definition of ∆ m 2 ee compared with other definitions of ∆m 2 ee in matter in section VI. Finally, we end with our conclusions in section VII, and some details are included in the appendices. II. DEFINING ∆ m 2 ee IN MATTER In this section we create a qualitative picture to derive the ∆ m 2 ee presented in the previous section. We then verify that it passes the necessary consistency checks. Figure 1 gives the neutrino mass squared eigenvalues in matter, m 2 i , as a function of the neutrino energy as well as the value of their electron neutrino content, \| U ei \| 2 . Neutrinos (anti-neutrinos) are positive (negative) energy in this figure and vacuum corresponds to E = 0. From the ν e content, it is clear that for energies greater than a few GeV that ∆ m 2 32 will dominate the L/E dependence of ν e disappearance and similarly ∆ m 2 31 will dominate for energies less than negative, a few GeV, that is, ∆ m 2 ee = m 2 3 − m 2 1 , a/∆m 2 21 −1 m 2 3 − m 2 2 , a/∆m 2 21 1 ,(5) where a = 2 √ 2EG F N e is the matter potential, G F is Fermi's constant, N e is the electron density, and the m 2 i /2E are the exact eigenvalues which are calculated in [9], see also appendix A. This is independent of mass ordering. We note that m 2 2 and m 2 1 are approximately constant for a/∆m 2 21 −1 and a/∆m 2 21 1, respectively. This suggests defining ∆ m 2 ee as follows 1 : ∆ m 2 ee ≡ m 2 3 − ( m 2 1 + m 2 2 − m 2 0 ) ,(6) where m 2 0 ≡ m 2 2 (a = −∞) = m 2 1 (a = +∞) = ∆m 2 21 c 2 12 (7) using the (convention dependent) asymptotic values for the eigenvalues shown in Table I. By construction, this reproduces eq. 5 for \|a/∆m 2 21 \| 1 and is applicable for both mass orderings. The sign of ∆ m 2 ee determines the mass ordering. It is also useful to note that m 2 0 can be written as m 2 0 = ∆m 2 ee − [m 2 3 − (m 2 1 + m 2 2 )] .(8) Then, as suggested by eq. 4, ∆ m 2 ee can also be written in the following simple and easy to remember form, ∆ m 2 ee −∆m 2 ee = ( m 2 3 −m 2 3 )−( m 2 1 −m 2 1 )−( m 2 2 −m 2 2 ) ,(9) where recovery of the vacuum limit is manifest. In the following sections we will address in more detail why the definition of eq. 4 works for all matter potentials including \|a/∆m 2 21 \| 1. Here we will use eq. 4 to re-write the m 2 i 's in matter as a function of the two relevant ∆ m 2 's: ∆ m 2 ee and ∆ m 2 21 . By properties of the trace of the Hamiltonian 2 , we have m 2 3 + m 2 2 + m 2 1 = ∆m 2 31 + ∆m 2 21 + a . Then together with eq. 6 above m 2 3 = ∆m 2 31 + 1 2 a + 1 2 (∆ m 2 ee − ∆m 2 ee ) , m 2 2 + m 2 1 = ∆m 2 21 + 1 2 a − 1 2 (∆ m 2 ee − ∆m 2 ee ) . (11) We make the typical definition ∆ m 2 21 ≡ m 2 2 − m 2 1 , then m 2 1 = 1 4 a − 1 4 (∆ m 2 ee − ∆m 2 ee ) − 1 2 (∆ m 2 21 − ∆m 2 21 ) m 2 2 = ∆m 2 21 + 1 4 a − 1 4 (∆ m 2 ee − ∆m 2 ee ) + 1 2 (∆ m 2 21 − ∆m 2 21 ) m 2 3 = ∆m 2 31 + 1 2 a + 1 2 (∆ m 2 ee − ∆m 2 ee ) ,(12) 1 Note that m 2 0 is identical to λ b = λ 0 from [10]. [11] or Table 4 of [10]. Adding the same constant to all entries in this table, does not effect oscillation physics. Our convention is that in vacuum m 2 1 = 0. which implies ∆ m 2 31 = ∆m 2 31 + 1 4 a + 3 4 (∆ m 2 ee − ∆m 2 ee ) + 1 2 (∆ m 2 21 − ∆m 2 21 ) (13) ∆ m 2 32 = ∆ m 2 31 − ∆ m 2 21 . We can also use ∆ m 2 ee to estimate ∆ m 2 21 except near a ≈ 0. For \|a/∆m 2 21 \| 1, either m 2 2 = m 2 0 or m 2 1 = m 2 0 . Then, ∆ m 2 21 ≈ \| m 2 2 + m 2 1 − 2m 2 0 \| ≈ ∆m 2 21 a 12 /∆m 2 21 − cos 2θ 12 + O(∆m 2 21 ) ,(14) where we have made the natural definition, a 12 ≡ 1 2 (a + ∆m 2 ee − ∆ m 2 ee )(15) as the effective matter potential for the 12 sector as was used in [12]. For this derivation eq. 11 is needed. The asymptotic eigenvalues in Table I, can also be used to obtain a simple approximate expression for ∆ m 2 ee , when \|a\| ∆m 2 ee : ∆ m 2 ee ≈ ∆m 2 ee a/∆m 2 ee − cos 2θ 13 .(16) These two asymptotic expressions for ∆ m 2 ee and ∆ m 2 21 , eqs. 16 and 14 respectively, which were obtained with only general information of the neutrino mass squareds in matter here, will be compared to the expressions obtained using the approximations of [11] & [10] in section IV. III. THREE-FLAVOR TO TWO-FLAVOR Instead of studying the asymptotic behavior of ∆ m 2 ee , we instead focus on explicitly connecting the threeflavor expression with the two-flavor expression. The exact three-flavor ν e disappearance probability in matter P a (ν e → ν e ) is given by 1 − P a = 4\| U e3 \| 2 \| U e1 \| 2 sin 2 ∆ 31 + \| U e2 \| 2 sin 2 ∆ 32 + 4\| U e1 \| 2 \| U e2 \| 2 sin 2 ∆ 21 = sin 2 2 θ 13 c 2 12 sin 2 ∆ 31 + s 2 12 sin 2 ∆ 32 + c 4 13 sin 2 2 θ 12 sin 2 ∆ 21 ,(17) where we have used s ij = sin θ ij and c ij = cos θ ij . As was shown in [13], eq. 17 can be rewritten without approximation, as 1 − P a (ν e → ν e ) = c 4 13 sin 2 2 θ 12 sin 2 ∆ 21 + 1 2 sin 2 2 θ 13 1 − 1 − sin 2 2 θ 12 sin 2 ∆ 21 cos(2 ∆ EE + Ω) ,(18) where Ω = arctan(cos 2 θ 12 tan ∆ 21 ) − ∆ 21 cos 2 θ 12 and ∆ m 2 EE is a new frequency defined by ∆ m 2 EE ≡ cos 2 θ 12 ∆ m 2 31 + sin 2 θ 12 ∆ m 2 32 .(19) For \|E\| greater than a few GeV, ∆ m 2 21 ∆m 2 21 (see fig. 1) and therefore θ 12 ≈ 0 or π/2, which makes 1 − sin 2 2 θ 12 sin 2 ∆ 21 ≈ 1 and Ω ≈ 0. Hence, 1 − P a (ν e → ν e ) ≈ sin 2 2 θ 13 sin 2 ∆ EE , in agreement with eq. 3 in this energy range 3 . Also in this energy region, it is clear that 4 ∆ m 2 EE ≈ ∆ m 2 31 , a ∆m 2 21 ∆ m 2 32 , a ∆m 2 21 .(20) Using the explicit results from [9], it is simple to show, without approximation, that ∆ m 2 EE = ( m 2 3 − m 2 a )( m 2 3 − m 2 1 )( m 2 3 − m 2 2 ) ( m 2 3 ) 2 − m 2 3 m 2 a − β + m 2 1 m 2 2 ,(21) where β ≡ ∆m 2 ee c 2 13 ∆m 2 21 c 2 12 = m 2 1 m 2 2 m 2 3 /a m 2 a ≡ a + ∆m 2 ee s 2 13 + ∆m 2 21 s 2 12 . 3 Note sin 2 2 θ 13 > c 4 13 sin 2 2 θ 12 except when \|E\| < 1.1 GeV, see fig. 6. We take ρ = 3 g/cc throughout the article. 4 This statement is made under the assumption that θ 12 → π/2 (0) as a → ∞ (−∞). In fact, there is a small correction to this assumption. In this limit, [14]. sin 2 θ 12 = 1 − O( 2 ) where 2 < 3 × 10 −4 , Note 5 that m 2 3 (a → ∞) → m 2 a and m 2 1 (a → −∞) → m 2 a . In the low energy limit, when \| m 2 3 \| \| m 2 j \| for j = (1, 2, a), a first order perturbative expansion in m 2 j / m 2 3 gives ∆ m 2 EE ≈ m 2 3 − ( m 2 1 + m 2 2 − m 2 0 ),(22) consistent with our previous definition, eq. 6. In fact, ∆ m 2 ee and ∆ m 2 EE differ by less than < 0.3% for all values of matter potential. In vacuum (E = 0), it is known that eq. 2 is an excellent approximation over the first couple of oscillations see e.g. [15], further verifying the use of this two-flavor approximation. The analysis of this paper can be trivially extend away from vacuum region using the matter oscillation parameters. IV. RELATION TO DMP APPROXIMATION While eq. 6 is a compact expression that behaves as we expect ∆ m 2 ee ought to, it is not simple due to the complicated expressions for the eigenvalues, in particular the cos( 1 3 cos −1 . . . ) part of each eigenvalue, see appendix A. In order to both verify the behavior of ∆ m 2 ee for \|a/∆m 2 ee \| 1 and provide an expression that is simple we look to approximate expressions of the eigenvalues. In refs. [11], [10] & [12] (DMP) simple, approximate, and precise analytic expressions were given for neutrino oscillations in matter. In the DMP approximation 6 through zeroth order, the definition of ∆ m 2 ee given in eq. 6 can be shown to be ∆ m 2 ee ≈ m 2 3 − ( m 2 1 + m 2 2 − m 2 0 ) ≡ ∆ m 2 ee , = cos 2 θ 12 ∆ m 2 31 + sin 2 θ 12 ∆ m 2 32 ,(23) = ∆m 2 ee (cos 2θ 13 − a/∆m 2 ee ) 2 + sin 2 2θ 13 , where θ 12 and θ 13 are excellent approximations for the matter mixing angles θ 12 and θ 13 and ∆ m 2 31 and ∆ m 2 32 are the corresponding approximate expressions for ∆ m 2 31 and ∆ m 2 31 from [10] and reproduced in appendix B below 7 . The approximation has corrections to the eigenvalues of O( 2 ) where = sin( θ 13 − 5 Also note that m 2 a is identical to λa from [10]. 6 In the notation of DMP, ∆ m 2 ee ≡ ∆λ +− = cos 2 ψ ∆λ 31 + sin 2 ψ ∆λ 32 , see eq. A.1.7 of [10]. Also, θ 12 = ψ and m 2 i = λ i in DMP; see [12]. 7 The notation is such that while both x and x are quantities in matter, x denotes the exact quantity and x denotes the zeroth order approximation from DMP, and x is an excellent approximation for x. θ 13 )s 12 c 12 ∆m 2 21 /∆m 2 ee . \| \| < 0.015 and is equal to zero in vacuum. Equation 23 provides a very simple means to modify the vacuum ∆m 2 ee to get the corresponding expression in matter. In the DMP approximation, all three expressions, eq. 23, for ∆ m 2 ee can be shown to be analytically identical. This is however not true for the exact eigenvalues and mixing angles in matter, there are small differences between these expressions (quote fractional differences.). We use the first line of eq. 23 for our definition ∆m 2 ee in matter, because this definition allows us a general understanding of the three neutrino eigenvalues in matter (see eqs. 12 and 13). We now verify that this definition of ∆m 2 ee in matter meets all the other criteria we need it to. First we see that by using the DMP zeroth order approximation, ∆ m 2 ee is just the matter generalization of the vacuum expression, ∆m 2 ee = cos 2 θ 12 ∆m 2 ee + sin 2 ∆m 2 32 and provides a connection to why the definition of eq. 6 works for \|a/∆m 2 21 \| < 1 also. Asymptotically, as \|a/∆m 2 ee \| 1, in this approximation scheme ∆ m 2 ee → ∆m 2 ee a/∆m 2 ee − cos 2θ 13 , in agreement with eq. 16. Similarly for ∆ m 2 21 , from DMP ∆ m 2 21 = ∆m 2 21 (cos 2θ 12 − a 12 /∆m 2 21 ) 2 + sin 2 2θ 12 cos 2 ( θ 13 − θ 13 ) 1/2 ,(25) again in agreement with eq. 14. So everything discussed in section II is consistent with the simple and compact DMP approximation. In the next section we will analytically and then numerically show that the fractional difference between the two expressions, ∆ m 2 ee and ∆ m 2 EE , are small. V. COMPARISON OF THE TWO EXPRESSIONS As previously shown the vacuum ∆m 2 ee can be written in two equivalent ways, ∆m 2 ee = c 2 12 ∆m 2 31 + s 2 12 ∆m 2 32 , = m 2 3 − m 2 1 − m 2 2 + m 2 0 . The two expressions can be seen as two choices for the how to relate these to the matter version: one is to elevate each eigenvalue to its matter equivalent (everything except m 2 0 ) and the other is to elevate each term including the mixing angles. We refer to the former as ∆ m 2 ee and the latter as ∆ m 2 EE . To understand how these expressions differ, we carefully examine their difference, ∆ Ee ≡ ∆ m 2 EE − ∆ m 2 ee = m 2 1 + c 2 12 ∆ m 2 21 − c 2 12 ∆m 2 21 .(28) We now quantify the difference between these expressions using DMP. If both expressions provide good approximations for the two flavor frequency in matter then the difference between them should be small. At zeroth order the difference is ∆ (0) Ee = m 2 1 + c 2 12 ∆ m 2 21 − c 2 12 ∆m 2 21 = 0 ,(29) so these expressions are exactly equivalent at zeroth order. At first order the eigenvalues receive no correction, but θ 12 does. From [14] we have that the first order correction is θ (1) 12 = − ∆m 2 ee t 13 s 2 12 ∆ m 2 31 + c 2 12 ∆ m 2 32 ,(30) where t ij = tan θ ij . This leads to a correction of, (31) As expected ∆ Ee ∝ a for small a. Also, we can verify that ∆ Ee /∆ m 2 ee is always small by seeing that a/∆ m 2 ee remains finite and the only case where t 13 ∝ a for a → ∞, but ∆ m 2 32 ∆ m 2 31 ∝ a 2 , thus the difference between the two expressions is always small. ∆ Ee provides an adequate approximation of the difference between ∆ m 2 ee and ∆ m 2 EE as shown in fig. 2. A precise estimate of the difference requires the second order correction to θ 12 given explicitly in [14] along with the second order corrections to the eigenvalues from DMP. This is because this difference ∆ Ee depends strongly on the asymptotic behavior of θ 12 which only becomes precise beyond the atmospheric resonance at second order. The result of this is also shown in fig. 2 which shows that first order is not sufficient to accurately describe the difference, but second order is. We see that for neutrinos the expressions agree to 0.3%, and the agreement is ∼ 3 orders of magnitude better for anti-neutrinos. In the next section we will investigate how well the twoflavor approximation, eq. 3, works numerically for both the depth and position over the first oscillation minimum for ν e disappearance for all values of the neutrino energy. VI. PRECISION VERIFICATION The goal of ∆ m 2 ee is to provide the correct frequency such that the two-flavor disappearance expression, eq. 3, is an excellent approximation for ν e disappearance over the first oscillation in matter. In particular, we want this expression to reproduce the position and depth of the first oscillation minimum at high E (small L) correctly compared to the complete three-flavor picture. A. Numerical Comparison Using the definition of ∆ m 2 ee given in eq. 6, we plot in fig. 3 ∆ m 2 ee ∆m 2 ee 2 (1 − P a (ν e → ν e )) verses ∆ ee , (32) for various values of the neutrino energy. Here P a (ν e → ν e ) is evaluated using the exact oscillation probability given in [9]. We see that this behaves like sin 2 ∆ ee as expected, with increasing precision for increasing energy. Note the approximate neutrino energy independence of this figure, demonstrating the universal nature of the approximation given in eq. 3 using our definition of ∆ m 2 ee . Next, we want to check that this two-flavor expression reproduces the first oscillation minimum at high E (small L) correctly compared to the complete three-flavor picture. The minimum occurs when the derivate of P is zero. We now have a choice: we can define the minimum when dP a /dL = 0 or dP a /dE = 0. Since both θ and ∆ m 2 ee are nontrivial functions of E, the correct option is to use dP a /dL = 0. In order to numerically test the various expressions, we find the location L of the first minimum by solving dP a /dL = 0 for a given E using the full three-flavor expressions. We then convert the (L, E) pair at the first minimum into the corresponding ∆ m 2 ee using ∆ m 2 ee L 4E = π 2 .(33) Next, we compare the difference between this numeric solution and the expressions presented in this paper, eqs. 4, 19, and 23. We also compare to the approximate analytic solution from [16] (HM), see appendix C. This comparison is shown in fig. 4. When determining the minimum from the exact expression, a two-flavor expression using only ∆ m 2 ee will get the ∆m 2 31 and ∆m 2 32 terms correct including matter effect, but will always be off by ∆m 2 21 terms. Thus in fig. 4 we don't include the effect of the 21 term which will affect any two-flavor approximation comparably. We see that for either eq. 6 or eq. 23 the agreement is excellent with relative error < 0.2%. In addition, the two expressions clearly agree with each other to a higher level of precision than is necessary. For the HM expression the agreement is good for anti-neutrinos and in the high energy limit, but is poor in a broad range near the atmospheric resonance for neutrinos. In addition, we have modified the HM expression by taking the absolute value so that the HM expression asymptotically returns to the correct expression past the atmospheric resonance for neutrinos. 4. We show the fractional error (δx/x) of various different ∆ m 2 ee expressions with the precise numerical one determined at the point where dPa/dL = 0, see eq. 33. For the exact numerical expression we ignore the ∆ m 2 21 term as no definition will get it correct. The ee curve uses the formula from eq. 4 and the EE curve uses the formula from eq. 19. The DMP curve uses the zeroth order expressions [10] in the same formula which leads to the simple expression shown in eq. 23. The HM curve uses the expression from [16] and takes the absolute value to get the sign correct for large E, see appendix C. We have fixed ρ = 3 g/cc and assumed the NO. E > 0 corresponds to neutrinos, E < 0 corresponds to antineutrinos, and E = 0 corresponds to the vacuum. We have also compared ∆ m 2 ee with the exact solution including the ∆m 2 21 term and found agreement to better than 1%. B. Analytic Comparison We now analytically estimate the precision of the twoflavor expression, for both the small E (large L) limit and the large E (small L) limit. First, if ∆ m 2 21 \|∆ m 2 ee \| then at the n th oscillation minimum the ratio of the 21 term to the ee term is well approximated by ∆m 2 21 ∆m 2 ee [(2n − 1)π/4] 2 ,(34) as derived in appendix D. For the first (second) oscillation peak this yields an error estimate of < 2% (16%); this two-flavor approach breaks down for n > 5 when the ratio is > 1. The second case is when ∆ m 2 21 \|∆ m 2 ee \|, which occurs away from vacuum (high E, low L), and the ratio of the 21 coefficient to the ee coefficient is c 4 13 sin 2 2 θ 12 sin 2 2 θ 13 = \| U e1 \| 2 \| U e2 \| 2 \| U e3 \| 2 (1 − \| U e3 \| 2 ) ,(35) which is small away from vacuum as desired. In particular, it is < 1 for \|E\| > 1 GeV. See appendix D for details and numerical confirmation of each region. VII. CONCLUSIONS In this paper, we have demonstrated that ∆ m 2 ee ≡ m 2 3 − ( m 2 1 + m 2 2 ) − [m 2 3 − (m 2 1 + m 2 2 )] + ∆m 2 ee (36) ≈ ∆m 2 ee (cos 2θ 13 − a/∆m 2 ee ) 2 + sin 2 2θ 13 , is the matter generalization of vacuum ∆m 2 ee that has been widely used by the short baseline reactor experiments Daya Bay and RENO and will be precisely measured (< 1%) in the medium baseline JUNO experiment. The exact and approximate expressions in the above equation differ by no more than 0.06%. Another natural choice called ∆ m 2 EE is numerically very close to ∆ m 2 ee but does not provide the ability to simply rewrite the eigenvalues as ∆ m 2 ee does. For ν e disappearance in matter the position of the first oscillation minimum, for fixed neutrino energy E, is given by L = 2πE ∆ m 2 ee ,(37) and the depth of the minimum is controlled by sin 2 2 θ 13 ≈ sin 2 2θ 13 ∆m 2 ee ∆ m 2 ee 2 ,(38) ≈ sin 2 2θ 13 (cos 2 2θ 13 − a/∆m 2 ee ) 2 + sin 2 2θ 13 . This two-flavor approximate expression is not only simple and compact, but it is precise to within < 1% precision at the first oscillation minimum 8 . The combination of ∆ m 2 ee and ∆ m 2 21 is very powerful for understanding the effects of matter on the eigenvalues and the mixing angles of the neutrinos. In this article we have illuminated the exact nature of ∆ m 2 ee and ∆ m 2 21 which were extensively used in DMP [10,12]. + 3 4 (∆ m 2 ee − ∆m 2 ee ) . (B5) The remaining two oscillation parameters, θ 23 = θ 23 and δ = δ, remain unchanged in this approximation. We note that for each parameter above x provides an excellent approximation for x. We also note two additional useful expressions, An alternate approximate expression was previously provided in [16], the expression from that paper is sin 2 θ 13 = sin 2θ 13 ∆m 2 ee ∆ m 2 ee ,(B6)∆ m 2 ee,HM = (1 − r A )∆m 2 ee + r A 2s 2 13 1 − r A ∆m 2 31 − s 2 12 ∆m 2 21 ,(C1) where r A ≡ a/∆m 2 31 . This expression clearly has a pole at a = ∆m 2 31 which is the atmospheric resonance for neutrinos. In addition, past the resonance, for a > ∆m 2 31 , the sign is incorrect as ∆ m 2 ee,HM < 0 for the NO. Thus we take the absolute value in our numerical studies. In fig. 2 of [16], the author compared eq. C1 with the minimum obtained via solving dP a /dE = 0 whereas we have argued in section VI that a better comparison is obtained by solving dP a /dL = 0 for fixed E. Appendix D: Precision in Different Ranges In this appendix we further expand upon the discussion in subsection VI B. 3) compared to the full three-flavor expression (P3 from eq. 17) in matter is shown in the solid curves for the first several oscillation minima. The dashed lines are the simple approximation from eq. 34. As expected eq. 34 performs well near vacuum at \|E\| few GeV. The exact three-flavor expression in matter from eq. 17 can be written as, 1 − P a = sin 2 2θ 13 ∆m 2 ee ∆ m 2 ee 2 sin 2 ∆ ee + C(E)c 4 13 sin 2 2 θ 12 sin 2 ∆ 21 , where C(E) 1 contains the correction between the first and second term. For the two-flavor approximation to be valid, the 21 term, C(E)c 4 13 sin 2 2 θ 12 sin 2 ∆ 21 must be small compared to the two-flavor ee term, sin 2 2 θ 13 sin 2 ∆ ee . As in section VI B, we consider two cases. First, if ∆ m 2 21 \|∆ m 2 ee \| then at the n th oscillation minimum the ratio R 1 of the 21 term to the ee term is sin 2 ∆ 21 ∆ 2 21 , where the approximation uses the DMP zeorth order expression, the θ 13 ≈ θ 13 approximation of eq. B6, and s 2 13 ≈ ∆m 2 21 /∆m 2 ee . The C(E) term contains the effect of combining the ∆ 31 and ∆ 32 terms and is just under one within a few GeV of the vacuum. Since all of the terms in the right square bracket are < 1, R 1 ≈ ∆m 2 21 ∆m 2 ee [(2n − 1)π/4] 2 .(D2) We numerically confirmed that eq. 34 is correct to within ∼ 10% near vacuum as shown in fig. 5. 1. See eq. 35 in the text. We also show the ratio of the mass squared differences in matter in red. The second case is when ∆ m 2 21 \|∆ m 2 ee \|, which occurs away from vacuum. In this case we compare the ratio R 2 of the coefficients which is R 2 = c 4 13 sin 2 2 θ 12 sin 2 2 θ 13 = \| U e1 \| 2 \| U e2 \| 2 \| U e3 \| 2 (1 − \| U e3 \| 2 ) .(D3) Away from vacuum, θ 12 π/2 (0) for neutrinos (antineutrinos) (see e.g. fig. 1 of [10]) which makes the numerator of R 2 very small. The remaining part is 1/(4 tan 2 θ 13 ). This part is large only when θ 13 → 0. Since θ 12 → 0 faster than θ 13 , we always have R 2 1 as desired. See fig. 6 for a numerical verification that R 2 is small away from the vacuum. ∆ m 2 2ee ≡ m 2 3 − ( m 2 1 + m 2 2 ) FIG. 1 . 1Upper panel: the eigenvalues as a function of energy for ρ = 3 g/cc and the NO. Positive energies refer to neutrinos while negative energies refer to anti-neutrinos; E = 0 refers to the vacuum. The νe content of each eigenvalue is shaded in orange, while the νµ and ντ content is shaded in black. The magenta (cyan) arrows indicate how ∆ m 2 ee (∆ m 2 21) changes with energy. Lower panel: the νe content of each mass eigenstate, \| Uei\| 2 , as a function of neutrino energy. 2 2Explicitly, in the flavor basis we have that 2E tr(H) = tr(U M U † + A) = tr(U U † M ) + tr(A) = ∆m 2 31 + ∆m 2 21 + a. In the matter basis the trace of the Hamiltonian is 2E tr(H) = tr( U M U † ) = tr( U U † M ) = i m 2 i . where a 12 ≡ 12(a + ∆m 2 ee − ∆ m 2 ee )/2 and cos 2 ( θ 13 − θ 13 ) = ∆ m 2 ee + ∆m 2 ee − a cos 2θ 13 2∆ m 2 ee. 2 ee − ∆ m 2 ee , FIG. 2 . 2The fractional difference between the two expressions is shown in the red solid curve. The green dashed curve shows the difference through first order, and the blue dash-dotted curve shows the difference through second order. Note that at zeroth order in DMP the difference is exactly zero. DMP2 is hard to see at it is on top of exact. FIG. 3 . 3Here we demonstrate the validity of the two-flavor approximation by plotting eq. 32 showing the expected sinusoidal dependence. Here Pa is the exact three flavor νe disappearance probability. Note the small deviations due to the 21 term that grow as the phase \| ∆ee\| increases for small energies. FIG. 4. We show the fractional error (δx/x) of various different ∆ m 2 ee expressions with the precise numerical one determined at the point where dPa/dL = 0, see eq. 33. For the exact numerical expression we ignore the ∆ m 2 21 term as no definition will get it correct. The ee curve uses the formula from eq. 4 and the EE curve uses the formula from eq. 19. The DMP curve uses the zeroth order expressions [10] in the same formula which leads to the simple expression shown in eq. 23. The HM curve uses the expression from [16] and takes the absolute value to get the sign correct for large E, see appendix C. We have fixed ρ = 3 g/cc and assumed the NO. E > 0 corresponds to neutrinos, E < 0 corresponds to antineutrinos, and E = 0 corresponds to the vacuum. sin 2 θ 12 = cos( θ 12 − θ 12 ) sin 2θ 12 FIG. 5 . 5The error of the two-flavor approximation (P2 from eq. FIG. 6 . 6An approximation of the size of the 21 term (numerator in blue, denominator in green) away from vacuum. For \|E\| 5 GeV we see that R2 TABLE I . IThe mass squareds in matter for various limits of a in the NO. See eqs. 5.3-5.4 of The 21 rotation yields ∆ m 2 21 = ∆m 2 21 (cos 2θ 12 − a 12 /∆m 2 21 ) 2 + cos 2 ( θ 13 − θ 13 ) sin 2 2θ 12where we similarly define a 12 ≡ (a + ∆m 2 ee − ∆ m 2 ee )/2. Finally, from eqs. B1 and B3 it is straightforward to show that1/2 , (B3) cos 2 θ 12 = ∆m 2 21 cos 2θ 12 − a 12 ∆ m 2 21 , (B4) ∆ m 2 31 = ∆m 2 31 + 1 4 a + 1 2 (∆ m 2 21 − ∆m 2 21 ) In eq. 38, the exact and second approximation differ in value by no more than 4 × 10 −4 and the fractional difference is smaller than 0.1% except for very large positive values of the energy where the fractional difference is however never larger than 1%. ACKNOWLEDGMENTSWe thank Hisakazu Minakata for comments on an earlier version of this manuscript.where w = ∆m 2 21 + ∆m 2 31 + a , x = ∆m 2 31 ∆m 2 21 + a ∆m 2 31 c 2 13 + ∆m 2 21 (c 2 13 c 2 12 + s 2 13 ) , y = a∆m 2 31 ∆m 2 21 c 2 31 c 2 12 ,Therefore,Using eq. 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[ "The artificial retina for track reconstruction at the LHC crossing rate", "The artificial retina for track reconstruction at the LHC crossing rate" ]	[ "A Abba \nPolitecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly\n", "F Bedeschi \nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n", "M Citterio \nPolitecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly\n", "F Caponio \nPolitecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly\n", "A Cusimano \nPolitecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly\n", "A Geraci \nPolitecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly\n", "P Marino \nScuola Normale Superiore\nPiazza dei Cavalieri 756127PisaItaly\n\nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n", "M J Morello \nScuola Normale Superiore\nPiazza dei Cavalieri 756127PisaItaly\n\nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n", "N Neri \nPolitecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly\n", "G Punzi \nUniversity of Pisa\nLungarno Pacinotti 4356126PisaItaly\n\nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n", "A Piucci \nUniversity of Pisa\nLungarno Pacinotti 4356126PisaItaly\n", "L Ristori \nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n\nWilson and Kirk Rd60510Fermilab, BataviaILUSA\n", "F Spinella \nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n", "S Stracka \nScuola Normale Superiore\nPiazza dei Cavalieri 756127PisaItaly\n\nINFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly\n", "D Tonelli \nCERN 385 Route de Meyrin\nGenevaSwitzerland\n" ]	[ "Politecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "Politecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly", "Politecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly", "Politecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly", "Politecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly", "Scuola Normale Superiore\nPiazza dei Cavalieri 756127PisaItaly", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "Scuola Normale Superiore\nPiazza dei Cavalieri 756127PisaItaly", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "Politecnico\nINFN-Milano\nVia Celoria 1620133MilanoItaly", "University of Pisa\nLungarno Pacinotti 4356126PisaItaly", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "University of Pisa\nLungarno Pacinotti 4356126PisaItaly", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "Wilson and Kirk Rd60510Fermilab, BataviaILUSA", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "Scuola Normale Superiore\nPiazza dei Cavalieri 756127PisaItaly", "INFN-Pisa\nL.go Bruno Pontecorvo 356127PisaItaly", "CERN 385 Route de Meyrin\nGenevaSwitzerland" ]	[ "Nuclear Physics B Proceedings Supplement" ]	We present the results of an R&D study for a specialized processor capable of precisely reconstructing events with hundreds of charged-particle tracks in pixel and silicon strip detectors at 40 MHz, thus suitable for processing LHC events at the full crossing frequency. For this purpose we design and test a massively parallel pattern-recognition algorithm, inspired to the current understanding of the mechanisms adopted by the primary visual cortex of mammals in the early stages of visual-information processing. The detailed geometry and charged-particle's activity of a large tracking detector are simulated and used to assess the performance of the artificial retina algorithm. We find that highquality tracking in large detectors is possible with sub-microsecond latencies when the algorithm is implemented in modern, high-speed, high-bandwidth FPGA devices.	10.1016/j.nuclphysbps.2015.09.434	[ "https://arxiv.org/pdf/1411.1281v1.pdf" ]	58,522,589	1411.1281	965b909d0776d638b2f96af16b9e9828bab38816	The artificial retina for track reconstruction at the LHC crossing rate 2014 A Abba Politecnico INFN-Milano Via Celoria 1620133MilanoItaly F Bedeschi INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly M Citterio Politecnico INFN-Milano Via Celoria 1620133MilanoItaly F Caponio Politecnico INFN-Milano Via Celoria 1620133MilanoItaly A Cusimano Politecnico INFN-Milano Via Celoria 1620133MilanoItaly A Geraci Politecnico INFN-Milano Via Celoria 1620133MilanoItaly P Marino Scuola Normale Superiore Piazza dei Cavalieri 756127PisaItaly INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly M J Morello Scuola Normale Superiore Piazza dei Cavalieri 756127PisaItaly INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly N Neri Politecnico INFN-Milano Via Celoria 1620133MilanoItaly G Punzi University of Pisa Lungarno Pacinotti 4356126PisaItaly INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly A Piucci University of Pisa Lungarno Pacinotti 4356126PisaItaly L Ristori INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly Wilson and Kirk Rd60510Fermilab, BataviaILUSA F Spinella INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly S Stracka Scuola Normale Superiore Piazza dei Cavalieri 756127PisaItaly INFN-Pisa L.go Bruno Pontecorvo 356127PisaItaly D Tonelli CERN 385 Route de Meyrin GenevaSwitzerland The artificial retina for track reconstruction at the LHC crossing rate Nuclear Physics B Proceedings Supplement 00201410.1016/S0168-9002(00)00676-8}Pattern recognition, Trigger algorithms We present the results of an R&D study for a specialized processor capable of precisely reconstructing events with hundreds of charged-particle tracks in pixel and silicon strip detectors at 40 MHz, thus suitable for processing LHC events at the full crossing frequency. For this purpose we design and test a massively parallel pattern-recognition algorithm, inspired to the current understanding of the mechanisms adopted by the primary visual cortex of mammals in the early stages of visual-information processing. The detailed geometry and charged-particle's activity of a large tracking detector are simulated and used to assess the performance of the artificial retina algorithm. We find that highquality tracking in large detectors is possible with sub-microsecond latencies when the algorithm is implemented in modern, high-speed, high-bandwidth FPGA devices. Introduction Higher LHC energy and luminosity increase the challenge of data acquisition and event reconstruction in the LHC experiments. The large number of interactions for bunch crossing (pile-up) greatly reduces the discriminating power of usual signatures, such as the high transverse momentum of leptons or the high transverse missing energy. Therefore real-time track reconstruction could prove crucial to quickly select potentially interesting events for higher level of processing. Performing such a task at the LHC crossing rate is a major challenge because of the large combinatorial and the size of the associated information flow and requires unprecedented massively parallel pattern-recognition algorithms. For this purpose we design and test a neurobiology-inspired pattern-recognition algorithm well suited for such a scope: the artificial retina algorithm. An artificial retina algorithm The original idea of an artificial retina tracking algorithm was inspired by the mechanism of visual receptive fields in the mammals eye [1]. Experimental studies have shown neurons tuned to recognize a specific shape on specific region of the retina ("receptive field") The strength of the response of each neuron to a stimulus is proportional to how close the shape of the stimulus is to the shape for which the neuron is tuned to. All neurons react to a stimulus, each with different strength, and the brain obtains a precise information of the received stimulus performing some sort of interpolation between the responses of neurons. The retina concepts can be geared toward track reconstruction. Assuming a generic tracking detector, the 3D charged particle trajectory is described by five parameter. The space of track parameters are discretized into cells, which mimic the receptive fields of the retina. The center of each cell identifies a track in the detector space, that intersects detector layers in spatial points that we call receptors. For each incoming hit, the algorithm computes the excitation intensity, i. e. the response of the receptive field, of each cell as follows: R = k, r exp − s 2 kr 2σ 2 ,(1) where s kr is the distance, on the layer k, between the hit and the receptor r. σ is a parameter of the retina algorithm, that can be adjusted to optimize the sharpness of the response of the receptors. After all hits are processed, tracks are identified as local maxima over a threshold in the space of track parameters. Averaging over nearby cells of the identified maximum provides track parameters with a significant better resolution than the available cell granularity. Retina algorithm in a real HEP experiment To evaluate the performances and the robustness of the algorithm in a real HEP detector, we focus on the upgraded LHCb detector. The upgraded LHCb detector [2], a single-arm spectrometer covering the pseudorapidity range 2 < η < 5, is a major upgrade of the current LHCb experiment, and it will run at the instantaneous luminosity of 3×10 33 cm −2 s −1 , with a beam energy of 7 TeV. All the sub-detectors will be read out at 40 MHz, allowing a complete event reconstruction at the LHC crossing rate. To benchmark the retina algorithm, we decided to perform the first stage of the upgraded LHCb detector tracking reconstruction [3], using the information of only two sub-detectors, placed upstream of the magnet: the vertex locator (VELO), a silicon-pixel detector [4] and the upstream tracker (UT) [5], a silicon microstrip detector. We used the last eight forward pixel layers of the VELO and the two axial layers of the UT. We arbitrarily chose to parametrize tracks with the following parameters: (u, v, d, z 0 , k). (u, v) are the the spatial coordinate of the intersection point of the track with a "virtual plane" perpendicular to the z-axis, placed to a distance z vp from the origin of the coordinate system. d is the signed transverse impact parameter, z 0 is the z-coordinate of the point of the closest approach to the z-axis. k is the signed curvature in the bending plane ( B = Bŷ). The detector geometry and magnetic field (negligible in the VELO and about 0.05 T in the UT), allow us to use only the (u, v) parameters to perform the pattern recognition, since the 5D tracks' parameters space can be factorized into (u, v) ⊗ (d, z 0 , k). Thus (u, v) are the "main" parameters where pattern recognition is performed, whereas (d, z 0 , k) are treated as "perturbation" of the main parameters (u, v) [6,7]. To evaluate the performances of the algorithm, we develope a detailed C++ simulation of the retina algorithm [8] able to process simulated events, interfaced with the default LHCb simulation. We discretize the main (u, v)-subspace into 22 500 cells, a granularity O(100) larger than the maximum expected number of tracks in a typical upgraded LHCb event. Generic collisions samples from the default LHCb simulation are used to assess the performances of the retina algorithm. The generic collisions are generated with beam energy of 7 TeV, and luminosities up to L = 3 × 10 33 cm −2 s −1 . A typical response of the retina algorithm is shown in fig. 1, where several clusters are clearly identifiable, and most of them reconstructed as tracks. All hits from simulated events from the default LHCb simulation are sent and processed by the retina. In order to evaluate tracking performances we considered only tracks in a region of the (u, v)-plane where they have full acceptance on the chosen layer configuration. In addition, cuts close to the ones applied to calculate the offline efficiency [3] are applied. For instance, we required at least three hits on VELO layers and two hits on UT layers, and also a momentum p > 3 GeV/c and a transverse momentum p T > 200 MeV/c. Tracks satisfying all these requirements are defined as reconstructable, and the tracking efficiency is defined as the number of reconstructed tracks over the number of reconstructable tracks. The efficiency of the retina is reported in figure 2 as function of p T , d parameters. We also report the efficiency of the offline LHCb track reconstruction algorithm, performing the same task as the retina [6]. The retina algorithm shows very high efficiencies in reconstructing tracks, about 95% for generic tracks, which is comparable to the offline tracking algorithm. The fake track rate is 8% at L = 2 × 10 33 cm −2 s −1 and 12% at L = 3 × 10 33 cm −2 s −1 , slightly higher than the fake rate of the offline algorithm. We also estimate the efficiency of the retina algorithm in recostruncting signal tracks from some benchmark decay modes, such as B 0 s → φφ, D * ± → D 0 π ± and B 0 → K * µµ for L = 2 × 10 33 cm −2 s −1 . The efficiency for these channels is about 97-98%. Resolutions on tracking parameters determined by the retina are comparable with those of the offline reconstruction. Hardware implementation To fully exploit the high-grade of parallelism of the algorithm, we developed the retina algorithm into FPGA chips [9]. The logic is implemented in VHDL language; detailed logic-gate placement and simulation on the high-bandwidth Altera Stratix V device model 5SGXEA7N2F45C2ES is achieved using Altera's proprietary software. Figure 3 shows an overview of the devices architetture. To achieve an efficient distribution of the hit information coming from the detector layers to the cells of the space of track parameters, we design an intelligent information delivery system that routes each hit in parallel to all and only those cells for which such hit is likely to contribute a significant weight. The switching network completes its processing in 30 clock cycles. Each cell in the tracks parameter space is defined as a logic module, the engine. The engine is implemented as a clocked pipeline, that calculate the excitations. The engine process takes 17 clock cycles. At the end, the logic that identifies the center-of-mass in the space of track parameters take 11 cycle of clock cycles along with another 10 cycles for fanout. With a clock frequencies of 350 MHz, the latency for reconstructing Figure 3. Illustration of the device's architecture tor to the processing engines that calculate the excitations. The need for a 40 MHz throughput with a flow of several Tbit/s of input data make this a nontrivial task. The other challenge is performing pattern recognition quickly enough to remain within the harsh latency constraints. Any solutions to either issues necessarily depends on the actual geometry of the tracking layout. The peculiar LHCb geometry with straight-line tracks traversing the vertex detector before being curved by the magnetic field in the downstream tracking stations allows an efficient solution for both challenges. Tracking performance sufficient for triggering can be achieved by doing a pattern recognition in a volume where the magnetic field is weak. In this regime, contiguous detectors hits correspond to contiguous regions in track parameter space, which simplifies significantly the switching task. A mapping between detector hits and parameters associated with tracks produced in the collisions is performed using simulation. The result is used to associate a "zip-code" with each possible detector hit, which is used by the nodes of the switching network to properly route the hit. The LHCb geometry allows factorizing pattern recognition into two steps. First, tracks are assumed to be straight lines originated from a common nominal interaction point and track-finding is performed in a two-dimensional primary plane transverse to the beam, whose intersection which each track identifies the track's two parameters. Then, the determination of the momentum and actual spatial origin of the charged particle are treated as small perturbations of the primary two-dimensional track. Figure 3 illustrates the architecture of the retina track processor. Switching network We design an intelligent and economical information-delivery system that routes each hit in parallel to all and only those engines for which such hit is likely to contribute a significant weight. Each online tracks is less than 0.5 µs. Each Stratix V can host up to 900 engines leaving approximately 25% of logic available for other uses, including a 15% of switching and the logic for center-of-mass calculation [6]. Conclusions We showed that high-quality tracking in large LHC detectors is possible at a 40 MHz event rate with subµs latencies, when appropriate parallel algorithms are used in conjunction with current high-end FPGA device. This opens the interesting possibility of designing high-rate experiments where track reconstruction happen transparently as part of the detector readout. Figure 1 : 1Left: response of the retina algorithm (only the (u, v)-plane, where the pattern recognition is made) to a generic collision from the default LHCb simulation, with instantaneous luminosity of L = 2 × 10 33 cm −2 s −1 . The hole at the center of the figure is due to the physical hole in the VELO layers. Right: a zoom of the retina response. Figure 2 : 2Tracking reconstruction efficiency of the retina algorithm (in red) and of the offline VELO+UT algorithm (in blue), as function of: (a) p T , (b) d. The distribution of the considered parameter is, also, reported in black. Luminosity of L = 3 × 10 33 cm −2 s −1 . Figure 3 : 3Illustration of the devices architetture. An artificial retina for fast track finding. L Ristori, {10.1016/S0168-9002(00)00676-8}Nuclear Instruments and Methods. 4531-2L. Ristori, An artificial retina for fast track finding, Nuclear In- struments and Methods 453 (1-2) (2000) 425 -429. doi:{http: //dx.doi.org/10.1016/S0168-9002(00)00676-8}. Framework TDR for the LHCb Upgrade. GenevaCERNTech. Rep. LHCb-TDR-12LHCb Collaboration, Framework TDR for the LHCb Upgrade: Technical Design Report, Tech. Rep. LHCb-TDR-12, CERN, Geneva (Apr 2012). VeloUT tracking for the LHCb Upgrade. E Bowen, B Storaci, GenevaCERNTech. Rep. CERN-LHCb-PUB-2013-023Bowen E., Storaci B., VeloUT tracking for the LHCb Upgrade, Tech. Rep. CERN-LHCb-PUB-2013-023, CERN, Geneva (Apr 2014). LHCb VELO Upgrade Technical Design Report. GenevaCERNTech. Rep. LHCB-TDR-013LHCb Collaboration, LHCb VELO Upgrade Technical De- sign Report, Tech. Rep. LHCB-TDR-013, CERN, Geneva (Nov 2013). LHCb Tracker Upgrade Technical Design Report. GenevaCERNTech. Rep. LHCB-TDR-015LHCb Collaboration, LHCb Tracker Upgrade Technical Design Report, Tech. Rep. LHCB-TDR-015, CERN, Geneva (Feb 2014). A specialized track processor for the LHCb upgrade. A Abba, LHCb-PUB-2014-026. CERN-LHCb-PUB- 2014-026GenevaCERNTech. Rep.A. Abba et al., A specialized track processor for the LHCb upgrade, Tech. Rep. LHCb-PUB-2014-026. CERN-LHCb-PUB- 2014-026, CERN, Geneva (Mar 2014). A specialized processor for track reconstruction at the LHC crossing rate. A Abba, Journal of Instrumentation. 9099001A. Abba et al., A specialized processor for track reconstruction at the LHC crossing rate, Journal of Instrumentation 9 (09) (2014) C09001. Simulation and performance of an artificial retina for 40 MHz track reconstruction. P Marino, arXiv:1409.0898P. Marino et al., Simulation and performance of an artificial retina for 40 MHz track reconstruction (2014). arXiv:1409.0898. The artificial retina processor for track reconstruction at the LHC crossing rate. D Tonelli, arXiv:1409.1565D. Tonelli et al., The artificial retina processor for track recon- struction at the LHC crossing rate (2014). arXiv:1409.1565.	[]
[ "Universality at the Edge for Unitary Matrix Models", "Universality at the Edge for Unitary Matrix Models" ]	[ "M Poplavskyi [email protected] \nMathematics Division\nB. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkivUkraine\n" ]	[ "Mathematics Division\nB. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkivUkraine" ]	[]	Using the results on the 1/n-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with n varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.	null	[ "https://arxiv.org/pdf/1306.6892v1.pdf" ]	119,673,639	1306.6892	75b0203725d1d91fd1ced16a8ef28c1478da7ebc	Universality at the Edge for Unitary Matrix Models 28 Jun 2013 M Poplavskyi [email protected] Mathematics Division B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave61103KharkivUkraine Universality at the Edge for Unitary Matrix Models 28 Jun 2013Received August 5, 2012unitary matrix modelslocal eigenvalue statisticsuniver- salitypolynomials orthogonal on the unit circle Mathematics Subject Classification 2010: 15B5242C05 Using the results on the 1/n-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with n varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval. Introduction We study a class of random matrix ensembles known as unitary matrix models. These models are defined by the probability law p n (U) dµ n (U) = Z −1 n,2 exp −nTrV U + U * 2 dµ n (U) ,(1.1) where U = {U jk } n j,k=1 is an n × n unitary matrix, µ n (U) is the Haar measure on the group U(n), Z n,2 is the normalization constant, and V : [−1, 1] → R is a continuous function called the potential of the model. Denote e iλ j the eigenvalues of the unitary matrix U. The joint probability density of λ j , corresponding to (1.1), is given by (see [1]) p n (λ 1 , . . . , λ n ) = 1 Z n 1≤j<k≤n e iλ j − e iλ k 2 exp −n The random matrix theory deals with several asymptotic regimes of the eigenvalue distribution. The global regime is centred around the weak convergence of NCM. It is well known (see e.g. [2]) that for some smooth conditions for the potential V there exists a measure N ∈ M 1 ([−π, π]) with a compact support σ such that N n converges to N in probability . Let p (n) l (λ 1 , . . . , λ l ) = p n (λ 1 , . . . , λ l , λ l+1 , . . . , λ n ) dλ l+1 . . . dλ n be the l -th marginal density of p n . The local regime of eigenvalue distribution describes the asymptotic behaviour of marginal densities when their arguments are on the distances of order of the typical distance between eigenvalues. The universality conjecture of marginal densities was suggested by Dyson (see [3]) in the early 60s. He supposed that their asymptotic behaviour depends only on the ensemble symmetry group and does not depend on other ensemble parameters. First rigorous proofs for the hermitian matrix models with non-quadratic V appeared only in the 90s. The case of general V which is locally C 3 function was studied in [4]. The case of real analytic V was studied in [5], where the asymptotic behaviour of orthogonal polynomials was obtained. For the unitary matrix models the bulk universality was proved for V = 0 (see [3]), and for the locally C 3 functions (see [6]). The edge universality was proved only in the case of the linear V (see [7]). In the present paper we prove the universality conjecture for UMM with a smooth potential V in the case of one-interval support σ of the limiting NCM. It was proved in [2] that the limiting measure can be obtained as a unique minimizer of the functional E[m] = π −π V (cos λ)m(dλ) − π −π log e iλ − e iµ m(dλ)m(dµ) in the class of unit measures on the interval [−π, π] (see [8] for the existence and properties of the solution). It is well known, in particular, that for smooth V ′ the equilibrium measure has a density ρ which is uniquely defined by the condition that the function u (λ) = V (cos λ) − 2 σ log e iλ − e iµ ρ (µ) dµ (1.3) takes its minimum value if λ ∈ σ = supp ρ. From this condition in the case of differentiable V one can obtain the following integral equation for the equilibrium density ρ: (V (cos λ)) ′ = v.p. σ cot λ − µ 2 ρ (µ) dµ, for λ ∈ σ. (1.4) We also use the weak convergence of the first marginal density ρ n (λ) = p (n) 1 proved in [2]. Proposition 1.1 For any φ ∈ H 1 (−π, π) , φ (λ) ρ n (λ) dλ − φ (λ) ρ (λ) dλ ≤ C φ 1/2 2 φ ′ 1/2 2 n −1/2 ln 1/2 n, (1.5) where · 2 denotes L 2 norm on [−π, π]. We consider here the case of one interval σ. Our main conditions on the potential V are Condition C1. The support σ of the equilibrium measure is a single symmetric subinterval of the interval [−π, π], i.e., σ = [−θ, θ] , with θ < π. R e m a r k 1.2 In fact, there is one more possibility to have one-interval σ. Another case is some left symmetric arc of the circle, i.e., [π − θ, π + θ]. In this case we replace V (cos x) in (1.2) by V (cos (π − x)). This replacement will rotate all eigenvalues on the angle π and we will have the support from condition C1. Condition C2. The equilibrium density ρ has no zeros in (−θ, θ) and ρ (λ) ∼ C \|λ ∓ θ\| 1/2 , for λ → ±θ, and the function u (λ) of (1.3) attains its minimum if and only if λ belongs to σ. R e m a r k 1.3 From this condition we obtain the necessary scaling for marginal densities at the edge of σ ∆ ρ (λ) dλ ∼ n −1 ⇒\| ∆ \|∼ n −2/3 , (1.6) hence the typical distance between eigenvalues is of order n −2/3 . Condition C3. V (cos λ) possesses four bounded derivatives on σ ε = [−θ − ε, θ + ε]. The following simple representation of ρ plays an important role in our asymptotic analysis (see [9]) Proposition 1.4 Under conditions C1-C3 the density ρ has the form ρ (λ) = 1 4π 2 χ (λ) P (λ) 1 σ , where χ (λ) = \|cos λ − cos θ\|, P (λ) = θ −θ (V (cos µ)) ′ − (V (cos λ)) ′ sin (µ − λ) /2 dµ χ (µ) . (1.7) The main result of the paper is the following theorem Theorem 1.5 Consider the unitary matrix ensemble of the form (1.1), satisfying conditions C1-C3 above. Then • for the endpoints θ ± = ±θ and any positive integer l the rescaled marginal density γn 2/3 −l n! (n − l)! p (n) l θ ± ± t 1 /γn 2/3 , . . . , θ ± ± t l /γn 2/3 (1.8) with the sign ± corresponding to θ ± and γ = tan 1/3 θ/2 P (θ) 4π 2/3 converges weakly, as n → ∞, to det {Q Ai (t j , t k )} l j,k=1 , where Q Ai (x, y) is the Airy kernel Q Ai (x, y) = Ai (x) Ai ′ (y) − Ai ′ (x) Ai (y) x − y ; (1.9) • if ∆ ⊂ R is a finite union of disjoint bounded intervals and E n (∆ n ) = P (∆ n does not contain eigenvalues of U) is the hole probability for ∆ n = θ ± ± ∆/γn 2/3 , then lim n→∞ E n (∆ n ) = 1 + ∞ l=1 (−1) l l! ∆ dt 1 . . . dt l det {K (t j , t k )} l j,k=1 , (1.10) i.e., the limit is the Fredholm determinant of the integral operator K ∆ defined by the kernel K on the set ∆. The paper is organized as follows. In Section 2 we give a brief outline of the orthogonal polynomials method. In Section 3 we prove the main Theorem 1.5 using some technical results. These results are proved in Section 4. Orthogonal Polynomials We prove Theorem 1.5, using the orthogonal polynomials technique. This method is based on a simple observation. Joint eigenvalue distribution (1.2) is expressed in terms of the Vandermonde determinant of powers of e iλ k , and therefore by the properties of determinants, can be written in terms of the determinant of any system of linearly independent trigonometric polynomials. We consider a system of polynomials orthogonal on the unit circle(OPUC) with a varying weight. Let w n (λ) = e −nV (cos λ) be the weight function for the system of polynomials. Then the system can be obtained from e ikλ ∞ k=0 if we use the Gram-Schmidt procedure in L (n) := L 2 ([−π, π] , w n (λ)) with the inner product f, g n = π −π f (x) g (x)w n (x) dx. Hence, for any n we get the system of trigonometric polynomials P which are orthonormal in L (n) . One can see from the Szegö's condition that the system P is not complete in L (n) . To construct the complete system one should also include polynomials with respect to e −iλ . Thus, following [10], we introduce the Laurent polynomials χ (n) 2k (λ) = e ikλ P (n) 2k (−λ) , χ (n) 2k+1 (λ) = e −ikλ P (n) 2k+1 (λ) . (2.1) It is easy to check (see, e.g., [10,11]) that the system χ (n) k (λ) ∞ k=0 is an orthonormal basis in L (n) . Moreover, it was proved in [10] that the functions χ (for the definition and properties see [9]). Denote by Θ (n) j = −α (n) j ρ (n) j ρ (n) j α (n) j , M (n) = E 1 ⊕ Θ (n) 2 ⊕ Θ (n) 4 ⊕ ..., L (n) = Θ (n) 1 ⊕ Θ (n) 3 ⊕ Θ (n) 5 ⊕ ..., C (n) = M (n) L (n) . (2.2) From the properties of the Verblunsky coefficients one can see that the semi-infinite matrices M (n) and L (n) are symmetric, three diagonal and unitary. C (n) is also a unitary five diagonal matrix. Finally, using the above notations, we can write the recurrence relations as e iλ − − → χ (n) = C (n) − − → χ (n) . Hence, C (n) is a matrix presentation of the multiplication operator by e iλ in the basis χ (n) k (λ) ∞ k=0 . The main advantage of the orthogonal polynomials technique is the determinant formulas which can be obtained in the same way as in [1], n! (n − l)! p (n) l (λ 1 , . . . , λ l ) = det K (n) n (λ j , λ k ) l j,k=1 , (2.3) where K (n) m (λ, µ) = m−1 k=0 χ (n) k (λ) χ (n) k (µ)w 1/2 n (λ) w 1/2 n (µ) (2.4) is the reproducing kernel of the system χ . Similarly to [12], the weak convergence of the kernel K (n) n to K as n → ∞ will prove Theorem 1.5. Proof of Theorem 1.5 To prove the weak convergence of the reproducing kernel (2.4), we use the lemma (see [12]) Lemma 3.1 Consider the sequence of functions K n : R × R → C and define for ℑζ, ξ = 0, F n (ζ, ξ) = ℑ 1 x − ζ ℑ 1 y − ξ \|K n (x, y)\| 2 dxdy. (3.1) Assume that there exists F (ζ, ξ) of the form F (ζ, ξ) = ℑ 1 x − ζ ℑ 1 y − ξ \|K (x, y)\| 2 dxdy,(3. 2) with K bounded uniformly in each compact in R 2 and such that for any fixed A > 0 uniformly on the set Ω A = {ζ, ξ : 1 ≤ ℑζ, ℑξ ≤ A, \|ℜζ, ℜξ\| ≤ A} (3.3) we have \|F n (ζ, ξ) − F (ζ, ξ)\| ≤ ε n , ε n → 0, as n → ∞. (3.4) Then for any intervals I 1 , I 2 ⊂ R lim n→∞ I 1 dx I 2 dy \|K n (x, y)\| 2 = I 1 dx I 2 dy \|K (x, y)\| 2 . The lemma helps to prove the convergence of \|K n \| 2 to \|K\| 2 . Similarly, we can check the convergence of K n (t 1 , t 2 ) K n (t 2 , t 3 ) . . . K n (t l , t 1 ) for any l ∈ N. To prove the second part of Theorem 1.5, we use another proposition from [12]. Proposition 3.2 Let ∆ ⊂ R be a system of disjoint intervals as in Theorem 1.5 and let K n : L 2 (∆) → L 2 (∆) be a sequence of positive definite integral operators with kernels K n (x, y) and K : L 2 (∆) → L 2 (∆) a positive definite integral operator with kernel K (x, y), such that for any l ∈ N, det {K n (t j , t k )} l j,k=1 → det {K (t j , t k )} l j,k=1 weakly as n → ∞. Assume also that for any ∆ there exists C ∆ such that ∆ K n (s, s) ds ≤ C ∆ . (3.5) Then, for the Fredholm determinants of K n and K we have lim n→∞ det (1 − K n ) = det (1 − K) . We are going to use Lemma 3.1 for the scaled reproducing kernel of the system of OPUC. Let K n (x, y) = n −2/3 K (n) n θ + xn −2/3 , θ + yn −2/3 1 \|x,y\|≤c θ n 2/3 (3.6) for some small enough θ-dependent constant c θ . This will be sufficient in view of the following lemma (the analogue of Theorem 11.1.4, [13]) Lemma 3.3 Let the model (1.1) satisfy conditions C1-C3. Then, for any n-independent ε > 0, there exists a constant d ε > 0 such that σ c ε K (n) n (λ, λ) dλ ≤ Ce −ndε . Since the polynomials χ (n) k are functions of e iλ , it is more convenient to define a little bit different from (3.1) transformation and estimate the difference between it and (3.1). Hence, we consider the following transformation: F n (z, w) = n −4/3 [−π,π] G (z − λ) G (w − µ) K (n) n (λ, µ) 2 dλdµ, (3.7) with G (z) = ℜg (z) , and g (z) = 1 + e iz 1 − e iz (3.8) being the analogues of the Poisson and the Herglotz transformations. Proposition 3.4 It follows from the definition of g (z) that g (z) = i cot z 2 , g (z − λ) = e iλ + e iz e iλ − e iz . For z = x + iy we have g (x + iy) = i sin x + sinh y cosh y − cos x , hence g (z) = −g (z). And for G (z) we get G (x + iy) = sinh y cosh y − cos x , G (z − λ) = ℑ cot λ − z 2 . Moreover, G (z) is a Nevanlinna function and \|g (z)\| 2 = −1 + 2 coth ℑz · G (z) . (3.9) The difference between the new transformation and the old one can be estimated in the following way: Proposition 3.5 Let z = θ + ζn −2/3 and w = θ + ξn −2/3 with \|ζ\| , \|ξ\| ≤ c θ n −2/3 and ℑζ, ℑξ ≥ 1. Then, \|F n (z, w) − 4F n (ζ, ξ)\| ≤ Cn −1/6 (\|F n (z, w)\| + 1) . (3.10) The next step is to prove the convergence of F n (z, w) to the transformation F (3.2) of the Airy kernel Q Ai (1.9). F can be calculated in terms of the Airy functions, thus we are concentrated on the calculations of F n . First, using the properties of CMV matrices, we present F n (z, w) in terms of the "resolvent" of C (n) . After that we use the asymptotic behaviour of the Verblunsky coefficients, obtained in [9], to get an approximation of the "resolvent". The approximation will be given in terms of the Airy functions. Then we will estimate the error of the "resolvent" approximation and prove the uniform bound (3.4). We start with a simple corollary from the spectral theorem and Proposition 3.4. Proposition 3.6 Let g (n) (z) = C (n) + e iz C (n) − e iz −1 , be the "resolvent" of the CMV matrix C (n) . Then, g (n) (z) † = −g (n) (z) , G (n) (z) := 1 2 g (n) (z) − g (n) (z) , g (n) (z) g (n) (z) † = −I + 2 cot ℑz · G (n) (z) and F n (z, w) = n −4/3 n−1 j,k=0 G (n) j,k (z) G (n) k,j (w) . (3.11) First of all, we would like to restrict the summation above by j, k ≤ M = Cn 1/2 log n with some constant C. Lemma 3.7 There exists V -depended constants C such that under the conditions of Theorem 1.5 uniformly in Ω A of (3.3) we have n −2/3 n j=M +1 G (n) n−j,n−j (z) ≤ Cn −1/12 log n. Now we present the approximation for the matrix elements G (n) n−j,n−k . Using the three-diagonal matrices expansion (2.2) of the C (n) , we can write the matrix g (n) as g (n) (z) = M (n) e −iz/2 + L (n) e iz/2 M (n) e −iz/2 − L (n) e iz/2 −1 . From the definitions of M (n) and L (n) one can find their matrix elements M (n) n+k,n+k−1 = d n+k ρ (n) n+k , M (n) n+k,n+k = d n+k α (n) n+k − d n+k+1 α (n) n+k+1 , L (n) n+k,n+k−1 = d n+k+1 ρ (n) n+k , L (n) n+k,n+k = d n+k+1 α (n) n+k − d n+k α (n) n+k+1 , where d k = (1 + s k ) /2 and s k = (−1) k . Denote C (n) ± (z) = M (n) e −iz/2 ± L (n) e iz/2 . At the first step we derive the representation for the matrix elements of the inverse matrix of C (n) − (z). Note that C (n) r − is three-diagonal and symmetric, and its entries are C (n) − n+k,n+k−1 (z) = s n+k ρ (n) n+k e n+k (z) , C (n) − n+k,n+k (z) = s n+k α (n) n+k e n+k (z) + s n+k α (n) n+k+1 e n+k+1 (z) with e k (z) = cos z 2 − is k sin z 2 . For the Verblunsky coefficients we use the result of [9]. Lemma 3.8 Consider the system of orthogonal polynomials and the Verblunsky coefficients defined above. Let the potential V satisfy conditions C1-C3 above. Then, for any k, α (n) n+k = (−1) k s (n) cos θ 2 − p θ x (n) k n −2/3 + O (ε n,k ) , ρ (n) n+k = sin θ 2 + cot θ 2 p θ x (n) k n −2/3 + O (ε n,k ) , where s (n) = 1 or s (n) = −1 and x (n) k = kn −1/3 , ε n,k = n −4/3 log 11 n 1 + x (n) k 2 1 \|k\|<n + 1 \|k\|≥n , with p θ = π √ 2 P (θ) and P defined in (1.7). To introduce the approximation for the resolvent, we define two "rotation" matrices which help to present the matrix C (n) r − in the form, similar to the discrete Laplacian matrix. Let U (n) and V (n) be two semi-infinite matrices with the entries U (n) n+j,n+k = is (n) 2nk−k−1 δ jk , V (n) n+j,n+k = is (n) 2nk−k δ jk and C (n) r ± (z) = U (n) C (n) ± V (n) , R (n) (ζ) = C (n) r − (z) −1 , where z = θ + ζn −2/3 . Then the entries of the new matrix are C (n) r − n+k,n+k−1 (z) = ρ (n) n+k e n+k (z) , C (n) r − n+k,n+k (z) = −is (n) s n α (n) n+k e n+k (z) + α (n) n+k+1 e n+k+1 (z) . Using the above definitions, we write g (n) (z) = I + 2L (n) V (n) R (n) (ζ) U (n) e iz/2 . (3.12) Now we prove that the matrix elements of R (n) (ζ) can be expressed in terms of the Airy functions. For this aim we present an approximation matrix R ⋆ and find the difference between R ⋆ and R (n) . Note that e iz/2 = e iθ/2 + ie iθ/2 ζn −2/3 + O \|ζ\| 2 n −4/3 , e n+k (z) = e n+k (θ) − is n+k e n+k (θ) ζn −2/3 + O \|ζ\| 2 n −4/3 , Let y (n) k = x (n) k − n −1/3 /2 and r (n) k,ζ = n −4/3 ε n,k + \|ζ\| 2 . Then C (n) r − n−k,n−k−1 (ζ) = sin θ 2 e n+k (θ) − cot θ 2 e n+k (θ) p θ y (n) k n −2/3 −is n+k sin θ 2 e n+k (θ) ζn −2/3 − 1 2 cot θ 2 e n+k (θ) p θ n −1 +n −4/3 O r (n) k,ζ ,(3.13)C (n) r − n−k,n−k (ζ) = − sin θ − 2 sin θ 2 p θ y (n) k n −2/3 − 2 cos 2 θ 2 ζn −2/3 − is n+k p θ cos θ 2 n −1 + n −4/3 O r (n) k,ζ . (3.14) The matrix elements of C (n) r − are similar to the matrix elements of the discrete Laplace operator with some potential in the n −1/3 scale, but offdiagonal elements contain alternating terms is n+k sin 2 θ 2 . Hence, we define the approximate resolvent in terms of the Airy function with some shift. Set δ (n) k = is n+k+1 δ, δ = 1 2 tan θ 2 , h = n −1/3 and R ⋆ n−k,n−j (ζ) = h −1 R ζ y (n) k + δ (n) k h, y (n) j + δ (n) j h ,(3.15) where R ζ (z, w) , defined by R ζ (z, w) = ab −1 π ψ − (z, ζ) ψ + (w, ζ) , ℜz ≤ ℜw, ψ + (z, ζ) ψ − (w, ζ) , ℜz ≥ ℜw (3.16) with ψ ± defined in the Appendix, is the extension of the resolvent of the operator L L [f ] (x) = a 3 f ′′ (x) − b 3 xf (x) (3.17) to the complex plane, where a 3 = sin θ and b 3 = 2p θ sin −1 (θ/2). For the properties, asymptotic behaviour, and the integral representation of R ζ see Appendix. Denote by D (n) the error of the approximation D (n) (ζ) = C (n) r − (ζ) R ⋆ (ζ) − I. (3.18) To present the bounds for D (n) n−k,n−j , we introduce the notations d (p) n−k,n−j = sup \|s\|≤δ+1 ∂ p ∂z p R ζ y (n) k + sh, y (n) j + δ (n) j h . One can see from the definition of R ζ that ∂ ∂z R ζ is not defined for z = w. In this case, by ∂ ∂z we denote the half of the sum of the left and the right derivatives 1 2 ∂ + ∂z + ∂ − ∂z . Then D (n) satisfies the following bound. Lemma 3.9 There exists constants C 1 , C 2 such that uniformly in k, j and ζ ∈ Ω A D (n) n−k,n−j (ζ) ≤ C 1 h 2 log C 2 n 1 + h 2 y (n) k 2 d (0) n−k,n−j + y (n) k + \|ζ\| d (1) n−k,n−j . (3.19) Now we are ready to analyse the r.h.s of (3.11). From (3.15), (3.12), and Lemma 3.9 one can see that G (n) n−k,n−j ≈ h −1 ℑR ζ y (n) k , y (n) j , and if we could neglect the remainder, then F n (ζ, ξ) ≈ h 2 ℑR ζ y (n) k , y (n) j ℑR ξ y (n) j , y (n) k . On the other hand, changing a double sum by the double integral and using (5.4), we obtain F [Q Ai ]. Hence, our main goal now is to estimate the remainder that appears after replacement of the "resolvent" of C (n) r − by the resolvent of the differential operator. We will do these calculations in several steps. We start from the proof of the bound for Σ M = n −2/3 M j=0 G (n) n−j,n−j (z) (3.20) with M = C 0 n 1/2 log n . It follows from (3.12) and the definition of G (n) that G (n) (z) = L (n) V (n) R (n) (ζ) e iz/2 − R (n) ζ e iz/2 U (n) . Using the definition of D (n) , we can write R (n) as R (n) (ζ) = R ⋆ − R (n) (ζ) D (n) (ζ) . Then, Σ M = n −2/3 M j=0 L (n) V (n) R ⋆ e (ζ) − R (n) e D (n) (ζ) U (n) n−j,n−j = Σ * M − Σ D (n) M , where R ⋆ e (ζ) = R ⋆ (ζ) e iz/2 − R ⋆ ζ e iz/2 and the same with R (n) and R M by some small factor which we get using the Cauchy inequality and the bounds (3.19) for D (n) n−k,n−j . Thus we obtain the quadratic inequality (3.23). Solving this inequality, we will obtain (3.20). Indeed, \|Σ * M \| ≤ C M j=0 \|k−j\|≤1 h ℑR ζ y (n) k + δ (n) k h, y (n) j + δ (n) j h (3.21) +h 3 R ζ y (n) k + δ (n) k h, y (n) j + δ (n) j h . (3.22) Using Proposition 5.5, we can estimate Σ ⋆ M as follows: \|Σ * M \| ≤ C. To estimate Σ D (n) M , we start with the relation L (n) V (n) R (n) e D (n) U (n) = L (n) V (n) R (n) e U (n) U (n) −1 D (n) U (n) = g (n) (z) − g (n) (z) D (n) , where D (n) entries have the same bounds as D (n) , and we will write below D (n) to simplify notations. Note that g (n) D (n) n−j,n−j = g (n) D (n) e n−j , e n−j = D (n) e n−j , g (n) † e n−j ≤ D (n) e n−j g (n) † e n−j = D (n) † D (n) 1/2 n−j,n−j g (n) † g (n) 1/2 n−j,n−j , and by the Cauchy inequality and (3.9), Σ D (n) M ≤ Cn −2/3 M j=0 D (n) † D (n) n−j,n−j 1/2 × M + 2 coth ℑz M 2 j=M 1 +1 G (n) n−j,n−j 1/2 = S 1/2 D (n) O n −5/6 log n + 2n −2/3 coth ℑζn −2/3 Σ M . Using Lemma 3.9, the Cauchy inequality, and Proposition 5.4, we estimate S D (n) as follows: Using the definition of G (n) , the sum in (3.11) can be splitted into four parts with different products of g (n) and g (n) . For each sum, the Cauchy inequality yields S D (n) = M j=0 D (n) † D (n) n−j,n−j ≤ C 1 n −4/3 log C 2 n M j=0 ∞ k=0 y (n) k 2 + \|ζ\| 2 d (1) n−k,n−j 2 + d (0) n−k,n−j 2 + h 4 y (n) k 4 + \|ζ\| 4 d (0) n−k,n−j 2 ≤ C 1 n −1 log C 2 n M j=0 1 + y (n) j 3/2 + h 4 1 + y (n) j 5/2 ≤ C 1 n −2/3 log C 2 n Mn −1/3 5/2 ≤ C 1 n −1/4 log C 2 n.n −4/3 j,k g (n) n−j,n−k (z) g (n) n−k,n−j (w) ≤ n −4/3 j g (n) g (n) † n−j,n−j (z) 1/2 × n −4/3 j g (n) g (n) † n−j,n−j (w) 1/2 , where each of the brackets is bounded because of (3.9) and (3.24). Changing the summation limits in the previous bound to j ∈ [M, n] and using Lemma 3.7, we obtain that under the conditions of Lemma 3.1 F n (z, w) = n −4/3 M j,k=0 G (n) n−k,n−j (z) G (n) n−j,n−k (w) + O n −1/24 log n . Now we use once more the identity G (n) = G ⋆ − G (n) D (n) . Repeating the above arguments, we obtain F n (z, w) = F ⋆ n (z, w) + F D (n) (z, w) , and F D (n) (z, w) ≤ C 1 n −1/8 log C 2 n. Since G ⋆ = L (n) V (n) R ⋆ e U (n) with R ⋆ e defined above, we have G ⋆ n−k,n−j = n 1/3 ℑR ζ y (n) k , y (n) j + r G ⋆ k,j , where r G ⋆ k,j contains terms with some derivatives of the R ζ multiplied by h in some non-negative power. Thus, from the boundness of the corresponded integrals (see proof of Proposition 5.4 for the arguments) h p+q M n −1/3 0 M n −1/3 0 ∂ p+q ∂x p ∂y q R ζ (x, y) 2 dxdy ≤ C p,q,r,s , we obtain that we can neglect terms from r G * k,j and F ⋆ n (z, w) = M n −1/3 0 M n −1/3 0 ℑR ζ (x, y) ℑR ξ (y, x) dxdy + O h 1/2 . Finally we note that by (5.7) and (5.8), ∞ M n −1/3 dx dy \|R ζ (x, y)\| 2 ≤ ∞ M n −1/3 ℑR ζ (x, x) dx ≤ Cn −1/12 log n, and ∞ 0 ∞ 0 ℑR ζ (x, y) ℑR ξ (y, x) dxdy ≤ C. Hence, F n (z, w) = ∞ 0 ∞ 0 ℑR ζ (x, y) ℑR ξ (y, x) dxdy + O Cn −1/24 log C n . (3.25) Estimate (3.25), integral representation (5.4), and the following relation (see [14]) Q Ai (x, y) = ∞ 0 Ai (x + t) Ai (y + t) dt imply (3.4) with K (x, y) = a −2 b −4 Q Ai a −1 b −2 x, a −1 b −2 y . Auxiliary Results P r o o f of Proposition 3.5. Using Lemma 3.3 with ε = 2c θ and inequality K (n) n (λ, µ) 2 ≤ K (n) n (λ, λ) K (n) n (µ, µ) , (4.1) we obtain λ∈σ c ε G (z − λ) K (n) n (λ, µ) 2 dλ ≤ Ce −nd(ε) sup λ∈σ c ε G (z − λ) K (n) n (µ, µ) . Due to the restrictions on λ and z we get G (z − λ) ≤ C ′ when λ ∈ σ c ε . Thus, σ c ε G (z − λ) G (w − µ) K (n) n (λ, µ) 2 dλdµ = e −cn O ℑ −1 z + ℑ −1 w . Changing the variables by the scaled ones in (3.7), we get F n (z, w) = n −4/3 ℑ cot ζ − x 2n 2/3 ℑ cot ξ − y 2n 2/3 \|K n (x, y)\| 2 dxdy + O e −cn . Finally we estimate the difference between F n and 4F n 4F n (ζ, ξ) − F n (z, w) = n −4/3 (I 1 (ζ, ξ) + I 2 (ζ, ξ) + I 2 (ξ, ζ)) + O e −cn with I 1 and I 2 of (4.2) and (4.3). It is easy to see that \|I 1 (ζ, ξ)\| = ℑ 2n 2/3 ζ − x − cot ζ − x 2n 2/3 ℑ 2n 2/3 ξ − y − cot ξ − y 2n 2/3 \|K n (x, y)\| 2 dxdy ≤ C \|K n (x, y)\| 2 dxdy ≤ Cn, (4.2) where we have used that for 0 < \|z\| ≤ 2c θ cot z − 1 z ≤ C. In addition, since the kernel K (n) n (λ, µ) 2 is positive definite, we can use the Cauchy inequality to get \|I 2 (ζ, ξ)\| = ℑ 2n 2/3 ζ − x − cot ζ − x 2n 2/3 ℑ cot ξ − y 2n 2/3 \|K n (x, y)\| 2 dxdy ≤ \|I 1 (ζ, ξ)\| 1/2 n 4/3 F n (z, w) 1/2 ≤ Cn 7/6 \|F n (z, w)\| 1/2 . (4.3) Finally, collecting the above bounds, we obtain \|F n (z, w) − F n (ζ, ξ)\| ≤ Cn −1/6 \|F n (z, w)\| 1/2 + C ′ n −1/3 , and using the Cauchy inequality, we get (3.10). P r o o f of Lemma 3.9. The proof is based on the direct calculations of the matrix elements D (n) n−j,n−k . We start with the case j = k. Then all derivatives of R ζ are well defined and the points y r − . These calculations are a little bit involved, so we present them in several steps. First, we calculate R ⋆ n−k∓1,n−j , R ⋆ n−k∓1,n−j = h −1 R ζ y (n) k ± h − δ (n) k h, y (n) j + δ (n) j h = h −1 R ζ y (n) k , y (n) j + δ (n) j h + ±1 − δ (n) k ∂ ∂z R ζ y (n) k , y (n) j + δ (n) j h + ±1 − δ (n) k 2 h ∂ 2 ∂z 2 R ζ y (n) k , y (n) j + δ (n) j h + h 2 O r ⋆ n−k,n−j (δ + 1) with the remainder r ⋆ n−k,n−j (d) = sup \|s\|<d ∂ 3 ∂z 3 R ζ y (n) k + s, y (n) j + δ (n) j h , where the last bound follows from differential equation (5.1) valid for the functions ψ ± . To simplify calculations for C (n) r − , we use the following notations: S k := C (n) r − n−k,n−k−1 + C (n) r − n−k,n−k+1 , D k := C (n) r − n−k,n−k−1 − C (n) r − n−k,n−k+1 . Then, combining the above expansion with (3.13)-(3.14), we obtain D (n) n−k,n−j = h −1 R ζ y (n) k , y (n) j + δ (n) j h S k + C (n) r − n−k,n−k + ∂ ∂z R ζ y (n) k , y (n) j + δ (n) j h D k − δ (n) k S k + δ (n) k C (n) r − n−k,n−k + h ∂ 2 ∂z 2 R ζ y (n) k , y (n) j + δ (n) j h 1 2 S k − δ (n) k D k − δ 2 2 S k + C (n) r − n−k,n−k + O r ⋆ n−k,n−j (δ + 1) ,(4.4) where for the last term we have used the uniform bound for elements C (n) r − n−j,n−k . Now it is sufficient to calculate every expression in the brackets. We start with S k and D k , S k = sin θ −2 cos θ 2 cot θ 2 p θ y (n) k h 2 −2 sin 2 θ 2 ζh 2 +is n+k p θ cos θ 2 h 3 +h 4 O r (n) k,ζ , D k = −2is n+k sin 2 θ 2 + 2is n+k cos θ 2 p θ y (n) k h 2 − is n+k sin θζh 2 − cos θ 2 cot θ 2 p θ h 3 + h 4 O r (n) k,ζ . Therefore, with an error of order h 4 O r (n) k,ζ we can write S k + C (n) r − n−k,n−k ≈ −2h 2 p θ sin −1 (θ/2) y (n) k + ζ , D k − δ (n) k S k + δ (n) k C (n) r − n−k,n−k ≈ −2δ (n) k h 2 p θ sin −1 (θ/2) y (n) k − ζ + is n+k p θ cos (θ/2) sin −2 (θ/2) h . Finally, combining the above relations and the equation for R ζ in the form sin θ ∂ 2 ∂z 2 R ζ y (n) k , y (n) j + δ (n) j h − 2p θ sin −1 θ/2y (n) k + ζ R ζ y (n) k , y (n) j + δ (n) j h = 0, we obtain the remainder in (4.4) with all terms of order less than h 2 . Gathering all these remainders and the remainder h 4 O r (n) k,ζ , we get (3.19). For j = k, the calculations can be performed similarly if we take into account jump condition (5.2). P r o o f of Lemma 3.7. We start with estimate of X n (ζ) = n −2/3 K n (x, x) G (ζ − x) n −2/3 dx, where K n is defined as in (3.6) but without any restriction. Let ζ = s + iε. Changing variables to z = θ + ζn −2/3 and using (3.6) with (3.8), we obtain X n (ζ) = n 1/3 ℜh n (z) , where h n (z) = π −π g (z − λ) ρ n (λ) dλ. For further estimates we use the "quadratic" equation obtained in [6], h 2 n (z) − 2iV ′ (ℜz) h n (z) − 2iQ n (z) − 1 = − 2 n 2 δ n (z) , with Q n (z) = π −π g (z − λ) (V ′ (λ) − V ′ (ℜz)) ρ n (λ) dλ, δ n (z) = π −π K (n) n (λ, µ) 2 (g (z − λ) − g (z − µ)) 2 dλdµ. Solving the "quadratic" equation, we get X n (ζ) = n 1/3 ℜ f n (s, ε) − 2n −2 δ n (z), where the function f n (s, ε) = −V ′2 θ + sn −2/3 + 2iQ n θ + (s + iε) n −2/3 + 1 is twice differentiable in both variables. Using the symmetry of the kernel K (n) n and (4.1), we can estimate δ n (z) as n −2 δ n (z) ≤ 4n −2 π −π K (n) n (λ, λ) \|g (z − λ)\| 2 dλ. Then the identity (3.9) yields n −2 δ n (z) ≤ 4n −1 +2n −4/3 coth εn −2/3 ·X n (ζ) ≤ Cn −2/3 n −1/3 + ε −1 X n (ζ) , as ε = O (1). Now we continue the estimation of Q n (z). For the density ρ n , we use the bound (see [6]) \|ρ ′ n (λ)\| ≤ C ψ (n) n−1 2 + ψ (n) n 2 + 1 , where ψ (n) k = P (n) k w 1/2 n are orthonormal functions. Hence, the density ρ n is uniformly bounded and therefore, similarly to (2.17) of [6], we have \|Q n (z) − Q n (ℜz)\| ≤ Cℑz \|log ℑz\| . The weak convergence (1.5) with φ (λ) = V ′ (λ) − V ′ θ + s/γn 2/3 cot λ − θ − s/γn 2/3 2 implies Q n θ + s/γn 2/3 − Q θ + s/γn 2/3 ≤ Cn −1/2 log 1/2 n if \|s\| ≤ c θ n 2/3 . Hence, combining the above relations, we obtain \|f n (s, ε) − f (s)\| ≤ Cn −2/3 log n \|log ε\| + n 1/6 , with f (s) := f (s, 0). The properties of the Herglotz transformation yield (see [6]) ρ (λ) = 1 2π lim ε→+0 ℜh (λ + iε) . Therefore, at the edge point θ we obtain f (0) = 0 and f ′ (0) < 0. Hence, by the differentiability of f (s) , we obtain X (ζ) = ℜ O (s + ε −1 X (ζ) + n 1/6 log n). (4.5) Solving the quadratic inequality, we estimate X (ζ) as follows: X (ζ) ≤ C ε −1 + s 1/2 + n 1/12 log 1/2 n . Now we write (4.5) more precisely X (ζ) = ℜ −Cs + ε −2 O 1 + εs 1/2 + εn 1/12 log 1/2 n . Below we need the estimate of X (ζ) for s > Cn 1/6 log n and ε = O (1). Hence we obtain X (ζ) ≤ C 1 s − C 2 n 1/6 log n −1/2 . Note that all constants in the above estimates depend only on V and can be bounded by some combination of sup \|V \|, sup \|V ′′ \| and sup \|V ′′′ \|. Now we return to the estimate of the sum in Lemma 3.7. By the spectral theorem, Then, by the same argument as above for model (1.1), we define the first marginal density I (M) = n −2/3 n j=M +1 G (n) n−j,n−j (z) = n −2/3 n−M −1 j=0 G (λ − z) χ (n) j (λ) 2 w n (λ) dλ.ρ (n−M ) n−M (λ) = 1 n − M n−M −1 j=0 χ (n) j (λ) 2 w n (λ) . On the other hand, this density can be considered as the first marginal density for model (1.1) with the potential V = n n − M V . Hence, I (M) = n −2/3 G (λ − z) K (n−M, V ) n−M (λ, λ) dλ = X V n−M (ζ) . But it follows from the result of [15] that the support of the equilibrium density for V is [θ M , θ M ] with θ M = θ − c V (Mn −1 ) + o (Mn −1 ) with some c V > 0. Hence, by (4.6), X V n−M ≤ Cn −1/12 , and Lemma 3.7 is proved. Appendix In this section we present the properties and the asymptotic analysis of the resolvent of the Airy operator. Denote by L the second order differential operator on the set of twice continuously differentiable functions on R, L [f ] (x) = a 3 f ′′ (x) − b 3 xf (x) . Let R ζ (x, y) be the kernel of the resolvent (L − ζI) −1 for ℑζ = 0. By the general principles (for example see [16], Section 72) Proposition 5.1 Let Ai (z) and Bi (z) be the standard Airy functions. Denote by ψ ± the following functions: ψ − (x, ζ) = Ci (X x,ζ ) , ψ + (x, ζ) = Ai (X x,ζ ) , with Ci (X) = iAi (X) − Bi (X) and X x,ζ = a −1 bx + a −1 b −2 ζ. Then these functions are the unique solutions of the differential equation a 3 ∂ 2 ∂x 2 ψ ± (x, ζ) − b 3 x + ζ ψ ± (x, ζ) = 0, (5.1) that are square integrable on the right (left) half axis and fixed by jump condition ψ − (x, ζ) d dx ψ + (x, ζ) − ψ + (x, ζ) d dx ψ − (x, ζ) = a −1 bπ −1 . (5.2) And the resolvent R ζ has two representations R ζ (x, y) = ab −1 π ψ − (x, ζ) ψ + (y, ζ) , x ≤ y, ψ + (x, ζ) ψ − (y, ζ) , x ≥ y, (5.3) R ζ (x, y) = a −2 b −1 1 t − ζ Ai a −1 bx + a −1 b −2 t Ai a −1 by + a −1 b −2 t dt. (5.4) The following asymptotic behaviour of the Airy functions can be found in [17]. The main term for the derivatives can be obtained by direct differentiation of the asymptotics. The last proposition and the definition of the functions ψ ± yield the asymptotic behaviour of them Proposition 5. 3 The functions ψ ± are entire in x and ζ and have the following asymptotic behaviour in x for ℑζ = ε > 0: (5.4) give the extra factor of order \|y\| 2 + \|ζ\| 2 to the integrand. Therefore, we start with I (s; 0). Since \|R ζ (x, y)\| ≤ C A e −c A \|x−y\| 1/2 for x ≥ −A and ζ ∈ Ω A , we split the integral from (5.5) into two parts I (s; 0) = \|y−x\|<2\|x\| + \|y−x\|>2\|x\| ≤ C s (x s + \|ζ\| s ) \|R ζ (x, y)\| 2 dy \|ψ + (x, ζ)\| = π −1/2 \|X x,ζ \| −1/4 1 + O \|X x,ζ \| −3/2      exp − 2 3 \|ℜX x,ζ \| 3/2 , x → ∞ exp a −1 b −2 ε \|ℜX x,ζ \| 1/2 , x → −∞ \|ψ − (x, ζ)\| = (4π) −1/2 \|X x,ζ \| −1/4 1 + O \|X x,ζ \| −3/2      exp 2 3 \|ℜX x,ζ \| 3/2 , x → ∞ exp −a −1 b −2 ε \|ℜX x,ζ \| 1/2 , x → −∞+ C A t>2\|x\| (t + x) s e −c A t 1/2 dt. (5.6) For the first integral we note that by the spectral theorem and the resolvent identity, ∞ −∞ \|R ζ (x, y)\| 2 dy = ℑR ζ (x, x) ℑζ . (5.7) The asymptotic behaviour of ψ ± from Proposition 5.3 implies \|R ζ (x, x)\| ≤ C A (1 + \|x\|) −1/2 , and \|ℑR ζ (x, x)\| ≤ C A (1 + \|x\|) −3/2 . (5.8) Combining (5.6) with (5.7) and (5.8), we obtain (5.5) with q = 0. In view of equation (5.1), it is sufficient to prove (5.4) only for q = 0, 1. If q = 1, similarly to the above argument, we split the integral into two parts. In the first term, integrating by parts, we have The r.h.s satisfies the necessary bound for q = 1, hence the proposition is proved. (1 + \|x j \|) −3/2 ≤ C. The second statement can be checked in a similar way. The proof of the third statement consists of several steps. First, we change z j by x j in (5.11). The error of this change is a combination of sums of higher derivatives with extra factors h. These sums are small, because for z j far from z k these derivatives admit the exponential bound, and for z j ∼ z k , in view of equation (5.1) and restriction \|z k \| ≤ Cn 1/6 log n, every two extra derivatives will give us the sum as in (5.11) with the factor of order n −1/2 log n. After the change of z j by x j , we obtain the sum which can be estimated by the integral C ∞ 0 x p ∂ d ∂z d R ζ x, z (2) k 2 dx, because of the smoothness and exponential decreasing of R ζ . And finally, the identity (5.7) and Proposition 5.4 yield (5.11). We used the identity (5.7) which is valid for real x, but it remains valid for complex x because the l.h.s and r.h.s of the (5.7) are entire functions equal at the real line. V (cos λ j ) . (1.2) Normalized Counting Measure of eigenvalues (NCM) is given by N n (∆) = n −1 ♯ λ (n) l ∈ ∆, l = 1, . . . , n , ∆ ⊂ [−π, π]. ⋆ M can be estimated immediately by using Proposition 5.5, and Σ D (n) Combining this inequality with the above estimate of Σ D (n) M , we obtain the inequality for Σ M \|Σ M \| ≤ C 1 + C 2 n −1/8 log C 3 n O n −5/6 log n + \|Σ M \| we are ready to find the limit of the r.h.s. of(3.11) G Proposition 3.2 implies that it is sufficient to check(3.5) to finish the proof of Theorem 1.5. We use an evident relation G (t + iε − s) the inequality valid for any s ∈ [a, b] (t + i − s) n −2/3 dt ≥ Cn 2/3 , with some absolute constant C. The last inequality, the positiveness of K n and G, and definition of G (n) Hence, by (3.24) for any finite ∆ ⊂ [−A + 1, A − 1] we obtain(3.5). V (cos λ j ) . Proposition 5. 4 4For any non-negative integers s, q and any A ∈ R + there exists a constant C A,s,q such that for any x ≥ −A and ζ ∈ Ω A I (s; q) = ∞ −∞ \|y\| s ∂ q ∂y q R ζ (x, y) 2 dy ≤ C A,s,q (1 + \|x\|) s+q−3/2 . (5.5) P r o o f of Proposition 5.4. In view of equation (5.1), two extra derivatives in 1 y + c 2 ζ) \|R ζ (x, y)\| 2 dy. Proposition 5. 5 − 5Let h = n −1/3 , M = C 0 n 1/2 log n . Also, denote by x j = jh the equidistant set and z any non-negative integer p, d = 0 or 1 and k ≤ Mh ∞ j=0 \|x j \| p ∂ d ∂z d R ζ z x j = O (h), \|ℑR ζ (x, x)\| ≤C (1 + \|x\|) −3/2 and derivatives of R ζ are bounded near the real line, we obtain that ℑR ζ z Acknowledgement. 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[ "ON NORMAL APPROXIMATIONS FOR THE TWO-SAMPLE PROBLEM ON MULTIDIMENSIONAL TORI", "ON NORMAL APPROXIMATIONS FOR THE TWO-SAMPLE PROBLEM ON MULTIDIMENSIONAL TORI" ]	[ "Solesne Bourguin ", "Claudio Durastanti " ]	[]	[]	In this paper, quantitative central limit theorems for U -statistics on the q-dimensional torus defined in the framework of the two-sample problem for Poisson processes are derived. In particular, the U -statistics are built over tight frames defined by wavelets, named toroidal needlets, enjoying excellent localization properties in both harmonic and frequency domains. The Berry-Esséen type bounds associated with the normal approximations for these statistics are obtained by means of the so-called Stein-Malliavin techniques on the Poisson space, as introduced by Peccati, Solé, Taqqu, Utzet (2011) and further developed byPeccati, Zheng (2010)andBourguin, Peccati (2014). Particular cases of the proposed framework allow to consider the two-sample problem on the circle as well as the local two-sample problem on R q through a local homeomorphism argument.2010 Mathematics Subject Classification. 60F05, 60G55, 62G09, 62G10, 62G20.	10.1016/j.jspi.2017.10.009	[ "https://arxiv.org/pdf/1604.01316v1.pdf" ]	88,520,827	1604.01316	b408235b8ca82ddf556ddfc3eb05930f397c1060	ON NORMAL APPROXIMATIONS FOR THE TWO-SAMPLE PROBLEM ON MULTIDIMENSIONAL TORI 5 Apr 2016 Solesne Bourguin Claudio Durastanti ON NORMAL APPROXIMATIONS FOR THE TWO-SAMPLE PROBLEM ON MULTIDIMENSIONAL TORI 5 Apr 2016arXiv:1604.01316v1 [math.PR] In this paper, quantitative central limit theorems for U -statistics on the q-dimensional torus defined in the framework of the two-sample problem for Poisson processes are derived. In particular, the U -statistics are built over tight frames defined by wavelets, named toroidal needlets, enjoying excellent localization properties in both harmonic and frequency domains. The Berry-Esséen type bounds associated with the normal approximations for these statistics are obtained by means of the so-called Stein-Malliavin techniques on the Poisson space, as introduced by Peccati, Solé, Taqqu, Utzet (2011) and further developed byPeccati, Zheng (2010)andBourguin, Peccati (2014). Particular cases of the proposed framework allow to consider the two-sample problem on the circle as well as the local two-sample problem on R q through a local homeomorphism argument.2010 Mathematics Subject Classification. 60F05, 60G55, 62G09, 62G10, 62G20. Introduction The aim of this paper is to establish quantitative central limit theorems, by means of Stein-Malliavin techniques, for wavelet-based U-statistics on q-dimensional tori arising in the context of the two-sample problem for Poisson processes. The two-sample (or homogeneity) problem for Poisson processes can be described as follows: let N 1 and N 2 denote two independent Poisson processes observed on a measurable space X, whose intensities with respect to some positive non-atomic σ-finite measure µ are denoted by f 1 and f 2 , respectively. Given the observation of N 1 and N 2 , the two sample problem aims at testing the null-hypothesis (H 0 ) : f 1 = f 2 versus the alternative hypothesis (H 1 ) : f 1 = f 2 , see for instance [FLRB13] for an in-depth description. In such a problem, two-sample U -statistics arise very naturally (see for instance [FLRB13;vdV98;Das08]) as they can be used to approximate both the distribution under the null as well as the alternative hypotheses (note that in the case of the alternative hypothesis, one has to deal with different underlying distributions for the Poisson processes, see e.g. [Lee90]). This paper assumes a slightly different, but equivalent framework in which the null-hypothesis (H 0 ) is that observations of a unique Poisson process N , sampled over two disjoint q-dimensional tori T q 1 and T q 2 , are distributed according to the same intensity with respect to µ (note when working under the null-hypothesis, this is equivalent to considering two independent Poisson processes over the same support). For this purpose, let {N t : t ≥ 0} be a Poisson process over a q-dimensional torus T q with control measure given by (1) µ t (dθ) := R t f (θ) dθ, where R t > 0 denotes, roughly speaking, the expected number of observations at time t > 0 and f is a density function over T q satisfying one of two possible mild regularity conditions on its spectral decomposition (see Conditions 1.1 and 1.2 in Subsection 1.2). Consider the estimator, given in the form of a U -statistic, defined by U j (t) := Kj k=1   T q 1 ψ j,k (θ) N t (dθ) − T q 2 ψ j,k (θ) N t (dθ) 2 − T q 1 ψ 2 j,k (θ) N t (dθ) − T q 2 ψ 2 j,k (θ) N t (dθ) ,(2) where, given a scale parameter B, the set {ψ j,k : j ≥ 0, k = 1, . . . , K j } is the set of the q-dimensional toroidal needlets for which the index j denotes the multiresolution level, while K j stands for the cardinality of needlets at a given resolution level j is fixed (a more detailed description can be found in Subsection 2.4). The strategy for deriving the upcoming quantitative normal approximation results for the two-sample problem on multidimensional tori will be to make use of the celebrated Stein-Malliavin techniques obtained in [Pec+10] through the combination of Stein's method with the Malliavin calculus of variations for Poisson functionals (see Section 1.2 for a self-contained description of these techniques). The Berry-Esséen type bounds associated with the limit theorems proved below only depend on the resolution level j of the needlets, on the expected number of observations R t sampled at time t, and on the spectral parameter α controlling the rate of decay of the density function f . As pointed out in Remark 1.5, these bounds, through their dependence on the regularity parameter α of the density function f appearing in the control measure of the Poisson process provides a new quantitative estimation of the impact of the regularity of f in the quality of the normal approximation of the estimator (2) (as well as the impact on the rate of convergence to the Gaussian distribution when such a convergence applies). A rigorous formulation of these quantitative bounds, as well as of the role of the regularity of f , is given in Theorem 1.4, and are the main findings of this paper, and to our knowledge the first quantitative bounds for the two-sample problem involving the regularity parameter of the density function f . 1.1. An overview in the literature. The comparison between two probability distributions has been an important and long-lasting subject with applications in a wide range of fields, such as biology, medicine, physics and cosmology (among a lot of others). Following the seminal papers [Cox53;PW40], dealing with homogeneous Poisson processes, two-sample test statistics were introduced within many different settings: some kernel-based procedures were proposed in [AHD94; Gre+08; HT02], while the study of asymptotic properties of U-statistics based on non-homogeneous Poisson processes was addressed, for instance, in [DMNN99;FLRB13]. The reader is also referred to [Lee90;vdV98] for further details and discussions on this topic. Stein-Malliavin techniques, first introduced in [NP09] have become increasingly popular within the scientific community. Initially used to establish Berry-Esséen type bounds for functionals of Gaussian fields, these techniques have been extended to framework of Poisson random measures (see e.g. [BP14; Pec+10; PZ10]). More recently, such methods have been generalized to the setting of spectral theory of general Markov diffusion generators (see e.g. [ACP14;Led12]). The reader is reffered to the text [NP12] for a detailed introduction to this topic. In the context of statistical analysis over the sphere, Stein-Malliavin methods have proven quite powerful to establish asymptotic normality of wavelet-based linear and nonlinear statistics (respectively in [Dur15b; DMP14] and [Bou+14]) built over the sphere by means of the so-called spherical needlets. Spherical needlets were introduced in [NPW06a; NPW06b] and form a tight frame on the sphere, so that a reconstruction formula (the counterpart of the harmonic expansion in the wavelet framework) holds. Furthermore, they enjoy remarkable localization properties in both harmonic and spatial domains, while their statistical properties when applied to the study of spherical random fields were investigated in [Bal+09b]. Further remarkable statistical applications can be found, for instance, in [Bal+09a; CM15; DLM13]. This paper makes use of wavelets on T q , built exactly as in [NPW06b], called toroidal needlets (see e.g. Subsection 2.4 for technical details). Other extensions on various manifolds such as the unit ball of R 3 and spherical spin fiber bundles can be found in [Dur+14] and [GM10], respectively. The monograph [MP11] should be noted to be a remarkable go-to reference for details and discussions on both the theoretical and applied aspects of wavelets and needlets. Finally, note that the choice of T q as the support of the Poisson processes under consideration in this work is very natural in view of the size of the literature dealing with statistical tests aimed at comparing two distributions over circular data, which corresponds the the case q = 1 of the framework of thhis paper (see for instance [Epl79; MJ09; RJS01] among many others). In this light, the framework of a q-dimensional tori T q can be viewed as a unifying generalization as it also encompasses the local two-sample problem over R q through the fact that R q and T q are locally homeomorphic and the spatial concentration of the toroidal needlets which ensures consistency of the normal approximation results for any local approximation of R q by T q . 1.2. Main results. The Poisson random measures considered in the framework of this paper are Poisson random measures over T q with control measures given by (1). The density function f can be viewed in terms of its harmonic expansion (see Section 2 for more details), that is, for any θ ∈ T q , it holds that (3) f (θ) = n∈Z q a n s n (θ) , where {s n : n ∈ Z q } is the set of eigenfunctions of the Laplace-Beltrami operator on T q , and forms an orthonormal system for T q . {a n : n ∈ Z q } are the corresponding Fourier (or harmonic) coefficients. Two sets of assumptions on the density function f (more precisely on the harmonic expansion of f ) will be used throughout the paper, namely Condition 1.1. For any n ∈ Z q , there exist two constants c > 0 and α ≥ 1 2 such that (4) a n = c (ℓ n + 1) −α , where ℓ n is the eigenfunction of the Laplace-Beltrami operator associated with the multiindex n. Condition 1.2. For any n ∈ Z q , there exist two constants c > 0 and α ≥ 1 2 such that (5) a n = c q m=1 (n m + 1) −α , where n m , m = 1, . . . , q, are the components of the multiindex n. Remark 1.3. Conditions 1.1 and 1.2 imply a strong interdependence and statistical independence between the components of the coefficients of the density function associated to the frequency n, respectively. The parameter α controls the rate of decay of the density function f . Note that the two conditions reduce to the same one if q = 1. Define U j (t) to be the normalized version of the U -statistic defined in (2), namely (6) U j (t) := U j (t) V ar (U j (t)) . The following theorem is the main result of this paper, providing quantitative normal approximation bounds for the two-sample problem over q-dimensional tori. Throughout the paper, the symbol L → denotes convergence in law, while the symbol ∼ denotes an asymptotic equivalent and the symbols and stand for asymptotically equivalent upper and lower bounds, respectively. Theorem 1.4. Let U j (t) be given by (6), Z ∼ N (0, 1) be a standard Gaussian random variable and d W denote the Wasserstein metric (see upcoming Definition 2.5). (i) Assume that Condition 1.1 prevails. Then, it holds that d W U j (t) , Z R − 1 2 t B j 4 + R − 1 2 t + B − j 2 if α ∈ [1, +∞) , R − 1 2 t B j 4 + R − 1 2 t B j(1−α) + B j(1−2α) if α ∈ 1 2 , 1 . Furthermore, if R − 1 2 t B j max( 1 4 ,1−α) → 0 as t → ∞, then U j (t) L → N (0, 1) . (ii) Assume that Condition 1.2 prevails. Then, it holds that d W U j (t) , Z R − 1 2 t B jq 4 + R − 1 2 t + B − jq 2 if α ∈ [1, +∞) , R − 1 2 t B jq 4 + R − 1 2 t B jq(1−α) + B jq(1−2α) if α ∈ 1 2 , 1 . Furthermore, if R − 1 2 t B jq max( 1 4 ,1−α) → 0 as t → ∞, then U j (t) L → N (0, 1) . Remark 1.5. Observe that the above results indicate a regime transition whenever the regularity parameter α of the density function f hits the value α = 1. For α < 1, the rates of convergence depends on α and are slower the closer α is to 1/2, whereas for α ≥ 1, the regularity of f does not influence the rates of convergence anymore. This phenomenon provides a new quantitative interpretation of the impact of the regularity of f in the normal convergence of the studied estimator. 1.3. Plan of the paper. The rest of the paper is organized as follows: Section 2 contains the needed elements of harmonic and wavelet analysis on the q-dimensional torus as well as the version of the Stein-Malliavin approximation results for Poisson random measures used here. Section 3 contains the proof of the main result, namely Theorem 1.4, while Section 4 gathers technical auxiliary results along with their proofs. Background and notation This section provides some background on Poisson random measures, quantitative estimates for normal approximations of Poisson multiple integrals, elements of Fourier analysis on T q (see e.g. [Gra08]) and the construction of toroidal needlets and their properties (see e.g. [NPW06a;NPW06b]). From now on, n ∈ Z q will denote the vector n = (n 1 , . . . , n q ) and ℓ n := n ℓ 2 (Z q ) = q i=1 n 2 i will denote the eigenvalue of the Laplace-Beltrami operator associated with the multiindex n. 2.1. Poison random measures and multiple Wiener-Itô integrals. Let (Ω, F , P) be a probability space, (X, X ) be a polish space and µ be a positive, σ-finite measure over (X, X ) with no atoms. Denote by X µ the class of those A ∈ X such that µ(A) < ∞. Then, a Poisson random measure with control µ on (X, X ) with values in (Ω, F , P) is a map N : X µ → Ω with the following properties: (1) For any set A in X µ , N (A) is a Poisson random variable with parameter µ(A); (2) If r ∈ N and A 1 , . . . , A r ∈ X µ are disjoint, then N (A 1 ) , . . . , N (A r ) are independent; (3) If r ∈ N and A 1 , . . . , A r ∈ X µ are disjoint, then N ( r i=1 A i ) = r i=1 N (A i ). If N is a Poisson random measure on (X, X ) with control µ, denote byN the associated centered Poisson random measureN (A) = N (A) − µ(A), A ∈ X µ . Note that one can regard Poisson random measures as acting on L 2 (µ) through the identifications N ( 1 A ) = N (A) andN (1 A ) = N (A) − µ(A), for all A ∈ X µ . For every deterministic function h ∈ L 2 (µ), write I 1 (h) =N (h) = X h(x)N (dx) to indicate the Wiener-Itô integral of h with respect toN . For every q ≥ 2 and every symmetric function f ∈ L 2 (µ q ), denote by I q (f ) the multiple Wiener-Itô integral of order q of f with respect toN . For any real constant c, set I 0 (c) = c and for any f ∈ L 2 (µ q ) (not necessarily symmetric), set I q (f ) = I q ( f ), where f (x 1 , . . . , x q ) = 1 q! π∈Sq f x π(1) , . . . , x π(q) denotes the symmetrization of f . The reader is referred for instance to [PT11, Chapter 5] for a complete discussion of multiple Poisson integrals and their properties. Proposition 2.1. The following equalities hold for every q, p ≥ 1 and every symmetric functions f ∈ L 2 (µ q ) and g ∈ L 2 (µ p ): (1) E (I q (f )) = 0; (2) E (I q (f )I p (g)) = q! f, g L 2 (µ q ) 1 {q=m} . The following definition introduces the star-contraction operation between two symmetric functions f ∈ L 2 (µ q ) and g ∈ L 2 (µ p ). Definition 2.2. Let q, p ≥ 1 and let f ∈ L 2 (µ q ) and g ∈ L 2 (µ p ) be symmetric functions. The star contraction of index (r, ℓ) between f and g, denoted by f ⋆ ℓ r g, is defined as the L 2 µ q+p−r−ℓ function given by f ⋆ ℓ r g (s 1 , . . . , s q−r , t 1 , . . . , t p−r , γ 1 , . . . , γ r−ℓ ) := X ℓ f (s 1 , . . . , s q−r , γ 1 , . . . , γ r−ℓ , z 1 , . . . , z ℓ ) g (t 1 , . . . , t p−r , γ 1 , . . . , γ r−ℓ , z 1 , . . . , z ℓ ) µ ⊗ℓ (dz 1 , . . . , dz ℓ ) . In particular, f ⋆ 0 0 g = f ⊗ g, f ⋆ q q f = f 2 L 2 (µ q ) and f ⋆ 0 q f = f 2 . The next statement contains an important product formula for Poisson multiple integrals. Note that the statement involves the star-contraction operators defined above. Proposition 2.3. Let q, p ≥ 1 and let f ∈ L 2 (µ q ) and g ∈ L 2 (µ p ) be symmetric functions. Then, I q (f )I p (g) = q∧p r=0 r! q r p r r ℓ=0 r ℓ I q+p−r−ℓ f ⋆ ℓ r g . From now on, let X = R + × T q and X = B (R + × T q ), where B (R + × T q ) denotes the class of Borel subsets of R + × T q . N will denote a Poisson random measure on R + × T q with control measure η = τ × µ, where τ is such that τ (ds) = R · λ (ds) with λ (ds) denoting the Lebesgue measure on R and R > 0, so that τ ([0, t]) = R · t = R t . µ is defined to be a probability measure on T q that is absolutely continuous with respect to the Lebesgue measure on T q , denoted by ν, so that µ (dθ) = f (θ) ν (dθ) . For any fixed t > 0, denote by N t the Poisson random measure over (T q , B (T q )) with control measure µ t := R t µ. Remark 2.4. As pointed out in [Bou+14; DMP14; Dur15a], t can be regarded as the time parameter, so that, for any A ∈ T q , µ t (A) represents the expected number of observations sampled on A at time t. 2.2. Stein-Malliavin bounds for Poisson multiple integrals. The Stein-Malliavin bounds that will be made use of in this paper will be stated for the specific Poisson random measure introduced above for notational convenience, but the result holds for any Poisson random measure as defined in Subsection 2.1. Before stating these bounds, one needs to introduce the so-called Wasserstein metric. Definition 2.5 (Wasserstein metric). Let Lip(1) denote the class of real-valued Lipschitz functions, from R to R, with Lipschitz constant less or equal to one, that is functions h that are absolutely continuous and satisfy the relation h ′ ∞ ≤ 1. Given two real-valued random variables X and Y , the Wasserstein distance between the laws of X and Y , written d W (X, Y ) is defined as d W (X, Y ) = sup h∈Lip(1) \|E [h (X)] − E [h (Y )]\| . The following statement is a version of Theorem 4.2 in [Pec+10] in the case of double Poisson integrals, stated within the framework of this paper (see also [Pec+10,Example 4 .4]). Proposition 2.6. Let Z ∼ N (0, 1) and assume that F j = I 2 (h j ), where the symmetric function h j ∈ L 2 µ 2 t is such that 2 h j 2 L 2 (µ 2 t ) = 1, h j ⋆ 1 2 h j ∈ L 2 (µ t ), h j ∈ L 4 µ 2 t , and T q T q h j (z, a) 4 µ t (da)µ t (dz) < ∞. Furthermore, if h j L 4 (µ 2 t ) + h j ⋆ 1 1 h j L 2 (µ 2 t ) + h j ⋆ 1 2 h j L 2 (µt) → 0 as j → ∞, then F j converges in law to Z and the following bound on the Wasserstein distance between F j and Z holds: d W (F j , Z) ≤ √ 8 h j ⋆ 1 1 h j L 2 (µ 2 t ) + 6 + 2 √ 2 h j ⋆ 1 2 h j L 2 (µt) + √ 8 h j 2 L 4 (µ 2 2. 3. Harmonic analysis on multidimensional tori. As is well-known in the literature (see for instance [Gra08]), the q-dimensional torus can be viewed as the direct product of q unit circles, i.e. T q = S 1 ×...×S 1 ⊂ C q . Let the generic coordinates over T q be given by θ = (θ 1 , . . . , θ q ) and let ν denote the uniform Lebesgue measure over T q , that is (7) ν (dθ) = q i=1 ρ (dθ i ) where ρ is the Lebesgue measure over the unit circle. If we denote ·, · the standard scalar product between q dimensional vectors, the set of functions s n : T q → C, n = (n 1 , ..., n q ) ∈ Z q , defined by (8) s n (θ) := (2π) − q 2 exp (i n, θ ) , forms an orthonormal basis for the functional space L 2 (T q , ν), see again [Gra08]. Indeed, T 1 can be also identified as an equivalence class of the quotient space R/2πZ: a coordinate system on T q is therefore provided by the canonical representation in [0, 2π) q . Furthermore, {s n : n ∈ Z q } also coincides with the set of eigenfunctions associated to ∇ T q , the Laplace-Beltrami operator on T q , given by ∇ T q = q i=1 ∂ 2 ∂x 2 i , so that ∇ T q + ℓ 2 n s n (θ) = 0. Hence, the following orthonormality property holds (9) T q s n1 (θ) s n2 (θ) ν (dθ) = δ n2 n1 . Any f ∈ L 2 (T q , ν) can be represented by its harmonic expansion (10) f (θ) = n∈Z q a n s n (θ) , θ ∈ T q , where, for all n ∈ Z q , the complex-valued coefficients a n are the so-called Fourier coefficients, given by (11) a n = T q f (θ) s n (θ) ν (dθ) . Toroidal needlets. Let us introduce needlet-like wavelets on the q-dimensional torus and describe some of their properties. As already mentioned in Section 1, needlets were introduced in the literature on the q-dimensional sphere by Narcowich et al. in [NPW06a;NPW06b] (see also [MP11] for further details). The construction of an analogous wavelet system on T q can be roughly viewed as a natural extension of the standard needlets on S 1 to T q , so that the technical details of the construction will be omitted here for the sake of brevity. Fix a resolution level j ∈ N. There exists a set of cubature points {ξ j,k : k = 1, . . . , K j }, ξ j,k ∈ T q , associated to a set of cubature weights {λ j,k : k = 1, . . . , K j } (see e.g. [NPW06b]). Roughly speaking, T q can be represented as a partition of K j subregions, named pixels. Each pixel is centered on the corresponding ξ j,k and its area is given by λ j,k . Fix a scale parameter B > 1. Then, the q-dimensional toroidal needlets are defined by ψ j,k (θ) = λ j,k n∈Z q b B −j ℓ n s n (θ) s n (ξ j,k ) . The so-called window function b : R → R + satisfies the following properties: (1) b has compact support in B −1 , B ; (2) b ∈ C ∞ (R); (3) the so-called partition of unity property holds: for any c > 1, j∈N b 2 B −j c = 1. As a consequence, needlets are characterized by the following pivotal properties. In view of (1), for any j ∈ N, b B −j ℓ n is different from zero only over a finite subset of Z q . Let Λ q j = n ∈ Z q : ℓ n ∈ B j−1 , B j+1 , so that it holds that (12) ψ j,k (θ) = λ j,k n∈Λ q j b B −j ℓ n s n (θ) s n (ξ j,k ) . From (2), toroidal needlets enjoy a quasi-exponential localization property in the spatial domain, stated as follows: for any θ ∈ T q , M > 0, there exists c M > 0 such that \|ψ jk (θ)\| ≤ c M B q 2 j 1 + B q 2 j d (θ, ξ jk ) M , where d (θ, ξ j,k ) is the geodesic distance over T q . Loosely speaking, this property ensures that each needlet ψ j,k (θ) is not negligible only if θ ∈ E j,k . As a consequence, the following bounds on the L p norms of the toroidal needlets hold (see e.g. [NPW06a]): for any p ∈ [ 1, ∞) , there exist positive constants c p , C p , depending solely on p, such that (13) c p B jq( 1 2 − 1 p ) ≤ ψ j,k L p (T q ,dν) ≤ C p B jq( 1 2 − 1 p ) Finally, from (3), one can infer that the needlet system {ψ j,k : j ≥ 0, k = 1, . . . , K j } is a tight frame over T q : for any f ∈ L 2 (T q , ν), let the needlet coefficients be given by (14) β j,k = T q f (θ) ψ j,k (θ) ν (dθ) . Then, it holds that j,k \|β j,k \| 2 = f 2 L 2 (T q ,ν) . Therefore, the following reconstruction formula holds in the L 2 -sense f (θ) = j≥0 Kj k=1 β j,k ψ j,k (θ) , θ ∈ T q . Proofs of the main results Observe that, by Lemma 4.1, U j (t) can be rewritten as a double Poisson integral of a needlet-based kernel, namely (15) U j (t) = 1 2 T h j (θ 1 , θ 2 )N t (dθ 1 )N t (dθ 2 ) , where h j is given by (18) and the integration domain T is given by (19). In the upcoming proof of Theorem 1.4, the strategy will be to apply Proposition 2.6 to the normalized version of the double integral representation of U j (t) given by (15). Proof of Theorem 1.4. Starting with the first part of the statement and hence assuming that Condition 1.1 holds, one can write, in view of (24) and Lemma 4.2, that h j 4 L 4 (µ 2 t ) = 2R 2 t T q ×T q n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni s n1 (θ 1 ) s n1 (θ 2 ) s n2 (θ 1 ) s n2 (θ 2 ) s n3 (θ 1 ) s n3 (θ 2 ) s n4 (θ 1 ) s n4 (θ 2 ) f (θ 1 ) f (θ 2 ) dθ 1 dθ 2 = 2R 2 t n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni a n1−n2+n3−n4 a n1−n2+n3−n4 ∼ R 2 t ℓn 1 ,ℓn 2 ℓn 3 ,ℓn 4 ∈[B j−1 ,B j+1 ] c ℓn 1 c ℓn 2 c ℓn 3 c ℓn 4 4 i=1 b 2 B −j ℓ ni (ℓ n1−n2+n3−n4 + 1) −2α . Fixing ℓ n1 , ℓ n2 , ℓ n3 and summing over ℓ n4 from zero to infinity yields ∞ ℓn 4 =0 (ℓ n1−n2+n3−n4 + 1) −2α = O (1) as α > 1 2 . Hence, h j 4 L 4 (µ 2 t ) R 2 t ℓn 1 ,ℓn 2 ℓn 3 ∈[B j−1 ,B j+1 ] 1 = R 2 t B 3j . In conclusion, using Lemma 4.4 yields h j Var (U j (t)) 4 L 4 (µ 2 t ) O R −2 t B j . In order to compute h j ⋆ 1 2 h j L 2 (µt) , (24) can be used once more to obtain h j ⋆ 1 2 h j 2 L 2 (µt) = T 3 h j (θ 1 , θ 2 ) h j (θ 1 , θ 2 ) h j (θ 1 , θ 3 ) h j (θ 1 , θ 3 ) µ t (dθ 2 ) µ t (dθ 3 ) µ t (dθ 1 ) = 2R 3 t T q ×T q ×T q n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni s n1 (θ 1 ) s n1 (θ 2 ) s n2 (θ 1 ) s n2 (θ 2 ) s n3 (θ 1 ) s n3 (θ 3 ) s n4 (θ 1 ) s n4 (θ 3 ) f (θ 2 ) f (θ 3 ) f (θ 1 ) dθ 2 dθ 3 dθ 1 = 2R 3 t n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni a n1−n2+n3−n4 a n3−n4 a n1−n2 ∼ R 3 t ℓn 1 ,ℓn 2 ,ℓn 3 ,ℓn 4 ∈[B j−1 ,B j+1 ] c ℓn 1 c ℓn 2 c ℓn 3 c ℓn 4 4 i=1 b 2 B −j ℓ ni (ℓ n1−n2+n3−n4 + 1) −α (ℓ n3−n4 + 1) −α1 (ℓ n1−n2 + 1) −α R 3 t ℓn 1 ,ℓn 2 ,ℓn 3 ,ℓn 4 ∈[B j−1 ,B j+1 ] (ℓ n3−n4 + 1) −α (ℓ n1−n2 + 1) −α . In the case where α > 1, fixing n 1 , n 3 and summing over n 2 , n 4 yields h j ⋆ 1 2 h j 2 L 2 (µt) R 3 t B 2j . On the other hand, if 1 2 < α < 1, we use a Riemann sum argument to get h j ⋆ 1 2 h j 2 L 2 (µt) R 3 t B 2j(2−α) . In conclusion, we have h j Var (U j (t)) ⋆ 1 2 h j Var (U j (t)) 2 L 2 (µt) R −1 t if α > 1, R −1 t B 2j(1−α) if 1 2 < α < 1. Finally, using (24) again, it holds that h j ⋆ 1 1 h j 2 L 2 (µt) = T 4 h j (θ 1 , θ 3 ) h j (θ 2 , θ 3 ) h j (θ 1 , θ 4 ) h j (θ 2 , θ 4 ) µ t (dθ 3 ) µ t (dθ 4 ) µ t (dθ 1 ) µ t (dθ 2 ) = 2R 4 t T q ×T q ×T q ×T q n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni s n1 (θ 1 ) s n1 (θ 3 ) s n2 (θ 2 ) s n2 (θ 3 ) s n3 (θ 1 ) s n3 (θ 4 ) s n4 (θ 2 ) s n4 (θ 4 ) f (θ 1 ) f (θ 2 ) f (θ 3 ) f (θ 4 ) dθ 1 dθ 2 dθ 3 dθ 4 = 2R 4 t n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni a n1+n3 a n2+n4 a n2−n1 a n3−n4 R 4 t ℓn 1 ,ℓn 2 ,ℓn 3 ,ℓn 4 ∈[B j−1 ,B j+1 ] c ℓn 1 c ℓn 2 c ℓn 3 c ℓn 4 (ℓ n1+n3 + 1) −α (ℓ n2+n4 + 1) −α (ℓ n2−n1 + 1) −α (ℓ n3−n4 + 1) −α . In the case where α > 1, using the same summability argument as before, one gets h j ⋆ 1 1 h j 2 L 2 (µt) R 4 t ℓn 1 ,ℓn 2 ,ℓn 3 ,ℓn 4 ∈[B j−1 ,B j+1 ] (ℓ n2+n4 + 1) −α (ℓ n2−n1 + 1) −α (ℓ n3−n4 + 1) −α R 4 t B j . In the case where 1 2 ≤ α ≤ 1, one can use a Riemann sum argument once again to get h j ⋆ 1 1 h j 2 L 2 (µt) R 4 t ℓn 1 ,ℓn 2 ,ℓn 3 ,ℓn 4 ∈[B j−1 ,B j+1 ] (ℓ n2+n4 + 1) −α (ℓ n2−n1 + 1) −α (ℓ n3−n4 + 1) −α R 4 t B 4j(1−α) . Summing up yields h j Var (U j (t)) ⋆ 1 1 h j Var (U j (t)) 2 L 2 (µt) B −j if α > 1, B 2j(1−2α) if 1 2 < α < 1. Applying Proposition 2.6 concludes the first part of the proof. For the second part of the statement, assume that Condition 1.2 holds and observe that for any function g (n) = q m=1 g m (n m ) , there exists a constant C > 0 such that n∈Λj b 2 B −j ℓ n g (n) ≤ q m=1 nm∈[B j−1 ,B j+1 ] sup x∈[B j−1 ,B j+1 ] b (x) g m (n m ) ≤ C q m=1 nm∈[B j−1 ,B j+1 ] g m (n m ) .(16) Using (16) and following a similar procedure to the one used to prove the first part of the statement, one obtains h j 4 L 4 (µ 2 t ) = 2R 2 t n1,n2,n3,n4∈Λj 4 i=1 b 2 B −j ℓ ni a n1−n2+n3−n4 a n1−n2+n3−n4 ∼ R 2 t q m=1 (\|n 1,m − n 2,m + n 3,m − n 4,m \| + 1) −2α . For any m = 1, . . . , q, fixing n 1,m , n 2,m , n 3,m and sum over n 4,m from zero to infinity yields ∞ n4,m=0 (\|n 1,m − n 2,m + n 3,m − n 4,m + 1) −2α = O (1) as α > 1 2 . Hence, h j 4 L 4 (µ 2 t ) R 2 t ℓn 1 ,ℓn 2 ℓn 3 ∈[B j−1 ,B j+1 ] 1 = R 2 t B 3jq . In conclusion, using Lemma 4.4 h j Var (U j (t)) 4 L 4 (µ 2 t ) R −2 t B jq . A similar procedure is used also to establish an upper bound for h j ⋆ 1 2 h j L 2 (µt) and h j ⋆ 1 1 h j L 2 (µt) , leading to h j Var (U j (t)) ⋆ 1 2 h j Var (U j (t)) 2 L 2 (µt) R −1 t if α > 1, R −1 t B 2j(1−α) if 1 2 < α < 1 and h j Var (U j (t)) ⋆ 1 1 h j Var (U j (t)) 2 L 2 (µt) B −j if α > 1, B 2j (1−2α) if 1 2 < α < 1. Applying Proposition 2.6 concludes the proof. Auxiliary results The following lemma shows that the estimator U j (t) defined in (2) can be represented as a double Poisson integral. Lemma 4.1. Let U j (t) be given by (2). Then, it holds that (17) U j (t) = 1 2 T h j (θ 1 , θ 2 )N t (dθ 1 )N t (dθ 2 ) , where the symmetric function h j ∈ L 2 µ 2 t is given by h j (θ 1 , θ 2 ) := Kj k=1 ψ j,k (θ 1 ) ψ j,k (θ 2 ) 1 {T q 1 ×T q 1 } (θ 1 , θ 2 ) − ψ j,k (θ 1 ) ψ j,k (θ 2 ) 1 {T q 1 ×T q 2 } (θ 1 , θ 2 ) −ψ j,k (θ 1 ) ψ j,k (θ 2 ) 1 {T q 2 ×T q 1 } (θ 1 , θ 2 ) + ψ j,k (θ 1 ) ψ j,k (θ 2 ) 1 {T q 2 ×T q 2 } (θ 1 , θ 2 ) (18) and where the integration domain T is defined as (19) T := (T q 1 × T q 1 ) ∪ (T q 1 × T q 2 ) ∪ (T q 2 × T q 1 ) ∪ (T q 2 × T q 2 ) . Proof. Centering the measures N t appearing in the expression of U j (t) given by (2) yields U j (t) = Kj k=1   T q 1 ψ j,k (θ 1 )N t (dθ 1 ) − T q 2 ψ j,k (θ 1 )N t (dθ 1 ) 2 − T q 1 ψ 2 j,k (θ 1 )N t (dθ 1 ) − T q 2 ψ 2 j,k (θ 1 )N t (dθ 1 ) − 2 ψ j,k 2 L 2 (µt) . From expanding the square and using the product formula for Poisson multiple integrals stated in Proposition 2.3, it follows that U j (t) = Kj k=1 T q 1 ×T q 1 ψ j,k (θ 1 ) ψ j,k (θ 2 )N t (dθ 1 )N t (dθ 2 ) + T q 2 ×T q 2 (ψ j,k (θ 1 ) ψ j,k (θ 2 ))N t (dθ 1 )N t (dθ 2 ) − T q 1 ×T q 2 ψ j,k (θ 1 ) ψ j,k (θ 2 )N t (dθ 1 )N t (dθ 2 ) − T q 2 ×T q 1 ψ j,k (θ 1 ) ψ j,k (θ 2 )N t (dθ 1 )N t (dθ 2 ) = 1 2 T h j (θ 1 , θ 2 )N t (dθ 1 )N t (dθ 2 ) , where h j (·, ·) is the kernel given by (18) and T the domain defined in (19), as claimed. The following result provides an alternate expression of the kernel h j be defined in (18) used to reduce the complexity of the domain T defined in (19). Lemma 4.2. Let h j be given by (18). For all a, b = 1, 2 and for θ 1 , θ 2 ∈ T q , define (20) h ab,j (θ 1 , θ 2 ) = n∈Λj b 2 n B j s n (θ 1 ) s n (θ 2 )1 {T q a ×T q b } (θ 1 , θ 2 ) . Then, it holds (21) h j (θ 1 , θ 2 ) = 2 a,b=1 (−1) a+b h ab,j (θ 1 , θ 2 ) . Proof. Observe that, by using (12), one can write Kj k=1 ψ j,k (θ 1 ) ψ j,k (θ 2 )1 {T q a ×T q b } (θ 1 , θ 2 ) = n1,n2∈Λj b ℓ n1 B j b ℓ n2 B j 1 {T q a ×T q b } (θ 1 , θ 2 ) s n1 (θ 1 ) s n2 (θ 2 ) × Kj k=1 λ j,k s n1 (ξ j,k )s n2 (ξ j,k ) . Using the fact that Kj k=1 λ j,k s n1 (ξ j,k )s n2 (ξ j,k ) = T q s n1 (θ)s n2 (θ) ν (dθ) yields Kj k=1 ψ j,k (θ 1 ) ψ j,k (θ 2 )1 {T q a ×T q b } (θ 1 , θ 2 ) = n1,n2∈Λj b ℓ n1 B j b ℓ n2 B j 1 {T q a ×T q b } (θ 1 , θ 2 ) s n1 (θ 1 ) s n2 (θ 2 ) T q s n1 (θ)s n2 (θ) ν (dθ) = n∈Λj b 2 ℓ n B j 1 {T q a ×T q b } (θ 1 , θ 2 ) s n (θ 1 − θ 2 ) , where the last equality comes from the orthogonality property (9). Recalling the definition of h j , given by (18), concludes the proof. The following lemma provides an explicit expression for the variance of U j (t), which will be crucial in deriving lower bounds for said variance in Lemma 4.4. Lemma 4.3. Let U j (t) be given by (2). It holds that E (U j (t)) = 0 and E U 2 j (t) = 8R 2 t n1,n2∈Λj b 2 B −j ℓ n1 b 2 B −j ℓ n2 \|a n1−n2 \| 2 . Proof. Using the representation of U j (t) given by Lemma 4.1, it is easily seen that E (U j (t)) = 0. On the other hand, the isometry property of Poisson multiple integrals (see Proposition 2.1) yields E U j (t) 2 = 2 h j 2 L 2 (µ 2 t ) , where, using Lemma 4.2, it holds that 2 h j 2 L 2 (µ 2 t ) = 2 T   2 a,b=1 (−1) a+b h ab,j (θ 1 , θ 2 )   2 µ t (dθ 1 ) µ t (dθ 2 ) := V 1 + V 2 , with V 1 = 2 T 2 a,b=1 h 2 ab,j (θ 1 , θ 2 ) µ t (dθ 1 ) µ t (dθ 2 ) and V 2 = 4 T h 11,j (θ 1 , θ 2 ) h 22,j (θ 1 , θ 2 ) µ t (dθ 1 ) µ t (dθ 2 ) + 4 T h 12,j (θ 1 , θ 2 ) h 21,j (θ 1 , θ 2 ) µ t (dθ 1 ) µ t (dθ 2 ) +4 T 2 b =a,a=1 h aa,j (θ 1 , θ 2 ) (h ba,j (θ 1 , θ 2 ) + h ab,j (θ 1 , θ 2 )) µ t (dθ 1 ) µ t (dθ 2 ) . Observe that V 2 = 0 as each θ i , i = 1, 2, can only belong to one of the disjoint tori T q 1 or T q 2 . On the other hand, using the Fourier expansion of the density function given by f (θ) = n∈Z q a n s n (θ) , we get V 1 = 8R 2 t n1,n2∈Λj n3,n4∈Z b 2 ℓ n1 B j b 2 ℓ n2 B j a n3 a n4 T q s n1 (θ 1 ) s n2 (θ 1 ) s n3 (θ 1 ) dθ 1 T q s n1 (θ 2 ) s n2 (θ 2 ) s n4 (θ 2 ) dθ 2 = 8R 2 t n1,n2∈Λj n3,n4∈Z b 2 \|n 1 \| B j b 2 \|n 2 \| B j a n3 a n4 δ n3 n1−n2 δ n4 n1−n2 = 8R 2 t n1,n2∈Λj b 2 ℓ n1 B j b 2 ℓ n2 B j \|a n1−n2 \| 2 , as claimed. Finally, under the assumptions stated in Conditions 1.1 and 1.2, the following result provides lower bounds for the variance of U j (t). (23) E U j (t) 2 R 2 t B qj . Proof. Recalling that under Condition 1.1, the Fourier coefficients satisfy a n ∼ (ℓ n + 1) −α , with α > 1 2 in order to guaranty that n∈Z q \|a n \| 2 < +∞, yields E U j (t) 2 ∼ R 2 t n1,n2∈Λj b 2 B −j ℓ n1 b 2 B −j ℓ n2 (ℓ n1−n2 + 1) −2α R 2 t n1,n2∈Λ ′ j b 2 B −j ℓ n1 b 2 B −j ℓ n2 (ℓ n1−n2 + 1) −2α , where Λ ′ j = n ∈ Z q : ℓ n ∈ (B ′ ) j−1 , (B ′ ) j+1 with B ′ < B such that b B −j n is bounded away from zero. Now, observe that, for any given real valued function g, there exists a set of coefficients {c ℓn } so that (24) n∈Λj g (ℓ n ) = ℓn∈[B j−1 ,B j+1 ] c ℓn g (ℓ n ) . Loosely speaking, 0 ≤ c ℓn < ∞ denotes the number of possible combinations of components of different n ∈ Λ j corresponding to the same ℓ n . Dropping the terms for which n 1 = n 2 in the above sum, we get E U j (t) 2 R 2 t B j , as claimed. Under Condition 1.2, the Fourier coefficients are such that a n ∼ q m=1 \|n (m) \| + 1 −α , with α > 1 2 . 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World Scientific, 2001. A W Van Der, Vaart, Asymptotic statistics. CambridgeA. W. van der Vaart. Asymptotic statistics. Cambridge, 1998. . address: [email protected] Mall. 111Boston University, Department of Mathematics and StatisticsBoston University, Department of Mathematics and Statistics, 111 Cummington Mall, Boston, MA 02215, USA E-mail address: [email protected]	[]
[ "N. CAI AND M. HAYASHI: SECURE NETWORK CODE FOR ADAPTIVE AND ACTIVE ATTACKS 1 Secure Network Code for Adaptive and Active Attacks with No-Randomness in Intermediate Nodes", "N. CAI AND M. HAYASHI: SECURE NETWORK CODE FOR ADAPTIVE AND ACTIVE ATTACKS 1 Secure Network Code for Adaptive and Active Attacks with No-Randomness in Intermediate Nodes" ]	[ "Fellow, IEEENing Cai ", "Fellow, IEEEMasahito Hayashi " ]	[]	[]	We analyze the security for network code when the eavesdropper can contaminate the information on the attacked edges (active attack) and can choose the attacked edges adaptively (adaptive attack). We show that active and adaptive attacks cannot improve the performance of the eavesdropper when the code is linear. Further, we give an non-linear example, in which an adaptive attack improves the performance of the eavesdropper. We derive the capacity for the unicast case and the capacity region for the multicast case or the multiple multicast case in several examples of relay networks, beyond the minimum cut theorem, when no additional random number is allowed as scramble variables in the intermediate nodes.IndexTerms secrecy analysis, secure network coding, adaptive attack, active attack I. INTRODUCTION Secure network coding is a method securely transmitting information from the authorized sender to the authorized receiver. Cai and Yeung [1], [2], [3] discussed the secrecy for the malicious adversary, Eve, wiretapping a subset E E of all channels in the network. The papers [4], [5], [6], [7], [8], [9], [10] developed several types of secure network coding. The papers [11], [12], [13], [14] showed the existence of a secrecy code that universally works for any types of eavesdroppers under the size constraint of E E . In particular, the papers [13], [14] constructed it by using the universal hashing lemma [15], [16], [17]. Further, the papers [11], [12],[18]evaluated errors when the information on a part of network is changed, but they evaluated the secrecy only when the information on a part of network is not changed or Eve did not know the replaced information. The recent paper[19]discussed the secrecy as well as the error when Eve contaminates the eavesdropped information and knows the replaced information. (For the detailed relation, see[19,Remark 8].) The effects of Eve's contamination depend on the type of the network code. When the code is linear, the contamination does not improve her performance. However, when the code is not linear, there exists only one example where the contamination improves her performance[19].Despite these developments, there are still some problems in existing studies. Although these existing studies achieved the optimal rate with secrecy condition, their optimality relies on the minimum cut theorem. That is, they assumed that the eavesdropper may choose any r-subset channels to access, and did not address another type of conditions for the eavesdropper. For example, the studies [11], [12], [13],[14]optimized only the codes in the source and terminal nodes and did not optimize the coding operations on the intermediate nodes. Also, in other existing studies, the intermediate nodes do not have as complicated codes as the source and terminal nodes. In this paper, to achieve the optimal rate beyond the minimum cut theorem, we address the optimization of the coding operations on the intermediate nodes as well as on the source and terminal nodes.Further, we consider a new type of attacks, adaptive attacks. Assume that distinct numbers are assigned to the edges, and the communication on edges are done in the decreasing order for the assigned numbers. Usually, Eve cannot decide the edges to be attacked depending on the previous observation. Now, we allow Eve to choose the edges to be attacked based on the previous observations. Indeed, the channel discrimination, it is known that such an adaptive strategy does not improve the asymptotic performance[20]. Then, we find two characteristics for adaptive attacks, which are similar to the case of active attacks. First, we find a non-linear code where an adaptive Ning Cai is with the N. CAI AND M. HAYASHI: SECURE NETWORK CODE FOR ADAPTIVE AND ACTIVE ATTACKS 2 attack significantly improves Eve's performance. Using this characteristic, we find an example of a non-asymptotic network model, which has no secure code for adaptive attacks, but has no secure code for conventional attacks. Second, we show that any adaptive attack cannot improve Eve's performance when the code is linear. Using this fact, we derive the asymptotic performance in several typical network models in the following way when Eve is allowed to use adaptive and active attacks.In this paper, we discuss the asymptotic securely transmittable rate over the above attacks not only for a unicast network but also for a multiple multicast network, in which, multiple senders are intended to send their different messages to different multiple receivers. Under these settings, we define the capacity and the capacity regions for given network models, and calculate them in several examples. For the definition, we define two types of capacity regions depending on the requirement on the code on the intermediate nodes. Usually, a secure network code employs scramble random numbers, which need to be physical random numbers different from pseudo random numbers. In the first capacity region, we allow each node to introduce new scramble random numbers unlimitedly. Here, the scramble random numbers of each node are not shared with other nodes and should be independent of random variables in other players and other nodes before starting the transmission. In the second capacity region, only source nodes are allowed to employ scramble random numbers due to the following reason. To realize physical random numbers as scramble random numbers, we need a physical device. If the physical random number has sufficient quality, the physical device is expensive and/or consumes a non-negligible space because it often needs high level quantum information technologies with advanced security analysis [21],[22]. It is not so difficult to prepare such devices in the source side. However, it increases the cost to prepare devices in the intermediate nodes because networks with such devices require more complicated maintenance than a conventional network. Therefore, from the economical reason, it is natural to impose this constraint to our network code. Unfortunately, only a few papers [23], [24], [25] discussed such a restriction. Hence, this paper addresses the difference between the capacities with and without such a restriction by introducing the no-randomness capacity and the full-randomness capacity. Further, as an intermediate case, by introducing the limited-randomness capacity, we can consider the case when the number of available scramble random numbers in each intermediate node is limited to a certain amount. Then, the relation between our capacities and the existing studies is summarized asTable I. In addition, for both types of capacities and capacity regions, we define the linear codes version, in which, our codes are limited to linear codes. We also show that the linear version of capacities and capacity regions are the same as the original capacities and capacity regions under the above examples because the optimal rate and rate regions in the original setting can be attained by linear codes.The remaining parts of this paper is organized as follows. Section II gives the formulation of our network model. Section III gives an example of network model, in which, an adaptive attack efficiently improves Eve's performance. To discuss the asymptotic setting, Section IV defines the capacity region. Section V discusses the relay network model and derives its capacity. Section VII discusses the homogenous multicast network model and derives its capacity. Section VIII discusses the homogenous multiple multicast network model and derives its capacity. In Section VI, we give an important lemma, which is used in the converse part in the above models.	10.1109/tit.2019.2957078	[ "https://arxiv.org/pdf/1712.09035v4.pdf" ]	13,791,997	1712.09035	c695c5f1a9fad67185224bcea69fe22a64278c80	N. CAI AND M. HAYASHI: SECURE NETWORK CODE FOR ADAPTIVE AND ACTIVE ATTACKS 1 Secure Network Code for Adaptive and Active Attacks with No-Randomness in Intermediate Nodes Fellow, IEEENing Cai Fellow, IEEEMasahito Hayashi N. CAI AND M. HAYASHI: SECURE NETWORK CODE FOR ADAPTIVE AND ACTIVE ATTACKS 1 Secure Network Code for Adaptive and Active Attacks with No-Randomness in Intermediate Nodes We analyze the security for network code when the eavesdropper can contaminate the information on the attacked edges (active attack) and can choose the attacked edges adaptively (adaptive attack). We show that active and adaptive attacks cannot improve the performance of the eavesdropper when the code is linear. Further, we give an non-linear example, in which an adaptive attack improves the performance of the eavesdropper. We derive the capacity for the unicast case and the capacity region for the multicast case or the multiple multicast case in several examples of relay networks, beyond the minimum cut theorem, when no additional random number is allowed as scramble variables in the intermediate nodes.IndexTerms secrecy analysis, secure network coding, adaptive attack, active attack I. INTRODUCTION Secure network coding is a method securely transmitting information from the authorized sender to the authorized receiver. Cai and Yeung [1], [2], [3] discussed the secrecy for the malicious adversary, Eve, wiretapping a subset E E of all channels in the network. The papers [4], [5], [6], [7], [8], [9], [10] developed several types of secure network coding. The papers [11], [12], [13], [14] showed the existence of a secrecy code that universally works for any types of eavesdroppers under the size constraint of E E . In particular, the papers [13], [14] constructed it by using the universal hashing lemma [15], [16], [17]. Further, the papers [11], [12],[18]evaluated errors when the information on a part of network is changed, but they evaluated the secrecy only when the information on a part of network is not changed or Eve did not know the replaced information. The recent paper[19]discussed the secrecy as well as the error when Eve contaminates the eavesdropped information and knows the replaced information. (For the detailed relation, see[19,Remark 8].) The effects of Eve's contamination depend on the type of the network code. When the code is linear, the contamination does not improve her performance. However, when the code is not linear, there exists only one example where the contamination improves her performance[19].Despite these developments, there are still some problems in existing studies. Although these existing studies achieved the optimal rate with secrecy condition, their optimality relies on the minimum cut theorem. That is, they assumed that the eavesdropper may choose any r-subset channels to access, and did not address another type of conditions for the eavesdropper. For example, the studies [11], [12], [13],[14]optimized only the codes in the source and terminal nodes and did not optimize the coding operations on the intermediate nodes. Also, in other existing studies, the intermediate nodes do not have as complicated codes as the source and terminal nodes. In this paper, to achieve the optimal rate beyond the minimum cut theorem, we address the optimization of the coding operations on the intermediate nodes as well as on the source and terminal nodes.Further, we consider a new type of attacks, adaptive attacks. Assume that distinct numbers are assigned to the edges, and the communication on edges are done in the decreasing order for the assigned numbers. Usually, Eve cannot decide the edges to be attacked depending on the previous observation. Now, we allow Eve to choose the edges to be attacked based on the previous observations. Indeed, the channel discrimination, it is known that such an adaptive strategy does not improve the asymptotic performance[20]. Then, we find two characteristics for adaptive attacks, which are similar to the case of active attacks. First, we find a non-linear code where an adaptive Ning Cai is with the N. CAI AND M. HAYASHI: SECURE NETWORK CODE FOR ADAPTIVE AND ACTIVE ATTACKS 2 attack significantly improves Eve's performance. Using this characteristic, we find an example of a non-asymptotic network model, which has no secure code for adaptive attacks, but has no secure code for conventional attacks. Second, we show that any adaptive attack cannot improve Eve's performance when the code is linear. Using this fact, we derive the asymptotic performance in several typical network models in the following way when Eve is allowed to use adaptive and active attacks.In this paper, we discuss the asymptotic securely transmittable rate over the above attacks not only for a unicast network but also for a multiple multicast network, in which, multiple senders are intended to send their different messages to different multiple receivers. Under these settings, we define the capacity and the capacity regions for given network models, and calculate them in several examples. For the definition, we define two types of capacity regions depending on the requirement on the code on the intermediate nodes. Usually, a secure network code employs scramble random numbers, which need to be physical random numbers different from pseudo random numbers. In the first capacity region, we allow each node to introduce new scramble random numbers unlimitedly. Here, the scramble random numbers of each node are not shared with other nodes and should be independent of random variables in other players and other nodes before starting the transmission. In the second capacity region, only source nodes are allowed to employ scramble random numbers due to the following reason. To realize physical random numbers as scramble random numbers, we need a physical device. If the physical random number has sufficient quality, the physical device is expensive and/or consumes a non-negligible space because it often needs high level quantum information technologies with advanced security analysis [21],[22]. It is not so difficult to prepare such devices in the source side. However, it increases the cost to prepare devices in the intermediate nodes because networks with such devices require more complicated maintenance than a conventional network. Therefore, from the economical reason, it is natural to impose this constraint to our network code. Unfortunately, only a few papers [23], [24], [25] discussed such a restriction. Hence, this paper addresses the difference between the capacities with and without such a restriction by introducing the no-randomness capacity and the full-randomness capacity. Further, as an intermediate case, by introducing the limited-randomness capacity, we can consider the case when the number of available scramble random numbers in each intermediate node is limited to a certain amount. Then, the relation between our capacities and the existing studies is summarized asTable I. In addition, for both types of capacities and capacity regions, we define the linear codes version, in which, our codes are limited to linear codes. We also show that the linear version of capacities and capacity regions are the same as the original capacities and capacity regions under the above examples because the optimal rate and rate regions in the original setting can be attained by linear codes.The remaining parts of this paper is organized as follows. Section II gives the formulation of our network model. Section III gives an example of network model, in which, an adaptive attack efficiently improves Eve's performance. To discuss the asymptotic setting, Section IV defines the capacity region. Section V discusses the relay network model and derives its capacity. Section VII discusses the homogenous multicast network model and derives its capacity. Section VIII discusses the homogenous multiple multicast network model and derives its capacity. In Section VI, we give an important lemma, which is used in the converse part in the above models. Active Adaptive Node Linearity attack attack randomness Papers [1], [2], [3], [6] not allowed not allowed not allowed scalar Paper [4] not allowed not allowed allowed non-linear Papers [9], [10], [23], [24], [25] not allowed not allowed allowed scalar Papers [5], [8] not allowed not allowed not allowed scalar Papers [13], [14], [11] not allowed not allowed not allowed vector Papers [12], [35] semi active attack not allowed not allowed vector Paper [19] allowed not allowed not allowed vector/ non-linear Our non-linear example not allowed allowed not allowed non-linear No-randomness capacity allowed allowed not allowed vector Limited-randomness capacity allowed allowed partially vector Full-randomness capacity allowed allowed allowed vector Node randomness expresses the random number generated in intermediate nodes, which is independent of the variables in other nodes and other players before starting the transmission. Linearity expresses whether the code is linear or not. When it is linear, the column expresses which linearity condition is imposed, scalar or vector linearity. These two kinds of linearity conditions are explained in Section V-E. Semi active attack means that Eve injects the noise in several nodes and eavesdrops several nodes, but she estimates the message only from the eavesdropped information on the node without use of the information of the noise. For the detailed relation for active attack, see Remark 8 of [19]. denote a network code by Φ. We denote the cardinality of the message M i,j by \|Φ\| i,j . When a = 1, we simply denote it by \|Φ\| j . In particular, when a = b = 1, we simply denote it by \|Φ\|. We denote the set of codes by C 0 . Now, we consider two conditions for our network code Φ. (C1) [Linearity] Any message, any scramble random number, and information on any edge can be given as elements of vector spaces over the finite field F q All of the conversions in source, intermediate, and terminal nodes are linear over F q , i.e., they are written as matrices whose entries are elements of F q . Then, the code is called linear with respect to F q 1 . Here, to apply the linearity condition, we choose a subset of X whose cardinality is a power of q. Then, the information on any edge can be given as an element of vector space over the finite field F q . While all edges sent the information on the same set X , the above subset might depend on the edge. This is because the dimension of the information to be sent depends on the edge in general. Since the cardinality of the set X is an arbitrary number, we can apply this linearity condition to the case when X is a given as the n-th power of a certain set. group is composed of one node, as a typical example, we assume that the node in i-th group can use γ i random numbers per transmission. Next, we define Eve's attack. The conventional attack is modeled by a collection A 0 of subset of [ ]. That is, in the conventional attack, Eve chooses a subset s ∈ A 0 , and eavesdrops the edges in the subset s. This types of attack is called a deterministic attack. Hence, the set of deterministic attacks is identified with A 0 . The following discussion depends on the collection A 0 of subset of [ ]. That is, our problem is characterized by the structure of network and the collection A 0 . Also, Eve can randomly choose her choice s. Such an attack is written as a probability distribution P S and is called a randomized attack. We denote the set of randomized attacks byĀ 0 . In this paper, we allow Eve to adaptively choose the edges to be eavesdropped. For simplicity, we assume that all subsets in the collection A 0 have the same cardinality ζ. While Eve is allowed to eavesdrop ζ edges, she can adaptively choose them as follows. She chooses the first edge α 1 ∈ [ ] to be eavesdropped, and obtains the information Z 1 ∈ X on the edge. Based on the information Z 1 , she chooses the second edge α 2 (Z 1 ) ∈ [ ] to be eavesdropped and obtains the information Z 2 ∈ X on the edge. In this way, based on the information Z 1 , . . . , Z j−1 , she chooses the j-th edge α j (Z 1 , . . . , Z j−1 ) ∈ [ ] to be eavesdropped and obtains the information Z j ∈ X on the edge. Since the choice of the set α = {α 1 , . . . , α ζ } of attacked edges is given as a function of ζ − 1 outcomes z 1 , . . . , z ζ−1 , it is often written as α(z 1 , . . . , z ζ−1 ) to clarify this point. Here, for any data z 1 , . . . , z ζ−1 , α(z 1 , . . . , z ζ−1 ) is required to belong to the family A 0 . This type of attack is called a general adaptive attack. In this type of attack, the order of eavesdropped edges has no relation with the numbers assigned to the edges. A general adaptive attack α = (α 1 , . . . , α ζ ) is called a time-ordered adaptive attack when α 1 < α 2 (z 1 ) < . . . < α ζ (z 1 , . . . , z ζ−1 ). Although a general adaptive attack has less practical meaning than a time-ordered adaptive attack, we consider a general adaptive attack due to its mathematical simplicity. We denote the sets of time-ordered adaptive attacks and general adaptive attacks by A 1 and A 2 , respectively. The sets of their randomizations are written as A 1 andĀ 2 , respectively. Now, we identify the set of deterministic attacks with the collection A 0 . Considering a constant function α, which does not depend on ζ − 1 outcomes z 1 , . . . , z ζ−1 , we can consider the collection A 0 as a subset of A 1 while A 1 ⊂ A 2 . Next, we consider a more powerful attack than a time-ordered adaptive attack α = (α 1 , . . . , α ζ ). Although Eve decides the eavesdropped edges in the same way as the time-ordered adaptive attack α = (α 1 , . . . , α ζ ), she is allowed to change the information Z j on the j-th eavesdropped edge α j (Z 1 , . . . , Z j−1 ) to β j (Z 1 , . . . , Z j ), which is a function of her observations Z 1 , . . . , Z j . This kind of attack is called an adaptive and active attack and is written as the pair (α, β) of α = (α 1 , . . . , α ζ ) and β = (β 1 , . . . , β ζ ). We denote the set of adaptive and active attacks (such functions) by A 3 . The sets of the randomizations are written asĀ 3 . When α does not depends on her observations Z 1 , . . . , Z ζ−1 , α is a deterministic attack and the pair (α, β) is called an active attack. Indeed, when active attack is made, the information on the network is changed. However, in this paper, we do not care about the correctness of the recovered information when active attack is made. We consider the correctness in the decoding only when no active attack is made, i.e., we discuss only the secrecy when active attack is made. Hence, we have the relations A 0 ⊂ A 1 ⊂ A 2 , A 0 ⊂ A 1 ⊂ A 3 , andĀ 0 ⊂Ā 1 ⊂Ā 2 ,Ā 0 ⊂Ā 1 ⊂Ā 3 . We also assume that there is no error in any edges except for the eavesdropped edge. Under a code Φ and an attack (α, β) ∈ A 3 , we denote the mutual information between the messages and Eve's observations Z = (Z 1 , . . . , Z ζ ) by I(M ; Z) Φ,(α,β) . Also, under an attack α ∈ A 2 we denote it by I(M ; Z) Φ,α . In addition, an attack P ∈Ā i with i = 0, 1, 2, 3, we denote it by I(M ; Z) Φ,P . Then, for any attack P ∈Ā i for i = 0, 1, 2, 3 and a network code Φ, we can choose an attack x ∈ A i such that I(M ; Z) Φ,P ≥ I(M ; Z) Φ,x . That is, we have min P ∈Ā i I(M ; Z) Φ,P = min x∈A i I(M ; Z) Φ,x(1) for i = 0, 1, 2, 3. First, we consider the case when the network code is not necessarily linear. Then, we have the following theorem 2 when Y i expresses the information on the edge i. Theorem 1. Assume that a network code Φ satisfies the following condition. Given an arbitrary element s = {s 1 , . . . , s ζ } ∈ A 0 , we have H(M \|Y s1 = z 1 , . . . , Y sζ = z ζ ) = H(M \|Y s1 , . . . , Y sζ ) s(2) for any element (z 1 , . . . , z ζ ). Then, any general adaptive attack α ∈ A 2 satisfies I(M ; Z) Φ,α ≤ max s∈A 0 I(M ; Z) s .(3) Theorem 1 will be shown in the next subsection. Since I(M ; Z) Φ,s = 0 for any s ∈ A 0 implies the condition (2), we have the following corollary. Corollary 1. When the relation I(M ; Z) Φ,s = 0(4) holds for an arbitrary element s ∈ A 0 , any general adaptive attack α ∈ A 2 satisfies I(M ; Z) Φ,α = 0.(5) This corollary guarantees that perfect security for any deterministic attack (4) implies perfect security for any general adaptive attack (5) without the linearity condition. Notice that the mutual information leaked to wiretapper is not zero in the counter example given in Section III. In the case of linear network codes, we have the following lemma, which will be shown in the next subsection. Lemma 1. Let M be the message and L be the scramble random variable. We assume that they are subject to the independent uniform distribution. For a linear function f 1 , we define the variable X := f 1 (M, L). We choose a linear function g = (g 1 , g 2 ) such that g(x) ∈ f −1 1 (x). Then, P M,X (m, x) = P M,X (m − g 1 (x), 0).(6) When the message M and the scramble random variable L are subject to the independent uniform distribution, applying Lemma 1 to the case when X = (Y s1 , . . . , Y sζ ), we have H(M \|Y s1 = z 1 , . . . , Y sζ = z ζ ) = H(M \|Y s1 = 0, . . . , Y sζ = 0),(7) which implies the condition (2). Hence, Theorem 1 guarantees the following theorem. Theorem 2. Assume that a network code Φ is linear with respect to a certain finite field F q . When the message M and the scramble random variable L are subject to the independent uniform distribution, any general adaptive attack α ∈ A 2 satisfies (3). Further, we have the following proposition. I(M ; Z) Φ,(α,β) = I(M ; Z) Φ,α .(8) Although the paper [19] shows Proposition 1 only for an active attack, the proof can be extended to an adaptive and active attack. That is, the reduction from an adaptive and active attack (α, β) ∈ A 3 to an adaptive attack α ∈ A 2 can be shown in the same way as [19,Theorem 1]. Therefore, when Φ is a linear code, combing the above fact and (1), we find the relations min α∈A 3 I(M ; Z) Φ,β = min (α,β)∈A 2 I(M ; Z) Φ,β = min s∈A 0 I(M ; Z) Φ,s .(9) That is, when a network code is linear, we can restrict Eve's attacks to deterministic attacks. Remark 1. Here, we remark the difference between our adaptive attack and the adaptive attack in [34]. The paper [34] considers the following attack when the code has block length n and the sender sends information to the receiver n times. The eavesdropper can change the nodes to be attacked on the i-th transmission by using the information obtained by the previous attacks. However, in our setting, the eavesdropper can change the node to be attacked during one transmission from the sender to the receiver. H(M \|Z) Φ,α = z1 z2 · · · zζ P Yα 1 ,Y2(z 1 ),...,Yα ζ (z 1 ,z 2 ,...,z ζ−1 ) H(M \|Y α1 = z 1 , Y α2(zs1) = z 2 , . . . , Y αζ(z1,z2,...,zζ−1) = z ζ ) = z1 z2 · · · zζ P Yα 1 ,Y2(z 1 ),...,Yα ζ (z 1 ,z 2 ,...,z ζ−1 ) H(M \|Y α1 , Y α2(zs1) , . . . , Y αζ(z1,z2,...,zζ−1) ) ≥ min s∈A 0 H(M \|Y s1 , . . . , Y sζ ).(10) This relation implies (3). Proof of Lemma 1: Given x, m, we have {l\|f 1 (m, l) = x} = {l\|f 1 (m − g 1 (x), l − g 2 (x)) = 0}.(11) So, we have \|{l\|f 1 (m, l) = x}\| = \|{l\|f 1 (m − g 1 (x), l) = 0}\|.(12) Hence, we have (6). III. NETWORK WITH POWERFUL ADAPTIVE ATTACK In this section, to consider when adaptive attack is more powerful than deterministic attack, we address the single shot setting, in which, the sender sends only one element of F p , which is called the scalar linearity. Although this section addresses the scalar linearity, Theorem 1 holds under vector linearity. It is known that there exists a linear imperfectly secure code over a finite field F q of a sufficiently large prime power q when Eve may access a subset of channels that does not contain a cut between Alice and Bob even when the linear code does not employ private randomness in the intermediate nodes [36] 3 . Theorem 1 guarantees that such a linear code is still imperfectly secure even for active and adaptive attack over the same network. However, it is not clear whether there exists such a linear imperfectly secure code over a finite field F p of prime p. We consider this problem over the finite field F 2 in order to investigate the importance of the linearity condition in Theorem 1. The previous paper [19,Section VII] showed that there exists no imperfectly secure code over active attacks under a toy network while there exists a imperfectly secure code over deterministic attacks. In that network model, non-linear code realizes the imperfect security over active attacks. In this section, we show that there exists no imperfectly secure code over adaptive attacks in the same network model. The toy network model given in [19, Section VII] is the network of (1), e(4)}}. We adopt a imperfect security criterion in this section. When Z E is Eve's information and I(M ; Z E ) < 1 for all of Eve's possible attacks, we say that the code is imperfectly secure. Otherwise, it is called insecure. That is, when there exists no functionψ such thatψ(Z E ) = M , our code is imperfectly secure. We consider the case when the sender transmits only the binary message M ∈ F 2 and any edge can transmit only a binary information. As shown in [19, Theorem 4 of Section VII], there is no imperfectly secure linear code over finite field F p with prime p for deterministic attacks. In other words, no linear code over finite field F p can realize the situation that Eve cannot recover the message M perfectly with deterministic attack. Now, we prepare the binary uniform scramble random variable L ∈ F 2 . We consider the following code. The encoder φ is given as Y 1 Y 3 Y 2 Y 4 source terminateY 1 := L, Y 2 := M + L.(13) Then, we consider non-linear code in the intermediate node as Y 3 := Y 1 (Y 2 + Y 1 ) = Y 1 (Y 2 + 1),(14)Y 4 := (Y 1 + 1)(Y 2 + Y 1 ) = (Y 1 + 1)Y 2 .(15) The decoder ψ is given as ψ(Y 3 , Y 4 ) := Y 3 + Y 4 . Since Y 3 and Y 4 are given as follows under this code; Y 3 = LM, Y 4 = LM + M,(16) the decoder can recover M nevertheless the value of L. The leaked information for the deterministic attack is calculated as follows. As shown in [19,Appendix B], the mutual information and the l 1 norm security measure of these cases are calculated to I(M ; Y 1 , Y 3 ) = I(M ; Y 1 , Y 4 ) =I(M ; Y 2 , Y 3 ) = I(M ; Y 2 , Y 4 ) = 1 2 ,(17)d 1 (M \|Y 1 , Y 3 ) = d 1 (M \|Y 1 , Y 4 ) =d 1 (M \|Y 2 , Y 3 ) = d 1 (M \|Y 2 , Y 4 ) = 1 2 ,(18) where the l 1 norm security measure d 1 (X\|Y ) is defined as d 1 (X\|Y ) := y x \| 1 \|X \| P Y (y) − P XY (xy)\| by using the cardinality \|X \| of the set of outcomes of the variable X. In this section, we choose the base of the logarithm to be 2. Therefore, we find that this code is secure for deterministic attacks. That is, we find that there exists a secure code over deterministic attacks. Further, as shown in [19, Lemma 3 of Section VII], when Eve cannot recover the message M perfectly with any deterministic attack in the code, the network code is limited to this code or a code equivalent to this code. This fact shows that there exists no imperfectly secure code over active attacks. Now, we show that there exists no imperfectly secure code even for adaptive attacks without active modification. Due to the above observation, it is sufficient to show that there exists an adaptive attack to recover the message M for the above given code. Here, we give two types of adaptive attacks to recover the message M as follows. (i) First, Eve eavesdrops e(1). When Y 1 = 1, she eavesdrops e(3). Then, she recovers M as Y 3 = Y 2 + 1 = Y 2 + Y 1 = M . When Y 1 = 0, she eavesdrops e(4) Then, she recovers M as Y 4 = Y 2 = Y 2 + Y 1 = M . (ii) First, Eve eavesdrops e(2). When Y 2 = 1, she eavesdrops e(4). Then, she recovers M as Y 4 = Y 1 + 1 = Y 1 + Y 2 = M . When Y 2 = 0, she eavesdrops e(3) Then, she recovers M as Y 3 = Y 1 = Y 1 + Y 2 = M . Therefore, we find that this code is not imperfectly secure even for adaptive attacks without active modification. That is, there exists no imperfectly secure code over adaptive attacks in this network model. This fact shows that an adaptive attack is powerful for this kind of non-linear code as an active attack even when it has no active modification. The discussion in this section is summarized as Table II. Next, given a network and the collection A 0 , we consider the capacity and the capacity region depending on the restrictions on the codes. Due to (1), in the following, we do not consider randomization of Eve's attack. We assume that each edge transmits {1, . . . , d} n when we use channel at n times, where the number n is called the block-length. Given integers n and d, we apply the formulation (including the linearity) given in Section II-A to the case when X is given as {1, . . . , d} n . In this sense, the linearity condition (C1) is defined with block-length n, and Theorem 1 can be applied in this discussion. Then, dependently of the block length n, we denote A i and C 0 by A i n and C 0 n , respectively, although the collection A 0 n does not depend on n. First, we focus only on an adaptive attack α ∈ A 2 n . Since there is no noise, we denote the decoding error probability depends only on our code Φ ∈ C 0 n . Hence, we denote it by P e (Φ). Then, we impose the following two conditions to our code Φ ∈ C 0 n . (C3) [Reliability] The relation P e (Φ) = 0. (C4) [Secrecy] The relation I(M ; Z) Φ,α = 0 holds for α ∈ A 2 n . We denote the set of codes satisfying the above two conditions by C 1 n . Additionally, we denote the set of codes satisfying the no-randomness condition (C2) as well as these two conditions by C 2 n . In the unicast case, i.e., the case with a = b = 1, we define the full-randomness capacity C 1 and the no-randomness capacity C 2 as C i := sup n sup Φ∈C i n 1 n log \|Φ\|, i = 1, 2.(19) Here, we should remark that we impose no linearity condition for our code. From the definition, we have the relation C 2 ≤ C 1 .(20) In the multiple multicast case, we define the full-randomness capacity region C 1 and the no-randomness capacity region C 2 as C i := sup n sup Φ∈C i n {( 1 n log \|Φ\| i,j ) i,j }, i = 1, 2.(21) Similar to (20), we have the relation C 2 ⊂ C 1 .(22) Next, we consider the case when each node has limited randomness, which is given as the condition (C2'). Since this generalized case is complicated, we discuss this generalized setting only with the unicast case. Further, we suppose that each group is composed of one node. Then, as in the condition (C2'), we assume that the node in i-th group can use γ i random numbers T i per transmission. We denote the set of codes satisfying this condition with length n by C n [(γ i ) i ]. Then, we define the capacity C[(γ i ) i ] with limited randomness as C[(γ i ) i ] := sup n sup Φ∈Cn[(γi)i] 1 n log \|Φ\|.(23) To clarify the effect by the linearity restriction, we denote the capacity and capacity region by C i,L and C i,L , respectively when the linearity restriction (C1) is imposed to our codes. Then, we have the relation C i,L ≤ C i and C i,L ⊂ C i . Also, the capacity with limited randomness with linearity restriction (C1) to our codes is denoted by C[(γ i ) i ] L . Restricting Eve's attack to the deterministic attacks A 0 n , we define the above type of capacities and capacity regions, which are denoted by C i,D , C i,L,D , C[(γ i ) i ] D , C[(γ i ) i ] L,D , C i,D and C i,L,D , respectively. Then, we have the relations C i,L,D = C i,L , C i,D ≥ C i , C i,L,D = C i,L , C i,D ⊃ C i , and the similar relations. Now, we address the case when an adaptive and active attack β ∈ A 3 n is allowed for Eve. In this case, we replace the condition (C4) by the following condition; (C4') [Secrecy] The relation I(M ; Z) Φ,β = 0 holds for β ∈ A 3 n . However, we do not replace (C3) by the following robustness condition; P e (Φ, β) = 0 for ∀β ∈ A 3 n ,(24) where P e (Φ, β) is the decoding error probability with our code Φ when Eve makes the attack β. This situation can be justified in the following way when free public channel with no error is available. In this case, to communicate each other securely, they need to share secret random variables. To generate secret random variables, they send secret random variables via the secure network coding. The secrecy of the generated random variables is guaranteed by the secrecy condition (C4). That is, condition (4) is definitely needed. However, the robustness condition (24) is not necessary because they can check whether the transmitted random number is correct by the error verification test with the public channel after the transmission [27, Section VIII] [28, Step 4 of Protocol 2]. Hence, we impose the condition (C3) instead of (24). Replacing the condition (C4) by the condition (C4'), we define the above type of capacities and capacity regions, which are denoted by C i,AC , C i,L,AC , C[(γ i ) i ] AC , C[(γ i ) i ] L,AC , C i,AC and C i,L,AC , respectively. Then, we have the relations C i,L,AC = C i,L , C i,D ≥ C i,AC , C i,L,AC = C i,L , C i,D ⊃ C i,AC , and the similar relations. In summary, for each i = 1, 2, we have C i,L,AC = C i,L = C i,L,D ⊂ C i,AC ⊂ C i ⊂ C i,D .(25) That is, when the equality C i,D = C i,L,D holds, all the capacities have the same value. In other cases, we have similar relations. Example 1. Now, as a typical example, we consider a single source acyclic network where Eve may choose any r-subset channels to access, which we call r-wiretap network [1], [2], [30], [31]. That is, A 0 is given as {s ⊂ [ ] : \|s\| = r}. To discuss the capacities of the given network, we introduce two kinds of minimum cuts. To define them, we define a pseudo source node as a node that has only out-going edges but has no original message to be transmitted. A pseudo source node is classified as an intermediate node because it is not the source node nor the terminal node. The first type of minimum cut mincut 1 is the minimum number of edges crossing a line separating the source node and the terminal node. The second type of minimum cut mincut 2 is the minimum number of edges crossing a line separating the source node and the terminal node with removing all edges out-going from pseudo source nodes. That, while edges out-going from pseudo source nodes are ignored in mincut 2 , they are counted in mincut 1 . For r-wiretap network, we have C 2,L,AC = C 2,L = C 2,L,D = C 2,AC = C 2 = C 2,D = mincut 2 −r. (26) mincut 2 −r ≤ C 1,L,AC = C 1,L = C 1,L,D ≤ C 1,AC ≤ C 1 ≤ C 1,D ≤ mincut 1 −r.(27) When the network has no pseudo source node, mincut 2 = mincut 1 , which implies the equalities in (27). For example, the network given in Fig. 2 shows a network has different rates mincut 1 and mincut 2 . This network has a linear code to realize mincut 1 −r when r = 1, which implies the equalities in (27). The relations (26) and (27) can be shown as follows. It was shown in [2, Section III] that the rate mincut 2 −r is achievable by a linear code where only source node generates randomness when Eve is allowed to use deterministic attack. However, any adaptive and active attack is reduced to deterministic attack under a linear code. Hence, we obtain C 2,L,D ≥ mincut 2 −r. Using a idea similar to [2, Section IV], we show C 1,D ≤ mincut 1 −r. For this aim, we choose edges crossing a line separating the source node and the terminal node such that these edges contains the r eavesdropped edges. Let Z be the variable on the r eavesdropped edges, and Y be the variable on the above edges crossing the separating line. Let M be the message to be securely transmitted. Due to the security condition, we have I(M ; Z) = 0 When an edge has an information with cardinality d, the receiver's information B satisfies I(M ; B) ≤ I(M ; Y ) = I(M ; Y Z) = I(M ; Z) + I(M ; Y \|Z) = I(M ; Y \|Z) ≤ H(Y \|Z) ≤ (mincut 1 −r) log d,(28)M+L 1 1 2 5 L 1 Alice Bob 3 4 L 2 L 2 M+L 2 L 1 Fig. 2. Network with equality in (27). Node 1 is the source node and Node 5 is the terminal node. Node 4 is a pseudo source node. Hence, mincut2 = 1 and mincut1 = 2. It also shows a linear code to achieve mincut1 −r when r = 1. The source node (Node 1) has the message M and a scramble variable L1. The pseudo source node (Node 4) has another scramble variable L2. Even when Eve wiretaps any one edge, she cannot obtain any information for the message M . which implies C 1,D ≤ mincut 1 −r. Therefore, using (20) and (25) and combining these facts, we obtain (27). When no intermediate node is allowed to generate randomness, any pseudo source node plays no role. Hence, the above discussion yields that C 2,D ≤ mincut 2 −r. Thus, we obtain (26). Example 2. Next, we consider the case when A 0 is given by using the following group structure of the intermediate nodes. The intermediate nodes are divided into c − 1 groups, from the first group to the c − 1-th group. Here, a source nodes and b terminal nodes are regarded as the 0-th group and the c-th group, respectively. For i = 1, . . . , c, there are several edges between the i − 1-th group and the i-th group. We call the set of these edges the i-th edge group. As seen later, this grouping of edges is essential to define the collection A 0 . Each intermediate node has incoming edges and outgoing edges. Eve is assumed to eavesdrop a part of edges from the i-th edge group. Eve's ability is characterized by the collection of subsets of the i-th edge group to be eavesdropped, which is called the i-th tapped-edge collection and is denoted by S i . When an intermediate node of i-th group is directly linked to an intermediate node of i + 2-th group, we consider that the intermediate node of i-th group is connected to intermediate node of i + 1-th group with an edge that is not contained in any member of the i + 1-th tapped-edge collection S i+1 . Similarly, when an intermediate node of i-th group is directly linked to an intermediate node of i + i -th group, we can apply the same reduction. Hence, without loss of generality, we can assume that an outgoing edge of an intermediate node of i-th group is linked only to an intermediate node of i + 1-th group. Hence, the collection A 0 is given to be S 1 × S 2 × · · · × S c . V. RELAY NETWORK A. Formulation and capacities Now, as a special case of Example 2, we consider the relay network given in Fig. 3 Here, we assume that Eve can eavesdrop r i edges Y i,si := (Y i,si(1) , . . . , Y i,si(ri) ) among k i edges Y i := (Y i,j ) j=1,...,ki between the i − 1 and i-th nodes. In this notation, the function s i expresses the edges eavesdropped by Eve. That is, she can eavesdrop c i=1 r i edges totally. In this paper, we allow stronger attacks for Eve than conventional attacks, i.e., adaptive attacks and active attacks. Then, we have the following capacity theorem. Theorem 3. Defining h 1 := k 1 , h j := min(k j , k j−1 − r j−1 k j−1 h j−1 + γ j ),(29) we have C 1 = C 1,L = C 1,D = C 1,L,D = C 1,AC = C 1,L,AC = log d min 1≤j≤c (k j − r j ),(30)C 2 = C 2,L = C 2,D = C 2,L,D = C 2,AC = C 2,L,AC = log d min 1≤j≤c (k j − r j ) (k j+1 − r j+1 ) · · · (k c − r c ) k j+1 · · · k c ,(31)C[(γ i ) i ] = C[(γ i ) i ] L = C[(γ i ) i ] D = C[(γ i ) i ] L,D = C[(γ i ) i ] AC = C[(γ i ) i ] L,AC = log d min 1≤j≤c k j − r j k j h j .(32) Here, we discuss the relation to existing results with respect to the difference between two capacities C 1 and C 2 . A larger part of studies discuss the capacity (or capacity region) with no restriction of randomness generated in intermediate nodes. For example, in r-wiretap network, which is a typical network model, as explained in Example 1, the capacity with no restriction can be achieved without use of randomness generated in intermediate nodes. However, the paper [23] showed an example, in which randomness generated in intermediate nodes improves the capacity. In this example, the source node is connected only with one edge. Usually, the secure transmission can be done by use of the difference between information on different edges connected to the same node. Hence, it is natural that randomness generated in intermediate nodes improves the capacity when each source node is connected only to one edge. The papers [24], [25] addressed the difference between the existence and non-existence of randomness generated in intermediate nodes in another network only for deterministic attacks. However, they did not derive the capacities C 1,D and C 2,D exactly. Their analysis depends on special codes. Therefore, our analysis is the first derivation of the difference between the capacities C 1,D and C 2,D except for the case when the source node is connected only with one edge. B. Converse part For any j = 1, . . . , c, the rate of secure transmission from the j − 1-th intermediate node to the j-th intermediate node is log d(k j − r j ). Taking the minimum with respect to j, we obtain C 1,D ≤ log d min 1≤j≤c (k j − r j ). Next, we consider (31). For the amount of leaked information, we have the following theorem. I(M ; Y 1,s1 , . . . , Y c,sc ) ≥H(M ) − (log d) min 1≤j≤c (k j − r j ) (k j+1 − r j+1 ) · · · (k c − r c ) k j+1 · · · k c .(33) Therefore, to realize the condition max s1,...,sc I(M ; Y 1,s1 , . . . , Y c,sc ) = 0,(34) the message M needs to satisfy the condition H(M ) ≤ log d min 1≤j≤c (k j − r j ) (k j+1 − r j+1 ) · · · (k c − r c ) k j+1 · · · k c .(35) When use the same network n times, the condition (34) requires the condition H(M ) ≤ n log d min 1≤j≤c (k j − r j ) (k j+1 − r j+1 ) · · · (k c − r c ) k j+1 · · · k c ,(36) which implies C 2,D ≤ log d min 1≤j≤c (k j − r j ) (kj+1−rj+1)···(kc−rc) kj+1···kc . Theorem 4 can be generalized to the limited randomness case as follows. Hence, it is sufficient to show Theorem 5. I(M ; Y 1,s1 , . . . , Y c,sc ) ≥H(M ) − log d min 1≤j≤c k j − r j k j h j .(37) Proof of Theorem 5: Now, we independently choose the sets S 1 , S 2 , . . . , S c subject to the uniform distribution. We denote the expectation is with respect to this random choice by E. We prove Theorem 5 by using Lemma 4, which will be shown in the latter section. Application of Lemma 4 to X = ( Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) shows the inequality EH( Y j,Sj \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) ≥ r j k j EH(Y j \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 )(38) for 1 ≤ j ≤ c. Then we have for 1 ≤ j ≤ c, EH(M \| Y c,Sc , Y c−1,Sc−1 . . . , Y 2,S2 , Y 1,S1 ) ≤EH(M \| Y j,Sj , Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) (a) ≤ EH(Y j \| Y j,Sj , Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) =EH( Y j,S c j \| Y j,Sj , Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) =EH(Y j \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) − EH( Y j,Sj \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) (b) ≤ k j − r j k j EH(Y j \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ),(39) where (b) follows from (38), and (a) follows from the fact that M is determined by the random variable Y j . Similarly, we have EH(Y j \| Y j−1,Sj−1 , Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) (a) ≤ EH(Y j−1 , K j \| Y j−1,Sj−1 , Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) =EH( Y j−1,S c j−1 , K j \| Y j−1,Sj−1 , Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) ≤EH( Y j−1,S c j−1 \| Y j−1,Sj−1 , Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) + γ j log d =EH(Y j−1 \| Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) − EH( Y j−1,Sj−1 \| Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) + γ j log d (b) ≤ k j−1 − r j−1 k j−1 EH(Y j−1 \| Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) + γ j log d,(40) where (a) follows from the fact that Y j is determined by the random variables Y j−1 , K j , and (b) follows from (38). Now, we show EH(Y j \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) ≤ h j log d(41) by induction with respect to j. Since H(Y 1 ) ≤ k 1 , (41) holds for j = 1. Assume that EH(Y j−1 \| Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) ≤ h j−1 log d. Then (40) implies that EH(Y j \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) = k j−1 − r j−1 k j−1 EH(Y j−1 \| Y j−2,Sj−2 , . . . , Y 2,S2 , Y 1,S1 ) + γ j log d ≤ k j−1 − r j−1 k j−1 h j−1 log d + γ j log d,(42) Also, we have EH(Y j \| Y j−1,Sj−1 , . . . , Y 2,S2 , Y 1,S1 ) ≤ H(Y j ) ≤ k j log d.(43) Combining (42) and (43), we have (41). Therefore, combining (39) and (41), we have EH(M \| Y c,Sc , Y c−1,Sc−1 . . . , Y 2,S2 , Y 1,S1 ) ≤ k j − r j k j h j log d,(44) which is equivalent to EI(M ; Y c,Sc , Y c−1,Sc−1 . . . , Y 2,S2 , Y 1,S1 ) ≥ H(M ) − k j − r j k j h j log d.(45) Hence, we obtain the desired statement. C. Code construction to achieve capacity C 1,L,D We give a code to achieve the capacity C 1,L,D . For simplicity, we assume that the integer d is a power q of a prime p. The general case will be discussed later. When we can make the desired code in the case with c = 1, we can employ the constructed code for the secure transmission code from the i − 1-th intermediate node to the i-th intermediate node because the i − 1-th intermediate node can employ scramble random numbers T i−1 . For this purpose, we prepare the following lemma. Lemma 2. For any prime power q, any two natural numbers k > r, there exist a natural integer n k,r and r vectors v 1 , . . . , v r ∈ F k q n k,r such that v i,j = δ i,j for j = 1, . . . , m and the r × r matrix (v i,s(j) ) i,j is invertible for any injective function t from {1, . . . , r} to {1, . . . , k}. This lemma might be shown in the context of the wiretap channel II introduced by Ozarow and Wyner [29]. In the model of wiretap channel II, a secrete message is encoded to a codeword in an n k,r -length code. A wiretapper may take any r components out of k parallel channels but may have no information about the message. A linear code, e.g., a Reed-Solomon code can serve as the code. It is called a (k, r) code for wiretap channel II, and satisfies the condition for Lemma 2. Also, this leamma also can be regarded as a very simple and special case of the code in [2, Section III]. For readers' convenience, we give its proof in Appendix A. Here, we make the desired code in the case with c = 1. We employ the finite filed F q with q = q nk,r . That is, we need finite field of large size, whose efficient construction is discussed in [32,Appendix D]. So, when we use the channel n := n n k,r times, our transmission can be regarded as n times transmission on F q , i.e., each edge can transmit up to n symbols in F q . In the following, all random variables are treated as random variables taking values in F q . According to Lemma 2, we choose r vectors v 1 , . . . , v r ∈ F k q n k,r . Using r additional scramble random numbers L 1 , . . . , L r , we can transmit k − r random variables M 1 , . . . , M k−r by encoding the random variable Y j for the j-th edge by Y j := L j when j ≤ r M j−r + r j =1 v j ,j L j when r + 1 < j ≤ k.(46) Then, Bob recovers the original messages M 1 , . . . , M k−r as M j := Y j+r − r j =1 v j ,j Y j .(47) Assume that Eve eavesdrops r edges, the s(1)-th edge, . . ., the s(r)-th edge. Due to the condition in Lemma 2, for any function s, the vectors (v j ,s(1) ) 1≤j ≤r , . . . , (v j ,s(r) ) 1≤j ≤r are linearly independent. So, r j =1 v j ,s(1) L j , . . . , r j =1 v j ,s(r) L j are r uniform random numbers even when we fixed the values of the random variables M 1 , . . . , M k−r . Eve cannot obtain any information for M 1 , . . . , M k−r . Repeating n times this procedure, we can extend this method to the case when we transmit (k − r)n random variables M 1 , . . . , M (k−r)n with rn additional scramble random numbers L 1 , . . . , L rn . Therefore, the transmission rate of this code is (k−r) log 2 q nk,r = (k − r) log q. Since (M 1 , . . . , M (k−r)n ) can be regarded as an element of a vector space over F q , this operation is a linear code with respect to the finite field F q . Therefore, since it satisfies the linearity condition (C1), the above security analysis over the deterministic attack guarantees the security over the adaptive and active attack due to Theorem 2 4 . Here, we make the desired code in the case with general c. Based on Lemma 2 with respective k i and r i , we choose n ki,ri . Then, we choose the finite filed F q with q = q max1≤i≤c nk i ,r i . Therefore, we can transmit the minimum rate log q min 1≤j≤c (k j − r j ). In this construction, the transmission on each step is given by a linear code over the finite field F q , the whole operation is also given as a linear code over the finite field F q . Therefore, since it satisfies the linearity condition (C1), Theorem 2 guarantees the security over the adaptive and active attack. Now, we consider the case that the integer d is not a power q of a prime p. In this case, we have the following lemma. Given a sufficiently large integer n, we choose a prime power q := argmax q: prime power {log q\|q ≤ d n }. We treat n uses of a channel as a single transmission of random variable taking values in F q . Due to Lemma 3, the code given above achieves the transmission rates log d min 1≤j≤c (k j − r j ) when n goes to infinity. D. Code construction to achieve capacity C[(γ i ) i ] L,D and C 2,L,D Since the capacity C 2,L,D is a special case of C[(γ i ) i ] L,D with γ i = 0, we construct only a code to achieve the capacity C[(γ i ) i ] L,D . Similar to the previous section, we choose the finite filed F q with q = q max1≤i≤c nk i ,r i , and we consider the case of n := n max 1≤i≤c n ki,ri uses of the channel, i.e., each edge can transmit up to n symbols in F q . In the following, all random variables are treated as random variables taking values in F q . For notational simplicity, we consider the case when single use of each edge transmits an element of F q . To achieve the above purpose, we give a linear code with respect to F q satisfying the following two conditions (D1) and (D2) by induction with respect to j. Since the code satisfies the linearity condition (C1), it is sufficient to consider the deterministic attack. (D1) The code securely transmits the message M of h j symbols per single use of channel to the j-th node from the source node, where h j := min 1≤j ≤j k j −r j k j h j . That is, I(M ; Y 1,s1 , . . . , Y j,sj ) = 0 for any (s 1 , . . . , s j ) ∈ S 1 × · · · × S j . (D2) The j-th node receives secure random number T j of h j − h j symbols per single use of channel, which contains the random numbers generated from the 1st node to the j − 1-th node, where h j := kj−rj kj h j . That is, the j-th node receives secure random number of h j symbols per single use of channel, i.e., I(M T j ; Y 1,s1 , . . . , Y j,sj ) = 0 for any (s 1 , . . . , s j ) ∈ S 1 × · · · × S j . Since h 1 = h 1 = (k 1 − r 1 ), the desired code with j = 1 was constructed in Subsection V-C. We show the existence of the desired linear code with respect to F q by induction. That is, we assume the existence in the case of j − 1 with block length n j−1 . We find that h j = min((k j − r j ), kj−rj kj (h j−1 + γ j )) and h j = min((k j − r j ), kj−rj kj (h j−1 + γ j ), h j−1 ) for j ≥ 2. We show the existence of such a code with j by classifying three cases. (1) Case of h j = h j = (k j − r j ): To achieve the desired task, the j − 1-th node needs to securely transmit the message M of (k j −r j ) symbols per single use of channel to the j-th node, which requires scramble random numbers T j of r j symbols per single use of channel at the j −1-th node. Since r j ≤ h j−1 +γ j −(k j −r j ), the j −1-th node has sufficient scramble random numbers for this purpose. We divide the scramble random numbers T j into two parts T j,1 and T j,2 , where T j,1 has γ j symbols per single of channel and T j,2 has (r j −γ j ) symbols per single of channel. Due to the assumption of induction, the sender securely transmits M and T j,2 to the j−1-th node by a linear code with block length n j−1 , where the first n j−1 (k j −r j ) symbols are M , the next n j−1 (r j −γ j ) symbols are T j,2 , and the remaining symbols are fixed to zero. That is, I(M T j,2 ; Y 1,s1 , . . . , Y j−1,sj−1 ) = 0 for any (s 1 , . . . , s j−1 ) ∈ S 1 × · · · × S j−1 . We choose block length n j to be n j−1 . Since T j,1 is composed of n j−1 γ j symbols and is independent of other random variables, We apply the code given in Subsection V-C with n = n j−1 to the message M and the scramble random number T j . Then, the j −1-th node securely transmits the message M to the j-th node by a desired linear code with respect to F q of block length n j . Therefore, I(M ; Y j,sj \| Y 1,s1 , . . . , Y j−1,sj−1 ) = 0 for any (j 1 , . . . , s j ) ∈ S 1 ×· · ·×S j . Hence, I(M ; Y 1,s1 , . . . , Y j,sj ) = 0 for any (j 1 , . . . , s j ) ∈ S 1 × · · · × S j . (2) Case of h j = h j = kj−rj kj (h j−1 + γ j ): To achieve the desired task, the j − 1-th node needs to securely transmit the message M of h j = h j symbols per single use of channel to the j-th node, which requires scramble random numbers T j of kj−rj kj h j symbols per single use of channel at the j − 1-th node. Since kj−rj kj h j = rj kj−rj kj−rj kj (h j−1 + γ j ) = h j−1 + γ j − kj−rj kj (h j−1 + γ j ) , the j − 1-th node has sufficient scramble random numbers for the above purpose. Therefore, similar to the case (1), we can show the existence of the desired linear code with respect to F q . (3) Case of h j = h j−1 < h j : Since h j is k j − r j or kj−rj kj (h j−1 + γ j ) , due to the discussion with the above two cases (1) and (2), the j-th node receives secure random number of h j symbols per single use of channel. To achieve the desired task, the j − 1-th node needs to securely transmit the message M of h j (≤ k j − r j ) symbols per single use of channel to the j-th node, which requires scramble random numbers T j of kj−rj kj h j symbols per single use of channel at the j − 1-th node. Since rj kj−rj h j = rj kj−rj h j−1 ≤ γ j , the j − 1-th node has sufficient scramble random numbers for this purpose. Therefore, similar to the case (1), we can show the existence of the desired linear code with respect to F q . Therefore, there exists a code that transmits the message with the rate h c to the source node from the source node. Remark 2. We consider how many uses of the channel can achieve the capacity when d is a prime power q and the intermediate node cannot use additional random number, i.e., γ i = 0. To answer this problem, we consider another proof in this special case. When we set n := k 2 · · · k c and n := n · max 1≤i≤c n ki,ri , we can achieve the capacity in the following way. That is, our transmission can be regarded as n times transmission on F q , i.e., n · max 1≤i≤c n ki,ri times transmission of the original channel. In the following construction, we employ k 1 · · · k c random variables. In this protocol, we securely transmit (k 1 − r 1 ) · · · (k i − r i )k i+1 · · · k c random variable to the i-th node. That is, in the transmission from the i − 1-th node to the i-th node, we transmit (k 1 − r 1 ) · · · (k i−1 − r i−1 )r i k i+1 · · · k c random numbers, in which, (k 1 − r 1 ) · · · (k i − r i )k i+1 · · · k c random numbers are securely transmitted and the remaining (k 1 − r 1 ) · · · (k i−1 − r i−1 )r i k i+1 · · · k c random variables are treated as scramble random variables. Such a transmission is possible by applying the method given Subsection V-C to the (k 1 − r 1 ) · · · (k i − r i )k i+1 · · · k c random variables, which are securely transmitted to the i − 1-th node. Using the above recursive construction, we can securely transmit c i=1 (k i − r i ) random variables. The single use of the channel between the i − 1-th node and the i-th node can securely transmit (k i − r i ) random variables. So, to realize this code, we need to use the channel between the i − 1-th node and the i-th node at (k1−r1)···(ki−ri)ki+1···kc ki−ri = (k 1 − r 1 ) · · · (k i−1 − r i−1 )k i+1 · · · k c times. That is, to realize this code, we need to use this relay channel max 1≤i≤c (k 1 − r 1 ) · · · (k i−1 − r i−1 )k i+1 · · · k c times. Overall, this code can transmit min 1≤i≤c c i=1 (k i − r i ) (k 1 − r 1 ) · · · (k i−1 − r i−1 )k i+1 · · · k c = min 1≤j≤c (k j − r j ) (k j+1 − r j+1 ) · · · (k c − r c ) k j+1 · · · k c(49) variables per single use of the relay channel. That is, the transmission rate of this code is log q min 1≤j≤c (k j − r j ) (kj+1−rj+1)···(kc−rc) kj+1···kc . Therefore, we can realize a code to satisfy the conditions (34) and (36) for the above given n. E. Scalar linearity Now, we show that this capacity cannot be attained under the scalar linearity condition. That is, we consider the special case to satisfy the following conditions. The intermediate node cannot use additional random number, i.e., γ i = 0. We can transmit only a single symbol of a finite filed F q in each channel. The coding operations are limited to linear operations over the finite filed F q . Since each channel can send only a scalar in F q , this kind of linearity is called the scalar linearity [26]. To distinguish the condition (C1) from the scalar linearity, the condition (C1) is often called the vector linearity [26]. Existing studies employ one of these constraints as Table I. Only a deterministic attack is allowed to the eavesdropper. Under the above condition, the number of symbols transmitted securely is not greater than max(k 1 − c j=1 r j , 0), which can be shown as follows. Due to the network structure, the sender can transmit only k symbols M 1 , . . . , M k in F q , where the k symbols M 1 , . . . , M k is given as linear functions of the message and the scramble random variable. First, we fix the linear coding operation on each nodes. In the first group of edges, Eve chooses r 1 edges such that the information on the r 1 edges are given as r1 i=1 t 1,i ,i M i with i = 1, . . . , r 1 and { t 1,i } is linearly independent, where t 1,i = (t 1,i ,i ) k i=1 for i = 1, . . . , r 1 . Similarly, when k 1 ≥ r 1 + r 2 , in the second group of edges, Eve chooses r 2 edges such that the information on the r 2 edges are given as r2 i=1 t 2,i ,i M i with i = 1, . . . , r 1 and { t 1,i } ∪ { t 2,i } is linearly independent, where t 2,i = (t 2,i ,i ) k i=1 for i = 1, . . . , r 2 . When k 1 < r 1 + r 2 , in the second group of edges, Eve chooses k 1 −r 1 edges such that the information on the k 1 −r 1 edges are given as k1−r1 i=1 t 2,i ,i M i with i = 1, . . . , r 1 and { t 1,i } ∪ { t 2,i } is linearly independent, where t 2,i = (t 2,i ,i ) k i=1 for i = 1, . . . , k 1 − r 1 . When k 1 > r 1 + r 2 , we repeat this process up to the c-th group or j -th group satisfying k 1 − j j=1 r j ≤ 0. Hence, the information with dimension max(k 1 , c j=1 r j ) is leaked to the eavesdropper. Therefore, the number of symbols transmitted securely is not greater than max(k 1 − c j=1 r j , 0). This fact shows the following effect. To achieve the capacity even with deterministic attacks, each channel needs to transmit several symbols in the finite field F q . That is, we need to handle the vector space over the finite field F q . Furthermore, as a special case, in the setting given in Section III, we find that we need to introduce a non-linear code to realize the situation that Eve cannot recover the message perfectly with deterministic attack. We often increase the size q of finite field F q in the scalar linearity while we fix the size q of finite field F q and increase the dimension of the vector space in the vector linearity. In the real communication, the data is given as a sequence of F 2 . In this case, when q = 2, the coding operation satisfying the vector linearity can be easily implemented because the vector linearity reflects the structure of the data. However, the coding operation satisfying the scalar linearity cannot be easily implemented unless q is a power of 2 because the scalar linearity does not reflect the structure of the data. Only when q is a power of 2, the scalar linearity not be easily implemented. However, even in this case, the scalar linearity has worse performance than the vector linearity due to the above discussion because the scalar linearity introduces a constraint that does not appear in the vector linearity. Hence, it is better to impose the vector linearity. VI. IMPORTANT LEMMAS Here, for the latter discussion, we prepare important lemmas. We denote the set {1, . . . , k} by [k], and denote the collection of subsets S ⊂ [k] with cardinality r by [k] r . Now, we consider the random variables X, Y 1 , . . . , Y k . For any subset S ⊂ [k], we denote the tuple of random variables ( Y s ) s∈S by Y S . We can show the following two lemmas. Lemma 4. We have S∈( [k] r ) H( Y [k] \| Y S c X) ≤ k − 1 r − 1 H( Y [k] \|X) = k − r k k k − r H( Y [k] \|X)(50) Remark 3. It is possible to prove Lemma 4 by using Baranyai's Theorem [33]. However, this paper shows Lemma 4 by using our invented lemma, Lemma 5. Proof of Lemma 5: We prove the lemma by induction in h. When h = 1, it is trivial. Assume that the lemma holds for h − 1. We pick a subcollection S : = {S 1 , S 2 , . . . , S f } ⊂ S h such that ∪ f i=1 S i = [k]. We define S i := S i ∩ (∪ i−1 j=1 S j ) and S h−1 = (S h \ S ) ∪ {S 2 , . . . , S f }. We can see that any element of [k] is contained in exactly h − 1 members of S h−1 , from the following lines. Assume that an element a ∈ [k] is contained in exactly b members of S . Notice that a is contained by S i , for each particular i, if and only if it is contained by exactly one of S i \ [∪ i−1 j=1 S j ] and S i . For any element a ∈ [k], there uniquely exists an integer i such that a ∈ S i \ [∪ i−1 j=1 S j ]. So, the element a is contained in exactly b − 1 members of {S 2 , . . . , S f }. Therefore, the element a is contained in exactly h − b + (b − 1) = h − 1 members of S h−1 = (S h \ S ) ∪ {S 2 , . . . , S f }. Therefore, S∈Sh H( Y S \|X) = S∈Sh\S H( Y S \|X) + f i=1 H( Y Si \|X) = S∈Sh\S H( Y S \|X) + f i=1 H( Y S i \|X) + f i=1 H( Y Si\S i \| Y S i X) (a) ≥ S∈Sh−1 H( Y S \|X) + f i=1 H( Y Si\(∪ i−1 j=1 Sj) \| Y ∪ i−1 j=1 Sj X) (b) ≥(h − 1)H( Y [k] \|X) + H( Y [k] \|X) = hH( Y [k] \|X),(52) where (a) follows from the relation S i ⊂ ∪ i−1 j=1 S j and (b) follows from the relation ∪ f i=1 S i = [k] and the induction hypothesis, the fact that S∈Sh−1 H( Y S \|X) ≥ (h − 1)H( Y [k] \|X). Proof of Lemma 4: Now, we show Lemma 4 by using Lemma 5. Any element a ∈ [k] is contained in exactly k−1 r−1 members of [k] r . So, we apply Lemma 5 to the case with S h = [k] r and h = k−1 r−1 . Hence, we have S∈( [k] r ) H( Y S c \|X) ≤ k − 1 r − 1 H( Y [k] \|X).(53) Thus, S∈( [k] r ) H( Y [k] \| Y S c X) = S∈( [k] r ) H( Y S c Y S \| Y S c X) = S∈( [k] r ) H( Y S \| Y S c X) = S∈( [k] r ) [H( Y S Y S c \|X) − H( Y S c \|X)] = k m H( Y [k] \|X) − S∈( [k] r ) H( Y S c \|X) ≤ k r H( Y [k] \|X) − k − 1 r − 1 H( Y [k] \|X) = k r − 1 H( Y [k] \|X).(54 VII. HOMOGENEOUS MULTICAST RELAY NETWORK A. Formulation and capacity regions Next, as a special case of Example 2, we consider the homogeneous multicast relay network defined as follows. This network has one source node and b terminal nodes. It has c − 1 groups of intermediate nodes. The i-th group has b i intermediate nodes, and the set of b terminal nodes is regarded as the c-th group, and the source node is regarded as the 0-th group. So, the numbers b 0 and b c are defined to be 1 and b. Each node of the i-the group is expressed as n(i, 1), . . . , n(i, b i ). Each node of the i − 1-th group is connected to every node of the i-th group with k i edges. That is, there are b i−1 b i k i edges from the i − 1-th group to the i-th group. For each node of the i-th group, Eve is assumed to wiretap r i edges among b i−1 k i edges connected to the node of the i-th group from nodes of the i − 1-th group. That is, Eve wiretaps r i b i edges among b i−1 b i k i edges between the i − 1-th group and the i-th group. Then, we have the following theorem for the no-randomness capacity region. Theorem 6. C 2 = C 2,L = C 2,D = C 2,L,D = C 2,AC = C 2,L,AC = (R 1 , . . . , R b ) b i =1 R i ≤ A 1 , R i ≤ A 2 for i = 1, . . . , b (55) where A 1 := (log d) min 1≤j≤c (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−1 k c − r c ) b j k j+1 · · · b c−1 k c(56)A 2 := (log d)(b c−1 k c − r c ).(57) For the full-randomness capacity region, we have the following theorems. Theorem 7. Assume that c = 2 and r 2 /k 2 is an integer. Then, we have Theorem 8. Assume that c = 3 and r 3 /k 3 is an integer. C 1 = C 1,L = C 1,D = C 1,L,D = C 1,AC = C 1,L,AC = (R 1 , . . . , R b ) b i =1 R i ≤ A 3 , R i ≤ A 2 for i = 1, . . . , b , (59) where A 3 := (log d) min (k 1 − r 1 )b 1 , min 2≤j≤3 (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−1 k c − r c ) b j k j+1 · · · b c−1 k c = (log d) min (k 1 − r 1 )b 1 , (b 1 k 2 − r 2 )b 2 b 2 k 3 − r 3 b 2 k 3 , (b 2 k 3 − r 3 )b 3 .(60) B. Converse part for Theorem 6 We consider the j-th group as one intermediate node, and the set of the b terminal nodes as one terminal node, which yields a relay network. Then, applying the relation (36) to this relay network, we obtain the condition b i =1 R i ≤ A 1 . Next, we consider the j-th group as one intermediate node, and focus only on the i-th terminal nodes, which yields another relay network. Then, applying the relation (36) to this relay network, we obtain the other condition R i ≤ A 2 . C. Code construction for Theorem 6 Here, by induction, we make a linear code to achieve the RHS of (59) when d is a prime power q. In the general case, we can construct the desired linear code by using the method in Lemma 3. The liner code construction with c = 1 is given from the code given in Subsection V-C. We construct the desired linear code by induction with respect to the number c. Assume that n is a multiple of n kcn,rc . Now, we assume that the source node can securely transmit b i =1 N i letters to each intermediate node in the c − 1-th group by n use of the channel. When N i ≤ nk c , under this assumption, we can transmit N i b c−1 − nr c letters from the source node to the i -th terminal node by n use of the channel as follows. Such a code will be called Code (N 1 , . . . , N b ). For j 2 = 1, . . . , b, j 1 = 1, . . . , k c , we denote the j2−1 i =1 N i + j 1 -th securely transmitted letter to j-th intermediate node in the c − 1-th group by X j2,j1+jN i . Then, for a given j 2 = 1, . . . , b, the source node prepares messages M j2,j3 for j 3 = 1, . . . , N j2 b c−1 − nr c and scramble random numbers L j2,j3 for j 3 = 1, . . . , nr c . Then, the source node makes conversion from the pair of M j2 and L j2 to X j2 such that there is no information leakage for M j2 even when any nr c letters of X j2 are eavesdropped. Such a code can be constructed by using the discussion in Subsection V-C. Now, we employ the assumption of induction. So, there exist an integer n and a code Φ n with block-length n such that the rate tuple is ( A4 bc−1 , . . . , A4 bc−1 ), where A 4 := min 1≤j≤c−1 (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−2 k c−1 − r c−1 ) b j k j+1 · · · b c−2 k c−1 .(61) Using this fact, we show the desired statement by classifying two cases. (1) Case of A4 bc−1 ≥ k c b c : In this case, the minimum in (56) is realized with j = c, which implies A 1 = b c A 2 . To attain the RHS of (59), it is sufficient to give a code with the rate tuple (A 2 , . . . , A 2 ) = (log d(b c−1 k c − r c ), . . . , log d(b c−1 k c − r c )). The required secure transmission from the source node to the c − 1-th group is possible as follows. Combining the assumption of induction and Code (nk c , . . . , nk c ). we obtain a linear code with the rate tuple (log d(b c−1 k c − r c ), . . . , log d(b c−1 k c − r c )). (2) Case of A4 bc−1 < k c b c : We have A 1 = A 4 bc−1kc−rc bc−1kc . To attain the RHS of (59), it is sufficient to give a code with the rate tuple (R 1 , . . . , R b ) satisfying conditions b i =1 R i ≤ A 1 and R i ≤ A 2 . Due to the assumption of induction, the source node can securely transmit n A4 bc−1 letters to each node in the c − 1-th group. Now, we choose n such that n A4 bc−1b is an integer, n is a multiple of n kcn,rc , and nR i is integer for i = 1, . . . , b. Therefore, using Code (nR 1 , . . . , nR b ), we obtain a linear code, in which, the source node can securely transmit to the i -th terminal with rate R i . Since this linear code construction requires only the conditions b i =1 R i ≤ A 1 and R i ≤ A 2 , the RHS of (59) is attained. D. Proof of Theorem 7 To show Theorem 7, it is sufficient to show the converse part, i.e., C 1,D ⊂ C 2,D . The i-th intermediate node can transmit information of k 2 symbols per single use of channel to the j-th terminal node. In order that the j-th terminal node recovers the original message M j , the j-th terminal node needs to recover a part of information M i,j with respect to the original message that is determined by the information received by the i-th intermediate node. That is, collecting the variables M 1,j , . . . , M b1,j , the j-th terminal node recovers M j . We choose an injective function s from {1, . . . , r 2 /k 2 } to {1, . . . , b 2 }. Now, we consider the case that Eve wiretaps all the channels from the s(i)-th intermediate node to the j-th terminal node for i = 1, . . . , r 2 /k 2 . When the s(i)-th terminal node introduces scramble random variables L s(i),j in the channel to the j-th terminal node, the j-th terminal node needs to recover M s(i),j . In this case, Eve also recovers M s(i),j . Then, there is no merit to introduce the scramble random variables L s(i),j in this channel. When the i -th terminal node introduces scramble random variables L i ,j in the channel to the j-th terminal node for i ∈ {1, . . . , b 2 } \ {s(1), . . . , s(r 2 /k 2 )}, the j-th terminal node needs to recover M i ,j . In this case, Eve has no access to this channel. Hence, there is no need to introduce the scramble random variables L i ,j in this channel. Therefore, considering this special case, there is no advantage to introduce scramble random variables in the intermediate nodes. That is, any code can be reduced to a code with the no-randomness condition (C2). E. Proof of Theorem 8 Due to the discussion in Subsection VII-D, the scramble random number introduced in intermediate nodes in the 2nd group does not work. Hence, we obtain the converse part, i.e., C D ⊂ (R 1 , . . . , R b ) b i =1 R i ≤ A 3 , R i ≤ A 2 for i = 1, . . . , b . Next, we construct a code to achieve the capacity region. Each intermediate node in the first group can securely transmit to each terminal node with the following capacity region: (R 1 , . . . , R b ) b i =1 R i ≤ A 5 b 1 , R i ≤ A 2 b 1 for i = 1, . . . , b(62) with A 5 := (log d) min (b 1 k 2 − r 2 )b 2 b 2 k 3 − r 3 b 2 k 3 , (b 2 k 3 − r 3 )b 3 .(63) Now, the source node can securely transmit information to each intermediate node in the first group with the rate (log d)(k 1 − r 1 ). Combining these discussions, the source node can securely transmit information to each terminal node via a specific intermediate node in the first group with the following capacity region: (R 1 , . . . , R b ) b i =1 R i ≤ A 3 b 1 , R i ≤ A 2 b 1 for i = 1, . . . , b(64) A. Formulation and capacity regions Next, as a special case of Example 2, we consider the homogeneous multiple multicast relay network defined as follows. This network has a source nodes and b terminal nodes. It has c − 1 groups of intermediate nodes. The i-th group has b i intermediate nodes, and the set of b terminal nodes is regarded as the c-th group, and the source node is regarded as the 0-th group. So, the numbers b 0 and b c are defined to be a and b. Each node of the i-the group is expressed as n(i, 1), . . . , n(i, b i ). Each source code is connected to each intermediate node in the first group with k 1 edges. For i ≥ 2, each node of the i − 1-th group is connected to every node of the i-th group with k i edges. That is, there are b i−1 b i k i edges from the i − 1-th group to the i-th group. For each node of the i-th group, Eve is assumed to wiretap r 1 edges among k 1 edges between each source node and each intermediate node in the first group. Totally, Eve wiretaps ab 1 r 1 edges among ab 1 k 1 edges between the 0-th group and the first group. For i ≥ 2, Eve is assumed to wiretap r i edges among b i−1 k i edges connected to the node of the i-th group from nodes of the i − 1-th group. That is, Eve wiretaps r i b i edges among b i−1 b i k i edges between the i − 1-th group and the i-th group. Then, we have the following theorem for the no-randomness capacity region. Theorem 9. C 2 = C 2,L = C 2,D = C 2,L,D = C 2,AC = C 2,L,AC = (R i,j ) 1≤i≤a,1≤j≤b i ,j R i ,j ≤ B 1 , j R i,j ≤ B 2 , i R i ,j ≤ B 3 for i = 1, . . . , a, j = 1, . . . , b ,(65) where B 1 :=(log d) min a(k 1 − r 1 )b 1 (b 1 k 2 − r 2 ) · · · (b c−1 k c − r c ) b 1 k 2 · · · b c−1 k c , min 2≤j≤c (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−1 k c − r c ) b j k j+1 · · · b c−1 k c (66) B 2 :=(log d) min (k 1 − r 1 )b 1 (b 1 k 2 − r 2 ) · · · (b c−1 k c − r c ) b 1 k 2 · · · b c−1 k c , min 2≤j≤c (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−1 k c − r c ) b j k j+1 · · · b c−1 k c(67)B 3 :=(log d)(b c−1 k c − r c ).(68) For the full-randomness capacity region, we have the following theorems. Theorem 10. Assume that c = 2 and r 2 /k 2 is an integer. Then, we have C 1 = C 1,L = C 2 = C 2,L = C 1,D = C 1,L,D = C 2,D = C 2,L,D = C 1,AC = C 1,L,AC = C 2,AC = C 2,L,AC .(69) Theorem 11. Assume that c = 3 and r 3 /k 3 is an integer. C 1 = C 1,L = C 1,D = C 1,L,D = C 1,AC = C 1,L,AC = (R i,j ) 1≤i≤a,1≤j≤b i ,j R i ,j ≤ B 4 , j R i,j ≤ B 5 , i R i ,j ≤ B 3 for i = 1, . . . , a, j = 1, . . . , b ,(70) where B 4 := (log d) min a(k 1 − r 1 )b 1 , min 2≤j≤3 (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−1 k c − r c ) b j k j+1 · · · b c−1 k c = (log d) min a(k 1 − r 1 )b 1 , (b 1 k 2 − r 2 )b 2 b 2 k 3 − r 3 b 2 k 3 , (b 2 k 3 − r 3 )b 3(71)B 5 := (log d) min (k 1 − r 1 )b 1 , min 2≤j≤3 (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−1 k c − r c ) b j k j+1 · · · b c−1 k c = (log d) min (k 1 − r 1 )b 1 , (b 1 k 2 − r 2 )b 2 b 2 k 3 − r 3 b 2 k 3 , (b 2 k 3 − r 3 )b 3 .(72) B. Converse part for Theorem 9 We consider the j-th group as one intermediate node, and the set of the b terminal nodes and the set of the a source nodes as one terminal node and one source node, respectively, which yields a relay network. Then, applying the relation (36) to this relay network, we obtain the condition b i ,j R i ,j ≤ B 1 . Applying the discussion in Subsection VII-B to the network from the i-th source node to the j-th group, we obtain the condition b j R i,j ≤ B 2 . Similarly, applying the discussion in Subsection VII-B to the network from the first group to the j-th terminal node, we obtain the condition i R i ,j ≤ B 3 . C. Code construction for Theorem 9 Here, by induction, we make a code to achieve the RHS of (65) when d is a prime power q. In the general case, we can construct the desired code by using the method in Lemma 3. The code construction with c = 1 is given from the code given in Subsection V-C. We construct the desired code by induction with respect to the number c. We choose a rate tuple (R i,j ) i,j satisfying the condition in the RHS of (65). As mentioned in the proof of Theorem 6, when we can securely transmit an unlimited number of messages from the source node to all of intermediate nodes in the c − 1-th group, using the code with block-length n constructed in Subsection V-C, we can transmit n(b c−1 k c − r c ) letters from the source node to each terminal node, in which, the source node securely transmits nk c letters to each intermediate node in the c − 1-th group. Therefore, the rate tuple (R i,j ) i,j can be realized by secure transmission with the rate R i,j := bc−1kc bc−1kc−rc j R i,j from the i-th source node to the j-th intermediate node in the c − 1-th group. The assumption of induction guarantees that the rate tuple (R i,j ) i,j is attainable in the network from the first group to the c − 1-th group because the rate tuple (R i,j ) i,j satisfies the conditions i ,j R i ,j ≤ B 1 , j R i,j ≤ B 2 , i R i ,j ≤ B 3 for i = 1, . . . , a, j = 1, . . . , b c−1 , where B 1 :=(log d) min 1≤j≤c−1 (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−2 k c−1 − r c−1 ) b j k j+1 · · · b c−2 k c−1(73)B 2 :=(log d) min (k 1 − r 1 )b 1 (b 1 k 2 − r 2 ) · · · (b c−2 k c−1 − r c−1 ) b 1 k 2 · · · b c−2 k c−1 , min 2≤j≤c−1 (b j−1 k j − r j )b j (b j k j+1 − r j+1 ) · · · (b c−2 k c−1 − r c−1 ) b j k j+1 · · · b c−2 k c−1 (74) B 3 :=(log d)(b c−2 k c−1 − r c−1 ).(75) Therefore, the rate tuple (R i,j ) i,j is achievable. D. Proof of Theorem 10 To show Theorem 7, it is sufficient to show the converse part C 1 ⊂ C 2 . As shown in the proof of Theorem 7, any code can be reduced to a code with the no-randomness condition (C2). Hence, we obtain C 1 ⊂ C 2 . E. Proof of Theorem 11 Similar to the proof of Theorem 8, the scramble random number introduced in intermediate nodes in the 2nd group do not work. Hence, we obtain the converse part. Next, we construct a code to achieve the capacity region. Each source node can securely transmit information to each intermediate node in the first group with the rate (log d)(k 1 − r 1 ). Combining this code and the codes given in (62) from each intermediate node in the first group to each terminal node, the set of source nodes can securely transmit information to each terminal node via a specific intermediate node in the first group with the following capacity region: (R i,j ) 1≤i≤a,1≤j≤b i ,j R i ,j ≤ B 4 b 1 , j R i,j ≤ B 5 b 1 , i R i ,j ≤ B 3 b 1 for i = 1, . . . , a, j = 1, . . . , b(76) because B 3 = A 3 , B 4 = min((log d)a(k 1 − r 1 )b 1 , A 5 ) and B 5 = min((log d)(k 1 − r 1 )b 1 , A 5 ). Summing up the above region with respect to intermediate nodes in the first group, we find that the rate region defined in the RHS of (59). IX. CONCLUSION We have studied active and adaptive attacks, and have investigated whether an adaptive attack improves Eve's ability. As our result, we have shown that an adaptive attack improves Eve's ability when our code is a linear code. However, when our code is not a linear code, we have found an example where an adaptive attack improves Eve's ability in Section III. Any linear code cannot realize the performance of the non-linear code given there under the setting of Section III when Eve is allowed to a deterministic attack. Hence, the improvement by the adaptive attack is essential in this setting. Next, we consider several types of network, in which there is restriction for randomness in the intermediate nodes. This kind of restriction is crucial in the secure network because randomness is required to realize the secrecy. In the latter part of this paper, we have addressed various types of relay networks in the asymptotic setting, where we employ liner codes, i.e., these codes are given as vector spaces over a finite field. In Section V, we have considered a typical type of unicast relay network and have derived the capacity under various restrictions for randomness in the intermediate nodes. To show the converse part, we have shown a notable lemma in Section VI. Our proof of the direct part follows from a lemma related to wiretap channel II. Also, in Subsection V-E, we have shown that the code does not work when it is given as a scalar of a finite field. Further, we have proceeded to more complicated networks, e.g., a typical type of multicast relay network and a typical type of multiple multicast relay network. Since their asymptotic performances are characterized as their capacity regions, in Sections VII and VIII, we have derived them under the condition that the intermediate nodes have no scramble random number by generalizing the method used in Section V. While our asymptotic results are limited to special networks, the minimum cut theorem does not work in these networks. Hence, our codes suggest a general theory for networks whose capacity cannot be shown by the minimum cut theorem. It is an interesting future study to establish such a theory. As explained in Section I, when the spaces of the intermediate nodes and/or the budget are limited, it might be better to avoid to equip scramble random variables in the intermediate nodes. The study with this constraint is much desired for the practical viewpoint. ACKNOWLEDGMENTS The authors are very grateful to Dr. Wangmei Guo for her helpful discussions and her hospitality during the authors' stay in Xidian University. The authors thank a reviewer of the previous version of this paper for explaining the network code given in Fig. 3 When k = r, it is trivial. When k = r +1, we do not need to make any algebraic extension because it is sufficient to choose m vectors v 1 , . . . , v r ∈ F k q such that v i,j with j = 1, . . . , r is δ i,j and v i,r+1 is 1. Now, we consider the case when k > r + 1. When q > p, we choose element e 1 , . . . , e t such that F q is given as F p [e 1 , . . . , e t ]. When t < k − 2, we make further algebraic extension F p [e 1 , . . . , e k−2 ] by adding elements e t+1 , . . . , e k−2 . Now, we denote 1 by e 0 . Then, we choose m vectors v 1 , . . . , v r ∈ F p [e 1 , . . . , e k−2 ] k by v i,j :=    δ i,j when j ≤ r 1 when j = r + 1 e i+j−r−2 when j > r + 1. (77) We can show that the r vectors v 1 , . . . , v r satisfy the required condition as follows. Choose the function s such that s(1) < . . . < s(r). It is sufficient to show that the vector (v 1,s(r) , . . . , v r,s(r) ) cannot be written as a linear combination of (v 1,s(1) , . . . , v r,s(1) ), . . . , (v 1,s(r−1) , . . . , v r,s(r−1) ). When s(r) = r or r + 1, it is trivial. So, we show the case when s(r) > r + 1. Since all entries of v i,s(j) belong to F p [e 1 , . . . , e s(r)−2 ], we choose coefficients α 1 , . . . , α r ∈ F p [e 1 , . . . , e s(r)−2 ] such that m i=1 α i v j,s(i) = 0 for j = 1, . . . , r. We show the desired statement by assuming α r = 1. For i = 1, . . . , r − 1, we divide the coefficient α i into r parts, i.e., we choose α i,j ∈ F p [e 1 , . . . , e s(r)+j−r−2 ] \ F p [e 1 , . . . , e s(r)+j−r−3 ] as α i = r j=1 α i,j . Since we have r−1 i=1 α i v j,s(i) = −e s(r)+j−r−2 ∈ F p [e 1 , . . . , e s(r)+j−r−2 ] for j = 1, . . . , r, we have r−1 i=1 α i,j v j,s(i) = 0 for j > j because α i,j / ∈ F p [e 1 , . . . , e s(r)+j−r−2 ] and v j,s(i) ∈ F p [e 1 , . . . , e s(r)+j−r−2 ]. That is, the vectors α(j ) := (α 1,j , . . . , α r,j ) T for j = 1, . . . , r and β(j) := (v j,s(1) , . . . , v j,s(r) ) T for j = 1, . . . , r − 1 satisfy the conditions: (α(j ), β(j)) = 0 for j > j (78) j j =1 (α(j ), β(j)) = −e s(r)+j−r−2 . Since j−1 j =1 (α(j ), β(j)) ∈ F p [e 1 , . . . , e s(r)+j−r−3 ], and −e s(r)+j−r−2 ( = 0) / ∈ F p [e 1 , . . . , e s(r)+j−r−3 ], (α(j), β(j)) = 0. We define γ(j) = β(j) − j−1 i=1 (α(i),β(j)) (α(i),β(i)) β(i). Then, for i = j we have (α(i), γ(j)) = 0 when i = j (α(j), γ(j)) = 0 when i = j. (80) Hence, we find that γ(1), . . . , γ(r − 1) are linearly independent. Since (α(r), γ(j)) = 0 for j = 1, . . . , m − 1, we have α(r) = 0, which implies e s(r)−2 = 0. So, we obtain contradiction. (C2) [No-randomness] All of intermediate nodes have no scramble random numbers. (C2') [Limited-randomness] Each limited intermediate node has limited scramble random numbers. When each Fig. 1 , 1whose edges are E = {e(1), e(2), e(3), e(4)}. Each edge e(i) is assumed to send the binary information Y i . No scramble random variable is allowed in the intermediate node, which is the condition (C2). Eve is allowed to attack two edges of E except for the pairs {e(1), e(2)} and {e(3), e(4)}. That is, A 0 = {{e(1), e(3)}, {e(2), e(3)}, {e(1), e(4)}, {e Fig. 1 . 1Non-linear code. Fig. 3 . 3Unicast relay network. as a generalization of the network of Fig 1. This network is a unicast network, and only one intermediate node in each intermediate group. That is, it has c−1 intermediate nodes. We have k i edges between the i−1 and i-th nodes. In one channel use, each edge e(i, j) can transmit the information Y i,j for i = 1, . . . , c and j = 1, . . . , k i that takes values on {1, . . . , d}. Theorem 4 . 4Under the condition (C2), we have max s1,...,sc Theorem 5 . 5Under the condition (C2'), we have max s1,...,sc q\|q ≤ d n } = log d. Fig. 4 . 4homogeneous multicast relay network C 1 = 1C 1,L = C 2 = C 2,L = C 1,D = C 1,L,D = C 2,D = C 2,L,D = C 1,AC = C 1,L,AC = C 2,AC = C 2,L,AC . because A 3 3= min((log d)(k 1 − r 1 )b 1 , A 5 ). Summing up the above region with respect to intermediate nodes in the first group, we find the relation C D,L ⊃ (R 1 , . . . ,R b ) b i =1 R i ≤ A 3 , R i ≤ A 2 for i = 1, . . . , b , which is the direct part. Fig. 5 . 5homogeneous multiple multicast relay network VIII. HOMOGENEOUS MULTIPLE MULTICAST RELAY NETWORK . MH was supported in part by the JSPS Grant-in-Aid for Scientific Research (A) No.17H01280, (B) No. 16KT0017, (C) No. 16K00014, and Kayamori Foundation of Informational Science Advancement. APPENDIX A PROOF OF LEMMA 2 TABLE I SUMMARY IOF COMPARISON WITH EXISTING RESULTS TABLE II SUMMARY IIFOR ONE HOP RELAY NETWORK (FIG. 1) WITH SINGLE SHOT SETTINGCode deterministic attack adaptive attack linear code over Fp with prime p insecure insecure linear code over Fq with imperfectly secure imperfectly secure sufficiently large prime power q non-linear code over F2 imperfectly secure insecure IV. ASYMPTOTIC FORMULATION This type of linear code is often called vector linear[26] because these random variables are given as elements of vector spaces over the finite field Fq. Although the paper[26] assumes that all the messages, the scramble random numbers, and the variables on the edges have the same dimension, we do not assume this condition. Even when the cardinality d of each channel is different from q, this theorem still holds. In contrast, the paper[11] discussed a similar code construction by increasing n (vector linearity) while it did not increase the size of q. 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[ "Automatic Accuracy Prediction for AMR Parsing", "Automatic Accuracy Prediction for AMR Parsing" ]	[ "Juri Opitz [email protected] \nResearch Training Group AIPHES Leibniz ScienceCampus \"Empirical Linguistics and Computational Language Modeling\" Department for Computational Linguistics\n69120Heidelberg\n", "Anette Frank [email protected] \nResearch Training Group AIPHES Leibniz ScienceCampus \"Empirical Linguistics and Computational Language Modeling\" Department for Computational Linguistics\n69120Heidelberg\n" ]	[ "Research Training Group AIPHES Leibniz ScienceCampus \"Empirical Linguistics and Computational Language Modeling\" Department for Computational Linguistics\n69120Heidelberg", "Research Training Group AIPHES Leibniz ScienceCampus \"Empirical Linguistics and Computational Language Modeling\" Department for Computational Linguistics\n69120Heidelberg" ]	[ "Proceedings of the Eighth Joint Conference on Lexical and Computational Semantics (*SEM)" ]	Meaning Representation (AMR) represents sentences as directed, acyclic and rooted graphs, aiming at capturing their meaning in a machine readable format. AMR parsing converts natural language sentences into such graphs. However, evaluating a parser on new data by means of comparison to manually created AMR graphs is very costly. Also, we would like to be able to detect parses of questionable quality, or preferring results of alternative systems by selecting the ones for which we can assess good quality. We propose AMR accuracy prediction as the task of predicting several metrics of correctness for an automatically generated AMR parse -in absence of the corresponding gold parse. We develop a neural end-to-end multi-output regression model and perform three case studies: firstly, we evaluate the model's capacity of predicting AMR parse accuracies and test whether it can reliably assign high scores to gold parses. Secondly, we perform parse selection based on predicted parse accuracies of candidate parses from alternative systems, with the aim of improving overall results. Finally, we predict system ranks for submissions from two AMR shared tasks on the basis of their predicted parse accuracy averages. All experiments are carried out across two different domains and show that our method is effective.	10.18653/v1/s19-1024	[ "https://www.aclweb.org/anthology/S19-1024.pdf" ]	119,302,659	1904.08301	867e131846bc967a3c4dde514ff8e3b250317ab0	Automatic Accuracy Prediction for AMR Parsing Association for Computational LinguisticsCopyright Association for Computational LinguisticsJune 6-7, 2019. 2019 Juri Opitz [email protected] Research Training Group AIPHES Leibniz ScienceCampus "Empirical Linguistics and Computational Language Modeling" Department for Computational Linguistics 69120Heidelberg Anette Frank [email protected] Research Training Group AIPHES Leibniz ScienceCampus "Empirical Linguistics and Computational Language Modeling" Department for Computational Linguistics 69120Heidelberg Automatic Accuracy Prediction for AMR Parsing Proceedings of the Eighth Joint Conference on Lexical and Computational Semantics (SEM) the Eighth Joint Conference on Lexical and Computational Semantics (SEM)MinneapolisAssociation for Computational LinguisticsJune 6-7, 2019. 2019212 Meaning Representation (AMR) represents sentences as directed, acyclic and rooted graphs, aiming at capturing their meaning in a machine readable format. AMR parsing converts natural language sentences into such graphs. However, evaluating a parser on new data by means of comparison to manually created AMR graphs is very costly. Also, we would like to be able to detect parses of questionable quality, or preferring results of alternative systems by selecting the ones for which we can assess good quality. We propose AMR accuracy prediction as the task of predicting several metrics of correctness for an automatically generated AMR parse -in absence of the corresponding gold parse. We develop a neural end-to-end multi-output regression model and perform three case studies: firstly, we evaluate the model's capacity of predicting AMR parse accuracies and test whether it can reliably assign high scores to gold parses. Secondly, we perform parse selection based on predicted parse accuracies of candidate parses from alternative systems, with the aim of improving overall results. Finally, we predict system ranks for submissions from two AMR shared tasks on the basis of their predicted parse accuracy averages. All experiments are carried out across two different domains and show that our method is effective. Introduction Abstract Meaning Representation (AMR) (Banarescu et al., 2013) represents the semantic structure of a sentence, including concepts, semantic operators and relations, sense-disambiguated predicates and their arguments. As a machine readable representation of the meaning of a sentence, AMR is potentially useful for many NLP tasks. Among other applications it has been used in machine translation (Jones et al., 2012), text (a / asbestos :polarity -:time (n / now) :location (t / thing :ARG1-of (p / produce-01 :ARG0 (w / we)))) Figure 1: Humanly produced AMR for: There is no asbestos in our products now. Numbered predicates refer to PropBank senses (Palmer et al., 2005). summarization (Liu et al., 2015;Dohare and Karnick, 2017) and question answering (Mitra and Baral, 2016). Since the introduction of AMR, many approaches to AMR parsing have been proposed: graph-based pipeline systems which rely on an alignment step (Flanigan et al., 2014(Flanigan et al., , 2016 or transition-based parsers relying on dependency annotation (Wang et al., 2015b(Wang et al., ,a, 2016a. In the following we will denote the former by JAMR and the latter by CAMR. More recently, endto-end neural systems have been proposed which produce linearized AMR graphs within characterbased (van Noord and Bos, 2017b) or word-based (Konstas et al., 2017) encoding models. Both approaches greatly profit from large amounts of silver training data. The silver data is obtained with self-training (Konstas et al., 2017) or the aid of additional parsers, where only parses with considerable agreement are chosen to extend the training data (van Noord and Bos, 2017b). Lyu and Titov (2018) formulate a neural model that jointly predicts alignments, concepts and relations. Their system -henceforth called GPLA (Graph Prediction with Latent Alignments) -defines the current state-of-the-art in AMR parsing. A system that can perform accuracy prediction for AMR parsing can be used in a variety of ways: (i) estimating the quality of downstream tasks that deploy AMR parses. E.g., in a document summarization scenario, we might expect lower qual-ity of a summary if the estimated quality of AMR parses used as a basis for the summary is low; (ii) AMR parsing accuracy estimation can be used to produce high-quality automatically parsed data: by filtering the outputs of single parsing systems in self-training, by selecting high-quality outputs from different parsing systems in a tri-parsing setting, or else by predicting overall rankings over alternative parsing systems applied to in-or outof-domain data; (iii) finally, AMR parse accuracy prediction could be used in the context of a parsersupported treebank construction process. E.g., in an active learning scenario, we can select useful targets for manual annotation based on their expected efficiency for parser improvement -the fine-grained evaluation measures predicted by our system can be used for targeted improvements. In the simplest case, we can provide the human annotator with automatic parses where only few flaws have to be mended. Hence, AMR accuracy prediction systems have the potential to tremendously reduce manual annotation cost and time. Contributions We define AMR accuracy prediction as the task of predicting a rich suite of metrics to assess various subtasks covered by AMR parsing (e.g. negation detection or semantic role labeling). To approach this task, we use the AMR evaluation suite suggested by Damonte et al. (2017) and develop a hierarchical multi-output regression model for automatically performing evaluation of 12 different tasks involved in AMR parsing (Sections §3 and §4; our code is publicly accessible 1 ). We perform experiments in three different scenarios on unseen in-domain and out-of-domain data and show that our model (i) is able to predict scores with significant correlation to gold scores and (ii) can be used to rank parses on a sentencelevel or to rank parsers on a corpus-level ( §5). Related Work Automatic accuracy prediction for syntactic parsing comes closest to what we are doing. Ravi et al. (2008) propose a feature-based SVM regression model with RBF kernel that predicts syntactic parser performance on different domains. Like us, they aim at a cheap and effective means for estimating a parser's performance. However, in contrast to their work, our method is domain and parser agnostic: we do not take into account characteristics of the domains of interest and do not provide any performance statistics of the competing parsing systems as features to our regressor. Biici (2016) addresses the task without any domain-dependent features, which results in a lower correlation to gold scores -even if additional features from a background language model are incorporated. In contrast to the prior systems that predict a single score, we predict an ensemble of metrics suitable for assessing AMR parse quality with respect to different linguistic aspects. Also, our system does not rely on externally derived features or complex pre-processing. Moreover, an AMR graph differs in important ways from a syntactic tree. Nodes in AMR do not explicitly correspond to words (as in dependency trees) or phrases (as in constituency trees). AMR structure elements can exist without any alignment to words in the sentence. To our knowledge, we are the first to propose an accuracy prediction model for AMR parsing, and offer the first general end-to-end parse accuracy prediction model that predicts an ensemble of scores for different linguistic aspects. Automatic accuracy prediction has also been researched for PoS-tagging (Van Asch and Daelemans, 2010) and in machine translation. For example, Soricut and Narsale (2012) predict BLEU scores for machine-produced translations. Under the umbrella of quality estimation researchers try to predict, i.a., the post-editing time or missing words in an automatic translation (Cai and Knight, 2013;Joshi et al., 2016;Chatterjee et al., 2018;Kim et al., 2017;Specia et al., 2013). The fact that manually creating an AMR graph is significantly more costly than a translation provides another compelling argument for investigating automatic AMR accuracy prediction techniques . 2 In recent work, Smith (2011, 2017); Jain et al. (2015); Rehbein and Ruppenhofer (2018) detect annotation errors in automatically produced dependency parses. The latter approach uses active learning and ensemble parsing in combination with variational inference. They predict edge labelling and attachment errors and use a back-and-forth encoding mechanism from non-structured to structured tree data in order to provide the variational inference model with the (a / asbestos (a / asbestos :time (n / now) :polarity -:polarity -:location (p / product) :location (p / product :time (n / now)) :poss (w / we))) __________________________ (a / asbesto metr.(F1)\| GP JA CA \| :polarity ----------\|--------------\| :ARG1 (w / we Smatch \| 70 \| 30 \| 67 \| :ARG1-of (p / product SRL \| 0 \| 14 \| 0 \| :mod (n / now)))) Concepts \| 67 \| 44 \| 50 \| IgnVars \| 55 \| 0 \| 60 \| Figure 2: Three AMR parses for: There is no asbestos in our products now, generated by GPLA (top), JAMR (bottom), CAMR (right). Light and severe errors are found in GPLA and JAMR parses; CAMR fails to provide we, the manufacturer of the product. Bottom right: F1 for Smatch and three example subtasks from evaluation against the gold parse (given in Figure 1). needed unstructured data. Their work differs from ours in three important aspects: firstly, they predict errors in specific edges or nodes, while we predict an accuracy score over the complete graph. Moreover, our model does not need several candidate parses as input -when several multiple parses are available, our model can be exploited for ranking (cf. Sections §5.2 & §5.3). Finally, our method is independent of live human feedback. Accuracy Metrics for AMR Parsing Automatic AMR parses are often deficient. Consider the examples in Figure 2. All parsers correctly detect the negation and its scope. The GPLA parse (top) provides a graph structure close to the gold annotation ( Figure 1). However, it does not correctly analyze the possessive our (product), which in the gold parse is represented as an object produced by the speaker (we). Instead it recognizes a location in the speaker's possession. JAMR (middle) fails to detect the concept in focus (asbestos), possibly due to a false-positive stemming mistake. Moreover, it fails to represent that asbestos is (not) in the product: it misses the :location-edge from asbestos to product. AMR accuracy metrics Usually, a predicted AMR graph G is evaluated against a gold graph G using triple matching based on a maximally scoring variable mapping. For finding the optimal variable mapping, Integer Linear Programming (ILP) can be used in the Smatch metric (Cai and Knight, 2013), which produces precision, recall and F1 score between G and G . While it is important to obtain a global measure of parse accuracy, we may also be interested in a quality assessment that focuses on specific subtasks or meaning aspects, such as entity linking, negation detection or word sense disambiguation (WSD). For example, if a parser commits a WSD error this might be less harmful than e.g., failing to capture negation, or missing or wrongly predicting a semantic role. However, the Smatch calculation would treat many of such errors with equal weight -a property which in some cases may be undesirable. To alleviate this issue, Damonte et al. (2017) proposed an extended AMR evaluation suite which allows parser performance inspection with regard to 11 additional subtasks captured by AMR. In total, 36 metrics can be computed (precision, recall and F1 for 12 tasks). F1 scores for three example metrics are displayed in Figure 2 (bottom, right): Smatch, SRL (Smatch computed on arg-i roles), IgnoreVars (triple overlap after replacing variables with concepts) and Concepts (F1 for concept identification). 3 GPLA produces the overall best parse but it is is outperformed by the other systems in SRL (JAMR) and IgnoreVars (CAMR). Task definition We adopt the proposed metrics by Damonte et al. (2017) and use them as target metrics for our task of AMR parse accuracy prediction. Given an automatic AMR graph G and a corresponding sentence S, we estimate precision, recall and F1 of the main task (Smatch) and of the subtasks, as they would emerge from comparing G with its gold counterpart G . One of our hypotheses is that predicting a wide range of accuracy metric scores for individual aspects of AMR structures will aid our model to better predict the global Smatch scores. We will therefore investigate a hierarchical model that builds on predicted subtask measures in order to predict the global smatch score. Being able to predict fine-grained quality aspects of AMR parses will also be useful to assess and exploit differences of alternative system outputs and provides a basis for guiding system development or targeted annotation in an active learning setting. Neural Accuracy Prediction Model We propose a neural hierarchical multi-output regression model for accuracy prediction of AMR Figure parses. Its architecture is outlined in Figure 3. Inputs Our model takes the following inputs: (i) a linearized AMR and a linearized dependency graph (implementation details in §5). The motivation for feeding the dependency parse instead of the original sentence is due to the moderate similarity of dependency and AMR structures. 4 We examine drawbacks and benefits of providing automatic dependency parses more closely in our ablation experiments ( §5.4). In addition, (ii) we produce alignments between sentence tokens and tokens in the sequential AMR structure, as well as between sentence tokens and the linearized dependency structure, and feed these sequences of pointers to our accuracy prediction model. The intuition of using pointers is to provide the model with richer information via shallow alignment between AMR, dependencies and the sequence of sentence tokens (see Section §5 for implementation details). Finally, (iii) we feed a sequence of PropBank sense indicators for AMR predicates. Joint encoding of AMR and dependency parses for metric prediction Embedding layers are shared between AMR/dependency pointers and AMR/dependency tokens. We embed the three sequences representing the AMR graph (tokens, pointers and senses) in three matrices and sum them up element-wise (indicated with + in Figure 3). The same procedure is applied to the linearized dependency graph (tokens and pointers). The resulting matrices are processed by two two-layered Bi-LSTMs to yield vectorized representations for (i) the AMR graph and (ii) the dependency tree (i.e., the last states of forward and backward reads are concatenated). Thereafter, we apply element-4 c.f. Groschwitz et al. (2018); Chen and Palmer (2017). wise multiplication, subtraction and addition to both vector representations and concatenate the resulting vectors (⊗ in Figure 3). The joint AMRdependency representation is further processed by a feed forward layer (FF) with sigmoid activation functions in order to predict, in total, 36 different metrics (green, Figure 3). Hierarchical prediction of multiple metrics The task naturally lends itself to be formulated in a hierarchical multi-task setup (orange, Figure 3). In this strand, we first compute the 33 fine-grained subtask metrics and on their basis we caclulate the Smatch scores (precision, recall, F1) as our primary metrics. In order to accomplish this, we collect the outputs from the subtask metric prediction layer in a vector and concatenate it with the previous layer's representation (⊕ in Figure 3). The resulting vector is fed through a last FF layer to predict the metrics for the task of main interest (Smatch). Our intuition is that the estimated quality of the parse with respect to the subtask metrics informs the model and allows it to better predict the overall quality. Loss In the non-hierarchical case, we denote our full model with f θ : X → [0, 1] d with parameters θ, where d describes the dimensionality of the score vector (one dimension represents one metric) and D = {(X i , y i )} N i=1 , y i ∈ [0, 1] d is our training data. In the non-hierarchical model, we minimize the mean squared error: (f θ ) = 1 dN N i=1 d j=1 (y i,j − f θ (X i ) j ) 2(1) For our hierarchical model, we have two functions, f θ : X → [0, 1] (d−k) which returns the output vector for the (d − k) subtask metrics and f θ : X → [0, 1] k which returns the output vector for our k main metrics (in our experiments, k = 3 for Smatch recall, precision and F1). Then, (f θ , f θ ) = λ 1 (d − k)N N i=1 d−k j=1 (y i,j − f θ (X i ) j ) 2 + λ 2 kN N i=1 d j=d−k+1 (y i,j − f θ (X i ) j−(d−k) ) 2 defines the total loss over the two entangled metric prediction models. Note that θ ⊂ θ , which means that by optimizing the parameters of f with gradient descent, we also concurrently optimize all parameters of f . By this construction, the hierarchical model instantiates a two-task model with shared parameters. For our experiments, we manually set the loss weights λ 1 = 0.2, λ 2 = 1. Experiments Data Since our goal is to predict the accuracy of an automatic parse, we need a data set containing automatically produced AMR parses and their scores, as they would emerge from comparison to gold parses. Our largest data set, LDC2015E86, comprises 19,572 sentences and comes in a predefined training, development and test split. We parse this data set with three parsers, JAMR (Flanigan et al., 2014(Flanigan et al., , 2016, CAMR (Wang et al., 2015b(Wang et al., ,a, 2016a and GPLA (Lyu and Titov, 2018). Since the three parsers have been trained on the training data partition, we naturally obtain more accurate parses for the training partition than for development and test data. Table 1, however, indicates that we still obtain a considerable amount of deficient parses for training. Based on the parser outputs we compute evaluations comparing the automatic parses with the gold parses by using amrevaluation-tool-enhanced 5 , a bug-fixed version of the script that computes the metrics of Damonte et al. (2017). This allows us to create full-fledged training, development and test instances for our accuracy prediction task. Each instance consists of a sentence and an AMR parse as input and a vector of metric scores as target. Our second data set, LDC2015R36, comprises submissions to the SemEval-2016 Task 8 (May, 2016). We have 1053 parses from each of the 11 team submissions (and 2 baseline systems). 6 Our 5 https://github.com/ChunchuanLv/ amr-evaluation-tool-enhanced 6 Riga ( Barzdins and Gosko, 2016), CMU (equal to JAMR) (Flanigan et al., 2016), Brandeis (Wang et al., 2016b), UofR (Peng and Gildea, 2016), ICL-HD (Brandt et al., 2016), M2L (Puzikov et al., 2016), UMD (Rao et al., 2016), third dataset, BioAMRTest is used as the test set in the SemEval-2017 Task 9 (May and Priyadarshi, 2017) and consists of 500 parses from each of the 6 teams. 7 The shared task organizers kindly made this data available for our experiments. Preprocessing For dependency annotation, we parse all sentences with spacyV2.0 8 . For sequentializing the AMR and dependency graph representations we take intuitions from van Noord and Bos (2017b) & Konstas et al. (2017) and output tokens by performing a depth-first-search over the graph. We replace the AMR negation token '-' and strings representing numbers with special tokens. The vocabularies (tokens, senses and pointers) are computed from our training partition of LDC2015E86 and comprise all tokens with a frequency ≥ 5 (tokens with lesser frequency are replaced by an OOV-token). PropBank senses of predicates are removed and collected in an extra list that is parallel to the tokens in the linearized AMR sequence. For each linearized AMR and dependency tree we generate a sequence with index pointers to tokens in the original sentence (-1 for tokens which do not explicitly refer to any token in the sentence, e.g. brackets, 'subj' or 'arg0' relations). Extraction of token-pointers from the dependency graph is trivial. For every concept in the linearized AMR we execute a search for the corresponding token in the sentence, looking for exact matches with surface tokens and lemmas. Training scribed in §4). We use the same single (hierarchical) model for all three evaluation studies, proving its applicability across different scenarios (a nonhierarchical model is only instantiated for the ablation experiments in Section §5.4). Correlation with Gold Accuracy The primary goal in our first experiment is to test whether the system is able to differentiate good from bad parses. This capacity is expressed by a high correlation of predicted accuracies with true accuracies on unseen data and by the ability to assign high scores to gold parses. We evaluate on the test partition of LDC2015E86 and BioAMRTest. Correlation results The results are displayed in Table 3. Over all metrics, in-domain and out-ofdomain, we achieve significant correlations with the gold scores (p < 0.005 for every metric). While on LDC2015E86 the model has learned to predict the KB linking F1 (ρ = 0.86) and negation detection F1 with high correlation to the gold scores (ρ = 0.87), Concept assessment poses the greatest challenge (ρ = 0.64). For the out-ofdomain data BioAMRTest, these two facts seem almost reversed: here, the assessment of KB linking poses difficulties (ρ = 0.23) while the Concept F1 predictions are better (ρ = 0.62). The main metrics of interest (Smatch precision, recall and F1) can be predicted with high correlation on indomain data (ρ ≥ 0.74, cf. also Figure 4) and solid correlation for out-of-domain data (ρ ≥ 0.41). Find the Gold AMR! Now, we want to test our system's capacity to reliably predict high Smatch F1 scores for unseen gold AMR parses. Ideally, the scores should be close or equal to 1. For in-domain data, it appears to work well: a large amount of Smatch predictions for gold AMR graphs are very close to one (Figure 5a). Evidently, our system also gets the ranking of the parsing systems right: the distribution of the state-of-the-art (GPLA) is shifted right towards higher predicted F1 scores, whereas the distribution of CAMR is shifted left towards lower scores. Also, more than 75% of gold parses have a predicted Smatch score of more than 0.99 (Table 4). On the other hand, finding gold parses in the BioAMRtest data is much harder: about 75% of Smatch scores get assigned a score of 0.83 or lower and only 1% of gold parses are predicted as perfect (Table 4). The estimated probability density function for gold parses (red solid line in Figure 5b) struggles to discriminate itself from the functions corresponding to the flawed parses of the automatic systems. Nevertheless, the prediction score density for gold parses is situated more on the right hand side than most others. In other words, we find that in the out-of-domain data gold parses tend to be assigned above-average scores. To sum up, our observations for the out-ofdomain data stand in some contrast to what we observe for the in-domain data. However, this outcome can be plausibly explained: assuming that the out-of-domain gold parses have some unfamiliar properties, a system that has never seen such parses cannot judge well whether they are gold or not. In fact, it can be interpreted positively that the system hesitates to assign maximum scores to gold parses from a domain in which the model is completely inexperienced. Additionally, bio-medial texts involve difficult concepts, naming conventions and complicated noun phrases which are hard to understand even for non-expert humans (e.g., "TAK733 led to a decrease in pERK and G1 arrest in most of these melanoma cell lines regardless of their origin, driver oncogenic mutations and in vitro sensitivity to TAK733".). Taking all this into account, the results for out-of-domain data may be not as bad as they perhaps appear at first glance. Application Study: AMR Parse Ranking Our automatic accuracy prediction method naturally lends itself for ranking parser outputs. For any sentence, provided automatic parses by competing systems can be ranked according to the scores predicted by our system. This scenario arises, e.g., when we run several AMR parsers over a large corpus with the aim of selecting the best parse for each sentence in order to collect silver training data. 9 In the worst case, we do not have any prior knowledge about a parser's performance (we may not even know the source of a parse). We use the test partition from LDC2015E86 and BioAMRTest to rank, for each sentence, the automatic candidate parses provided by the different parsers. In LDC2015E86 we assume not to be agnostic about the parsers as their performances on the development data of this data set are known (in terms of their sentence-average F1 Smatch score). Consider that we are given a sentence and three automatic parses. We select the maximum-score parse, where the score is defined by predicted Smatch F1 plus the average Smatch F1 of the parse-producing parser on the development data. As baselines in this scenario we (i) randomly choose a parse from the three options or (ii) always choose the parse of GPLA. On BioAMRTest, however, we have no prior information about the submitted systems. We select from 6 automatic parses for each sentence. Since now we are completely parser agnostic, the baseline is to randomly select a parse from the candidate set. Results The results are displayed in Table 5 Table 6: Results of different parse-ranking systems with respect to sentence-level parse rankings.ρ: average Pearson-r on a sentence level. %pos: ratio of predicted rankings with positive ρ to gold ranking. the best parse according to our model's predicted accuracy score improves over all individual parser results: the obtained average Smatch F1 per sentence increases (i) slightly by 0.2 pp. compared to always choosing outputs from GPLA and (ii) observably by 5.7 pp. compared to randomly selecting a parse from the competing system outputs. The difference compared to always choosing GPLA seems negligible which perhaps can be explained by the fact that GPLA has been shown to be on par or better than doubly-blind human annotators. 10 The oracle that always selects the best parse (upper-bound in Table 5) shows little room for improvement: it achieves 2.1 pp. Smatch F1 increase compared to our model. This margin is small and further success might also be hampered by peculiarities in the manual annotations. On BioAMRTest, no prior information about the systems is available. Using our model's predicted scores to select from the alternative system outputs, we can boost Smatch F1 by 5.2 pp. compared to randomly selecting a parse. Compared to always selecting the parses of the best submitted system (in-hindsight), we lag behind by 3.9 pp. Since our data comprises outputs from several parsers with varying performance, we can study the performance of our approach in combination with different parsers (Figure 6). When only choosing among CAMR and JAMR outputs, on LDC2015E86, our system boosts the F1 by 2.7 pp. compared to randomly selecting a parse, and by 0.6 pp. compared to always choosing the parse from the better system (determined on dev, here: JAMR). Choosing from CAMR and GPLA or JAMR and GPLA makes little difference: in most cases our system selects the GPLA parse and the difference to only choosing GPLA parses is 10 GPLA (Lyu and Titov, 2018) achieves a high 74.4% corpus-level Smatch F1 (primarily news texts), while a prior annotation study (Banarescu et al., 2013) reported doubly blind annotation corpus-level F1 of 0.71 (for web texts). CAMR/JAMR CAMR/GPLA JAMR/GPLA Figure 6: Using our model to predict the best parse out of two candidate parses, each from a different system. marginal. Moreover, across both test sets, the majority of rankings assigned by our method have positive correlations with the true rankings ( Table 6): 77% of all assigned rankings have a positive correlation with the true ranking (70% for biomedical). In sum, we can draw two conclusions from this experiment: given a sentence, ranking AMR parser outputs using our accuracy prediction model, on in-domain and out-of-domain unseen data (i) clearly improves performance when non state-of-the-art parsers are applied or if we are not informed about the parsers' performances and (ii) does not worsen results in other cases. Application Study: Predict System Ranks In our final case study, we use our accuracy prediction model to predict a ranking over systems. We use our model to rank the unseen submitted system parses of the SemEval-2017 Task 9 (evaluated on BioAMRTest) and SemEval-2016 Task 8 (evaluated on LDC2015R36) according to average predicted F1 Smatch scores. Again, we assume a parser-agnostic setting, meaning we have no prior knowledge of the submitted systems (i.e. we just consider their outputs). In this setting, we do not rank individual parses given a sentence, but rank the system outputs, according to estimated average Smatch F1 per sentence. We evaluate against the final team rankings of the two shared tasks. Results The results are displayed in Table 7. On BioAMRTest we have a good, albeit non statistically significant correlation with the true team ranking. On the in-domain LDC2015R36 test set we see a significant correlation of ρ = 0.645 (p 1,2 < 0.05). In this shared task, many teams were competitive and differences between the best teams were marginal. For example, in the true ranking, places 1 to 6 achieved between 0.60 and 0.62 Smatch F1. Notably, the first four teams ac-Rank LDC2015R36 Rank BioAMRTest rank r rankr rank r rankr DANGNT --1 3 Oxford --2 1 TMF-2 --3 2 RIGOTRIO --4 5 TMF-1 --5 4 JAMR 7 7 6 6 RIGA 1 4 --Brandeis 2 3 --CU-NLP 3 1 --UCL+Sheffield 4 2 --ICL-HD 5 8 --M2L 6 10 --JAMR-base 8 12 --UofR 9 11 --TMF 10 5 --UMD 11 6 --DynamicPower 12 13 -det. baseline 13 9 -ρ 0.645 (p1 = 0.017, p2 = 0.011) 0.771 (p1 = 0.072, p2 = 0.051) Table 7: True rank r (given corpus-Smatch) and predicted rankr (based on sentence average Smatch computed using our model). p 1 : probability of noncorrelation. p 2 : probability that a randomly produced ranking achieves equal or greater ρ (estimated over 10 6 random rankings). For team names, see fn. 6 & 7. cording to the true ranking and the first four teams according to our predicted ranking fall into the same group. This shows that our model successfully assigned high ranks to low error submissions. Ablation Experiments We finally perform ablation experiments to evaluate the impact of individual model components. We experiment with five different setups. (i) instead of stacking two Bi-LSTMs, we use only one Bi-LSTM (one-lstm, Table 8). (ii) instead of the dependency tree, we feed the words in the order as they occur in the sentence (no-dep). (iii) nopointers: we remove the token-pointers from our model. (iv), instead of using the hierarchical setup, we predict all metrics on the same level (green in Figure 3, no-HL in Table 8) and (v), no-HMTL: we optimize the non-hierarchical model only with respect to Smatch, disregarding the AMR subtasks. Remarkably, the dependency tree greatly helps the model on in-domain data over all measures (-37 total ∆ without dependencies) but hurts the model on out-of-domain data (+27 total ∆). A possible explanation is the degradation of the dependency parse quality: bio-medical data not only poses a challenge for our model, but also for the dependency parser. With special regard to the main AMR evaluation measure, Smatch F1, the learned pointer embeddings provide useful input on the indomain test data (-4 ∆ without pointers). Conclusion AMR parser evaluation with human gold annotation is very costly. Our main contributions in this work are two-fold: Firstly, we introduced the concept of automatic AMR accuracy prediction. Given only an automatic parse and the sentence, from whence it was derived, the goal is to predict evaluation metrics cheaply and possibly at runtime. Secondly, we framed the task as a multiple-output regression task and developed a hierarchical neural model to predict a rich suite of AMR evaluation metrics. We presented three case studies proving (i) the feasibility of automatic AMR accuracy prediction in general (significant correlation with gold scores on unseen indomain and out-of-domain data) and (ii) the applicability of our model in two use cases. In the first study, we ranked different automatic candidate parses per sentence, outperforming the random selection baseline by 5.7 pp. average Smatch F1 (in-domain) and 5.2 pp. (out-of-domain). In the second study, we ranked team submissions to two AMR shared tasks and our method was able to reproduce rankings similar to the true rankings. 3 : 3Our model: green: Evaluation metrics computed in a non-hierarchical fashion. orange: Main evaluation metric is computed on top of secondary metrics. For the optimization of the accuracy prediction model we use only the development and training sections of LDC2015E86 and the corresponding automatic parses together with the gold scores. Details on the training cycle can be found in the Supplemental Material §A (the loss is de-DynamicPower(Butler, 2016), TMF(Bjerva et al., 2016), UCL+Sheffield(Goodman et al., 2016) and CU-NLP(Foland and Martin, 2016). 7 TMF-1 and TMF-2 (van Noord and Bos, 2017a), DAN-GNT(Nguyen and Nguyen, 2017), Oxford(Buys and Blunsom, 2017), RIGOTRIO(Gruzitis et al., 2017) and JAMR(Flanigan et al., 2016) 8 https://spacy.io/ Figure 4 : 4Predicted (y-axis) & gold (x-axis) Smatch F1. Figure 5 : 5Probability density function estimations for predicted F1 Smatch scores using Scott's method (Scott, 2012) with respect to candidate parses from different systems. Table 2 : 2Statistics of data sets used in this work. Table 3 : 3Pearson correlation coefficient (ρ) over various metrics and across domains. Explanations of the metrics and AMR subtasks are in Section §3 and fn. 3 Table 4 : 4Various percentiles of Smatch F1 predictions for gold graphs. Table 5 : 5Results (sentence averages) of different AMR parsing (bottom part) and ranking (top part) systems on two test sets. Upper part: results when selecting from alternative parses: lower-bound (upper-bound): oracle selecting the worst (best) AMR parse; ours: results when selecting the best parse according to our models' accuracy prediction (hierarchical model). Table 8 : 8ρ correlation (F1) differences over different setups (columns), test sets (out-of-domain, in-domain) and subtasks (rows). ±x: plus and minus x pp.ρ. https://gitlab.cl.uni-heidelberg.de/ opitz/quamr Creating an AMR graph requires trained linguists and takes on average 8 to 13 minutes, cf.Banarescu et al. (2013) The other subtasks are: Unlabelled (Smatch after edge label removal), No WSD (Smatch after PropBank sense removal), NS frames (PropBank frame identification without sense), Wikification (entity linking), NER (named entity recognition), Reentrancy (Smatch over re-entrant edges). In a self-training scenario, we also could set a threshold of minimum predicted accuracy to select confident parses. AcknowledgmentsThis work has been supported by the German Research Foundation (grant no. GRK 1994/1) and the Leibniz Association (grant no. SAS-2015-IDS-LWC) and the Ministry of Science, Research, and Art of Baden-Württemberg. We are grateful to the NVIDIA corporation for donating the GPU used in this research.A Supplemental Material Hyper parameters and weights initializationWe initialize all parameters of the model randomly. Embedding vectors of dimension 128 are drawn from U (0.05, 0.05) and the LSTM weights (neurons: 128) and weights of the feed forward output layers are sampled from a Glorot uniform distribution(Glorot and Bengio, 2010). For future work, initializing the embedding layer with pre-trained vectors could further increase the performance. In this work, however, we learn all parameters from the given data. We fit our model using Adam (Kingma and Ba, 2014) (learning rate: 0.001) on the training data over 20 epochs with mini batches of size 16. We apply early stopping according to the maximum Pearson's ρ (with regard to Smatch F1) on the development data.(y i −ȳ) 2 quantifies the linear relationship between predicted scores (x 1 , ..., x n ) and true scores (y 1 , ..., y n ). Abstract meaning representation for sembanking. Laura Banarescu, Claire Bonial, Shu Cai, Madalina Georgescu, Kira Griffitt, Ulf Hermjakob, Kevin Knight, Philipp Koehn, Martha Palmer, and Nathan SchneiderLaura Banarescu, Claire Bonial, Shu Cai, Madalina Georgescu, Kira Griffitt, Ulf Hermjakob, Kevin Knight, Philipp Koehn, Martha Palmer, and Nathan Schneider. 2013. Abstract meaning representation for sembanking. Riga at semeval-2016 task 8: Impact of smatch extensions and character-level neural translation on amr parsing accuracy. 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Asso- ciation for Computational Linguistics.	[ "https://github.com/ChunchuanLv/" ]
[ "On the Linear Belief Compression of POMDPs A re-examination of current methods", "On the Linear Belief Compression of POMDPs A re-examination of current methods" ]	[ "Zhuoran Wang ", "Paul A Crook ", "Wenshuo Tang ", "· Oliver Lemon " ]	[]	[]	Belief compression improves the tractability of large-scale partially observable Markov decision processes (POMDPs) by finding projections from high-dimensional belief space onto low-dimensional approximations, where solving to obtain action selection policies requires fewer computations. This paper develops a unified theoretical framework to analyse three existing linear belief compression approaches, including value-directed compression and two nonnegative matrix factorisation (NMF) based algorithms. The results indicate that all the three known belief compression methods have their own critical deficiencies. Therefore, projective NMF belief compression is proposed (P-NMF), aiming to overcome the drawbacks of the existing techniques. The performance of the proposed algorithm is examined on four POMDP problems of reasonably large scale, in comparison with existing techniques. Additionally, the competitiveness of belief compression is compared empirically to a stateof-the-art heuristic search-based POMDP solver and their relative merits in solving large-scale POMDPs are investigated.	null	[ "https://arxiv.org/pdf/1508.00986v1.pdf" ]	15,867,863	1508.00986	bcea229cb13dc6286f36aab6d07fa7f3bd42178b	On the Linear Belief Compression of POMDPs A re-examination of current methods Zhuoran Wang Paul A Crook Wenshuo Tang · Oliver Lemon On the Linear Belief Compression of POMDPs A re-examination of current methods Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Belief Compression · POMDP · Nonnegative Matrix Factorisation Belief compression improves the tractability of large-scale partially observable Markov decision processes (POMDPs) by finding projections from high-dimensional belief space onto low-dimensional approximations, where solving to obtain action selection policies requires fewer computations. This paper develops a unified theoretical framework to analyse three existing linear belief compression approaches, including value-directed compression and two nonnegative matrix factorisation (NMF) based algorithms. The results indicate that all the three known belief compression methods have their own critical deficiencies. Therefore, projective NMF belief compression is proposed (P-NMF), aiming to overcome the drawbacks of the existing techniques. The performance of the proposed algorithm is examined on four POMDP problems of reasonably large scale, in comparison with existing techniques. Additionally, the competitiveness of belief compression is compared empirically to a stateof-the-art heuristic search-based POMDP solver and their relative merits in solving large-scale POMDPs are investigated. reward or cost of the intermediate action, but also consideration of the long term effect of a sequence of choices made in the future. If the true states of the system can be perfectly identified, Markov decision processes (MDPs) can be used to handle such planning problems efficiently. However, in realworld applications, the true system states are not always fully observable. Uncertainties may come from different sources, e.g. noisy sensors in a robotic system, or speech recognition errors in a spoken dialogue system, etc., which motivates the utilisation of probabilistic techniques to track the system states. The partially observable Markov decision process (POMDP) has been proven to be a powerful tool for modelling sequential decision making problems under uncertainty. It generalises the standard MDP to the case where an agent cannot directly observe the underlying states but has to maintain a probability distribution (called a belief) over all possible states based on noisy observations. The optimal policy of a POMDP then specifies an action for each possible belief to maximise expected discounted future reward. However, the exact solution for the policy optimisation problem of a POMDP is computationally intractable (Cassandra, 1998). Polynomial-time approximation algorithms can be achieved using point-based value iteration (PBVI) techniques (Pineau et al, 2003), which successively estimate the value function by updating the value and its gradient only at the points of a witness point set. In this case, the dimensionality of the belief space dominates the efficiency of the algorithms (see (Pineau et al, 2003;Smith and Simmons, 2005;Spaan and Vlassis, 2005)). Factored POMDPs explored in various previous studies (see below) provide a general direction for improving the tractability of large-scale problems via dimension reduction. The essential idea behind the factorisation is to decompose the original instantiations of the state, action and observation variables in a POMDP into their respective smaller sets of factor variables. Then conditional independence or context-specific independence among those factor variables can be exploited to achieve a more compact representation, using corresponding techniques such as dynamic Bayesian networks (Hoey et al, 2010;Thomson and Young, 2010;Williams et al, 2005), decision trees (Boutilier and Poole, 1996;Boutilier et al, 2000) or algebraic decision diagrams (Hansen and Feng, 2000;Shani et al, 2008). Unfortunately, such factored representations do not necessarily result in efficient policy implementations. Although for particular types of POMDP problems we will also be able to express the transition, observation and reward functions in a compact form with respect to their respective factor variables and optimise the policies in lower-dimensional spaces (Ong et al, 2010;Poupart, 2005;Sim et al, 2008), this method does not generalise to all domains by default. Belief compression provides an alternative solution to reduce the computation cost for POMDP policy optimisation by projecting the high-dimensional belief space into a low-dimensional one, using an automatically obtained projection basis. Main contributions in this area include exponential family principal component analysis (EPCA)-based compression (?), value-directed compression (VDC) (Poupart, 2005;Poupart and Boutilier, 2002) and nonnegative matrix factorisation (NMF)-based compression (Li et al, 2007;Theocharous and Mahadevan, 2010). The EPCA approach achieves a non-linear compression by exploiting the sparsity of the belief space based on sampled beliefs, whilst VDC produces a linear projection in a value-directed manner such that a belief and its compression will obtain an (approximately) identical value. Due to the nature of linear projection, VDC has the advantage that the piecewise linear and convex (PWLC) property of the value function remains after the compression, hence PBVI algorithms can be directly applied to solve the compressed POMDPs. Benefiting from the insights behind both EPCA and VDC, the NMF-based algorithms seek a projection basis (also based on sampled beliefs) that yields a low-rank approximation of the belief space, and use it to construct a linear compression to preserve convenience for PBVI. In this paper, after reviewing some background knowledge of POMDPs ( §2), we develop a unified theoretical framework to analyse linear belief compression algorithms in general ( §3). To the best of our knowledge, such an analysis has not been reported before. After this, the results are employed to examine three existing linear POMDP compression algorithms separately, including VDC (Poupart, 2005), orthogonal NMF (O-NMF) compression (Li et al, 2007), and locality preserving NMF (LP-NMF) compression (Theocharous and Mahadevan, 2010) (which are the only three linear belief compression methods that we are aware of). Our findings show that all the three existing models have their own critical deficiencies. For VDC ( §4), not only can the compressed value function violate the contractive property of a valid Bellman recursion (Bellman, 1957), and therefore can diverge to infinity in the worst case (even when the compression error is extremely small), but also the lack of nonnegativity constraints on its compression basis can confuse the pruning procedure in PBVI and drive the algorithm to an ill converging point. On the other hand, both the O-NMF and LP-NMF approaches share the common drawback that compression error does not directly relate to value loss ( §5), which results in good compressions not necessarily leading to promising policies. Therefore, a novel projective NMF belief compression algorithm (P-NMF) is proposed ( §6), aiming to revise the deficiencies of the existing techniques. Experimental results on four POMDP problems of reasonably large scale show that the proposed model outperforms the existing techniques ( §7.1). In addition, we also investigate the practical effectiveness of belief compression in solving large-scale POMDPs, in comparison with a state-of-the-art (uncompressed) POMDP solver called SARSOP (Kurniawati et al, 2008) ( §7.2), before we conclude ( §8). POMDP Basics A POMDP is a tuple S, A, Z, T, Ω, R, η , where the components are defined as follows. S, A and Z are the sets of states, actions and observations respectively. The transition function T (s \|s, a) defines the conditional probability of transiting from state s ∈ S to state s ∈ S after taking action a ∈ A. The observation function Ω(z\|s, a) gives the probability of the occurrence of observation z ∈ Z in state s after taking action a. R(s, a) is the reward function specifying the immediate reward of a state-action pair. Whilst, 0 < η < 1 is a discount factor. In this paper, we will focus on POMDPs with discrete state, action, and observation spaces. A standard POMDP operates as follows. At each time step, the system is in an unobservable state s, for which only an observation z can be received. A distribution over all possible states is therefore maintained, called a belief, denoted by b, where the probability of the system being in state s is b(s). Based on the current belief, the system selects an action a, receives a reward R(s, a) and transits to a new (unobservable) state s where it receives an observation z . Then the belief is updated to b based on z and a as follows: b (s ) = Pr(s \|z , a, b) = 1 Pr(z \|a, b) Ω(z \|s , a) s T (s \|a, s)b(s)(1) where Pr(z \|a, b) = s Ω(z \|s , a) s T (s \|a, s)b(s) is a normalisation factor. Policy and Value Function A policy π is defined as a mapping that maps each belief b to an action a = π(b). The value function of a given policy π and given starting point b 0 is the expected sum of discounted rewards, calculated as: V π (b 0 ) = E n t=0 η t r π(bt) (b t )(2) where n is the planning horizon (possibly inifinite) and r π(bt) (b t ) is the immediate reward obtained at time t using policy π. The objective of POMDP-based planning is to determine an optimal policy π * = arg max π V π (b) that maximises the value function. The value function corresponding to π * is usually denoted by V * . Value Iteration Commonly used policy optimisation algorithms include value iteration, policy iteration and linear programming. In this paper, we will focus on value iteration related techniques. Exact value iteration recursively computes the optimal value function V * as the sequence of value functions V n starting from an initial V 0 : V n+1 (b) = max a∈A R(b, a) + η z∈Z Pr(b \|b, a, z)Pr(z\|a, b)V n (b )(3) where Pr(b \|b, a, z ) is an indicator of b updating to b on action a and observation z . If we represent R in matrix form, where R(b, a) = b R ·,a , and define the mapping T a,z such that T a,z ij = T (s j \|a, s i )Ω(z\|a, s j ), Eq. (3) can be re-written as: V n+1 (b) := max a∈A b R ·,a + η z∈Z V n (b T a,z ) (4) Such a recursion is usually expressed in functional form: V n+1 = HV n(5) where H is the Bellman backup operator (Bellman, 1957). Eq. (5) is also called a Bellman equation or a Bellman recursion. The above recursion converges to V * , and can be represented as a piecewise linear convex (PWLC) function: V (b) = max α∈Γ b α (6) where Γ is a set of vectors called α-vectors, with each α-vector associated with an action, such that the action corresponding to the α-vector maximising the value function at the current belief b is the one executed by the underlying policy π. Value iteration can then be implemented as a dynamic programming procedure to iteratively construct the α-vectors. Furthermore, the exact Bellman backup operator H can be approximated with tractable computations by considering only a finite set of sampled belief points instead of the entire reachable belief space. This is known as point-based value iteration (PBVI). Detailed introductions to PBVI algorithms are omitted in this paper, but some commonly used techniques can be found in (Pineau et al, 2003;Smith and Simmons, 2005;Spaan and Vlassis, 2005). Linear Belief Compression We start the discussion from an ideal case where we assume that a lossless compression is achievable. Linear belief compression can be summarised as finding a linear function F ∈ R n×k such that: R = FR and T a,z F = FT a,z ∀a ∈ A, z ∈ Z(7) where n = \|S\| is the dimension of the state space in the original POMDP, k n is the compressed state space size, andR andT a,z are the compressed reward and transition matrices, and can be computed as: R = F † R andT a,z = F † T a,z F ∀a ∈ A, z ∈ Z(8) where F † ∈ R k×n is some certain form of 'inverse' of F . Here we temporarily keep this notation general, and leave its specifications for different models explained in later sections. Letb be the compressed belief, and: b = b F(9) For a given policy π, the value functionṼ π defined for the compressed problem can then be written as: V π (b) =b R ·,π(b) + η zṼ π (b T π(b),z )(10) The underlying theory behind lossless linear belief compression was initially proposed by Poupart and Boutilier (2002). We quote their theorem and proof here for convenience of further discussion. Theorem 1 (Poupart and Boutilier) Let B denote the set of all reachable beliefs for a POMDP. Let F ,R and T a,z satisfy Eq. (7), then V π (b) =Ṽ π (b), ∀π, b ∈ B. Proof Base case: let V π 0 (b) = b R ·,π(b) andṼ π 0 (b) =b R ·,π(b) , then V π 0 (b) = b R ·,π(b) = b FR ·,π(b) =b R ·,π(b) =Ṽ π 0 (b) Induction: let V π n (b) =Ṽ π n (b) with n stages-to-go, then V π n+1 (b) = b R ·,π(b) + η z V π n (b T π(b),z ) = b R ·,π(b) + η zṼ π n (b T π(b),z F ) : V π n (b ) =Ṽ π n (b ) = b FR ·,π(b) + η zṼ π n (b FT π(b),z ) : substituting Eq. (7) =b R ·,π(b) + η zṼ π n (b T π(b),z ) =Ṽ π n+1 (b) : substituting Eq. (9) Theorem 1 shows that if the conditions in Eq. (7) hold, all policies have identical values with respect to the compressed and uncompressed POMDPs, i.e. the compression is lossless. A Complementary Theory of Lossless Belief Compression Recall the α-vector representation of the value function in Eq. (6). For a given policy π, we can express V π (b) = b V π = b α π , where we use α π to denote the α-vector specified by π in computing the value of the current b. A similar representation is applicable toṼ π as well. (For instanceṼ π (b) =b Ṽ π = b α π .) Then by substituting Eq. (8) and Eq. (9) into Eq. (10), the value function of the compressed POMDP can be explicitly expressed in the following form:Ṽ π (b) =b Ṽ π = b FṼ π (11) = b FR ·,π(b) + η z b FT π(b),zṼ π = b F F † R ·,π(b) + η z b F F † T π(b),z FṼ π(12) Let V π = FṼ π and A = F F † , and substitute it into Eq. (11) and (12). We obtain: V π (b) = b V π = b AR ·,π(b) + η z b AT π(b),z V π(13) Similar to the definition of the original value function V , the recursive function V can also be written in functional form as: V n+1 = H V n(14) It means that the linear belief compression defined by Eq. (8) and (9) will actually result in a modified value function with the original Bellman backup operator H replaced by an approximated backup operator H. After this, we can obtain the following theorem, which is more relaxed than Theorem 1. Theorem 2 Let A be a low-rank square matrix that can be factored into the product of two rectangular matrices (of rank k) as A = F F † , andR andT a,z be defined as in Eq. (8). If there exists such an A that satisfies either (i) R = AR and T a,z = AT a,z , ∀a ∈ A, z ∈ Z, or (ii) b = b A, ∀b ∈ B, then V π (b) =Ṽ π (b), ∀π, b ∈ B. Proof By defining V π 0 (b) = b AR ·,π(b) and V π 0 (b) = b R ·,π(b) , and substi- tuting either condition (i) or condition (ii), we can obtain V π 0 (b) = V π 0 (b). It is straightforward to induce that if V π n (b) = V π n (b) with n stages-to-go, then V π n+1 (b) = V π n+1 (b), by substituting either condition (i) or condition (ii) into Eq. (13). Finally, substituting the definition of V π into Eq. (11) gives V π n+1 (b) = V π n+1 (b). Convergence of Lossy Belief Compression The above discussions are essentially based on the ideal assumption that a lossless compression exists, which is usually not the case in practical POMDP problems. However, for many problems lossy belief compression can still be employed to reduce the computational complexity of policy training (and execution). Lossy belief compression is designed to seek a projection matrix F by minimising some loss criteria, but as a consequence errors will exist betweeñ V π n (b) and V π n (b). Moreover, such errors may propagate during value iteration, and result in a significant loss in the quality of obtained policy. The error propagation problem has been studied in depth for MDPs under reinforcement learning scenarios in previous literature (Antos et al, 2008;Farahmand et al, 2010;Munos, 2007). However, their results do not directly transfer to POMDP problems due to more complex backup procedures. Hence, in this paper we only study a basic problem: sufficient conditions of the value function Eq. (10) under lossy compression being a valid Bellman equation that converges monotonically to a fixed point. Such convergence implies a bounded loss between the original and the compressed value functions. SinceṼ π (b) and V π (b) return the same value ifb is the compression of b, it is much easier to investigate the latter, which only involves adding an extra linear operator A to the original value function. (5) and (14) Lemma 1 Let b = (b A) andb = b b 1 (i.e. normalised b), For V and V defined in Eq.respectively, H V (b) = b A 1 H V (b). Proof H V (b) = max a, α b AR ·,a + ηb A z T a,z α = b A 1 max a, α b R ·,a + ηb z T a,z α = b A 1 H V (b) After this, Lemma 1 can be used to prove the following lemma. (13) and (14) is contractive, i.e. for two given value functions U 1 and U 2 and the recursion H it holds that Lemma 2 If η A ∞ < 1, V defined in Eq.HU 1 − HU 2 ∞ ≤ β U 1 − U 2 ∞ with 0 < β < 1 and · ∞ the supreme norm. Proof HU 1 − HU 2 ∞ = HU 1 (b) − HU 2 (b) ∞ = b A 1 HU 1 (b) − HU 2 (b) ∞ ≤ A ∞ HU 1 − HU 2 ∞ ≤ η A ∞ U 1 − U 2 ∞ where we utilise the matrix norm property bA 1 ≤ b 1 A ∞ , and the facts that b 1 = 1 and for a standard Bellman recursion, HU 1 − HU 2 ∞ ≤ η U 1 − U 2 ∞ . The contraction property ensures that the vector space defined by the compressed value function is complete. Therefore, the space of such value functions together with the supreme norm form a Banach space, and the Banach fixedpoint theorem ensures that a single fixed point exists, to which the value recursion always converges (Puterman, 2005). (13) and (14) is isotonic, i.e. for two given value functions U 1 and U 2 and the recursion H it holds that Lemma 3 V defined in Eq.U 1 ≤ U 2 ⇒ HU 1 ≤ HU 2 Proof Let HU 1 (b) = H a1 U 1 (b) and HU 2 (b) = H a2 U 2 (b) , with a 1 and a 2 denoting the actions maximising HU 1 and HU 2 at point b, respectively. Using this definition, we have H a1 U 2 (b) ≤ H a2 U 2 (b). Let b a,z = (b AT a,z ) , and α b,a,z 1 and α b,a,z 2 be the α-vectors maximising the value function U 1 and U 2 at b a,z respectively. The following holds U 1 ≤ U 2 ⇒ U 1 ( b a1,z ) ≤ U 2 ( b a1,z ), ∀b, z ⇒ b AT a1,z α b,a1,z 1 ≤ b AT a1,z α b,a1,z 2 , ∀b, z ⇒ b AR ·,a1 + η z b AT a1,z α b,a1,z 1 ≤ b AR ·,a1 + η z b AT a1,z α b,a1,z 2 , ∀b ⇒ H a1 U 1 (b) ≤ H a1 U 2 (b) ≤ H a2 U 2 (b), ∀b ⇒ HU 1 (b) ≤ HU 2 (b), ∀b ⇒ HU 1 ≤ HU 2 The isotonic property of the value function guarantees that value iteration converges monotonically. Nevertheless, the above lemmas only consider exact value backup, which is intractable in practice. If approximate backup are taken into account, pruning is an inevitable procedure to ensure an efficient size of the α-vector set (Zhang and Zhang, 2005). Moreover, working in the compressed belief space, the backup operation is based onṼ instead of V . Since an essential step in pruning is to check the domination of an α-vector by others, the following lemma gives a further constraint for a compressed POMDP to be efficiently solvable. Lemma 4 Letb be a compressed belief with respect to a compression function F , as defined in Eq. (9). For two arbitrary α-vectorsα 1 andα 2 (corresponding to the compressed value function),α 1 ≥α 2 ⇒b α 1 ≥b α 2 , ∀b, holds for an arbitrary valid POMDP, iff F ≥ 0. The proof is straightforward, hence is omitted here. Value-Directed Compression Lossless VDC is designed to seek a linear compression function F that satisfies Eq. (7), which is a more restricted condition compared to our Theorem 2. Poupart (2005) proposed that such an F can be obtained by iteratively exploring the Krylov subspace Kr({T a,z } a∈A,z∈Z , R) to find a Krylov basis (with k linearly independent column vectors). We can summarise this process as (i) initialising F to the linearly independent columns of R, and (ii) in each iteration, multiplying every column (F i ) of F with every T a,z (as T a,z F i ), and Algorithm 1: Lossless/Lossy Value-Directed Compression (Poupart, 2005, Chapter 4) 1: input: R, {T a,z } a∈A,z∈Z , k /truncation parameter, lossy VDC only/ 2: F ← ∅ 3: C ← R /candidate column set/ 4: repeat 5a: c ← C[first] /lossless VDC/ 5b: c ← arg max y∈C minx y − F x /lossy VDC/ 6: F ← [F, c] 7: for each (a, z) ∈ A × Z 8: C ← [C, T a,z c] 9: remove columns linearly dependent to F from C 10: until C = ∅ or length(F ) = k /truncation, lossy VDC only/ 11: solve Eq. (7) /lossless VDC/ or Eq. (15) /lossy VDC/ to obtainR and {T a,z } a∈A,z∈Z 12: return F ,R, {T a,z } a∈A,z∈Z appending the obtained vector to F if it is linearly independent of the columns of the current F . The process ends when no more columns can be added. Clearly, a lossless compression is achievable only if the Krylov subspace is low-rank. In a more general case, it was suggested that one can greedily select k basis vectors in a forward-search manner to approximately minimise the residual errors of their predictions on the remaining vectors in the Krylov subspace, which is known as lossy VDC (Poupart, 2005). After obtaining F , R andT a,z can be computed by either solving Eq. (7) for lossless VDC, or the following regression problem for lossy VDC. R = arg min R R−FR F andT a,z = arg min T a,z T a,z F −FT a,z F ∀a ∈ A, z ∈ Z(15) where · F denotes the Frobenius norm. For the convenience of future discussions, we list the pseudo-code for lossless and lossy VDC in Algorithm 1. Note here, we adapt the representation of the lossless VDC algorithm to make it more comparable to lossy VDC, but it works exactly in the same way as the original algorithm presented in (Poupart, 2005). In standard numerical computation libraries (e.g. LAPACK (Anderson et al, 1999)), an overdetermined linear equation is usually solved as a least-squares problem, which suggests that Eq. (7) and Eq. (15) are treated the same in practice, and both lead to a solution in the form of Eq. (8), with F † being the pseudo-inverse of F in this case. Therefore, comparing the two VDC algorithms, one can see that the essential difference between them is their column selection strategy. Deficiency of VDC Firstly, VDC does not guarantee a nonnegative F unless R is nonnegative. Therefore, according to Lemma 4, if PBVI is applied to the compressed problem, some α-vectors may be mistakenly pruned out, resulting in the algorithm converging to a non-optimal policy. Secondly, as F is unregularised in VDC, F F † ∞ can be arbitrarily large. In the worst case, the compressed value function may diverge to infinity. These arguments seem to contradict the proofs in (Poupart, 2005), so we give more detailed explanations as follows. Theorem 1 is derived by assuming that the compression is lossless. However, due to numerical errors in practical computations, a totally 'error-free' solution never really exists. Especially for lossless VDC, because of the lack of consideration of system conditioning, it tends to be less robust to numerical errors than lossy VDC. 1 In other words, residual errors always occur in the compressed value function. Furthermore, although Poupart (2005) also developed an error bound for lossy VDC, that: V * − FṼ * ∞ ≤ 1 1 − η R + η T 1 − η \|Z\| Ṽ * ∞(16) where R = R − FR ∞ , T = max a,z T a,z F − FT a,z ∞ , in the case wherẽ V * is a diverging function, the bound itself is infinitely large. Empirical Evidence To support our argument on the deficiencies of VDC, we investigated its performance on two benchmark problems, Coffee (Boutilier and Poole, 1996) and Hallway2 (Littman et al, 1995). In the implementation of VDC, we experiment with two ways of judging the linear dependence between a new obtained vector c and the existing columns of F . The first method is to set a threshold τ for the least-squares residual r = c − F w 2 , where w = arg maxŵ c − Fŵ 2 . Concretely, c will be appended to F only if r ≥ τ . The second way of doing this is to check whether rank([F, c]) > rank(F ) with rank(·) denoting the numerical rank of a matrix. The quantities R and T as defined in Eq. (16) are taken as measures of compression error. The compression quality of VDC with respect to different residual thresholds τ (for lossless VDC) and truncation levels k (for lossy VDC) on the two benchmark problems are illustrated in Figure 1. Note that in the Coffee problem, the Krylov iteration for the rank-based lossless VDC finishes when 201 columns in F are obtained, however, there is still a significant residual error T at this point. This is an example of the numerical instability issue of VDC. Some example points in Figure 1 are selected to examine the policy quality obtained from the corresponding compressed POMDPs. To roughly ensure the same compression level for lossless and lossy VDC, we choose lossless VDC with τ = 10 −4 (221 dimensions) and lossy VDC with k = 200 for Coffee, and lossless VDC with τ = 10 −6 (40 dimensions), and lossy VDC with k = 40 for Hallway2. In addition, we evaluate the rank-based VDC (201 dimensions for Coffee and 52 dimensions for Hallway2) and the original POMDPs for both problems as well. Perseus (Spaan and Vlassis, 2005) is employed to solve the compressed and uncompressed POMDPs here. The the expected value growth for each algorithm during the value iteration procedures are shown in Figure 2, where all the experiments are based on 5000 sampled belief points. We also sample 1000 decision trajectories for each learned policy to compute an average reward, which gives an insight into the actual quality of the obtained policy, since the expected values for compressed problems can be unreliable according to our discussion in Section 4.1. The above policy learning and evaluation procedure is repeated five times for each task, and Table 1 shows the means and standard deviations of the average rewards. These two problems demonstrate all the deficiencies of VDC mentioned above. Firstly, as shown in Figure 2, the rank-based VDC results in a diverging value function in both tasks. Especially in Hallway2, even when the compression error is in the 10 −10 level, it may still cause a significant (possibly unbounded) loss in the value function. Secondly, as can be seen in the Coffee problem, the policy learning for lossless VDC with τ = 10 −4 converges to an unreasonably high value, but the corresponding average sampled reward (in Table 1) is extremely low. This is due to the pruning procedure being confused by the negative elements in F . More concretely speaking, some α-vectors that should be dominated by all the others are mistaken as the dominating ones. The above theoretical and empirical analysis suggests that the performance of VDC is not guaranteed in practical usage, hence we stop further experiments with it in this paper. Orthogonal NMF for Belief Compression Orthogonal NMF (O-NMF) based belief compression, introduced by Li et al (2007), explores an alternative direction by taking advantage of the possible low-rankness of the reachable belief space B defined by a POMDP. It seeks a nonnegative factorisation of a sampled set of beliefs B = [b 1 , b 2 , . . . , b m ] subject to an orthogonal constraint, such as: B ≈ FB s.t. F F = I, F ≥ 0 andB ≥ 0(17) whereB denotes the set of the compressed beliefs and I is the identity matrix. The compressed reward and transition matrices are then constructed by substituting F † = F into Eq. (8). Deficiency of O-NMF Belief Compression An obvious deficiency of the above formulation is that the orthogonal constraint can never be satisfied in practice, since the compression matrix F is of low-rank. Therefore, the algorithm proposed in (Li et al, 2007) actually only solves the following optimisation problem. min F ≥0,B≥0 B − FB 2 F + λ I − F F 2 F (18) where λ ≥ 0 is a coefficient balancing the weights of the two loss functions in the objective. 2 Nevertheless, when compared to VDC, O-NMF belief compression has the superiority that the nonnegative F ensures that PBVI works properly (Lemma 4). Moreover, minimising the Frobenius norm difference between F F and I approximately controls the scale of F F ∞ due to the equivalence of norms in finite dimension, which preserves the convergence of the compressed value function (Lemma 2). The fact that essentially makes O-NMF belief compression work can be understood as follows. Assume we have a compressed belief b such that the original belief b = Fb. Then the compressed value function for a given policy π can be computed as: V π (b) =b R ·,π(b) + η zṼ π (b T π(b),z ) = b R ·,π(b) + η zṼ π (b T π(b),z F ) = b R ·,π(b) + η z V π (b T π(b),z )(19) Note here, although the compressed beliefb here is not obtained as in Eq. (9), the correspondingṼ can still be related to V . Letting b π(b),z = b T π(b),z , we have: V π (b) −Ṽ π (b) = η z [V π (b π(b),z ) − V π (b π(b),z )](20) Going further, we can prove the following bound on the difference between V and V . Theorem 3 For value functions V and V defined in Eq. (4) and Eq. (13) respectively, if η A ∞ < 1, then the following bound holds. V − V ∞ ≤ I − A ∞ 1 − η A ∞ ( R ∞ + η\|Z\| V * ∞ ) Proof V − V ∞ = max π V π − V π ∞ ≤ max α, α,a R ·,a + η z T a,z α − AR ·,a − η z AT a,z α ∞ = max α, α,a (I − A)R ·,a + η z T a,z α − η z AT a,z α + η z AT a,z α − η z AT a,z α ∞ ≤ (I − A)R ∞ + max α,a η(I − A) z T a,z α ∞ + max α, α,a ηA z T a,z (α − α) ∞ ≤ I − A ∞ R ∞ + η\|Z\| I − A ∞ V * ∞ + η A ∞ V − V ∞ ≤ I − A ∞ 1 − η A ∞ ( R ∞ + η\|Z\| V * ∞ ) where we apply Lemma 2 and the facts that T a,z ∞ ≤ 1 and V ∞ ≤ V * ∞ . Theorem 3 implies that the O-NMF method (with A = F F ) minimises the upper bound of the value loss caused by compression (as I − A ∞ ≤ √ n I − A F ). However, such a bound can be quite loose in practice. The drawback of O-NMF belief compression is that it does not directly relate the error in compressing a belief b to the loss in its corresponding compressed value function, because Eq. (20) indicates that even if b = Fb holds, V π (b)−Ṽ π (b) = 0 since V π (b ) = V π (b ) does not hold in general for the successor beliefs b transited to from b. Theocharous and Mahadevan (2010) proposed an extension to the O-NMF method, which is formulated as follows. Firstly, the sampled belief set B is subsampled into a smaller set B by only including those points in the original B that are at least δ apart in terms of Euclidean distance, where δ is a pre-fixed threshold. Then, in finding a factorisation of B , the original Frobenius norm loss is replaced by an unnormalised KL-divergence loss between B and FB . After this, an extra risk is introduced, which measures the distance (with respect to symmetric KL-divergence) between each pair of the compressed beliefs weighted by a neighbourhood graph among the original belief points. The neighbourhood graph is constructed by connecting every belief point with its K-nearest neighbourhood (KNN). Finally, similar to O-NMF, F † can be obtained by approximating I ≈ F F † . The insight behind this algorithm is that the neighbourhood graph inspired risk function forces two compressed beliefs to be close to each other if their corresponding original beliefs are so. Hence, it is named "locality preserving" NMF (LP-NMF) belief compression. On the Locality Preserving NMF Belief Compression According to our discussion in Section 5.1, LP-NMF shares both the advantages and the drawbacks of the O-NMF method. In addition, the locality preserving property is only intuitively motivated, as the initial closeness among the belief points will be lost soon during value iteration due to recursive influence of T a,z . Furthermore, KL-divergence does not tend to be a good measure for selecting the linear compression matrix F . As shown in (Poupart and Boutilier, 2000), the quality of the policies resulting from approximate belief monitoring can be significantly lower than the original policy even when the KL-divergence remains fairly small, whilst policy quality can be unaffected when KL-divergence is large. Projective NMF for Belief Compression To address the drawbacks of O-NMF belief compression, we propose a novel projective NMF belief compression algorithm motivated by Theorem 2 and Lemma 2. Again, we start from an ideal case. Assume that for a POMDP, the reachable belief space B is row-rank (say rank(B) = k n). Let B be a sampled set of beliefs, and B ⊂ span(B) (i.e. rank(B) = k). If we can find a matrix F ∈ R n×k and F ≥ 0, such that: B = F F B and F F ∞ < 1 η(21) letting F † = F and substituting it into Eq. (8) gives a lossless compression of the original POMDP. The correctness of this proposition is easy to check. As B contains a linear basis of B, there exists a weight vector w such that b = Bw, ∀b ∈ B. Therefore, F F b = F F Bw = Bw = b, ∀b ∈ B holds. Moving to the imperfect case, if for a subset of the sampled beliefs B ⊂ B, B = F F B is achievable, then ∀b ∈ span( B), b = F F b. Hence, the insight here is that we attempt to reduce the compression error directly from the value function, instead of minimising a loose error bound as O-NMF belief compression does. Intuitively, the optimisation problem to seek the F for the proposed method could be formulated as: min F ≥0 B − F F B 1 s.t. F F ∞ ≤ 1 η(22) However, we suggest using the Frobenius norm instead of the L 1 /L ∞ matrix norms for efficiency purposes, as minimising the latter requires repeatedly solving linear programming problems, which is computationally much more expensive. Hence, the problem becomes: min F ≥0 1 2 B − F F B 2 F + λ 2 F F 2 F(23) where λ is a regularisation coefficient to approximately control the scale of F F . Our preliminary experiments indicate that λ can be empirically selected and that η F F ∞ being slightly greater than 1 usually does not affect the convergence of value iteration in practice. Eq. (23) forms a regularised projective NMF problem. As suggested by Yang and Oja (2010), this class of problems are convex and thus can be solved via gradient descent as follows. Define the objective function: g(F ) = 1 2 B − F F B 2 F + λ 2 F F 2 F(24) The the constrained gradient of g for F is given by: ∂g ∂F i,j = −2(BB F ) i,j + (F F BB F ) i,j + (BB F F F ) i,j + 2λ(F F F ) i,j(25) After this, we can construct the additive update rule for minimisation: F i,j ← F i,j − ζ i,j ∂g ∂F i,j(26) where ζ i,j is a positive step size. In order to keep F i,j staying nonnegative, ζ i,j can be selected as: ζ i,j = F i,j (F F BB F ) i,j + (BB F F F ) i,j + 2λ(F F F ) i,j(27) Substituting Eq. (27) into Eq. (26), we have: F i,j ← F i,j 2(BB F ) i,j (F F BB F ) i,j + (BB F F F ) i,j + 2λ(F F F ) i,j(28) We will refer to this method as P-NMF belief compression in the remainder of this paper. Experimental Results In previous literature, belief compression is usually conceptually demonstrated on small-scale benchmark problems. Here, we are interested in seeing its actual performance on POMDPs with large numbers of states. We use empirical methods to investigate the following two questions: 1. whether P-NMF could improve over O-NMF and LP-NMF in policy quality as expected, since it is more focused on error reduction in the value function; 2. whether belief compression in general is a preferable technique to state-ofthe-art POMDP solvers, or under what situation it is so. Experiments on Benchmark Problems The P-NMF, O-NMF and LP-NMF belief compression methods are now compared on four POMDP problems, including UnderwaterNavigation (Kurniawati et al, 2008), LifeSurvey (Smith et al, 2007), RockSample [7,8] (Smith and Simmons, 2004), and a spoken dialogue system problem (ComplexGoal-Dialog) that is an updated version 3 of the complex user goal dialogue problem described in (Crook and Lemon, 2011). Perseus (Spaan and Vlassis, 2005) is employed to solve the compressed POMDPs. In addition, SARSOP (Kurniawati et al, 2008) is used to give a measure of the rewards that can be achieved for uncompressed problems, as Perseus tends to be unable to solve a POMDP with more than a few thousand states in acceptable time. SARSOP is a state-of-the-art PBVI POMDP solver that achieves its efficiency by restricting itself to only sample belief points near the subset of those reachable from the initial belief under optimal sequences of actions, where the sampling region is controlled by heuristic exploration of sampling paths. Hence, its performance here also gives us an insight into the practical competitiveness of belief compression in solving large-scale POMDPs. In the following experiments, the P-NMF, O-NMF and LP-NMF belief compression algorithms are implemented in Matlab. The parameters for P-NMF and LP-NMF are empirically tuned based on preliminary experiments. For O-NMF, we follow Li et al's (2007) method of automatically selecting λ to enforce the orthogonal constraint (though the orthogonality is unachievable). The compression and policy optimisation processes are based on 20,000 belief points randomly sampled by Perseus, in all the tasks except RockSample [7,8] where 100,000 beliefs are sampled to maintain proper performance of the belief compression methods. The computing time for all algorithms is measured using CPU time (seconds) on a computer with 2×Six-Cores 2.4GHz CPUs and 128GB memory. The compression levels k are selected empirically in consideration of both compression errors and computational complexities. We sample 1000 decision trajectories to compute an average reward for each policy obtained in each task. The policy learning and reward sampling procedures are repeated five times for each task to calculate a mean and standard deviation, as summarised in Table 2. It can be found that the proposed P-NMF method consistently outperforms O-NMF and LP-NMF with respect to the rewards obtained in all the four tasks. The performance of LP-NMF compares unfavourably to that of the other two belief compression methods in most of the tasks, and is sometimes very unstable, e.g. large variances in its average sampled rewards can be observed in the UnderwaterNavigation problem. Note that, for Rocksample [7,8] regard O-NMF and LP-NMF as unable to solve the problem properly, as only 1 α-vector is produced in their policies. Nevertheless, none of the above belief compression methods can guarantee to work on all POMDPs. For example, we also apply them to another two benchmark problems, Homecare (Kurniawati et al, 2008) and Fourth CIT (Cassandra, 1998), where they all fail to produce a meaningful policy. The failures could partially due to the limitations of Perseus, e.g. for Fourth CIT (\|S\| = 1052, \|A\| = 4, \|Z\| = 28), Perseus itself fails to solve it, even though 100,000 belief points are sampled. Another reason could be the lack of low-rankness in the reachable belief space of a problem, e.g. for Homecare (\|S\| = 5408, \|A\| = 9, \|Z\| = 928), if we sample a B of 10,000 belief points and analyse its singular values, it can be found that there are about 2,000 of them with non-trivial values. Therefore, a too low-dimensional compression cannot sufficiently approximate the problem, whilst a compression with too many dimensions is computationally intractable for both Perseus and the compression algorithm itself. Moreover, considering either time expense or policy quality, belief compression in general tends to be less competitive than SARSOP, which motivates further investigation in the next section. Belief Compression vs. the State-of-the-art: A Further Comparison Although SARSOP (Kurniawati et al, 2008) demonstrates its impressive efficiency and effectiveness in the above experiments, it can usually be noticed that its intermediate belief sampling procedure (a heuristic search-based sampling strategy) results in considerably higher space complexities than standard PBVI algorithms (such as Perseus). This situation tends to be worse for problems with more actions and observations. Hence, it suggests a potential advantage for belief compression, as SARSOP may easily run out of memory for some large-scale problems. To further explore the relative merits of SARSOP and belief compression methods, we investigate two further dialogue problems as follows. POMDP Construction Firstly, a hand-crafted rule-based spoken dialogue system was set up in a laboratory environment. The system's domain is restaurant search and it supports complex user goals (Crook and Lemon, 2010). Each simple goal (which forms part of a complex user goal) contains two pieces of information (2 attributes: food-type and location, with 4 and 3 values respectively). The combination of the attribute values forms 4 × 3 = 12 distinct goals. Each state of the system is drawn from the power-set of the 12 possible goals, i.e. 2 12 = 4096 states in total. We define 64 system actions and 807 user observations, with respect to different expressions of the state information. After this, 161 dialogues were collected from volunteers, and were manually transcribed and annotated. Then we used the collected data to train a POMDP. As the rule-based system assumes the user goal to be unchangeable during a dialogue, we preserve this setting in the first version of the POMDP (called "Dialog/Identity") by setting the transition matrices to the identity matrix for all system actions, except those for the initial state s 0 . (For the POMDP model trained here we take the state with an empty goal set as s 0 .) For s 0 the transition probabilities to other states for all actions are defined to be proportional to the probability mass of a Poisson distribution over the number of goals contained in the target state. A non-zero reward is only given to those actions that present goal information to the user, where R(s, a) = 10 if the information presented by a is fully contained in s, R(s, a) = −10 otherwise. The observation probabilities are modelled using the exponential family, for which the parameter is trained by maximising the regularised log-likelihood on the data. To be concrete, we use feature vectors to represent the states, actions and observations, as φ s (s), φ a (a) and φ z (z) respectively, where binary values are used to indicate the occurrences of certain attribute values, and extra fields are introduced for actions and observations indexing the dialogue act type. Then we let the joint feature representation of a (z, s, a)-tuple be the tensor product of the corresponding individual feature vectors as φ(z, s, a) = φ s (s) ⊗ φ a (a) ⊗ φ z (z), and formulate the observation probabilities as where w is the parameter to be trained on the data. Finally, we eliminate those observation probabilities below the threshold 10 −6 and re-normalise the distributions, to achieve a sparser problem for space efficiency purposes. SARSOP n/a -∞ n/a Table 3 SARSOP vsḃelief compression on two challenging dialogue problems. P-NMF compressed problems are solved using Perseus, with 100,000 sampled belief points for compression and policy optimisation. Results The performance of SARSOP and P-NMF belief compression for this problem is compared in the upper half of Table 3 (Dialog/Identity), where SARSOP runs out of memory 4 within its first 30 iterations (after the initialisation step), but a promising policy is still obtained. A possible reason for this could be the identity transition matrices, which make the problem converge easily. Therefore, we slightly modified the previous setting to provide a further realistic challenge, in the problem "Dialog/Browsing". In this second version of the POMDP, instead of keeping the user goal fixed, we assume that a user will be able to request for alternative goals when its current goal is correctly presented by the system, and this process can last for an infinite number of turns (i.e. an infinite-horizon planning problem). This corresponds to dialogues where users are browsing through the possible entities (e.g. find out what Thai restuarants there are, and then search for the closest restaurant). To enable such a setting, we re-define those transition probabilities T (·\|a, s) that have R(s, a) = 10 to be identical to T (·\|a, s 0 ). The performance of SAR-SOP and the P-NMF method on this modified POMDP (Dialog/Browsing) is shown in the lower half of Table 3. This time, SARSOP fails to finish in acceptable time, as it takes more than 15 hours to run each iteration when initialising the fast informed bound (Hauskrecht, 2000). On the contrary, the efficiency of P-NMF is much more preferable in this case. Furthermore, by looking into the sampled dialogue trajectories of the compressed problem, we found that the ratio between its correct decisions (reward 10) and incorrect decisions (reward -10) is approximately 3:1, which suggests a reasonable quality of the policy. Conclusion This paper introduces a theoretical framework to analyse linear belief compression techniques, under which the deficiencies of three existing algorithms are presented. The findings indicate that policy quality reduction resulting from the compression can be relieved if those deficiencies are properly revised, as demonstrated by a new proposed P-NMF model. However, the overall performance of belief compression techniques tends to be less competitive in com-parison with a state-of-the-art POMDP solver, such as SARSOP. However, we show that under particular situations SARSOP may fail due to time or space complexities, whilst belief compression could provide a feasible solution. A further question posed here would be whether there exists a way of combining SARSOP and belief compression to achieve further efficiencies. Unfortunately, our preliminary answer is negative, as various underlying theories (e.g. the fast informed bound) in SARSOP's heuristics rely on beliefs being distributions. The possibility of applying alternative heuristics in solving compressed problems still requires further investigation. Fig. 1 Fig. 2 12Compression errors of lossless and lossy VDC on Coffee and Hallway2. Expected value during value iterations on Coffee and Hallway2. Table 1Average sampled rewards on Coffee and Hallway2.Coffee Lossless/rank Lossless/τ = 10 −4 Lossy/k = 200 Uncompressed Reward −5.52 ± 2.11 −14.36 ± 2.34 10.88 ± 0.17 11.16 ± 0.14 Hallway2 Lossless/rank Lossless/τ = 10 −6 Lossy/k = 40 Uncompressed Reward 0.01 ± 0.00 0.23 ± 0.02 0.24 ± 0.04 0.34 ± 0.02 one canTask Algorithm k Reward Time (P) Time (C) ComplexGoalDialog P-NMF 100 23.5±0.3 500 7.3 × 10 3 (\|S\| : 4097, \|A\| : 23, \|Z\| : 49) O-NMF 100 23.1±0.2 500 1.6 × 10 3 LP-NMF 100 13.2±0.3 500 4.2 × 10 3 SARSOP n/a 24.2±0.3 600 n/a LifeSurvey P-NMF 200 89.9±5.8 1,500 1.9 × 10 4 (\|S\| : 7841, \|A\| : 7, \|Z\| : 28) O-NMF 200 73.3±12.9 600 6.5 × 10 3 LP-NMF 200 (−8.1 ± 1.2) × 10 3 150 1.4 × 10 4 SARSOP n/a 105.7±1.1 200 n/a UnderwaterNavigation P-NMF 100 530.3±37.8 180 2.9 × 10 3 (\|S\| : 2653, \|A\| : 6, \|Z\| : 103) O-NMF 100 354.5±31.4 300 1.0 × 10 3 LP-NMF 100 489.7±283.2 300 0.9 × 10 3 SARSOP n/a 733.5±5.0 160 n/a RockSample[7,8] P-NMF 100 13.2±0.3 6,000 2.9 × 10 5 (\|S\| : 12545, \|A\| : 13, \|Z\| : 2) O-NMF 100 7.4±0.0 60 4.4 × 10 4 LP-NMF 100 7.4±0.0 70 3.0 × 10 3 SARSOP n/a 20.9±0.5 130 n/a Table 2 2Performance of belief compression on benchmark problems. Time (P) and Time (C) stand for policy optimisation time and compression time (averaged in five trials), re- spectively. (\|S\| : 4096, \|A\| : 64, \|Z\| : 807) (\|S\| : 4096, \|A\| : 64, \|Z\| : 807)Task Algorithm k Reward Time (P) Time (C) Dialog/Identity P-NMF 100 41.3 3,000 3.0 × 10 4 SARSOP n/a 49.0 2,712 n/a Dialog/Browsing P-NMF 100 23.1 3,000 3.0 × 10 4 Detailed analysis on the numerical stabilities of VDC is out of the main scope of this paper, but can be found in the supplementary material. AlthoughLi et al (2007) suggested that in particular way of selecting λ, Eq. (18) is equivalent to Eq. (17), it is well-known that F F = I is impossible for any matrix F ∈ R n×k with k < n, as rank(F F ) = rank(F ) ≤ k < n = rank(I). 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[ "Co-evolution of density and topology in a simple model of city formation", "Co-evolution of density and topology in a simple model of city formation" ]	[ "Marc Barthélemy \nDépartement de Physique Théorique et Appliquée BP12\nCEA-Centre d'Etudes de Bruyères-le-Châtel\n91680Bruyères-Le-ChâtelFrance\n\nCentre d'Analyse et Mathématique Sociales (CAMS\nUMR 8557 CNRS-EHESS)\nEcole des Hautes Etudes en Sciences Sociales, 54 bd. RaspailF-75270Paris Cedex 06France\n", "Alessandro Flammini \nSchool of Informatics\nE. Tenth st\nIndiana University\n901, 47408BloomingtonIN\n" ]	[ "Département de Physique Théorique et Appliquée BP12\nCEA-Centre d'Etudes de Bruyères-le-Châtel\n91680Bruyères-Le-ChâtelFrance", "Centre d'Analyse et Mathématique Sociales (CAMS\nUMR 8557 CNRS-EHESS)\nEcole des Hautes Etudes en Sciences Sociales, 54 bd. RaspailF-75270Paris Cedex 06France", "School of Informatics\nE. Tenth st\nIndiana University\n901, 47408BloomingtonIN" ]	[]	We study the influence that population density and the road network have on each others' growth and evolution. We use a simple model of formation and evolution of city roads which reproduces the most important empirical features of street networks in cities. Within this framework, we explicitely introduce the topology of the road network and analyze how it evolves and interact with the evolution of population density. We show that accessibility issues -pushing individuals to get closer to high centrality nodes-lead to high density regions and the appearance of densely populated centers. In particular, this model reproduces the empirical fact that the density profile decreases exponentially from a core district. In this simplified model, the size of the core district depends on the relative importance of transportation and rent costs.PACS numbers:	10.1007/s11067-008-9068-5	[ "https://arxiv.org/pdf/0810.1376v1.pdf" ]	15,869,708	0810.1376	28feb795b17d90ad4cccad1c232ac1c574f3e7ea	Co-evolution of density and topology in a simple model of city formation 8 Oct 2008 (Dated: October 8, 2008) Marc Barthélemy Département de Physique Théorique et Appliquée BP12 CEA-Centre d'Etudes de Bruyères-le-Châtel 91680Bruyères-Le-ChâtelFrance Centre d'Analyse et Mathématique Sociales (CAMS UMR 8557 CNRS-EHESS) Ecole des Hautes Etudes en Sciences Sociales, 54 bd. RaspailF-75270Paris Cedex 06France Alessandro Flammini School of Informatics E. Tenth st Indiana University 901, 47408BloomingtonIN Co-evolution of density and topology in a simple model of city formation 8 Oct 2008 (Dated: October 8, 2008)PACS numbers: We study the influence that population density and the road network have on each others' growth and evolution. We use a simple model of formation and evolution of city roads which reproduces the most important empirical features of street networks in cities. Within this framework, we explicitely introduce the topology of the road network and analyze how it evolves and interact with the evolution of population density. We show that accessibility issues -pushing individuals to get closer to high centrality nodes-lead to high density regions and the appearance of densely populated centers. In particular, this model reproduces the empirical fact that the density profile decreases exponentially from a core district. In this simplified model, the size of the core district depends on the relative importance of transportation and rent costs.PACS numbers: I. INTRODUCTION It has been recently estimated that more than 50% of the world population lives in cities and this figure is bound to increase [1]. The migration towards urban areas has dictated a fast and short-term planned urban growth which needs to be understood and modelled in terms of socio-geographical contingencies, and of the general forces that drive the development of cities. Previous studies [2,3,4] about urban morphology have mostly focused on various geographical, historical, and socialeconomical mechanisms that have shaped distinct urban areas in different ways. A recent example of these studies can be found in [3], where the authors study the process of self-organization of transportation networks with a model that takes into account revenues, costs and investments. The goal of the present study is to model the coupling between the evolution of the transportation network and the population density. More precisely, the question we aim to answer is the following. Given the pattern of growth of the entire population of a given city, how is the local density of population changing within the boundaries of the city itself, and how the road network's topology is modified in order to accommodate these changes? There are in principle a huge number of potentially relevant factors that may influence the growth and shape of urban settlements, first and foremost the social, economical and geographical conditions that causes the population of a given city to increase in a particular moment of its history. We neglect in the present study this class of factors and consider the overall growth in the number of inhabitants as an exogenous variable. In order to achieve conclusions that have a good degree of generality, and, at the same time, to maintain the number of assumptions as limited as possible, we focus on two main features only: the local density of population and the structure of the road network. Population density and the topology of the network constitute two different facets of the spatial organization of a city, and from a purely qualitative point of view it is not hard to believe that their evolution is strongly correlated. Indeed, Levinson, in a recent case study [5] about the city of London in the 19 th and 20 th centuries has demonstrated how the changes in population density and transportation networks deployment are strictly and positively correlated. Obviously, the road network tends to evolve to better serve the changing density of population. In turn, the road network influences the accessibility and governs the attractiveness of different zones and thus, their growth. However, attractiveness leads to an increase in the demand for these zones, which in turn will lead to an increase of prices. High prices will eventually limit the growth of the most desirable areas. It is the mutual interaction between these processes that we aim to model in the present work. Although there are many other economical mechanisms (type of land use, income variations, etc.) which govern the individual choice of a location for a new 'activity' (home, business, etc), we limit ourselves to the two antagonist mechanisms of accessibility and housing price. These loosely defined notions can be taken into account when translated in term of transportation and rent costs. We note that in the context of the structure of land use surrounding cities, von Thünen [6] already identified the distance to the center (a simple measure of accessibility) and rent prices as being the two main relevant factors. At first we will discuss separately the two mechanisms of road formation and location choice. In particular, we explicitely consider the shape of the network and model its evolution as the result of a local cost-optimization principle [7]. In classical models used in urban economics, transportation costs are usually described in a very sim-plified fashion in order to avoid the description of a separate transportation industry [4]. Also, when space is explicitly taken into consideration, the shape of the transportation networks is rarely considered and transportation costs are computed according to the distance to a city center (as it is the case in the classical von Thünen's [6] or Dixit-Stiglitz's [8] models). In these approaches transportation networks are absent, and displacements of goods and individuals are assumed to take place in continuous space. On one side this allows for a more detailed description of the economical processes at play during the shaping of a city. On the other side, these approaches often rely on the hypothesis that the processes shaping a city are slow enough to allow the balancing of the different forces that contribute to these processes, allowing as a consequence the achievement of the global minimum of some opportune cost function. The point of view inspiring our work, instead, is that the evolution of a city is inherently an 'out-ofequilibrium' process where the city evolves in time to adapt to continuously changing circumstances. If some sort of optimization or 'planning' is driving the growth, it has to be continuously redefined in order to take into account the ever-changing economic and social conditions that are ultimately responsible for the evolution of urban areas. We do not, therefore, assume the optimization of a global cost (or utility) function. Finally, we would like to mention that our goal is not to be as realistic as possible but to consistently reproduce a set of coarse grained and very general features of real cities under a minimal set of plausible assumptions. Alternative explanations might also be possible and it would be interesting to compare our results with those produced in the same spirit. We hope that this simplified model could serve as a first step in the direction of designing more elaborated models. This paper is organized in three main parts. In the first part, we briefly establish the framework to describe the model and discuss the empirical evidences that motivated it. In the second part, we address the issue of how the growth of the local density affects the growth of the road network. In the third part we will study how the road network affects the potential for density growth in different areas. We finally integrate all these elements in the fourth section, where we study the full model and discuss our results. II. THE MODEL: EMPIRICAL EVIDENCES AND DEFINITION A. Framework In our simplified approach, we represent cities as a collection of points scattered on a two dimensional area (a square of linear size L throughout this study), and connected by a urban road network. The description of the street network adopted here consists of a graph whose links represent roads, and vertices represent roads' intersections and end points. Although the primary interest here is on roads' networks [10,12], it is worth mentioning that transportations networks appear in variety of different contexts including plant/leaves morphology [13], rivers [16], mammalian circulatory systems [17], commodity delivery [18], and technological infrastructures [19]. Indeed, networks are the most natural and possibly simplest representation of a transportation system [20,21]. It would be impossible to review, even schematically, the approaches and the insights gained in the specific fields mentioned above, but it is at least worth to mention a few studies which attempted to connect the evolution of networks to an optimization principle (as it is the case of the present work). Maybe is it not surprising that man-made transportation networks have been designed with the goal to serve efficiently and cost little [18], but relevant examples occur in natural sciences as well. It is remarkable, for example, how the Kirchoff law, that determines the current in the edges of a resistor network can be derived assuming the minimization of the dissipated energy [22]. More recently, optimization principles have been successfully applied to the study of the transportation of nutrients through mammalian circulatory systems in order to explain the allometric scaling laws in biology [17,23]. River networks constitute a further example where relevant features of the network organization can be derived from an optimality principle [16,24]. A last example worth mentioning is that of metabolic networks, where it has been found that specific pathways appear if conditions for optimal growth are assumed (see e. g. [25]). It is interesting to notice that there have been attempts to put some of the examples discussed above in the context of a single framework (see [26]). In addition, let us mention that there is a huge mathematical literature that studies optimal networks and the flow they support; Minimal Spanning Trees [27], Steiner Trees [28], and Minimum Cost Network Flows [29] are just three examples. Although the present study assumes a notion of optimality, there are some important aspects that differentiate it from the works discussed above. The first is that the principle of optimality is at work only locally: there is no global cost function that our road networks are supposed to minimize. The second, and possibly more important, is that we attempt to establish a connection between the evolution of the network and that of the quantity that such network is supposed to transport, i.e. population. Transportation networks, as shown from the example cited above, can generally display a large variety of patterns. However, recent empirical studies [10,11,30,31,32,33,34,35,36,37,38] have shown that roads' networks, despite the peculiar geographical, historical, social-economical processes that have shaped distinct urban areas in different ways, exhibit unexpected quantitative similarities, suggesting the possibility to model these systems through quite general and simple mechanisms. In the following subsection we intersections and centers) for data from [10] (circles) and from [12] (squares). In the inset, we show a zoom for a small number of nodes. Right: Total length versus the number of nodes. The line is a fit which predicts a growth as √ N (data from [10]). present the evidences that support the previous statement. B. Empirical results The degree distribution of planar networks decays very fast for large degrees as it usually happens for networks embedded in Euclidean space or with strong physical constraints which prohibits the emergence of hubs [39]. The degree distribution is therefore strongly peaked around its average (over the whole city) k = 2E/N ≡ 2e (E is the number of edges-the roads-and N is the number of nodes-the intersections). Concerning the average degree of random planar networks, little is known: For oneand two-dimensional lattices e = 1 and e = 2, respectively, and a classical result shows that for any planar network k ≤ 6, implying e ≤ 3 (see e.g. [40]). It has also been recently shown that planar networks obtained from random Erdos-Renyi graphs over a randomly planedistributed set of points upon rejection of non-planar occurrences, have e > 13/7 [41]. These facts are summarized in Fig. 1 (left), together with empirical data from 20 cities in different continents [10]. A first important empirical observation is that 1.05 ≤ e emp ≤ 1.69 , in a range lying between trees and 2d lattices, and the average degree over all these cities is k ≈ 2.87. The strongly peaked degree distribution suggests that a quasi-regular lattice could give a fair account of the road network topology. This suggestion is reinforced if one considers the cumulative length ℓ of the roads. With this picture in mind, one would expect that for a given average density ρ = N/L 2 , the typical inter distance between nodes is ℓ 1 ∼ 1 √ ρ . The total length is then the number of edges times the typical inter-distance which leads to ℓ ∼ Eℓ 1 ∼ k 2 L √ N (1) This behavior is reproduced in fig. (1b), where a fit of the empirical data in [10] with a function of the form aN 1/2 gives a emp ≈ 1.46 ± 0.04 (a fit with a function of the form aN τ leads to a ≈ 1.51 ± 0.24 and τ ≈ 0.49 ± 0.03). The value a emp = 1.46 has to be compared with the average degree over all cities. One finds 2.87/2 ≈ 1. 44 and considering statistical errors, it is hard to reject the hypothesis of a slightly perturbed lattice as a model for the road network. The two empirical facts above lend credibility to the simple picture that city streets are described by a quasiregular lattice with an essentially constant degree (equal to approximately 3) and constant road length (ℓ 1 ∼ 1/ √ ρ). There is however a further empirical fact which forces us to reconsider this simple picture. The roads' network define a tessellations of the surface and the authors of [35] measured the distribution of the area A of the polygons delimited by the edges of the network. Surprisingly, they found a power law behavior of the form P (A) ∼ A −α(2) with α ≃ 1.9 (the standard error is not available in [35]). This fact contradicts the simple model of an almost regular lattice since the latter would predict a distribution P (A) very peaked around a value of the order of ℓ 2 1 . The authors of [35] also measured the distribution of the form factor given by the area of a cell divided by the area of the circumscribed circle (for this value they use the largest distance D between nodes of the cell, a convention that we adopted): φ = 4A/(πD 2 ). They found that most cells have a form factor between 0.3 and 0.6 indicating a large variety of cell shapes. A first challenge is therefore to design a model for planar networks that can reproduce quantitatively these featurees and which is based on a plausible (small) set of assumptions. The simple indicators discussed above show that one cannot model the network by either lattices, Voronoi tessellation, random planar Erdos-Renyi graphs, all these networks having a peaked distribution of areas and form factors. Let's note that the scale-invariant distribution for cell sizes can be obviously reproduced by assuming by the fractal model of [9] which assumes a self-similar process of road generation. The power law distribution for cell sizes automatically follows from this assumption. In the following, we present a model that relies on a simple plausible mechanism, does not assume self-similarity and quantitatively accounts for the empirical facts presented above. III. THE MODEL OF ROAD FORMATION We first discuss the part of our model that describes the evolution of the road network. Our main assumption is that the network grows by trying to connect to a set of points -the 'centers'-in an efficient and economic way. These centers can represent either homes, offices or businesses. This parameter free model is based on a principle of local optimality and has been proposed in [7]. For the sake of self-consistency and readability, we first describe this model in detail. The application of optimality principles to both natural and artificial transportation u B B A M M ′ u A FIG. 2: The nearest road to the centers A and B is M . The road will grow to point M'. The proposed minimum expenditure principle suggests that the next point M' will be such that the variation of the total distance to the two points A and B is maximal. networks has a long tradition [42,43]. The rationale to invoke a local optimality principle in this context is that every new road is built to connect a new location to the existing road network in the most efficient way [44]. During the evolution of the street network, the rule is implemented locally in time and space. This means that at each time step the road network is grown by looking only at the current existing neighboring sites. This reflects the fact that evolution histories greatly exceed the timehorizon of planners. The self-organized pattern of streets emerges as a consequence of the interplay of the geometrical disorder and the local rules of optimality. In this regard our model is quite different from approaches to transportation networks where an equilibrium situation is assumed and which are based on either (i) minimization of an average quantity (e.g. the total travel time), or (ii) on the inclusion of many different socio-economical factors ( e.g. land use). A. Network growth When new centers (such as new homes or businesses) appear, they need to connect to the existing road network. If at a given stage of the evolution a single new center is present, it is reasonable to assume that it will connect to the nearest point of the existing road network. When two or more new centers are present (as in Fig. 2) and they want to connect to the same point in the network, we assume that economic considerations impose that a single road -from the chosen network's point -is built to connect both of them. In the example of figure 2, the nearest point of the network to both new centers A and B is M . We grow a single new portion of road of fixed length dx from M to a new point M ′ in order to grant the maximum reduction of the cumulative distance of A and B from the network. This translates in the requirement that δd = d(M, A) + d(M, B) − [d(M ′ , A) + d(M ′ , B)] (3) is maximal (dx being fixed). A simple calculation shows that the maximization of δd leads to d − −− → M M ′ ∝ u A + u B(4) where u A ( u B ) is the unitary vector from M to A (B). The procedure described above is iterated until the road from M reaches the the line connecting A and B, where a singularity occurs: d − −− → M M ′ = 0. From there two independent roads to A and B need to be built to connect to the two new centers. The rule Eq. (4) can be easily generalized to the case of n new centers, and, interestingly, was proposed in the context of visualization of leafs' venation patterns [13]. The growth scheme described so far leads to tree-like structures and we implement ideas proposed in [13] in order to create networks with loops. Indeed, even if treelike structures are economical, they are hardly efficient: the length of the path along a minimum spanning tree network for example, scales as a power 5/4 of the Euclidean distance between the end-points [14]. Better accessibility is then granted if loops are present. In order to obtain loops, we assume, following [13], that a center can affect the growth of more than one single portion of road per time step and can stimulate the growth from any point in the network which is in its relative neighborhood, a notion which has been introduced in [15]. In the present context a point P in the network is in the relative neighborhood of a center C if the intersection of the circles of radius d(P, C) and centered in P and C, respectively, contains no other centers or point of the network [15]. This definition rigorously captures the loosely defined requirement that, for v to belong to the relative neighborhood of s, the region between s and v must be empty. At a given time step, a generic center C then stimulates the addition of new portions of road (pointing to P) from all points in the network that are in its relative neighborood, naturally creating loops. When more than one center stimulates the same point P the prescription of (4) is applied and the evolution ends when the list of stimulated points is exhausted (We refer the interested reader to [7] for a detailed exposition of the algorithm). The formula above can be straigthforwardly extended to the case of centers with non-uniform weight η. This leads to a modified version of Eq. (4), where the sum of distances to be minimized is weighted by η and leads to d − −− → M M ′ ∝ η A u A + η B u B .(5) where η A and η B can be different. Simulations with nonuniform centers weights show that -as far as the location of 'heavy' and 'light' centers is uniformly distributed in space, uncorrelated and not broad -that the structure of the network is locally modified, but that its large scale properties are virtually unchanged. In the algorithm presented above, once a center is reached by all the roads it stimulates, it becomes inactive. An interesting variant of this model assumes that centers can stay active indefinetively. In this case we expect a larger effect of the weights' heterogeneity. We will leave this problem for future studies. In the following, we study networks resulting from the growth process described above. We assume that the appearance of new centers is given exogenously and is independent from the existing road network and from the position and number of the centers already present. The model accounts quantitatively for a list of descriptors -the ones discussed above in an empirical contextthat characterize at a coarse grained level the topology of street patterns. At a more qualitative level, the model leads to the presence of perpendicular intersections, and also reproduces the tendency to have bended roads even if geographical obstacles are absent. We show in Fig. 3 examples of patterns obtained at different times. The model gives information about the time evolution of the road network: at earlier times, the density is low and the typical inter-distance between centers is large (see Fig. 3). As time passes, the density increases and the typical length to connect a center to the existing road network becomes shorter. Since the number of points grows with time, the simple assumption that the typical road length is given by 1/ √ ρ leads to ℓ 1 ∼ 1/ √ t which is indeed what the model predicts. Beyond visual similarities, the model allows quantitative comparisons with the empirical findings The ratio e = E/N , initially close to 1 (indicating that the corresponding network is tree-like), increases rapidly with N , to reach a value of order 1.25 which is in the ballpark of empirical findings. The cumulative length of the roads produced by the model (Fig. 4a) shows a behavior of the form a √ N with a ≈ 1.90, in good agreement with the empirical measurements a emp ≈ 1.87). The form factor distribution (Fig. 4b) has an average value φ = 0.6 and values essentially contained in the interval [0.3, 0.7] in agreement with the results in [35] for 20 German cities. B. Effect of the center spatial distribution An important feature of street networks is the large diversity of cell shapes and the broad distribution of cell areas. So far, we have assumed that centers are distributed uniformly across the plane. Within this assumption, the model predicts a cell area distribution following an exponential (with a large cut-off however) as shown in Fig. 4(d) and ters in real cities, however, is not accurately described by an uniform distribution but decreases exponentially from the center [31,32]. We thus use such an exponential distribution P (r) = exp(−\|r\|/r c ) for the center spatial location and measure the areas formed by the resulting network (in the last section of this point is further discussed). Although most quantities (such as the average degree and the total road length) are not very sensitive to the center distribution, the impact on the area distribution is drastic. In Fig. 5 a power law with exponent equal to 1.9 ± 0.1 is found, in remarkable agreement with the empirical facts reported in [35] for the city of Dresden. Although we cannot claim that this exponent is the same for all cities, the appearance of a power law in good agreement with empirical observations confirms the fact that the simple local optimization principle is a possible candidate for the main process driving the evolution of city street patterns. This result also demonstrates that the centers' distribution is crucial in the evolution process of a city. The optimization process described above has several interesting consequences on the global pattern of the street network when geographical constraints are imposed, as illustrated by the following example. We simulated the presence of a river assuming that new centers cannot appear on a stripe of given width (and are otherwise uniformly distributed). The resulting pattern is shown in Fig. 6. The local optimization principle naturally creates a small number of bridges that are roughly equally spaced along the river and organizes the road network. To conclude, it is worth noting that in the present framework we didn't attempt the modelization of planning efforts. Simulations show that, at the present simplified stage, the presence of a skeleton of "planned" large roads has the effect of partitioning the plane in different regions where the growth of the network is dominated by the mechanism described above, and reproducing on a smaller scale the structures shown in fig. 3. C. Hierarchical structure of the traffic Finally, we discuss now the presence of hierarchy in the network generated by the model. Indeed, geographers have recognized for a long time (see e.g. [2]) that many systems are organized in a hierarchical fashion. Highways are connected to intermediate roads which in turn dispatch the traffic through smaller roads at smaller spatial scales. In order to test for the existence of such a hierarchy in our model, we use the edge betweenness centrality as a simple proxy for the traffic on the road network. For a generic graph, the betweenness centrality g(e) of an edge e [48,49,50] is the fraction of shortest paths between any pair of nodes in the network that go through e. Allowing the possibility of multiple shortest paths between two points, one has g(e) = s =t σ st (e) σ st(6) where σ st is the number of shortest paths going from s to t and σ st (e) is the number of shortest paths going from s to t and passing through e. Central edges are therefore those that are most frequently visited if shortest paths are chosen to move from and to arbitrary points. We computed this quantity for all edges of the road network generated by our algorithm. It appears that this quantity g(e) is broadly distributed and varies over more than 6 orders of magnitude. In order to get a simple representation of this quantity we arbitrarily group the edges in three classes: [1, 10 4 ], [10 4 , 10 5 ], [10 5 , ∞] and plot them with different thicknesses. In particular, we see in Fig 7, that edges with the largest centrality (represented by the thickest line) form almost a tree of large arteries. Proportionality between traffic and edge-centrality, as defined above, is virtually equivalent to assuming: i) a uniform origin-destination matrix, ii) everybody choose the shortest path to reach a destination, and iii) roads are "large" enough to support the traffic generated by i) and ii) without congestion effects. Under these assumption one indeed observes a hierarchy of smaller roads and streets with a decreasing typical length and the existence of a hierarchical structure of arteries, roads and streets. IV. LOCATION OF CENTERS: EFFECT OF DENSITY AND ACCESSIBILITY In the simple version of the model presented above, the location of centers are independent from the topology of the road network. In real urban systems, this is however unlikely to happen. There is an extensive spatial economics literature (see [4] and references therein) that focuses on the several factors that may potentially influence the choice location for new businesses, homes, factories, or offices (see also [45] and references therein). Empirical evidences suggest a strong correlation between transportation networks and density increase have been recently provided by Levinson [5]. Our goal here is to discuss, based on very simple and plausible assumptions, the coupled evolution of the road network and the population density. We first divide the city in square sectors of area S, and we assume that the choice of a location for a new center is governed by a probability P (i) that one (i) of these sectors is chosen. This probability, which reflects the attractiveness of a location, depends a priori on a large number of factors such as accessibility, renting prices, income distribution, number and quality of schools, shops, etc. A key observation made by previous authors (see for example [46] and references therein) is that commuting cost differences must be balanced by differences in living spaces prices. We will follow this observation and we will thus focus on two factors which are the rent price and the accessibility (which we will reduce to commuting costs). A. Rent price and accessibility The housing price of a given location is probably determined by many factors comprising tax policies, demography, etc., and is by itself an important subject of study (see for example [47]). We will make here the simplifying assumption that the rent price is an increasing function of the local density (for each grid sector) ρ(i) = N (i)/S, where N (i) is the number of centers in the sector (i), and in particular that the rent price C R is directly proportional to the local density of population (which can be seen as the first term of an expansion of the price as a function of the density) C R (i) = Aρ(i)(7) where A is some positive prefactor corresponding to the price per density. We note here that a more general form of the type C R = Aρ(i) τ could be used. A preliminary study suggests that as long as the rent cost is an increasing function of the density, our results remain qualitatively unchanged. It would be however very interesting to measure this function empirically. The second important factor for the choice of a location is its accessibility. Locations which are easily accessible and which allow to reach easily arbitrary destinations are more attractive, all other parameters being equal. Also, for a new commercial activity, high traffic areas can strongly enhance profit opportunities. In terms of the existing network, the best locations are therefore the most central and standard models of city formation (see for example [4]) indeed integrate the distance to the center and its associated (commuting) cost as a main factor. Euclidean distance, however, can be a poor estimator of the effective accessibility of a given location, if this location is poorly connected to the transportation network. This is why the notion of centrality has to be considered not only in geographical terms, but also from the point of view of the network that grants mobility. The possibility to easily reach an arbitrary location when movement is constrained by a network is nicely captured in quantitative terms by the notion of node betweenness centrality. In the previous section, we defined the edge betweenness centrality and here we need a similar quantity defined for nodes rather than for edges. The node betweenness centrality g(v) [48,49,50] of a node v is defined as the fraction of shortest paths between any pair of points in the network that go through v. The mathematical expression of this quantity is then g(v) = 1 N (N − 1) s =t σ st (v) σ st (8) where σ st is the number of shortest paths going from s to t and σ st (v) is the number of shortest paths going from s to t and passing through v. Betweenness centrality was initially introduced as a natural substitute for geometric centrality in graphs that are not embedded in Euclidean space (see fig. 8). Betweenness also naturally serves our purpose to quantify accessibility on planar graphs, especially in our simplified framework where an explicit distinction between resources and users has been sacrified to the sake of simplicity. It is important to note that betweenness centrality, on planar graphs, is strictly correlated to other, more common, measures of centrality. The two first panels of fig 9 show the contour and the 3D plot of node-betweeness for a Manhattan-like grid of 25 blocks per side and clearly show how central nodes are those that are the most frequently visited if shortest paths are chosen to move from and to arbitrary points. The third panel of the same figure shows the relation between the betweenness of a node and its average distance from all other nodes (along the same square grid). This plot demonstrates that the larger the betweenness of a node is, the shorter is its average distance from a generic node. The betweenness thus ex- tends the concept of geographical centrality to networks whose structure is not a lattice or planar. In our simple model, we will assume that accessibility reduces here to the facility of reaching quickly any other location in the network. This can also be seen as the average commuting cost which in previous models [4] was assumed to be proportional to the distance to the center. The natural extension for a network is then to take the transportation cost depending on the betweenness centrality. For each sector S i of the grid, we first compute the average betweenness centrality as g(i) = 1 N (i) v∈Si g(v).(9) where the bar represents the average over all nodes (centers and intersections) which belong in a given sector. Transportation costs are a decreasing function of the betweenness centrality and we will assume here that the transportation cost C T (i) for a center in sector (i) is given by C T (i) = B(g m − g(i))(10) where B and g m are positive constants (other choices, as long as the cost decreases with centrality, linearly or not, give similar qualitative results). Finally, we will assume (as it is frequently done in many models, see for example [46] and references therein) that all new centers have the same income Y (c) = Y . This assumption is certainly a rough approximation, as demonstrated by effects such as urban segregation, but in order to not overburden our model, we will neglect income disparities in the present study. The net income of a new center c in a sector (i) is then K(i) = Y − C R (i) − C T (i).(11) The higher the net income K(i) and the more likely the location (i) will be chosen for the implantation of a new business, home, etc. In urban economics the location is usually chosen by minimizing costs, and we relax this assumption by defining the probability that a new center will choose the sector (i) as its new location under the form P (i) = e βK(i) j e βK(j) .(12) This expression rewritten as P (i) = e β(λg(i)−ρ(i)) j e β(λg(j)−ρ(j))(13) where βA is redefined as β and where λ = B/A. For numerical simulations, the local density is normalized by the global density N/L 2 , in order to have the density and centrality contributions defined in the same interval [0, 1]. The relative weight between centrality and density is then described by λ and the parameter β implicitly describes in an 'effective' way all the factors (which could include anything from individual taste to the presence of schools, malls, etc) that have not been explicitly taken into account, and that may potentially influence the choice of location. If β ≈ 0, cost is irrelevant and new centers will appear uniformly distributed across the different sectors: P (i) ∼ 1 N (i) .(14) In the opposite case, β → ∞, the location with the minimal cost will be chosen deterministically. P (i) = 1 for i such that K(i) is minimum P (i) = 0 for all other sectors (15) The parameter β can thus be used in order to adjust the importance of the cost relative to that of other factors not explicitly included in the model. V. CO-EVOLUTION OF THE NETWORK AND THE DENSITY We finally have all the ingredients needed to simulate the simultaneous evolution of the population density and the road network. Before introducing the full model, analogously to what we have done for the first part of the model, it is worth to study this second part separately. To do that, we consider a toy -one dimensionalcase, where the network plays no role, since a single path only exists between each pair of nodes. Despite the simplicity of the setting, it is possible to draw some general conclusion. A. One-dimensional model We assume that the centers are located on a onedimensional segment [−L, L]. Since only a single path exists between any two points, the calculation of centrality is trivial. In the continuous limit, and for a generic location x it can be written as the product of the number of points that lie at the right and left of the given location g(x) = x −L ρ(y, t)dy N − x −L ρ(y, t)dy(16) where ρ(x, t) is the density at x. The equation for the density therefore reads: ∂ t ρ(x, t) = e β » λ R x −L ρ(y,t)dy N (N − R x −L ρ(y,t)dy) N − ρ(x,t) N -(17) where N = L −L ρ(y, t)dy. The numerical integration of Eq. (17) shows that, after a transient regime, the process locks in a pattern of growth in which the total population grows at a constant rate N = L −L ρ(y, t)dy ∝ t.(18) This suggests that a solution for large t can be found via the separation of variables under the form ρ(x, t) = αf (x)t(19) where one can set L −L f (x)dx = 1 without loss of generality. Plugging the expression (19) into Eq. (17) one gets αf (x) = e β[λ R x −L f (y)dy(1− R x −L f (y)dy)−f (x)](20) where α is an integration constant to be determined. An explicit solution for the inverse f −1 (x) can be achieved via the Lambert function (Lambert's function is the principal branch of the inverse of z = w exp w ), but the expression is not particularly illuminating and it is therefore not presented here. Several facts can however be understood using a direct numerical integration of Eq. (17) or the simulation of the relative stochastic process: • At large times, population in different location grows with a rate f (x) that depends on the location but not explicitely on time. This is a direct consequence of Eq. (19) and is obviously a different behavior from uniform growth. The ratio of population density in two different locations x 1 and x 2 stabilizes in the long run to f (x 1 )/f (x 2 ) • Although β models the 'noise' in the choice of location and λ the relative importance of centrality as compared to density, they have similar effects on the expected density in a given location. An increase in β and λ corresponds a concentration of density in the areas of large centrality and a steeper decay of density towards the periphery, as shown in fig. 10a. This can intuitively be understood by looking e.g. at the role played by the parameter β in Eq. (10): as β decreases to 0, the differences in rate of growth in different locations becomes negligible • Eq. (17) describes the average (or expected) behavior of the population density over time. Numerical simulations of the corresponding stochastic process show fluctuations from the above mentioned expected value. Such fluctuations increases as noise increases (ie. when β decreases). • Numerical integration of Eq. (17) suggests that, as λ increases, the decay of density assumes a power law form whose exponent depends on β and λ and approaches −1 as λ gets very large. This can be explained by assuming f (x) ≈ γx −r , and using Eq. (20) and its derivative both computed in L. The derivative of f (x) is: f ′ (x) = βλf (x) 1 − 2 x −L f (y)dy f (x).(21) The above expression can be computed in x = L, taking into account that that L −L f (x)dx = 1 and the assumed algebraic functional form of f (x). This leads to γrL −1 = γβL −r (λγ − rγL −1 ).(22) In the limit of large L one can keep only the leading orders in L, and match the power and the coefficient of the leading order on the two sides of the This simple one-dimensional model thus allowed us to understand some basic features of the model that will be discussed in their full generality in the next section. B. Two-dimensional case: existence of a localized regime We now apply the probability in Eq. (13) to the growth model described in the first part of this paper. The process starts with a 'seed' population settlement (few centers distributed over a small area) and a small network of roads that connects them. At any stage, the density and the betweenness centrality of all different subareas are computed, and a few new centers are introduced. Their location in the existing subareas is determined according to the probability defined in Eq. (13). Roads are then grown until the centers that just entered the scene are connected to the existing network. This process is iterated until the desired number of centers has been introduced and connected. In the two panels of figure 11 we show the emergent pattern of roads that is obtained when λ is small and very large, respectively. When λ is small the density plays the dominant role in determining the location of new centers. New centers appear preferably where density is small, smoothing out the eventual fluctuations in density that may occur by chance and the resulting density is uniform. On the other hand, when λ is very large, centrality plays the key role, leading to a city where all centers are located in the same small area. The centrality has thus an effect opposite to that of density and tends to favor concentration. We will now describe in more details the transition between the two regimes described above. On the right, we show the network obtained for λ = 8. In this case, the centrality is the most important factor leading to a few dominant areas with high density. We compute, in the two cases, the following quantity (previously introduced in a different context [51,52]): Y 2 = i N (i) N 2(23) where the sum runs over all sectors which number is N s . In the uniform case, all the N (i) are approximately equal and one obtains Y 2 ∼ 1/N s , which is usually small. In contrast, when most of the population concentrates in just a few sectors which represent a finite fraction of the total population, we obtain Y 2 ∼ 1/n where n represents the order of magnitude of these highly-populated sectorsthe 'dominating sectors'. The quantity σ = 1 Y 2 N s(24) gives therefore the fraction of dominating sectors. The behavior of σ vs. λ is shown in Fig. 12. We observe that σ decreases very fast when λ increases, signaling that a phenomenon of localization sets in as soon as transportation costs are involved. We conclude this section discussing the role played by the parameter β. Analogously to what happens in the one dimensional case, the concentration effect is weakened by a small values of β. The parameter β describes the overall importance of the cost-factors with respect to other factors that have not been explicitly taken into account, or, equivalently, the possibility of choice. Indeed, when β is very large, the location which maximizes the cost is chosen. In contrast, when the parameter β is small, the cost differences are smoothen out and a broader range of choices is available for new settlements. Figure 12 illustrates the importance of choice. In particular, the appearance of large-density zones (controlled by the importance of transportation accessibility) is counterbalanced by the possibility of choice and the resulting pattern is more uniform. C. Density profile: the appearance of core districts In this last part, we describe the effect of the interplay of transportation and rent costs on the decay of population density from the city center. In the following, the core district is identified as the sector with the largest density. The whole plane is then divided in concentric shells with internal radius r and width dr. The density profile ρ(r) is given by the ratio of the number δn of centers in a shell to its surface δS(r) ρ(r) = δn δS (25) For small λ, the density is uniform, as expected. In figure 13 we show the density profile ρ(r) in the case of λ large, where we observe a fast exponential decay of the form exp −r/r c , in agreement with empirical observations [31]. This behavior is the signature of the appearance of a well-defined core district of typical size r c , whose typical size r c decreases with λ. This simplified model predicts, therefore, the existence of a highly populated central area whose size can be estimated in terms of the relative importance of transport and rent costs. VI. DISCUSSION AND PERSPECTIVES We presented a basic model that describes the impact of economical mechanisms on the evolution of the population density and the topology of the road network. The interplay between rent costs and demand for accessibility leads to a transition in the population spatial density. When transportation costs are moderate, the density is approximately uniform and the road network is a typical planar network that does not show any strong heterogeneity. In contrast, if transportation costs are higher, we observe the appearance of a very densely populated area around which the density decays exponentially, in agreement with previous empirical findings. The model also predicts that the demand for accessibility easily prevails on the disincentive constituted by high rent costs. A very important ingredient in modeling the evolution of a city is how individuals choose the location for a new business or a new home. We isolated in this work the two important factors of rent price and transportations costs. For these costs, we assume some reasonable forms but it is clear that large scale empirical measures are needed. In particular, it would be interesting to characterize empirically how the rent price varies with the density and how transportation costs varies with the centrality. Possible outcomes to these studies would be to give an idea of the value of the parameter λ (and possibly also β) and thus to determine how much the city is centralized. As it happens in every modeling effort, a satisfactory compromise between realism and feasibility must be found and we opted for sacrificing some important economic considerations in order to be able to explicitly take into account the topology of the road transportation network and not the distance to a center only, as it is usually assumed in most models. Our model predicts so far the appearance of a core center, but it is however known that cities present a large diversity in their structure, ranging from a monocentric organization to different levels of polycentrism. In addition, interesting scaling relations between different parameters (total wages, walking speed, total traveled length, etc) and population size were recently found [53,54] showing that beyond the apparent diversity, there are some fundamental processes driving the evolution of a city. These different results appear as various facets of the process of city formation and evolution and it is at this stage not clear how to connect the scaling to the structural organization of a city and more generally, how to reconciliate the different existing results in a unified picture. We believe that the present model-modified or generalized-could help for future studies in this direction. FIG. 1 : 1Left: Numbers of roads versus number of nodes (ie. FIG. 3 : 3Snapshots of the network at different times of its evolution: for (a) t = 1, 000, (b) t = 2, 000, (c) t = 3, 000, (d) t = 4, 000. At short times, we have a tree structure and loops appear for larger density values obtained at larger times. For t = 4, 000, we have approximately 1, 700 nodes connected by 2, 000 roads. FIG. 4 : 4Results of the model (averaged over 1000 configurations). (a) Total length of roads versus the number of nodes. The dotted line is a square root fit. (b) Structure factor distribution showing a good agreement with the empirical results of [35]. (c-d) Rescaled distributions of the perimeter (c) and of the areas (d) of the cells displaying an exponential behavior. Fig. 5 .FIG. 5 : 55The Upper left plot: Uniform distribution of points (1000 centers, 100 configurations). In this case, the area distribution is exponentially distributed (bottom left). Upper right plot: Exponential distribution of centers (5000 centers, 100 configurations, exponential cut-off rc = 0.1). In this case, we observe a power law (bottom right). The line is a power law fit which gives an exponent ≈ 1.9. FIG. 6 : 6In the presence of an obstacle (here a 'river' delimited by the two dotted line) in which the centers are not allowed to be located, the local optimization principle leads to a natural solution with a small number of bridges. FIG. 7 : 7Traffic map of the network. The edge centrality is computed and divided in three different groups and the thickness of the edge is plotted according to the group (from thin to thick for increasing edge centrality. FIG. 8 : 8Illustration of the notion of node betweenness centrality for a non planar graph FIG. 9 : 9Top panel: Contour plot of node betweenness centrality -not normalized-for a square grid of linear size 25 (A light color indicates a high value of the betweenness). Middle panel: 3D plot of node betweenness centrality (not normalized) for the same square grid. Bottom panel: Plot of the node betweenness of a generic node vs its average distance from all the other nodes. FIG. 10 : 10The stationary growth rate for different values of the parameters.(a) Large values of β and λ implies larger degree of centralization and a faster decay of density from center to periphery. (b) At large values of λ the decay of density becomes algebraic for location away from the center. The exponent approaches −1 and f (x) is approximated in that region by 1/(βλ/x). equation above. This gives r = 1 and γ = 1/(βλ). The validity of this argument can be verified looking at fig. 10b, where ln(f ) is plotted vs ln(x) to highlight the power law behavior of f (x) and where the line 1/(βλx) has been plotted as a reference for the case β = 10 and λ = 10. FIG. 11 : 11Networks obtained for different values of λ (and for N = 500 and β = 1). On the left, λ = 0 and only the density plays a role and we obtain a uniform distribution of centers. FIG. 12 : 12Fraction of dominating sectors (obtained for 500 centers and averaged over 100 configurations). When λ is small, the center distribution is more uniform and σ is large (close to 100%).When λ increases, we see the appearance of a few sectors dominating and concentrating most of the population. This effect is smoothen out for smaller values of β corresponding to the possibility of choice. 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[ "The four-dimensional helicity scheme and dimensional reconstruction", "The four-dimensional helicity scheme and dimensional reconstruction" ]	[ "Radja Boughezal *[email protected]†[email protected] ", "Kirill Melnikov ", "Frank Petriello ‡[email protected] ", "\nHigh Energy Physics Division\nDepartment of Physics and Astronomy\nArgonne National Laboratory\n60439ArgonneILUSA\n", "\nDepartment of Physics & Astronomy\nJohns Hopkins University\nBaltimoreMDUSA\n", "\nHigh Energy Physics Division, Argonne National Laboratory, Argonne\nNorthwestern University\n60208, 60439EvanstonIL, ILUSA, USA\n" ]	[ "High Energy Physics Division\nDepartment of Physics and Astronomy\nArgonne National Laboratory\n60439ArgonneILUSA", "Department of Physics & Astronomy\nJohns Hopkins University\nBaltimoreMDUSA", "High Energy Physics Division, Argonne National Laboratory, Argonne\nNorthwestern University\n60208, 60439EvanstonIL, ILUSA, USA" ]	[]	The four-dimensional helicity regularization scheme is often used in one-loop QCD computations. It was recently argued in Ref.[1] that this scheme is inconsistent beyond the one-loop order in perturbation theory. In this paper, we clarify the reason for this inconsistency by studying the perturbative expansion of the vector current correlator in one-flavor QED through three-loop order. We develop a simple, practical way to fix the four-dimensional helicity scheme using the idea of dimensional reconstruction, and demonstrate its application in several illustrative examples.	10.1103/physrevd.84.034044	[ "https://arxiv.org/pdf/1106.5520v1.pdf" ]	119,208,087	1106.5520	408ed8fb8d1aced8ee4cd80ca2e83e36b4b43120	The four-dimensional helicity scheme and dimensional reconstruction 27 Jun 2011 Radja Boughezal [email protected]†[email protected] Kirill Melnikov Frank Petriello ‡[email protected] High Energy Physics Division Department of Physics and Astronomy Argonne National Laboratory 60439ArgonneILUSA Department of Physics & Astronomy Johns Hopkins University BaltimoreMDUSA High Energy Physics Division, Argonne National Laboratory, Argonne Northwestern University 60208, 60439EvanstonIL, ILUSA, USA The four-dimensional helicity scheme and dimensional reconstruction 27 Jun 2011 The four-dimensional helicity regularization scheme is often used in one-loop QCD computations. It was recently argued in Ref.[1] that this scheme is inconsistent beyond the one-loop order in perturbation theory. In this paper, we clarify the reason for this inconsistency by studying the perturbative expansion of the vector current correlator in one-flavor QED through three-loop order. We develop a simple, practical way to fix the four-dimensional helicity scheme using the idea of dimensional reconstruction, and demonstrate its application in several illustrative examples. The four-dimensional helicity (FDH) regularization scheme [2,3] is one of several regularization schemes [4][5][6] based on the idea that consistent definitions of quantum field theories can be achieved through analytic continuation in the number of space-time dimensions [4]. The modern use of the FDH scheme is largely restricted to one-loop computations [7]. The motivation for FDH arose from on-shell methods for loop computations [8], which seek to reconstruct higher-loop scattering amplitudes from tree-amplitudes through their unitarity cuts. The tree amplitudes in massless QCD have a remarkably simple form [9] if a four-dimensional concept -the spinor-helicity formalism [10] -is employed in their evaluation. If this simplification is to be used in loop computations, the spin degrees of freedom for virtual particles must be treated as four-dimensional, in contrast to their momenta. This distinction is made manifest in the FDH scheme. Until recently, very little work was done to extend the FDH scheme beyond one-loop computations. To the best of our knowledge, the majority of complete multiloop computations in non-supersymmetric theories are performed using conventional dimensional regularization (CDR) [29]. In contrast to this, the known higher-order FDH results include some two-loop scattering amplitudes (see Ref. [3] for examples), that have not been used for a computation of any physical quantity. [email protected] † [email protected] ‡ [email protected] There is a good reason for this state of affairs. The CDR scheme is a natural scheme to use if inclusive quantities such as cross-sections and decay rates are computed using the optical theorem. Since computations of multiloop integrals in those cases are mostly based on the integration-by-parts identities [11,12], it is important to set up calculations in such a way that D-dimensional Lorentz invariance, where D is the space-time dimension of CDR, is explicit in all stages of the computation. The fact that intermediate observable states are more naturally described by four-, rather than D-dimensional quantum fields, is immaterial within such an approach. The distinction between observable and unobservable states is accomplished indirectly, by taking the imaginary part of an appropriate Green's function at the very end of the calculation. However, CDR is also used in existing fully differential next-to-next-to-leading order (NNLO) computations [13][14][15][16][17][18][19]. Its use in such situations is much less natural. CDR necessitates calculations of multi-parton matrix elements to higher orders in ǫ = (4 − D)/2. If FDH were extended to the two-loop order, this step could be avoided, leading to increased efficiency in computations of quantities with rich multi-parton kinematics. In fact, FDH is a scheme of choice in many calculations that address next-to-leading order (NLO) QCD corrections to kinematic distributions at hadron colliders (see Ref. [20] for a recent summary of results). Having in mind that extension of perturbative computations for some basic LHC processes to NNLO is desirable for several reasons, it is interesting to understand if the FDH scheme can be used beyond one loop in non-supersymmetric theories. In a recent work [1], Kilgore made a step in this direction by studying the application of several regularization schemes to higher-order calculations: CDR, FDH, and the dimensional reduction approach [6] commonly used in supersymmetric theories. He considered the imaginary part of the correlator of two vector currents and pointed out that the FDH scheme as formulated in Ref. [3] becomes inconsistent at higher orders. The goal of this article is to elucidate the reasons behind this inconsistency, and see if they can be fixed. We find that the problem with the FDH scheme follows from the fact that the gauge invariance of the full theory is broken by the restriction of the loop momenta to a smaller dimensionality than the spin dimensions. As the result of this, the Ward identities are not satisfied for the additional spin degrees of freedom. The situation with the FDH scheme becomes very similar to that of the dimensional reduction. However, we also find that the ills of FDH at NNLO can be cured very simply using "dimensional reconstruction": if the one-loop result for an observable is known for an arbitrary number of spin dimensions, then the (incorrect) two-loop FDH result can be fixed once the one-loop renormalization constants of the theory are known for two different integer numbers of spin dimensions. No two-loop CDR computation is necessary, so that this set-up preserves some of the simplicity of the original FDH. The spirit of this fix is similar to a technique employed to reconstruct the rational parts of one-loop amplitudes in generalized D-dimensional unitarity [21]. We would like to illustrate this idea by considering as simple a set-up as possible. We choose to study Quantum Electrodynamics (QED) with a single massless fermion field. The Lagrangian of the theory reads L =ψ(i∂ µ γ µ + eA µ γ µ )ψ − 1 4 F µν F µν ,(1) where F µν = ∂ µ A ν − ∂ ν A µ . The vector current in this theory J µ =ψγ µ ψ is conserved, ∂ µ J µ = 0, and does not require renormalization. We study the correlator of two vector currents, ω µν (q)Π(q 2 ) = −i d 4 xe iqx 0\|T J µ (x)J ν (0)\|0 ,(2) where we used ω µν (q) = (q 2 g µν − q µ q ν ). It is important to stress that, as it is customary in FDH, the external Lorentz indices µ and ν are taken to be four-dimensional. Conservation of the vector current requires that the imaginary part of the correlator Im[Π(q 2 )] is finite without any renormalization if the gauge-invariant subset of diagrams with closed fermion loops (the only contribution to the renormalization of the electric charge e) are discarded. It is this absence of any renormalization that makes this quantity an ideal laboratory for our investigation into various regularization schemes. We note that Kilgore studied the correlator of the two conserved vector currents in QCD [1], where disentangling the coupling constant renormalization is possible, but more difficult. We believe that focusing on the QED aspect of the problem allows us to illustrate the main issue very sharply. It is straightforward to compute Π(q 2 ) to three-loops using various regularization schemes. We have done this in the variety of ways, including utilizing Mincer [22] or Air [23], as well as using in-house implementations of the Laporta algorithm [24] established previously [25]. We find complete agreement with the result reported in Ref. [1] when it is truncated to QED and all terms that are proportional to the number of lepton flavors are dropped. The imaginary parts of the correlator computed in CDR and FDH are Im Π(q 2 ) CDR = 1 12π 1 + 3 4 α π − 3 32 α π 2 , Im Π(q 2 ) FDH = 1 12π 1 + 3 4 α π − 15 32 α π 2 .(3) The difference is striking. It implies that the computation of a finite quantity, that does not require any renormalization, leads to different results when two different regularization schemes are applied. Moreover, we emphasize that both CDR and FDH computations are consistent with the conservation of the vector current J µ in four dimensions so there is nothing at this point that makes either of the two results in Eq. (3) obviously incorrect. One could have suspected that a finite shift in the coupling constant -familiar from the application of the FDH scheme in one-loop QCD computations [7]-can account for the difference of the two results. However, the known shift of the coupling constant is purely non-abelian [7]. Since it vanishes in the abelian (QED) limit, it is not possible to reconcile the two results shown in Eq. (3) by existing means. To understand the reason behind the difference, we review the rules [3] that are used in the FDH computation. We begin by considering QED in a D s -dimensional space, but with all momenta restricted to a D-dimensional subspace of this D s -dimensional space. This arrangement requires D s > D. Upon performing spin algebra in all contributing diagrams, we take D s → 4, keeping D fixed. The limit D → 4 is taken at the end of the calculation. We note that the CDR scheme can be formulated in a similar way, making it explicit that the two schemes differ by the order of limit-taking. Indeed, to arrive at the CDR result, we take D s → D for fixed D, and then take the limit D → 4. The origin of the differences in CDR and FDH results can be best understood by presenting Π(q 2 ) in a form where D s is kept fixed, while the limit D → 4 − 2ǫ is taken. We find Im Π(q 2 ) = 1 12π 1 + 3 4 − 3 8 δ s α π + − 15 32 − 3 16 δ s ǫ − 3 32 δ 2 s ǫ + O(δ s ) α π 2 ,(4) where δ s = D s −4. The FDH result is obtained by setting δ s = 0 in Eq. (4), while the CDR result corresponds to setting δ s = −2ǫ and taking the limit ǫ → 0. O(δ s ) terms that are not enhanced by inverse powers of ǫ are present at NNLO but are irrelevant for both CDR and FDH. A similar term at NLO is also irrelevant for both CDR and FDH but, as we will see, it is important for understanding differences at NNLO between the two schemes. For this reason, it is shown explicitly in Eq. (4). It follows from Eq. (4) that the difference between the CDR and FDH schemes appears at NNLO because terms of the form α 2 δ s /ǫ are present in that order of the perturbative expansion. Those terms either contribute to the final result (CDR), or are set to zero by convention (FDH). Note that no δ s /ǫ term appears at NLO. Therefore, to understand the difference between CDR and FDH schemes, we must explain why divergent terms proportional to the number of "extra-dimensional" degrees of freedom appear at NNLO. The reason becomes very clear if we set D s to an integer value greater than four. For the sake of argument, we take D s = 5. It immediately follows from Eq. (4) that Im Π(q 2 ) is divergent. To see why this divergence occurs, we must go back to the QED Lagrangian in Eq. (1) and ask what theory arises if we set D s to five but keep all space-time coordinates four-dimensional. We begin by extending the Dirac algebra to five dimensions by taking Γ µ = γ µ , µ = 0, .., 3 and Γ 4 = iγ 5 , so that Γ M Γ N + Γ N Γ M = 2g MN , M, N = 0, ..4.(5) The fermion fields are not analytically continued, so the number of independent fermion helicities remains two. The gauge field A M is split into a four-dimensional gauge field and a scalar field, A M = (A µ , φ). The QED Lagrangian of Eq. (1) written in terms of four-dimensional fields reads L =ψ(iγ µ ∂ µ + eA µ γ µ )ψ − ig φψ γ 5 ψφ − 1 4 F µν F µν + 1 2 ∂ µ φ∂ µ φ.(6) Note that in Eq. (6) we introduced a new coupling constant g φ , to parameterize the interaction of the field φ and the pseudoscalar fermion current. Because the Lagrangian in Eq. (6) originates from the five-dimensional QED Lagrangian in Eq. (1), g φ = e. However, since we use four-dimensional momenta and coordinates, this equality of the coupling constants can not be protected by the full D s -dimensional gauge invariance. This implies that in Eq. (6), the coupling constant g φ requires renormalization, while the electric charge e is not renormalized and is protected by the four-dimensional gauge invariance. For D s = 5, the result shown in Eq. (4) corresponds to the calculation of Im[Π(q 2 )] in a theory defined by Eq. (6) in terms of bare charges, e and g φ . The bare electric charge e coincides with the physical charge because of the four-dimensional gauge invariance, but renormalization is required for g φ to make the correlator of the two vector currents explicitly finite in D s = 5. Such renormalization has not been performed in Eq. (4). This is the reason for the 1/ǫ divergences present there. We conclude that the "divergences" in Eq. (4) -crucial for understanding the CDR/FDH difference -can be related to the renormalization of the coupling constant g φ for finite D s . Below we describe the details of this relation. Because the scalar field φ contributes to the correlator of the two vector currents only at NLO, through NNLO we only need the one-loop renormalization of the coupling constant g φ . It is easy to obtain this renormalization constant by considering the Green's function 0\|Tψ(x)φψ(x)\|0 . We find α bare φ \| Ds=5 = α 1 − 3 4ǫ α π ,(7) where α φ = g 2 φ /4π is introduced. Rewriting Eq. (4) for D s = 5 (δ s → 1) and separating the two couplings at NLO explicitly, we find Im Π(q 2 ) Ds=5 = 1 12π 1 + 3 4 α π − 3 8 α bare φ π + − 15 32 − 9 32ǫ α π 2 .(8) Removing the bare coupling from Eq. (8) using Eq. (7), we see that the divergence in Eq. (8) disappears. This proves our assertion about the origin of the divergent δ s /ǫ terms in Eq. (4). Having understood the origin of divergences in Eq. (4), we must find a way to calculate the difference between Im[Π(q 2 )] in the FDH and CDR schemes without performing a complete three-loop computation. We observe in Eq. (4) that only the O(δ s /ǫ) term contributes to the CDR/FDH difference; the O(δ 2 s /ǫ) term is not relevant. However, since Eq. (4) is a second-degree polynomial in δ s , it is not possible to isolate the desired term by performing the computation in a single integer-dimensional space. Two such calculations are required. A similar need occurs when attempting to reconstruct the rational parts of one-loop amplitudes using tree-level amplitudes in higher integer dimensions [21], albeit for a different reason. We have already discussed the case D s = 5. The case D s = 6 is qualitatively similar, but different in detail. The D s = 6 QED Lagrangian of Eq. (1) deconstructs to L =Ψ(iγ µ ∂ µ + eA µ γ µ )Ψ − g φ √ 2 Ψ γ 5 σ + Ψφ + h.c − 1 4 F µν F µν + ∂ µ φ∂ µ φ * ,(9) whereΨ = (ū,d) is the lepton "doublet", σ + = (σ 1 + iσ 2 )/2, σ 1,2,3 are the Pauli matrices and φ is a complex scalar field. We define the conserved vector current as J µ = (ūγ µ u +dγ µ d)/ √ 2, where the normalization factor is chosen for convenience. The corresponding result for the imaginary part of the polarization operator follows from Eq. (4), where we isolate the contribution due to scalar degrees of freedom at one-loop: Im Π(q 2 ) Ds=6 = 1 12π 1 + 3 4 α π − 3 4 α bare φ π + − 15 32 − 3 4ǫ α π 2 .(10) The divergence is removed by the renormalization of the bare coupling g φ which, for D s = 6, is computed from the "flavor-changing" Green's function 0\|Tdφu\|0 . We find α bare φ \| Ds=6 = α 1 − α πǫ .(11) It is clear from Eq.(10) that this renormalization of the coupling constant makes Im Π(q 2 ) finite. Since we understand the structure of ultraviolet divergences for two values of D s , it is easy to find a relation between the FDH and CDR schemes. We imagine that a one-loop computation is performed, and the D sdependence of the one-loop result is established. We assume that the two-loop FDH result is also known. The result for general D s reads Im Π(q 2 ) δs = Im Π(q 2 ) FDH + 1 12π − 3 8 δ s α bare φ π + c 1 δ s ǫ + c 2 δ 2 s ǫ α π 2 .(12) The CDR result corresponds to setting δ s → −2ǫ in Eq. (12) and neglecting all O(ǫ) terms. Doing so, we find Im Π(q 2 ) CDR = Im Π(q 2 ) FDH − c 1 6π α π 2 .(13) The connection between the two schemes requires knowledge of the coefficient c 1 . As we discussed earlier, both c 1 and c 2 are related to the renormalization constants of the couplings of pseudoscalar fields, that appear as the result of dimensional deconstruction, to fermion bi-linears. Hence, it is a simple matter to find c 1 . We require that Eq. (12) becomes finite for D s = 5, 6 if the renormalization of the coupling constant g φ is performed. Doing so for both values of D s leads to a system of two equations that can be solved for c 1 and c 2 . We find c 1 = 3π 4α ǫ δZ 5 − 1 2 δZ 6 .(14) In Eq. (14), we have introduced the one-loop renormalization constants for the couplings of the scalar fields to fermions in the compactification of D s -dimensional QED to four-dimensional space-time: α bare φ \| Ds = α (1 + δZ Ds ) .(15) Explicit expressions for δZ Ds for D s = 5, 6 follow from Eqs. (7,11). Using those results, we find 12πIm Π(q 2 ) CDR − Π(q 2 ) FDH = 12 32 α π 2 ,(16) in agreement with the explicit computations of Eq. (3). As advertised, we are able to obtain the correct NNLO result from the FDH result without dealing with D sdimensional spin degrees of freedom at NNLO. There are several possible directions that one can explore at this point, including how this picture generalizes to more complicated theories (QCD, massive QED, etc.) or more complicated observables. Except for a few comments, in this paper we restrict ourselves to QED but we study observables that depend on the mass of the lepton. We show that the procedure we introduced in the context of the vector current correlator is general and remains valid also in those cases. We work with one massive fermion flavor in both examples. • We begin by computing the mass renormalization constant in FDH in the on-shell scheme, and ask if we can relate it to the mass renormalization constant in CDR. The mass renormalization constant is defined as m 0 = Z m m,(17) where m 0 is the bare fermion mass and m is pole mass of a lepton. One can easily read off the MS mass renormalization constant from Eq. (17) because the lepton pole mass is an infra-red finite quantity. We compute Z m through two-loop order in QED. We consistently neglect the contribution of the fermion loops, so that the electric charge does not need to be renormalized. The mass renormalization constant takes the form Z m = 1 + a 0 Z (1) m + a 2 0 Z (2) m ,(18) where a 0 = α/πΓ(1 + ǫ)/(4π) −ǫ m −2ǫ and Z (1) m = − 3 4ǫ − 5 4 − δ s 8 1 ǫ + 1 + O(ǫ), Z (2) m = 1 ǫ 2 9 32 + δ s 16 − δ 2 s 128 + 1 ǫ 53 64 + δ s 16 − 13δ 2 s 256 + 219 128 − 5π 2 16 + π 2 2 ln 2 − 3 4 ζ 3 + O(ǫ).(19)µ dm(µ) dµ = m (2ǫα + β(α)) ∂ ∂α ln Z m .(20) Taking Z FDH m from Eq. (19) and setting β(α) = 0, we find µ dm(µ) dµ = mγ(a) = m 1 + ∞ i=1 γ i a i(21) which implies γ FDH 1 = − 3 2 , γ FDH 2 = 53 16 .(22) To find the anomalous dimension in the CDR scheme, we write a relation between the FDH renormalization constant and the renormalization constant at arbitrary D s Z m (δ s )−Z FDH m = −a δ s 8ǫ +a 2 c 21 δ s ǫ 2 + c 22 δ 2 s ǫ 2 +...,(23) where the ellipses stands for other terms that do not affect the anomalous dimension. To find Z CDR m , we need c 21 , since it leads to divergent contribution in the limit δ s = −2ǫ. Repeating what we did for the photon vacuum polarization, we must consider the theory at finite D s , so that c 21 and c 22 contribute to the leading two-loop divergence of the fermion self-energy. Since such divergence is entirely fixed by the lowest-order mass anomalous dimension and the β-functions for the coupling constants, we can find an equation for c 21 and c 22 . We note that the β-functions appear because of the need to renormalize the scalar-fermion couplings, as described in Eqs. (7,11). The relevant condition is that 2ǫα ∂ ∂α ln Z m (δ s ) + β(α φ ) ∂ ∂α φ ln Z m (δ s )(24) is free from 1/ǫ singularities for any value of δ s . In practice, we choose D s = 5 and D s = 6. The β-functions follow from Eqs. (7,11). We write them here for completeness: β(α φ ) = −3/4a 2 for D s = 5, and β(α φ ) = −a 2 for D s = 6. We finally find c 21 = 1/16 and c 22 = −1/128, in agreement with Eq. (19). The mass anomalous dimensions in the CDR scheme follows immediately. Finally, one can imagine that the difference between on-shell Z m factors in different schemes Eq. (19) can be understood completely, by going beyond the MS renormalization of the g φ coupling constants as in Eqs. (7,11) and insisting that the two couplings g φ and e are equal to each other, including the finite renormalization. We did not pursue this question in this paper but it is an interesting avenue for further studies. • As the final example we compute the two-loop QED corrections to the electron anomalous magnetic moment and show that the correct result can be obtained using the FDH scheme and the procedure outlined above. We begin by writing the amplitude for the electron scattering off the electromagnetic field as iM = −ieū(p 2 )Γu(p 1 ), Γ =ǫF 1 (q 2 ) + iσ µν ǫ µ q ν 2m F 2 (q 2 ).(25) In Eq. (25), ǫ µ is the "polarization vector" of the external field,ǫ = γ µ ǫ µ , and q = p 2 −p 1 is the momentum transfer from the electron to the field. The anomalous magnetic moment is given by a e = (g −2)/2 = F 2 (0). The one-loop result for arbitrary δ s is given by a (1) e = α 2π 1 − δ s 2 .(26) The two-loop result for g − 2 requires the on-shell wavefunction and mass renormalization constants for the electron at the one-loop order. The mass renormalization constant Z m is given in Eq. (19). The wave-function renormalization constant Z 2 coincides with Z m in QED at this order in both CDR and FDH schemes. We find the following results for the two-loop contribution to the electron anomalous magnetic moment in the CDR and FDH schemes a (2),CDR e = α π 2 − 31 16 + 3 4 ζ 3 − π 2 2 ln 2 + 5π 2 12 , a (2),FDH e = α π 2 − 35 16 + 3 4 ζ 3 − π 2 2 ln 2 + 5π 2 12 .(27) Our CDR result matches well-known results in the literature [26,27], when fermion-loop contributions are neglected. The FDH result is new. We now illustrate how to use dimensional reconstruction to obtain the CDR result, given the δ s -dependent 1-loop result in Eq. (26) and the 2-loop FDH result. We proceed as we did for the current correlator by writing the result for arbitrary δ s as a δs e = a FDH e − δ s 4 α bare φ π + c 1 δ s ǫ + c 2 δ 2 s ǫ α π 2 .(28) The CDR result is obtained by taking δ s = −2ǫ: a CDR e = a FDH e − 2c 1 α π 2 .(29) To obtain c 1 , we compute Eq. (28) for D s = 5, 6 and demand the result be finite after renormalizing α bare φ . We obtain c 1 = π 4α ǫ (2 δZ 5 − δZ 6 ) .(30) Inserting this into Eq. (29), we derive the correct (CDR) result for g − 2. Hence, the procedure that we developed by studying the correlator of two conserved currents appears to be valid in a more general context. Before concluding, we comment on two possible venues for the extension of this analysis, namely its extension to QCD and to its application to less inclusive observables. The first comment concerns the well-established procedure for applying the FDH scheme in one-loop QCD computations. As explained in Ref. [7], it is possible to use FDH in one-loop computations consistently provided that a finite renormalization of the strong coupling constant, α FDH s = α MS s 1 + C A 6 α s 2π ,(31) is performed. We can easily understand this result using dimensional reconstruction idea. Dimensional reconstruction in QCD leads to the appearance of color-octet massless scalars that interact with both fermions and "four-dimensional" gluons. Tree-level computations involve four-dimensional fields by definition, and therefore all one-loop amplitudes are proportional to the "fourdimensional" version of the strong coupling constant α s . Massless QCD is made finite by the coupling constant renormalization which, in the case of dimensional reconstruction, involves the contribution of color-octet scalars. Because we only need the divergent contribution of massless color-octet scalar to the renormalization of α s , we can find it by inspecting the QCD β-function, β 0 = 11/3C A −2/3N f −C A /6N s and focusing on the contribution of the color-octet scalars (the term proportional to N s ). As expected, the required shift in the coupling constant in Eq. (31) and the contribution of the coloroctet scalars to QCD β-functions are appropriately correlated. By studying the FDH scheme in one-loop QCD in terms of dimensional reconstruction, it is obvious that finite renormalization of the coupling constant in Eq. (31) is the only thing needed to perform self-consistent computations in FDH [30]. As a second comment, it is interesting to ask what the dimensional reconstruction procedure outlined in this paper implies for exclusive computations. For the sake of argument, consider again the correlator of two vector currents in QED. Its imaginary part is directly related to the inclusive decay rate of a vector boson. But how should a decay rate be treated if we require a certain number of "jets", borrowing from the QCD terminology? At NNLO, it is possible to have four, three and two jets in the final state. The four-jet rate is finite at this order. The three-jet rate is only needed through NLO, and therefore FDH can be used straightforwardly. The two-jet rate is needed at NNLO, which makes it obvious that this is the place where corrections to the inclusive rate must be accommodated. Moreover, since the phasespace for the two-jet configuration can be driven arbitrarily close to the two-parton kinematics by appropriate adjustments in the jet selection criteria, the correction to the inclusive cross-section is the finite renormalization of the Born matrix element. This argument applies to processes which possess infra-red finite total cross-sections, but it needs further refinements for consistent application of the FDH scheme to exclusive processes at hadron colliders beyond one-loop. To conclude, in this paper we explored the fourdimensional helicity scheme at NNLO, following an interesting observation in Ref. [1] that it becomes inconsistent at that order in perturbation theory. To avoid the complications of renormalization, we studied QED corrections to the imaginary part of the correlator of two conserved currents. We found that the differences between the FDH and the CDR schemes are related to the fact that, upon continuing QED to a space-time of higher dimensionality while restricting all the loop momenta to lower-dimensional space-times, (D s − 4) components of the gauge fields turn into scalar fields and become unprotected by full D s -dimensional gauge invariance. As a result, divergences are introduced that require additional renormalization. They are removed in the FDH scheme by simply ignoring these additional degrees of freedom. In the CDR scheme, terms of the form (D s − 4)/ǫ give additional finite contributions. One can argue for the correctness of the CDR result over FDH result by simply stating that the former respects gauge invariance of the theory in D s -dimensions, while the latter only respects four-dimensional gauge invariance. Restoring the D s -dimensional gauge invariance in the FDH scheme is possible using dimensional reconstruction: if the oneloop result is known for arbitrary δ s , and the two-loop FDH result is known, then the two-loop result in the CDR scheme can be obtained by studying the one-loop renormalization of the theory in D s = 5, 6. This gives a simple, practical prescription for maintaining the simplifying features of FDH while still getting the answer right [31]. We demonstrated the applicability of this procedure by reconstructing the two-loop CDR results from FDH for three examples: the correlator of two vector currents in QED, the mass anomalous dimension of a fermion in QED, and the electron anomalous magnetic moment. Further studies are required to extend these ideas to QCD and develop them to a point where computations of fully exclusive observables through NNLO in the FDH scheme are possible and transparent. This remains an interesting problem for the future. 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V. Tarasov, Comput. Phys. Commun. 71, 193-205 (1992). . A Signer, D Stockinger, Nucl. Phys. 808A. Signer, D. Stockinger, Nucl. Phys. B808, 88-120 (2009). In what follows, we do not distinguish between the t'Hooft-Veltman regularization scheme and CDR. In what follows, we do not distinguish between the t'Hooft-Veltman regularization scheme and CDR. For processes with hadrons in the initial state, parton distribution functions in the CDR scheme are typically employed. Because of that, FDH results are usually translated to CDR at the end of the calculation. The rules for such a translation are given in Refs. 7, 28For processes with hadrons in the initial state, parton distribution functions in the CDR scheme are typically employed. Because of that, FDH results are usually trans- lated to CDR at the end of the calculation. The rules for such a translation are given in Refs. [7, 28]. We note that this prescription will have to be extended for yet higher orders in perturbation theory, since the N k LO result for a particular observable is a rank-k polynomial in Ds. We note that this prescription will have to be extended for yet higher orders in perturbation theory, since the N k LO result for a particular observable is a rank-k poly- nomial in Ds.	[]
[ "ON UNIPOTENT AND NILPOTENT PIECES FOR CLASSICAL GROUPS", "ON UNIPOTENT AND NILPOTENT PIECES FOR CLASSICAL GROUPS" ]	[ "Ting Xue " ]	[]	[]	We show that the definition of unipotent (resp. nilpotent) pieces for classical groups given by Lusztig (resp. Lusztig and the author) coincides with the combinatorial definition using closure relations on unipotent (resp. nilpotent) classes. Moreover we give a closed formula for a map from the set of unipotent (resp. nilpotent) classes in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent (resp. nilpotent) pieces.	10.1016/j.jalgebra.2011.10.009	[ "https://arxiv.org/pdf/0912.3820v4.pdf" ]	66,041,823	0912.3820	204699b193ea7ce74ecc61a1429876a6cf298fab	ON UNIPOTENT AND NILPOTENT PIECES FOR CLASSICAL GROUPS 2 May 2011 Ting Xue ON UNIPOTENT AND NILPOTENT PIECES FOR CLASSICAL GROUPS 2 May 2011 We show that the definition of unipotent (resp. nilpotent) pieces for classical groups given by Lusztig (resp. Lusztig and the author) coincides with the combinatorial definition using closure relations on unipotent (resp. nilpotent) classes. Moreover we give a closed formula for a map from the set of unipotent (resp. nilpotent) classes in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent (resp. nilpotent) pieces. Introduction Let G be a symplectic or special orthogonal group defined over an algebraically closed field of characteristic exponent p ≥ 1 and let g be the Lie algebra of G. Denote U G (resp. N g ) the set of unipotent (resp. nilpotent) elements in G (resp. g). In [5,6], Lusztig defines a partition of U G into smooth locally closed G-stable pieces, called unipotent pieces (see [5] for symplectic groups and [6] for special orthogonal groups). In [7], Lusztig proposes another way to define unipotent pieces and shows that the new definition unifies the definitions in [5,6]. In Appendix A of [7], Lusztig and the author define an analogue partition of N g into smooth locally closed G-stable pieces, called nilpotent pieces. The unipotent or nilpotent pieces are indexed by unipotent classes in the group over C of the same type as G, and in many ways depend very smoothly on p. In section 4, we show that one can define pieces combinatorially using closure relations on classes (for unipotent pieces this definition is first considered by Spaltenstein [11]) and that the combinatorial definition gives rise to the same pieces as in [7] (for unipotent pieces in symplectic groups this follows from [5,7]). In section 5, we determine which classes lie in the same piece (for pieces in symplectic groups this follows from [5,7]; for unipotent pieces in special orthogonal groups another computation using different methods is given in [9]). In section 6, we define a partition of U G (resp. N g ) into special pieces as in [4] (where p = 1) and show that a special piece is a union of unipotent (resp. nilpotent) pieces (for U G this follows implicitly from [11,III], see [9]). Notations and recollections 2.1. Let P(n) denote the set of all partitions λ = (λ 1 ≥ λ 2 ≥ · · · ≥ 0) such that \|λ\| := λ i = n. For λ ∈ P(n), define λ * j = \|{λ i \|λ i ≥ j}\| and m λ (j) = λ * j − λ * j+1 . For λ, µ ∈ P(n), we say that λ ≤ µ if the following equivalent conditions hold (a) j∈ [1,i] λ j ≤ j∈ [1,i] µ j , for all i ≥ 1, (a ′ ) j∈ [1,i] λ * j ≥ j∈ [1,i] µ * j , for all i ≥ 1. 1 Let P 2 (n) denote the set of all pairs of partitions (α, β) such that \|α\|+\|β\| = n. For (α, β) ∈ P 2 (n), α = (α 1 ≥ α 2 ≥ · · · ), β = (β 1 ≥ β 2 ≥ · · · ), we set (1) A i = j∈ [1,i] (α j + β j ), B i = j∈ [1,i−1] (α j + β j ) + α i , i ≥ 1. For (α, β), (α ′ , β ′ ) ∈ P 2 (n), we say that (α, β) ≤ (α ′ , β ′ ) if A i ≤ A ′ i , B i ≤ B ′ i , for all i ≥ 1. 2.2. Let W be the Weyl group of G and W ∧ the set of irreducible characters of W over C. If W is of type B n (or C n ), n ≥ 1, then W ∧ is parametrized by ordered pairs of partitions (α, β) ∈ P 2 (n) (see [2]). We identify W ∧ with P 2 (n) where (n, −) is the trivial character and (−, 1 n ) is the sign character. If W is of type D n , n ≥ 2, then W ∧ is parametrized by unordered pairs of partitions {α, β} with \|α\| + \|β\| = n, where each pair {α, α} corresponds to two (degenerate) elements of W ∧ . We identify W ∧ with the set {(α, β) ∈ P 2 (n)\|β 1 ≤ α 1 }, where each pair (α, α) is counted twice. 2.3. Denote Ω p G and Ω p g the set of unipotent classes in U G and nilpotent classes in N g respectively. Recall that we have injective maps (see [13,3,14]) (2) γ p G : Ω p G → W ∧ , γ p g : Ω p g → W ∧ which map a class c to the irreducible character of W corresponding to the pair (c, 1) under Springer correspondence. We denote Λ p G (resp. Λ p g ) the image of the map γ p G (resp. γ p g ). We may write Ω p to denote either Ω p G or Ω p g and similar conventions apply for Λ p , γ p . When p = 2, there are natural identifications of the sets Ω p G , Ω p g with Ω 1 G , the sets Λ p G , Λ p g with Λ 1 G , and the maps γ p G , γ p g with γ 1 G . We have (see [2,10,12,15]) Λ 1 SO(2n+1) = {(α, β) ∈ P 2 (n)\|α i+1 ≤ β i ≤ α i + 2}, Λ 1 Sp(2n) = {(α, β) ∈ P 2 (n)\|α i+1 − 1 ≤ β i ≤ α i + 1}, Λ 1 SO(2n) = {(α, β) ∈ P 2 (n)\|α i+1 − 2 ≤ β i ≤ α i }; Λ 2 SO(2n+1) = Λ 2 Sp(2n) = {(α, β) ∈ P 2 (n)\|α i+1 − 2 ≤ β i ≤ α i + 2}, Λ 2 SO(2n) = {(α, β) ∈ P 2 (n)\|α i+1 − 4 ≤ β i ≤ α i }; Λ 2 so(2n+1) = {(α, β) ∈ P 2 (n)\|β i ≤ α i + 2}, Λ 2 sp(2n) = P 2 (n), Λ 2 so(2n) = {(α, β) ∈ P 2 (n)\|β i ≤ α i }, where for G = SO(2n) , each pair (α, α) is counted twice in the sets Λ. Note that (see also [8]) Λ 1 G ⊂ Λ 2 G ⊂ Λ 2 g . 2.4. Assume p = 2 and G = Sp(N ) (resp. SO(N )). The Sp(N ) (resp. O(N ))-conjugacy class of u ∈ U G is characterized by the partition λ ∈ P(N ) given by the sizes of Jordan blocks of u − 1. We can identify Ω 1 Sp(N ) with the set {λ ∈ P(N )\| m λ (i) is even if i is odd}, Ω 1 SO(N ) with the set {λ ∈ P(N )\| m λ (i) is even if i = 0 is even}, where in the case of SO(2n), each λ with all parts even corresponds to two (degenerate) classes. Assme λ = (λ 1 ≥ λ 2 ≥ · · · ) ∈ Ω 1 G and γ 1 G (λ) = (α, β), α = (α 1 ≥ α 2 ≥ · · · ), β = (β 1 ≥ β 2 ≥ · · · ). Recall that λ and (α, β) are related as follows [2]. If G = SO(2n + 1), then λ 2i−1 = 2α i + 1 + δ i , λ 2i = 2β i − 1 + θ i , where δ i =    1 if β i = α i + 2 −1 if α i = β i−1 (i ≥ 2) 0 otherwise , θ i =    1 if β i = α i+1 −1 if β i = α i + 2 0 otherwise ; (3) if G = Sp(2n), then λ 2i−1 = 2α i + δ i , λ 2i = 2β i + θ i , where δ i =    1 if β i = α i + 1 −1 if α i = β i−1 + 1 (i ≥ 2) 0 otherwise , θ i =    1 if β i = α i+1 − 1 −1 if β i = α i + 1 0 otherwise ; if G = SO(2n), then λ 2i−1 = 2α i − 1 + δ i , λ 2i = 2β i + 1 + θ i , where δ i =    1 if β i = α i −1 if α i = β i−1 + 2 (i ≥ 2) 0 otherwise , θ i =    1 if β i = α i+1 − 2 −1 if β i = α i 0 otherwise (note that γ 1 SO(2n) (λ) = (α, α) if and only if all λ i are even; the two degenerate classes corresponding to λ are mapped to the two degenerate elements of W ∧ corresponding to (α, α) respectively under γ 1 SO(2n) ). 2.5. Assume p = 2 and G = Sp(2n) (resp. SO(2n)). The Sp(2n) (resp. O(2n))-conjugacy class of u ∈ U G is characterized by the partition λ ∈ P(2n) given by the sizes of Jordan blocks of u − 1 and a map ε : N → {ω, 0, 1} satisfying the following conditions (see [11, I 2.6 ]) (a) ε(i) = ω, if i is odd, or if i ≥ 1 and m λ (i) = 0, (b) ε(i) = 1, if i = 0 is even and m λ (i) is odd, (c) ε(i) = ω, if i is even and m λ (i) > 0. (d) ε(0) = 1 (resp. ε(0) = 0). If G = SO(2n), then λ * 1 is even. We have a natural bijection (4) Ω 2 Sp(2n) → Ω 2 SO(2n+1) , (λ, ε) → (λ ′ , ε ′ ) given by the special isogeny SO(2n + 1) → Sp(2n), where λ = (λ i ), λ ′ = (λ ′ i ); if λ i−1 > 0 and λ i = 0, then λ ′ i = 1, ε ′ (λ ′ i ) = ω; otherwise, λ ′ i = λ i , ε ′ (λ ′ i ) = ε(λ i ). We identify Ω 2 G with the set of all (λ, ε) as above, where in the case of SO(2n) there are two (degenerate) classes corresponding to each (λ, ε) such that ε(λ i ) = 0 for all λ i with m λ (λ i ) > 0. Assume (λ, ε) ∈ Ω 2 G , λ = (λ 1 ≥ λ 2 ≥ · · · ), and γ 2 G ((λ, ε)) = (α, β), α = (α 1 ≥ α 2 ≥ · · · ), β = (β 1 ≥ β 2 ≥ · · · ). Recall that (λ, ε) and (α, β) are related as follows [2]. If G = Sp(2n), then (5) λ 2i−1 = 2α i + δ i , λ 2i = 2β i + θ i , ε(λ 2i−1 ) = ε(δ i ), ε(λ 2i ) = ε(θ i ), where δ i =            2 if β i = α i + 2 1 if β i = α i + 1 −2 if α i = β i−1 + 2 (i ≥ 2) −1 if α i = β i−1 + 1 (i ≥ 2) 0 otherwise , θ i =            2 if β i = α i+1 − 2 1 if β i = α i+1 − 1 −2 if β i = α i + 2 −1 if β i = α i + 1 0 otherwise , (6) ε(δ i ) =    0 if δ i = ±2 ω if δ i = ±1 1 otherwise , ε(θ i ) =    0 if θ i = ±2 ω if θ i = ±1 1 otherwise ; if G = SO(2n), then λ 2i−1 = 2α i − 2 + δ i λ 2i = 2β i + 2 + θ i , ε(λ 2i−1 ) = ε(δ i ), ε(λ 2i ) = ε(θ i ), where δ i =            2 if β i = α i 1 if β i = α i − 1 −2 if α i = β i−1 + 4 (i ≥ 2) −1 if α i = β i−1 + 3 (i ≥ 2) 0 otherwise , θ i =            2 if β i = α i+1 − 4 1 if β i = α i+1 − 3 −2 if β i = α i −1 if β i = α i − 1 0 otherwise , ε(δ i ) =    0 if δ i = ±2 ω if δ i = ±1 1 otherwise , ε(θ i ) =    0 if θ i = ±2 ω if θ i = ±1 1 otherwise (note that γ 2 SO(2n) ((λ, ε)) = (α, α) if and only if for all i, ε(λ i ) = 0; the two degenerate classes corresponding to (λ, ε) are mapped to the two degenerate elements of W ∧ corresponding to (α, α) respectively under γ 2 SO(2n) ). 2.6. Assume p = 2 and G = Sp(N ) (resp. SO(N )). The Sp(N ) (resp. O(N ))-class of x ∈ N g is characterized by the partition λ ∈ P(N ), λ = (λ 1 ≥ λ 2 ≥ · · · ), given by the sizes of the Jordan blocks of x and a map χ : {λ i } i≥1 → N satisfying the following conditions (see [1]) (a) 0 ≤ χ(λ i ) ≤ λ i 2 (resp. [ λ i +1 2 ] ≤ χ(λ i ) ≤ λ i ), (b) χ(λ i ) ≥ χ(λ i+1 ), λ i − χ(λ i ) ≥ λ i+1 − χ(λ i+1 ), (c) χ(λ i ) = λ i 2 (resp. χ(λ i ) = λ i ), if m λ (λ i ) is odd. If G = SO(N ), then {λ i = 0\|m λ (λ i ) is odd} = {a, a − 1} ∩ N for some a ∈ N. We identify Ω 2 g with the set of all (λ, χ) as above, where in the case of SO(2n) there are two (degenerate) classes corresponding to each (λ, χ) with χ(λ i ) = λ i /2 for all i ≥ 1. Assume (λ, χ) ∈ Ω 2 g , λ = (λ 1 ≥ λ 2 ≥ · · · ), and γ 2 g ((λ, χ)) = (α, β), α = (α 1 ≥ α 2 ≥ · · · ), β = (β 1 ≥ β 2 ≥ · · · ). Recall that (λ, χ) and (α, β) are related as follows [12,15]. If G = Sp(2n), then λ 1 = α 1 + β 1 if α 1 < β 1 2α 1 if α 1 ≥ β 1 , χ(λ 1 ) = α 1 , λ 2i =    α i+1 + β i if β i < α i+1 α i + β i if β i > α i 2β i if α i+1 ≤ β i ≤ α i , χ(λ 2i ) = α i if β i > α i β i if β i ≤ α i , λ 2i+1 =    α i+1 + β i if α i+1 > β i α i+1 + β i+1 if α i+1 < β i+1 2α i+1 if β i+1 ≤ α i+1 ≤ β i , χ(λ 2i+1 ) = α i+1 if α i+1 ≤ β i β i+1 if α i+1 > β i , i ≥ 1; if G = SO(2n + 1), let k ≥ 0 be the largest integer such that β k > 0, then λ 2i−1 =    α i + β i if i < k + 1 α i + 1 if i = k + 1 α i if i > k + 1 χ(λ 2i−1 ) = α i + 1 if i ≤ k + 1 α i if i > k + 1 λ 2i = α i + β i if i < k + 1 α i if i ≥ k + 1 χ(λ 2i ) = α i + 1 if i < k + 1 α i if i ≥ k + 1 , i ≥ 1; if G = SO(2n), then λ 2i−1 = λ 2i = α i + β i , χ(λ 2i−1 ) = χ(λ 2i ) = α i , i ≥ 1 (note that γ 2 so(2n) ((λ, χ)) = (α, α) if and only if for all i, χ(λ i ) = λ i /2; the two degenerate classes corresponding to (λ, χ) are mapped to the two degenerate elements of W ∧ corresponding to (α, α) respectively under γ 2 so(2n) ). i = (λ c i , ε i ) ∈ Ω 2 G (resp. c i = (λ c i , χ i ) ∈ Ω 2 g ), i = 1, 2. Recall that Lemma ( [7]). The classes c 1 and c 2 lie in the same piece if and only if λ c 1 = λ c 2 . 2.8. Let V be a vector space of dimension N equipped with a nondegenerate quadratic form Q. The orthogonal group O(V ) = {g ∈ GL(V )\|Q(gv) = v, ∀ v ∈ V }. The special orthogonal group SO(V ) is the identity component of O(V ). Assume G = SO(V ). Let c ∈ Ω p G (resp. Ω 2 g ) and u ∈ c (resp. x ∈ c). Let V * = (V ≥a ) (with V ≥a+1 ⊂ V ≥a ) be the canonical Q-filtration of V associated to u (resp. x) (see [7, 2.7(a), A.4(a)]). We define gr a (V * ) = V ≥a /V ≥a+1 and set f a = dim gr a (V * ). Then f a = 0 for finitely many a, and the set of numbers {f a } depends only on c and not on the choice of u ∈ c (resp. x ∈ c); we denote this set by Υ c . Let c 1 , c 2 ∈ Ω p . If G = SO(2n), we assume that c 1 , c 2 are not two degenerate classes conjugate under O(2n). Recall that Let T = u − 1 (resp. T = x). If p = 2, then (see [7]) (a) V ≥a = j≥max(0,a) T j (ker T 2j−a+1 ). Now assume p = 2. If T = 0, then V ≥a = 0 for all a ≥ 1 and V ≥a = V for all a ≤ 0. Assume from now on that T = 0. Let e be the smallest integer such that T e = 0 and f the smallest integer such that QT f = 0. Let m be the largest integer such that gr m (V * ) = 0. We have (see [7]) m = max(e − 1, 2f − 2); V ≥−m+1 = {v ∈ V \|T e−1 v = 0} if e ≥ 2f, V ≥−m+1 = {v ∈ V \|T e−1 v = 0, Q(T f −1 v) = 0} if e = 2f − 1, V ≥−m+1 = {v ∈ V \|Q(T f −1 v) = 0} if e < 2f − 1. Let V ′ = V ≥−m+1 /V ≥m . Then Q induces a nondegenerate quadratic form on V ′ and u (resp. x) induces an element u ′ ∈ U SO(V ′ ) (resp. x ′ ∈ N so(V ′ ) ), where SO(V ′ ) is defined with respect to Q ′ . Let c ′ be the class of u ′ (resp. x ′ ) in SO(V ′ ) (resp. so(V ′ )). Assume Υ c = {f a } and Υ c ′ = {f ′ a }. We have (see [7]) (b) f a = f ′ a , for all a ∈ [−m + 1, m − 1]. 2.9. Let c, c ′ ∈ Ω p G (resp. Ω p g ). We say that c ≤ c ′ , if c is contained in the closure of c ′ in G (resp. g); and that c < c ′ , if c ≤ c ′ and c = c ′ . In the following if G = SO(2n), we assume that c and c ′ are not two degenerate classes conjugate under O(2n) (such two classes are incomparable with respect to the partial order ≤). [11,II 8.2]), namely, the following conditions hold Assume c = λ, c ′ = λ ′ ∈ Ω 1 G . We have c ≤ c ′ if and only if λ ≤ λ ′ (see [11, II 8.2]). Assume c = (λ, ε), c ′ = (µ, φ) ∈ Ω 2 G . We order the set {ω, 0, 1} by ω < 0 < 1. Then c ≤ c ′ if and only if (λ, ε) ≤ (µ, φ) (see(a) λ ≤ µ, (b) j∈[1,i] λ * j − max(ε(i), 0) ≥ j∈[1,i] µ * j − max(φ(i), 0), for all i ≥ 1, (c) if j∈[1,i] λ * j = j∈[1,i] µ * j and λ * i+1 − µ * i+1 is odd, then φ(i) = 0, for all i ≥ 1. 3. Reformulation of closure relations on unipotent and nilpotent classes 3.1. Let c, c ′ ∈ Ω p (if G = SO(2n) , we assume that c, c ′ are not two degenerate classes conjugate under O(2n)). Proposition. We have c ≤ c ′ if and only if γ p (c) ≤ γ p (c ′ ). If c, c ′ ∈ Ω 2 g , the proposition is a result of Spaltenstein [12]. The proofs for c, c ′ ∈ Ω p G when p = 2 and p = 2 are given in subsections 3.2 and 3.3 respectively. 3.2. Assume c = λ, c ′ = λ ′ ∈ Ω 1 G , γ 1 G (c) = (α, β) and γ 1 G (c) = (α ′ , β ′ ). We show that (a) λ ≤ λ ′ iff (α, β) ≤ (α ′ , β ′ ). We prove (a) for G = SO(2n + 1). The proofs for Sp(2n) and SO(2n) are entirely similar and omitted. For (α, β) ∈ Λ 1 SO(2n+1) , let A i , B i be as in (1) and let ∆ i = j∈ [1,i] (δ j +θ j ), Θ i = ∆ i−1 +δ i , where δ j , θ j are as in (3). One can easily verify that ∆ i = 1 if β i = α i+1 0 otherwise , Θ i = 1 if β i = α i + 2 0 otherwise .(7) We have 2i j=1 λ j = 2A i + ∆ i and 2i−1 j=1 λ j = 2B i + Θ i + 1. Assume λ ≤ λ ′ . It follows from 2.1 (a) and (7) that A i ≤ A ′ i and B i ≤ B ′ i . Hence (α, β) ≤ (α ′ , β ′ ) (see 2.1). Assume (α, β) ≤ (α ′ , β ′ ). Then A i ≤ A ′ i and B i ≤ B ′ i for all i. We show that A i = A ′ i implies ∆ i ≤ ∆ ′ i . Assume otherwise, A i = A ′ i , ∆ i = 1, ∆ ′ i = 0. Then β i = α i+1 and β ′ i > α ′ i+1 . Since B i = A i − β i ≤ B ′ i = A ′ i − β ′ i and B i+1 = A i + α i+1 ≤ B ′ i+1 = A ′ i + α ′ i+1 , we have β i ≥ β ′ i and α i+1 ≤ α ′ i+1 which is a contradiction. Similarly B i = B ′ i implies Θ i ≤ Θ ′ i . Hence λ ≤ λ ′ . 3.3. Assume c = (λ, ε), c ′ = (µ, φ) ∈ Ω 2 G , γ 2 G (c) = (α, β) and γ 2 G (c ′ ) = (α ′ , β ′ ). We show that (a) (λ, ε) ≤ (µ, φ) iff (α, β) ≤ (α ′ , β ′ ). We prove (a) for G = Sp(2n) and then in view of (4), (a) follows for G = SO(2n + 1). The proof for SO(2n) is entirely similar and omitted. Since (1) and let ∆ i = j∈ [1,i] j>i λ * j = j∈[1,λ * i+1 ] (λ j − i) and, for i large enough, j∈[1,i] λ * j = j∈[1,i] µ * j , we have (b) j∈[1,i] λ * j = j∈[1,i] µ * j iff j∈[1,λ * i+1 ] (λ j − i) = j∈[1,µ * i+1 ] (µ j − i). We show that (c) if λ ≤ µ and j∈[1,k] λ * j = j∈[1,k] µ * j , then j∈[1,λ * k+1 ] λ j = j∈[1,λ * k+1 ] µ j , (d) if λ ≤ µ and j∈[1,m] λ j = j∈[1,m] µ j , then j∈[1,µm] λ * j = j∈[1,µm] µ * j . By 2.1 (a ′ ), the assumptions in (c) imply that λ * k ≤ µ * k , λ * k+1 ≥ µ * k+1 . It follows that µ j = k for j ∈ [µ * k+1 + 1, λ * k+1 ] and thus j∈[1,µ * k+1 ] (µ j − k) = j∈[1,λ * k+1 ] (µ j − k). Now (c) follows from (b). By 2.1 (a), the assumptions in (d) imply that λ m+1 ≤ µ m+1 and λ m ≥ µ m . Let k = µ m . Then j∈[1,λ * k ] (λ j − k) = j∈[1,m] (λ j − k), j∈[1,µ * k ] (µ j − k) = j∈[1,m] (µ j − k) (since λ i = k, i ∈ [m + 1, λ * k ]; µ i = k, i ∈ [m + 1, µ * k ]). Now (d) follows from (b). For (α, β) ∈ Λ 2 Sp(2n) , let A i , B i be as in(δ j + θ j ), Θ i = ∆ i−1 + δ i , where δ j and θ j are as in (6). One can easily verify that we have ∆ i =    2 if β i = α i+1 − 2 1 if β i = α i+1 − 1 0 otherwise , Θ i =    2 if β i = α i + 2 1 if β i = α i + 1 0 otherwise .(8) Using (5) one can easily check that λ 2i−1 = λ 2i iff β i ≥ α i , or (if i ≥ 2) β i = β i−1 , α i = α i+1 and β i ≤ α i+1 − 1; and λ 2i = λ 2i+1 iff β i ≤ α i+1 , or β i = β i+1 , α i = α i+1 and β i ≥ α i + 1. It then follows that (e) if β i = α i + 2, then λ * λ 2i +1 is even, (f) if α i = β i−1 + 2 (i ≥ 2), then λ * λ 2i−1 +1 is odd. We have j∈ [1,2i] [1,2i] λ j = j∈ [1,2i] µ j . In the latter case, we have β i ≥ α i+1 , β ′ i = α ′ i+1 − 2, λ 2i+1 ≤ µ 2i+1 and λ 2i ≥ µ 2i . Then µ 2i = µ 2i+1 and φ(µ 2i ) = 0 (we use (5)). Let k = µ 2i . By (d), we have j∈ [1,k] λ j = 2A i + ∆ i and j∈[1,2i−1] λ j = 2B i + Θ i . Assume (λ, ε) ≤ (µ, φ). It follows from λ ≤ µ and (8) that A i ≤ A ′ i except if ∆ i = 0, ∆ ′ i = 2 and j∈λ * j = j∈[1,k] µ * j . By (f), µ * k+1 is odd. If λ 2i > k, then λ * k+1 = 2i is even, which contradicts 2.9 (c). Hence λ 2i = k and thus ε(k) = 0 (we use 2.9 (b) and k even). It follows that β i = α i + 2 (since β i ≥ α i+1 ) and thus λ * k+1 is even by (e), which again contradicts 2.9 (c). Hence A i ≤ A ′ i . Similarly we have B i ≤ B ′ i . Hence (α, β) ≤ (α ′ , β ′ ). Assume (α, β) ≤ (α ′ , β ′ ). We show that if A i = A ′ i then ∆ i ≤ ∆ ′ i . Assume otherwise, A i = A ′ i , ∆ i = 1 (resp. 2), ∆ ′ i = 0 (resp. 1 or 0). Then as in the proof of 3.2 (a), we have β i ≥ β ′ i and α i+1 ≤ α ′ i+1 , which contradicts to ∆ i ≤ ∆ ′ i (we use (8)). Similarly one can show if B i = B ′ i then Θ i ≤ Θ ′ i . It follows that λ ≤ µ. We verify 2.9 (b). Assume ε(k) = 1, φ(k) ≤ 0, and j∈ [1,k] λ * j = j∈[1,k] µ * j . Let m = λ * k+1 . Then λ m+1 = µ m+1 = k (since µ * k+1 ≤ λ * k+1 < λ * k ≤ µ * k ). By (c), we have j∈[1,m] λ j = j∈[1,m] µ j . Suppose m = 2i. Note ε(λ m+1 ) = 1 implies that δ i+1 = 0, ∆ i = 0 and thus λ m+1 = 2α i+1 . Since A i ≤ A ′ i and 2A i + ∆ i = 2A ′ i + ∆ ′ i , we have A i = A ′ i and ∆ ′ i = 0. Together with φ(µ m+1 ) ≤ 0, this implies that µ m+1 > 2α ′ i+1 . Hence α i+1 > α ′ i+1 , which contradicts B i+1 ≤ B ′ i+1 . Suppose m = 2i − 1. Note ε(λ m+1 ) = 1 implies that θ i = 0, Θ i = 0 and thus λ m+1 = 2β i . Since B i ≤ B ′ i and 2B i + Θ i = 2B ′ i + Θ ′ i , we have B i = B ′ i and Θ ′ i = 0. Together with φ(µ m+1 ) ≤ 0 this implies that µ m+1 > 2β ′ i . Hence β i > β ′ i , which contradicts A i ≤ A ′ i . It remains to verify 2.9 (c). Assume j∈ [1,k] λ * j = j∈[1,k] µ * j , λ * k+1 − µ * k+1 is odd, and φ(k) = 0. Let λ * k+1 = m. Then µ m = k (since µ * k+1 < λ * k+1 ≤ λ * k ≤ µ * k ). By (c), j∈[1,m] λ j = j∈[1,m] µ j . Suppose m = 2i. Note that ε(µ m ) = 0 implies that θ ′ i = 2, ∆ ′ i = 2 (β ′ i = α ′ i+1 − 2) or θ ′ i = −2, ∆ ′ i = 0 (β ′ i = α ′ i + 2). If β ′ i = α ′ i+1 − 2, then ∆ ′ i = 2 and 2A i + ∆ i = 2A ′ i + ∆ ′ i imply that A i = A ′ i , ∆ i = 2 and thus β i = α i+1 − 2, θ i = 2. Since λ m = 2β i + θ i > µ m = 2β ′ i + θ ′ i , we have β i > β ′ i and thus α i+1 > α ′ i+1 , which contradicts B i+1 ≤ B ′ i+1 . If β ′ i = α ′ i + 2, then µ * k+1 is even (see (e)), which contradicts the fact that λ * k+1 − µ * k+1 is odd. Suppose m = 2i − 1. Note that φ(µ m ) = 0 implies that δ ′ i = 2, Θ ′ i = 2 (β ′ i = α ′ i + 2) or δ ′ i = −2, Θ ′ i = 0 (α ′ i = β ′ i−1 + 2). If β ′ i = α ′ i + 2, then Θ ′ i = 2 and 2B i + Θ i = 2B ′ i + Θ ′ i imply Θ i = 2, B i = B ′ i and thus β i = α i + 2, δ i = 2. Since λ m = 2α i + δ i > µ m = 2α ′ i + δ ′ i , we have α i > α ′ i and thus β i > β ′ i , which contradicts A i ≤ A ′ i . If α ′ i = β ′ i−1 + 2, then µ * k+1 is odd, which contradicts the fact that λ * k+1 − µ * k+1 is odd. This completes the proof of (a). Combinatorial definition of unipotent and nilpotent pieces 4.1. Letc ∈ Ω 1 G and let c ∈ Ω p be such that γ p (c) = γ 1 G (c). Define Σ p c to be the set of all classes c ′ ∈ Ω p such that c ′ ≤ c and c ′ c ′′ for any c ′′ < c with γ p (c ′′ ) ∈ Λ 1 G . We show that (a) {Σ p c }c ∈Ω 1 G form a partition of U G or N g (see 4.2 (a)) and that (b) each set Σ p c is a piece defined in [7] (see 5.1 (b)). The definition of unipotent pieces using closure relations is first considered by Spaltenstein and (a) for U G is shown in [11]. For completeness, we include here a different proof which applies for both unipotent and nilpotent pieces. We define a map Φ : Λ p → Λ 1 G , (α, β) → (α,β) as follows. When p = 2, Φ is the identity map. For (α, β) ∈ Λ 2 Bn , defineα 1 = α 1 and α i = [ α i +β i−1 2 ] if α i > β i−1 α i if α i ≤ β i−1 , i ≥ 2;β i = [ α i+1 +β i +1 2 ] if β i < α i+1 β i if β i ≥ α i+1 , i ≥ 1. For (α, β) ∈ Λ 2 Cn , definẽ α 1 = [ α 1 +β 1 2 ] if β 1 > α 1 + 1 α 1 if β 1 ≤ α 1 + 1 ,β 1 =    [ α 1 +β 1 +1 2 ] if β 1 > α 1 + 1 β 1 if α 2 − 1 ≤ β 1 ≤ α 1 + 1 [ α 2 +β 1 2 ] if β 1 < α 2 − 1 , α i =    [ α i +β i 2 ] if β i > α i + 1 [ α i +β i−1 +1 2 ] if β i−1 < α i − 1 α i if β i ≤ α i + 1 and β i−1 ≥ α i − 1 , i ≥ 2, β i =        [ α i +β i +1 2 ] if β i > α i + 1 [ α i+1 +β i 2 ] if β i < α i+1 − 1 β i if α i+1 − 1 ≤ β i ≤ α i + 1 and β i−1 ≥ α i − 1 or β i ≥ α i+1 − 1 and β i−1 < α i − 1 , i ≥ 2. For (α, β) ∈ Λ 2 Dn , defineα 1 = α 1 and α i = [ α i +β i−1 +2 2 ] if α i > β i−1 + 2 α i if α i ≤ β i−1 + 2 , i ≥ 2, β i = [ α i+1 +β i −1 2 ] if β i < α i+1 − 2 β i if β i ≥ α i+1 − 2 , i ≥ 1, (note that Φ((α, β)) = (α,α) if and only if (α, β) = (α,α); we define Φ to be the identity map on the set of degenerate elements of W ∧ ). It is easy to verify that in each case we get a well-defined element (α,β) ∈ Λ 1 G . In this subsection we show that for eachc ∈ Ω 1 G , (a) γ p (Σ p c ) = Φ −1 (γ 1 G (c)).(c) For any (α ′ ,β ′ ) ∈ Λ 1 such that (α, β) ≤ (α ′ ,β ′ ), we have Φ(α, β) ≤ (α ′ ,β ′ ). The first assertion in (b) follows from the definition of Φ. Suppose Φ((α, β)) = (α,β). Let A i , B i , A i ,B i ,Ã ′ i ,B ′ i be defined for (α, β) , (α,β), (α ′ ,β ′ ) respectively as in (1). We prove (b) and (c). (i) Assume (α, β) ∈ Λ 2 Bn . Note that we haveβ i +α i+1 = β i + α i+1 , B 1 =B 1 and thus B i =B i . Moreover, A i ≤Ã i , and A i <Ã i if and only if β i < α i+1 . Hence (α, β) ≤ (α,β). Assume there exists (α ′ ,β ′ ) ∈ Λ 1 Bn such that (α, β) ≤ (α ′ ,β ′ ) and (α,β) (α ′ ,β ′ ). Sincẽ B j = B j ≤B ′ j for all j, there exists an i such thatÃ ′ i <Ã i . It follows that β i < α i+1 (since A i <Ã i ) and thusβ i ≤α i+1 + 1 by the definition of Φ. On the other hand,β i >β ′ i ≥α ′ i+1 >α i+1 (we useB j = B j ≤B ′ j , j = i, i + 1, and the fact that (α ′ ,β ′ ) ∈ Λ 1 Bn ), which is a contradiction. (ii) Assume (α, β) ∈ Λ 2 Cn . We show by induction on i that (d) if β i > α i + 1, thenB i > B i ,Ã i = A i ; if α i+1 − 1 ≤ β i ≤ α i + 1, thenÃ i = A i ,B i = B i ; if β i < α i+1 − 1, thenB i = B i ,Ã i > A i . It then follows that (α, β) ≤ (α,β). It is easy to verify that (d) holds when i = 1. We have the following subcases: (ii-1) β i+1 > α i+1 + 1. Thenα i+1 > α i+1 andα i+1 +β i+1 = α i+1 + β i+1 . Since β i > α i+1 + 1, by induction hypothesis,Ã i = A i . It follows thatB i+1 >B i+1 andÃ i+1 = A i+1 . (ii-2) α i+2 − 1 ≤ β i+1 ≤ α i+1 + 1. If β i ≥ α i+1 − 1, thenÃ i = A i (by induction hypothesis) and α i+1 = α i+1 ; if β i < α i+1 − 1, thenB i = B i (by induction hypothesis) andβ i +α i+1 = β i + α i+1 . It follows thatB i+1 = B i+1 . Sinceβ i+1 = β i+1 , we haveÃ i+1 = A i+1 . (ii-3) β i+1 < α i+2 − 1. If β i ≥ α i+1 − 1, thenÃ i = A i andα i+1 = α i+1 ; if β i < α i+1 − 1, theñ B i = B i andβ i +α i+1 = β i + α i+1 . It follows thatB i+1 = B i+1 . Sinceβ i+1 > β i+1 , we havẽ A i+1 > A i+1 . (d) is proved. Assume there exists (α ′ ,β ′ ) ∈ Λ 1 Cn such that (α, β) ≤ (α ′ ,β ′ ) and (α,β) (α ′ ,β ′ ). Suppose that there exists an i such thatÃ ′ i <Ã i . Then it follows from (d) that β i < α i+1 − 1 (since A i <Ã i ) and thusβ i ≤α i+1 ;B i = B i ,B i+1 = B i+1 and thusβ i ≥β ′ i + 1 ≥α ′ i+1 ≥α i+1 + 1, which is a contradiction. Then there exists an i such thatB ′ i <B i . It follows from (d) that β i > α i + 1 and thusβ i ≥α i ;Ã i = A i ≤Ã ′ i ,Ã i−1 = A i−1 ≤Ã ′ i−1 , and thusβ i ≤β ′ i − 1 ≤α ′ i ≤α i − 1, which is again a contradiction. (iii) Assume (α, β) ∈ Λ 2 Dn . We haveβ i +α i+1 = β i + α i+1 , B 1 =B 1 and thus B i =B i . Moreover, A i ≤Ã i , and A i <Ã i if and only if β i < α i+1 − 2. Hence (α, β) ≤ (α,β). Assume there exists (α ′ ,β ′ ) ∈ Λ 1 Dn such that (α, β) ≤ (α ′ ,β ′ ) and (α,β) (α ′ ,β ′ ). Theñ B j = B j ≤B ′ j for all j and there exists an i such that A i ≤Ã ′ i <Ã i . It follows thatβ i <α i+1 , and β i >β ′ i ≥α ′ i+1 − 2 >α i+1 − 2, which is a contradiction. This completes the proof of (b) and (c). Explicit description of pieces 5.1. We define a map Ψ p G : Ω p G → Ω 1 G (resp. Ψ 2 g : Ω 2 g → Ω 1 G ) as follows. Let Ψ p G be the natural identification map between Ω p G and Ω 1 G if p = 2. If G = Sp(2n), let (9) Ψ 2 G ((λ, ε)) = λ (resp. Ψ 2 g ((λ, χ)) = λ). If G = SO(N ), define Ψ 2 G ((λ, ε)) =λ = (λ 1 ≥λ 2 ≥ · · · ) as follows, where λ = (λ 1 ≥ λ 2 ≥ · · · ). If λ 2i−1 is even, ε(λ 2i−1 ) = 1, and λ 2i−1 < λ 2i−2 (when i ≥ 2), thenλ 2i−1 = λ 2i−1 + 1; if λ 2i is even, ε(λ 2i ) = 1 and λ 2i > λ 2i+1 , thenλ 2i = λ 2i − 1. Otherwiseλ j = λ j . Note that Ψ 2 SO(2n) ((λ, ε)) =λ withλ i all even, if and only if λ =λ and ε(λ i ) = 0 for all i; for the two degenerate classes c 1 , c 2 corresponding to (λ, ε), we define Ψ 2 G (c i ) =c i by γ 2 G (c i ) = γ 1 G (c i ), i = 1, 2. If G = SO(N ) and (λ, χ) ∈ Ω 2 g , λ = (λ 1 ≥ λ 2 ≥ · · · ), let k ≥ 0 be the unique integer such that λ 2k+2 = λ 2k+1 − 1 (when N is odd), and k = ∞ (when N is even). Define Ψ 2 g ((λ, χ)) =λ as follows. λ 1 = 2χ(λ 1 ) − 1 if χ(λ 1 ) ≥ λ 1 2 + 1 λ 1 if χ(λ 1 ) ≤ λ 1 +1 2 , λ 2i =                λ 2i − χ(λ 2i ) + χ(λ 2i+1 ) if χ(λ 2i ) ≥ λ 2i − χ(λ 2i+1 ) + 1 and i ≤ k 2(λ 2i − χ(λ 2i )) + 1 if λ 2i 2 + 1 ≤ χ(λ 2i ) ≤ λ 2i − χ(λ 2i+1 ) and i ≤ k λ 2i if χ(λ 2i ) ≤ λ 2i +1 2 and i ≤ k λ 2i+1 if i ≥ k + 1 , λ 2i+1 =            λ 2i − χ(λ 2i ) + χ(λ 2i+1 ) if χ(λ 2i+1 ) ≥ λ 2i − χ(λ 2i ) + 1 and i ≤ k 2χ(λ 2i+1 ) − 1 if i ≤ k and λ 2i+1 2 + 1 ≤ χ(λ 2i+1 ) ≤ λ 2i − χ(λ 2i ) λ 2i+1 if χ(λ 2i+1 ) ≤ λ 2i+1 +1 2 or i > k , i ≥ 1. Note that Ψ 2 so(2n) ((λ, χ)) =λ withλ i all even, if and only if λ =λ and χ(λ i ) = λ i /2 for all i; for the two degenerate classes c 1 , c 2 corresponding to (λ, χ), we define Ψ 2 g (c i ) =c i by γ 2 g (c i ) = γ 1 G (c i ), i = 1, 2. Using the definition of Ψ p G (resp. Ψ 2 g ) and the description of the maps γ p in 2.4, 2.5 and 2.6, one verifies that (a) γ 1 G • Ψ p G = Φ • γ p G (resp. γ 1 G • Ψ 2 g = Φ • γ 2 g ) . Proposition 5.1. Two classes c, c ′ ∈ Ω p G (resp. Ω 2 g ) lie in the same unipotent (resp. nilpotent) piece as defined in [7] if and only if Ψ p G (c) = Ψ p G (c ′ ) (resp. Ψ 2 g (c) = Ψ 2 g (c ′ )). Proposition is clear when p = 2. If G = Sp(2n), the proposition follows from [7] (see Lemma 2.7 and (9)). The case where p = 2 and G = SO(N ) is proved in subsection 5.2. Note that the proposition computes the pieces in classical groups explicitly. Another computation of the unipotent pieces is given in [9]. Now in view of (a) and 4.2 (a), it follows from Proposition 5.1 that (b) each set Σ p c ,c ∈ Ω 1 G , defined in 4. 1 is an unipotent (resp. a nilpotent) piece defined in [7]. 5.2. Let G = SO(V ) be as in 2.8. We prove Proposition 5.1 by induction on dim V . Assume Ψ 2 G (c) =c (resp. Ψ 2 g (c) =c). We show that (a) Υ c = Υc. Suppose that Υ c = {f a }, Υc = {f a }. If G = SO(2n), then it follows from (a) that c is degenerate if and only if f 0 = 0, and then from [7] that each degenerate class itself is a piece. Now Proposition 5.1 follows from Lemma 2.8. If c = {0}, (a) is obvious. We assume c = {0}. Let u ∈ c (resp. x ∈ c) and let V ′ , u ′ (resp. x ′ ) be as in 2.8. Let c ′ be the class of u ′ (resp. x ′ ) in G ′ = SO(V ′ ) (resp. g ′ = so(V ′ )) and letc ′ = Ψ 2 G ′ (c ′ ) (resp. Ψ 2 g ′ (c ′ )). Suppose that Υ c ′ = {f ′ a }, Υc′ = {f ′ a }. Let m, e, f be defined for T = u − 1 (resp. x) as in 2.8. Assumec =λ, whereλ = (λ 1 ≥λ 2 ≥ · · · ). We show that (b) m =λ 1 − 1,f a =f ′ a for all a ∈ [−m + 1, m − 1], and f m =f m . Since dim V ′ < dim V , by induction hypothesis, Υ c ′ = Υc′. It then follows from (b) and 2.8 (b) that f a =f a for all a, since f a = f −a =f a =f −a = 0 for all a ≥ m + 1. Hence (a) holds. We prove (b) for Ψ 2 G . Assume c = (λ, ε), c ′ = (λ ′ , ε ′ ), andc ′ =λ ′ . Using definition of Ψ 2 G and 2.8, we can compute c ′ ,c,c ′ , f m = dim V ≥m = dim(V ≥−m+1 ∩ Q −1 (0)) andf m = mλ(λ 1 ) in various cases as follows. (I) ε(λ 1 ) = 1 and m λ (λ 1 ) = 2m 1 . We have e = λ 1 = 2f − 2 and thus m = λ 1 , f m = 1; λ ′ i = λ 1 , i ∈ [1, 2m 1 − 2], ε ′ (λ 1 ) ≤ 0, λ ′ 2m 1 −1 = λ ′ 2m 1 = λ 1 − 1, and λ ′ j = λ j , ε ′ (λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 1. Moreoverλ 1 = λ 1 + 1,λ i = λ 1 , i ∈ [2, 2m 1 − 1] andλ 2m 1 = λ 1 − 1;λ ′ i = λ 1 , i ∈ [1, 2m 1 − 2], λ ′ 2m 1 −1 = λ 1 − 1 andλ ′ j =λ j , j ≥ 2m 1 . Thusf m = 1. (II) ε(λ 1 ) = 0 and m λ (λ 1 ) = 2m 1 . We have e = λ 1 = 2f and thus m = λ 1 − 1, f m = 2m 1 . Let m λ (λ 1 − 1) = 2m 2 ≥ 0. We haveλ i = λ 1 , i ∈ [1, 2m 1 ],λ 2m 1 +i = λ 1 − 1, i ∈ [1, 2m 2 ]. We have the following cases: (i) ε(λ 1 − 2) < 1. Then λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ], ε ′ (λ 1 − 2) = 0, and λ ′ j = λ j , ε ′ (λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 2m 2 + 1. Moreover,λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ], andλ ′ j =λ j , j ≥ 2m 1 + 2m 2 + 1. Thusf m = 2m 1 . (ii) ε(λ 1 − 2) = 1. Then λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 1], ε ′ (λ 1 − 2) = 1, and λ ′ j = λ j , ε ′ (λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 2m 2 + 2. Moreover,λ 2m 1 +2m 2 +1 = λ 1 − 1, thusf m = 2m 1 ;λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 + 1],λ ′ 2m 2 +1+i = λ 1 − 2, i ∈ [1, 2m 1 ] andλ ′ j =λ j , j ≥ 2m 1 + 2m 2 + 2. (III) m λ (λ 1 ) = 2m 1 + 1. We have e = λ 1 = 2f − 2 and thus m = λ 1 , f m = 1. Moreover, λ 1 = λ 1 + 1,λ i = λ 1 , i ∈ [2, 2m 1 + 1]. Let m λ (λ 1 − 1) = 2m 2 ≥ 0. Then λ ′ i = λ 1 , ε ′ (λ 1 ) ≤ 0, i ∈ [1, 2m 1 ], λ ′ 2m 1 +i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 1 +2m 2 +1 = λ 1 − 2, ε ′ (λ 1 − 2) = 1, and λ ′ j = λ j , ε(λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 2m 2 + 2. Moreoverλ 2m 1 +1+i = λ 1 − 1, i ∈ [1, 2m 2 ];λ ′ i = λ 1 , i ∈ [1, 2m 1 ],λ ′ 2m 1 +1 = λ 1 − 1, andλ ′ j =λ j , j ≥ 2m 1 + 2. Thusf m = 1. (IV) ε(λ 1 ) = ω. Then m λ (λ 1 ) = 2m 1 . We have e = λ 1 = 2f − 1 and thus m = λ 1 − 1. Moreover, λ i = λ 1 , i ∈ [1, 2m 1 ]. We have the following cases: (i) m λ (λ 1 − 1) = 2m 2 and ε(λ 1 − 1) = 1 (note then m 2 > 0). Then f m = 2m 1 + 1; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 − 2], ε ′ (λ 1 − 1) ≤ 0, λ ′ 2m 2 −2+i = λ 1 − 2, i ∈ [1, 2m 1 + 2], and λ ′ j = λ j , ε ′ (λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 2m 2 + 1. Moreoverλ 2m 1 +1 = λ 1 ,λ 2m 1 +1+i = λ 1 − 1, i ∈ [1, 2m 2 − 2], and λ 2m 1 +2m 2 = λ 1 − 2, thusf m = 2m 1 + 1;λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 − 2],λ ′ 2m 2 −2+i = λ 1 − 2, i ∈ [1, 2m 1 + 1] andλ ′ j =λ j , j ≥ 2m 1 + 2m 2 . (ii) m λ (λ 1 −1) = 2m 2 and ε(λ 1 −1) ≤ 0. Then f m = 2m 1 ; λ ′ i = λ 1 −1, i ∈ [1, 2m 2 ], ε ′ (λ 1 −1) ≤ 0, λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ], and λ ′ j = λ j , ε ′ (λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 2m 2 + 1. Moreover λ 2m 1 +i = λ 1 − 1, i ∈ [1, 2m 2 ];λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ] andλ ′ j =λ j , j ≥ 2m 1 + 2m 2 + 1. Thusf m = 2m 1 . (iii) m λ (λ 1 − 1) = 2m 2 + 1. Then f m = 2m 1 + 1; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], ε ′ (λ 1 − 1) ≤ 0, λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ], λ 2m 1 +2m 2 +1 = λ 1 − 3, ε ′ (λ 1 − 3) = 1, and λ ′ j = λ j , ε ′ (λ ′ j ) = ε(λ j ) for all j ≥ 2m 1 + 2m 2 + 2. Moreoverλ 2m 1 +1 = λ 1 ,λ 2m 1 +1+i = λ 1 − 1, i ∈ [1, 2m 2 ];λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 1] andλ ′ j =λ j , j ≥ 2m 1 + 2m 2 + 2. Thusf m = 2m 1 + 1. In each case it is now easy to see that (b) holds. We prove (b) for Ψ 2 g . Assume c = (λ, χ), c ′ = (λ ′ , χ ′ ), andc ′ =λ ′ . For (λ, χ) ∈ Ω 2 g , we extend χ to a function χ : N → N as follows: χ(i) = max j ([λ j ; χ(λ j )](i)), where [p; l](i) = max(0, min(i − p + l, l)). Using definition of Ψ 2 g and 2.8, we compute c ′ ,c,c ′ , f m = dim V ≥m = dim(V ≥−m+1 ∩ Q −1 (0)) and f m = mλ(λ 1 ) in various cases as follows. (I) χ(λ 1 ) = λ 1 /2 and λ 1 = 2. Write m λ (2) = 2m 1 and m λ (1) = m 2 . We have e = 2, f = 1 and thus m = 1, f m = 2m 1 . Then λ ′ i = 1, i ∈ [1, m 2 ], λ ′ i = 0 for all i ≥ m 2 + 1. Moreover,λ i = 2, i ∈ [1, 2m 1 ] andλ 2m 1 +i = 1, i ∈ [1, m 2 ];λ ′ i = 1, i ∈ [1, m 2 ],λ ′ i = 0, i ≥ m 2 + 1. Thusf m = 2m 1 . (II) χ(λ 1 ) = λ 1 /2 and λ 1 ≥ 4. Then m λ (λ 1 ) = 2m 1 and m λ (λ 1 − 1) = 2m 2 (m 2 ≥ 0). We have e = λ 1 , f = λ 1 /2 and thus m = λ 1 − 1, f m = 2m 1 . Moreover,λ i = λ 1 , i ∈ [1, 2m 1 ],λ 2m 1 +i = λ 1 − 1, i ∈ [1, 2m 2 ]. We have the following cases: (i) χ(λ 1 − 2) = λ 1 /2. Then λ ′ i = λ 1 − 1, χ ′ (λ 1 − 1) = λ 1 /2, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, χ ′ (λ 1 − 2) = λ 1 /2, i ∈ [1, 2m 1 + 1]. Moreover,λ 2m 1 +2m 2 +1 = λ 1 − 1, thusf m = 2m 1 ;λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 + 1],λ ′ 2m 2 +1+i = λ 1 − 2, i ∈ [1, 2m 1 ] andλ ′ j =λ j for all j ≥ 2m 1 + 2m 2 + 2. (ii) χ(λ 1 − 2) < λ 1 /2. Then λ ′ i = λ 1 − 1, χ ′ (λ 1 − 1) = λ 1 /2, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, χ ′ (λ 1 − 2) = λ 1 /2 − 1, i ∈ [1, 2m 1 ]. Moreover,λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ] andλ ′ j =λ j for all j ≥ 2m 1 + 2m 2 + 1. Thusf m = 2m 1 . (III) χ(λ 1 ) = λ 1 +1 2 . Then m λ (λ 1 ) = 2m 1 . We have e = λ 1 , f = λ 1 +1 2 and thus m = λ 1 − 1;λ i = λ 1 , i ∈ [1, 2m 1 ]. There exists a unique j ≥ 0 such that χ(λ 1 − j) = λ 1 +1 2 and χ(λ 1 − j − 1) < λ 1 +1 2 . Assume m λ (λ 1 − i) = 2m i+1 , i ∈ [1, j − 1] . We have the following subcases: (i) j = 0. Then χ(λ 1 − 1) = λ 1 −1 2 , m λ (λ 1 − 1) = 2m 2 (since λ 1 −1 2 < λ 1 − 1), and f m = 2m 1 . We have λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ], χ ′ (λ 1 − k) = λ 1 −1 2 , k ∈ [1, 2], and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ 2m 1 + 2m 2 + 1. Moreover,λ 2m 1 +i = λ 1 − 1, i ∈ [1, 2m 2 ]; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 ] andλ ′ i =λ i for all i ≥ 2m 1 + 2m 2 + 1; thus f m = 2m 1 . (ii) j = 1 and m λ (λ 1 − 1) = 2m 2 . Then m 2 > 0 and f m = 2m 1 + 1. We have λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 − 2], λ ′ 2m 2 −2+i = λ 1 − 2, i ∈ [1, 2m 1 + 2], χ ′ (λ 1 − k) = λ 1 −1 2 , k ∈ [1, 2], and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ 2m 1 +2m 2 +1. Moreover,λ 2m 1 +1 = λ 1 ,λ 2m 1 +1+i = λ 1 −1, i ∈ [1, 2m 2 −2]; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 − 2],λ ′ 2m 2 −2+i = λ 1 − 2, i ∈ [1, 2m 1 + 1] andλ ′ i =λ i for all i ≥ 2m 1 + 2m 2 . Thusf m = 2m 1 + 1. (iii) j = 2 and m(λ 1 − 2) = 2m 3 . Then m 3 > 0 and f m = 2m 1 + 1. We have 3], and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ 2m 1 + 2m 2 + 2m 3 + 1. λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 2m 3 − 2], λ ′ 2m 1 +2m 2 +2m 3 −1 = λ ′ 2m 1 +2m 2 +2m 3 = λ 1 − 3, χ ′ (λ 1 − k) = λ 1 −1 2 , k ∈ [1,Moreover,λ 2m 1 +1 = λ 1 ,λ 2m 1 +1+i = λ 1 − 1, i ∈ [1, 2m 2 ],λ 2m 1 +2m 2 +1+i = λ 1 − 2, i ∈ [1, 2m 3 − 2]; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 1] , andλ ′ i =λ i for all i ≥ 2m 1 + 2m 2 + 2; thusf m = 2m 1 + 1. (iv) j ≥ 3 and m λ (λ 1 − j) = 2m j+1 . We have f m = 2m 1 + 1; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 2m 3 ], λ ′ a∈[1,k] 2ma+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [3, j − 1], λ ′ a∈[1,j] 2ma+i = λ 1 −j, i ∈ [1, 2m j+1 −2], λ ′ a∈[1,j+1] 2ma−2+i = λ 1 −j−1, i ∈ [1, 2], χ ′ (λ 1 −k) = λ 1 −1 2 , k ∈ [1, j + 1], and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ a∈[1,j+1] 2m a + 1. Moreover,λ 2m 1 +1 = λ 1 , λ a∈[1,k] 2ma+1+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [1, j − 1],λ a∈[1,j] 2ma+1+i = λ 1 − j, i ∈ [1, 2m j+1 − 2]; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 1], andλ ′ i =λ i for all i ≥ 2m 1 + 2m 2 + 2. Thusf m = 2m 1 + 1. (v) j = 1 and m λ (λ 1 − 1) = 2m 2 + 1. It follows that λ 1 = 3 and m λ (1) = 2m 3 + 1. We have f m = 2m 1 + 1; λ ′ i = 2, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = 1, i ∈ [1, 2m 1 + 2m 3 + 1], χ ′ (2) = 1, λ ′ i = 0, for all i ≥ a∈[1,3] 2m a + 2. Moreover,λ i = 3, i ∈ [1, 2m 1 + 1],λ 2m 1 +1+i = 2, i ∈ [1, 2m 2 ], λ 2m 1 +2m 2 +1+i = 1, i ∈ [1, 2m 3 ];λ ′ i = 2, i ∈ [1, 2m 2 ],λ ′ 2m 2 +i = 1, i ∈ [1, 2m 1 + 1] andλ ′ i =λ i for all i ≥ 2m 1 + 2m 2 + 2. Thusf m = 2m 1 + 1. (vi) j ≥ 2 and m λ (λ 1 − j) = 2m j+1 + 1. It follows that λ 1 − j = λ 1 +1 2 and m λ (λ 1 − j − 1) = 2m j+2 + 1. We have f m = 2m 1 + 1; λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ], λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 2m 3 ], λ ′ a∈[1,k] 2ma+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [3, j], λ ′ a∈[1,j+1] 2ma+i = λ 1 − j − 1, i ∈ [1, 2m j+2 + 1], λ ′ a∈[1,j+2] 2ma+2 = λ 1 − j − 2, χ ′ (λ 1 − k) = λ 1 −1 2 , k ∈ [1, j + 1], χ ′ (λ 1 − j − 2) = λ 1 −3 2 and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ a∈[1,j+2] 2m a + 3. Moreover,λ 2m 1 +1 = λ 1 ,λ a∈[1,k] 2ma+1+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [1, j + 1];λ ′ i = λ 1 − 1, i ∈ [1, 2m 2 ] ,λ ′ 2m 2 +i = λ 1 − 2, i ∈ [1, 2m 1 + 1] andλ ′ i =λ i for all i ≥ 2m 1 + 2m 2 + 2; thusf m = 2m 1 + 1. (IV) χ(λ 1 ) > λ 1 +1 2 and m λ (λ 1 ) = 2m 1 . There exists a unique j ≥ 0 such that χ(λ 1 − j) = χ(λ 1 ) and χ(λ 1 − j − 1) < χ(λ 1 ). Assume m λ (λ 1 − i) = 2m i+1 , i ∈ [1, j − 1]. We have e = λ 1 , f = χ(λ 1 ) and thus m = 2χ(λ 1 ) − 2, f m = 1. (i) j = 0. We have λ ′ i = λ 1 , i ∈ [1, 2m 1 − 2], λ ′ 2m 1 −1 = λ ′ 2m 1 = λ 1 − 1, χ ′ (λ 1 ) = χ ′ (λ 1 − 1) = χ(λ 1 ) − 1, and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ 2m 1 + 1. Moreover,λ 1 = 2χ(λ 1 ) − 1,λ i = λ 1 , i ∈ [2, 2m 1 − 1]. If χ(λ 1 ) = λ 1 +2 2 , thenλ ′ i = λ 1 , i ∈ [1, 2m 1 − 2],λ ′ 2m 1 −1 = λ 1 − 1, andλ ′ i =λ i for all i ≥ 2m 1 ; if χ(λ 1 ) ≥ λ 1 +3 2 , thenλ ′ 1 = 2χ(λ 1 ) − 3, andλ ′ i =λ i for all i ≥ 2. Thusf m = 1. (ii) j ≥ 1 and m λ (λ 1 −j) = 2m j+1 . We have λ ′ a∈[1,k] 2ma+i = λ 1 −k, i ∈ [1, 2m k+1 ], k ∈ [0, j −1], λ ′ a∈[1,j] 2ma+i = λ 1 − j, i ∈ [1, 2m j+1 − 2], λ ′ a∈[1,j+1] 2ma−2+i = λ 1 − j − 1, i ∈ [1, 2], χ ′ (λ 1 − k) = χ(λ 1 ) − 1, k ∈ [0, j + 1], and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ a∈[1,j+1] 2m a + 1. Moreover, λ 1 = 2χ(λ 1 ) − 1,λ a∈[1,k] 2ma+1+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [0, j − 1],λ a∈[1,j] 2ma+1+i = λ 1 − j, i ∈ [1, 2m j+1 − 2]. If χ(λ 1 ) = λ 1 +2 2 , thenλ ′ i = λ 1 , i ∈ [1, 2m 1 ],λ ′ 2m 1 +1 = λ 1 − 1, andλ ′ i =λ i for all i ≥ 2m 1 + 2; if χ(λ 1 ) ≥ λ 1 +3 2 , thenλ ′ 1 = 2χ(λ 1 ) − 3, andλ ′ i =λ i for all i ≥ 2. Thusf m = 1. (iii) j ≥ 1 and m λ (λ 1 − j) = 2m j+1 + 1. It follows that λ 1 − j = χ(λ 1 ) and m λ (λ 1 − j − 1) = 2m j+2 + 1. We have λ ′ a∈ [1,k] [1,j+2] 2ma+2 = λ 1 −j −2, χ ′ (λ 1 −k) = χ(λ 1 )−1, k ∈ [0, j +1], χ ′ (λ 1 −j −2) = χ(λ 1 ) − 2, and λ ′ i = λ i , χ ′ (λ ′ i ) = χ(λ i ), for all i ≥ a∈ [1,j+2] 2m a + 3. Moreover,λ 1 = 2χ(λ 1 ) − 1, λ a∈[1,k] 2ma+1+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [0, j + 1]. If χ(λ 1 ) = λ 1 +2 2 ,λ ′ i = λ 1 , i ∈ [1, 2m 1 ], λ ′ 2m 1 +1 = λ 1 − 1, andλ ′ i =λ i for all i ≥ 2m 1 + 2; if χ(λ 1 ) ≥ λ 1 +3 2 ,λ ′ 1 = 2χ(λ 1 ) − 3, andλ ′ i =λ i for all i ≥ 2. Thusf m = 1. 2ma+i = λ 1 − k, i ∈ [1, 2m k+1 ], k ∈ [0, j], λ ′ a∈[1,j+1] 2ma+i = λ 1 − j − 1, i ∈ [1, 2m j+2 +1], λ ′ a∈ (V) m λ (λ 1 ) = 2m 1 + 1, then χ(λ 1 ) = λ 1 > λ 1 +1 2 and m λ (λ 1 − 1) = 2m 2 + 1. We have e = f = λ 1 and thus m = 2λ 1 − 2, f m = 1. Moreover,λ 1 = 2λ 1 − 1,λ i = λ 1 , i ∈ [2, 2m 1 + 1]. We have λ ′ i = λ 1 , i ∈ [1, 2m 1 ], λ ′ 2m 1 +i = λ 1 − 1, i ∈ [1, 2m 2 + 1], λ ′ 2m 1 +2m 2 +2 = λ 1 − 2, χ ′ (λ 1 ) = χ ′ (λ 1 − 1) = λ 1 − 1, χ ′ (λ 1 − 2) = λ 1 − 2, λ ′ i = λ i = χ(λ i ) for all i ≥ 2m 1 + 2m 2 + 3. Moreover, if λ 1 = 2, thenλ ′ i = λ 1 , i ∈ [1, 2m 1 ],λ ′ 2m 1 +1 = λ 1 − 1, andλ ′ i =λ i for all i ≥ 2m 1 + 2; if λ 1 ≥ 3, thenλ ′ 1 = 2λ 1 − 3, and λ ′ i =λ i for all i ≥ 2. Thusf m = 1. In each case it is now easy to see that (b) holds. Proposition 5.1 is proved. special pieces We say that a unipotent or nilpotent class c is special if γ p (c) is a special character of W (see [2,4]). If G is of type B n or C n , then (α, β) ∈ P 2 (n) is special if and only if α i+1 ≤ β i ≤ α i + 1 for all i ≥ 1; if G is of type D n , then (α, β) ∈ W ∧ is special if and only if α i+1 − 1 ≤ β i ≤ α i for all i ≥ 1, in particular, each degenerate character is special (see [2]). Let c be a special unipotent (resp. nilpotent) class in G (resp. g). We define the corresponding special piece S c to be the subset of U G (resp. N g ) consisting of all elements in the closure of c which are not in the closure of any special unipotent (resp. nilpotent) class c ′ < c (see [4] when p = 1). We show that a special piece is a union of unipotent (resp. nilpotent) pieces (for unipotent case, see also [9]). Hence U G (resp. N g ) is partitioned into special pieces S c indexed by special unipotent (resp. nilpotent) classes c (when p = 2, this follows from [4]). In the remainder of this subsection assume p = 2. Let c be a special class and let S c be the corresponding special piece. Letc ∈ Ω 1 G be such that γ 2 (c) = γ 1 G (c). Assume the corresponding special piece Sc (in the unipotent variety of the group over C of the same type as G) is a union of the special classc :=c 0 and non-special classes c 1 , . . . ,c m . We show that (a) S c = ⊔ i∈[0,m] Σ 2 c i . Assume γ 1 G (c) = (α,β). Let c * ∈ S c and assume γ 2 (c * ) = (α, β). Then (α, β) ≤ (α,β) and for any special (α ′ ,β ′ ) < (α,β), (α, β) (α ′ ,β ′ ). Assume Φ(α, β) = (α * ,β * ). It follows from 4.2 (b) that (α * ,β * ) (α ′ ,β ′ ) and from 4.2 (c) that (α * ,β * ) ≤ (α,β). Hence (α * ,β * ) = (α i ,β i ) for some i ∈ [0, m] and thus c * ∈ Σ 2 c i (note if c is a degenerate class, then m = 0 (see [2,4]) and c * = c). This shows that S c ⊂ ⊔ i∈[0,m] Σ 2 c i . Now if c is a degenerate class, then m = 0 and r.h.s of (a) is {c} ⊂ S c (see 4.2 (a)). Assume c is not a degenerate class and assume γ 1 G (c j ) = (α j ,β j ), j ∈ [0, m]. Assume c * ∈ Σ 2 c j and γ 2 (c * ) = (α j , β j ). 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[ "Nonlinear Effects in Pulse Propagation through Doppler-Broadened Closed-Loop Atomic Media", "Nonlinear Effects in Pulse Propagation through Doppler-Broadened Closed-Loop Atomic Media" ]	[ "Robert Fleischhaker \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany\n", "Jörg Evers \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany\n" ]	[ "Max-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany", "Max-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany" ]	[]	Nonlinear effects in pulse propagation through a medium consisting of four-level double-Λ-type systems are studied theoretically. We apply three continous-wave driving fields and a pulsed probe field such that they form a closed interaction loop. Due to the closed loop and the finite frequency width of the probe pulses the multiphoton resonance condition cannot be fulfilled, such that a time-dependent analysis is required. By identifying the different underlying physical processes we determine the parts of the solution relevant to calculate the linear and nonlinear response of the system. We find that the system can exhibit a strong intensity dependent refractive index with small absorption over a range of several natural linewidths. For a realistic example we include Doppler and pressure broadening and calculate the nonlinear selfphase modulation in a gas cell with Sodium vapor and Argon buffer gas. We find that a selfphase modulation of π is achieved after a propagation of few centimeters through the medium while the absorption in the corresponding spectral range is small.	10.1103/physreva.77.043805	[ "https://arxiv.org/pdf/0711.4037v1.pdf" ]	118,544,656	0711.4037	4a9055092a0730fed3f26c44096b79e51c7f52b0	Nonlinear Effects in Pulse Propagation through Doppler-Broadened Closed-Loop Atomic Media 26 Nov 2007 (Dated: February 2, 2008) Robert Fleischhaker Max-Planck-Institut für Kernphysik Saupfercheckweg 1D-69117HeidelbergGermany Jörg Evers Max-Planck-Institut für Kernphysik Saupfercheckweg 1D-69117HeidelbergGermany Nonlinear Effects in Pulse Propagation through Doppler-Broadened Closed-Loop Atomic Media 26 Nov 2007 (Dated: February 2, 2008)numbers: 4250Gy4265Sf4265An3280Wr Nonlinear effects in pulse propagation through a medium consisting of four-level double-Λ-type systems are studied theoretically. We apply three continous-wave driving fields and a pulsed probe field such that they form a closed interaction loop. Due to the closed loop and the finite frequency width of the probe pulses the multiphoton resonance condition cannot be fulfilled, such that a time-dependent analysis is required. By identifying the different underlying physical processes we determine the parts of the solution relevant to calculate the linear and nonlinear response of the system. We find that the system can exhibit a strong intensity dependent refractive index with small absorption over a range of several natural linewidths. For a realistic example we include Doppler and pressure broadening and calculate the nonlinear selfphase modulation in a gas cell with Sodium vapor and Argon buffer gas. We find that a selfphase modulation of π is achieved after a propagation of few centimeters through the medium while the absorption in the corresponding spectral range is small. I. INTRODUCTION A main interest in laser driven atomic media is the study of their coherence properties. Coherence effects like electromagnetically induced transparency (EIT) [1], coherent population trapping [2], lasing without inversion [3], and others [4,5] are examples where the optical properties of an atomic medium are influenced with coherent fields. The interference of different excitation channels is the main underlying principle here. A particular class of systems in which quantum mechanical interference plays a major role are the so-called closed-loop systems [6,7,8,9,10,11,12,13,14,15,16,17]. In these systems the laser-driven transitions form a closed interaction loop such that photon emission and absorption can take place in a cycle. This leads to interference of indistinguishable transition pathways between different states. One consequence of this is that it can render the system dependent on the relative phase of the driving fields. At the same time, however, the investigation of closed-loop systems is made difficult by the fact that the interfering pathways typically prevent the system from reaching a time-independent steady state. Such a stationary state in general is only reached when the so-called multiphoton resonance condition on the detunings of the different driving field is fulfilled, which was therefore assumed in most previous studies. For general laser field detunings, a time-dependent analysis is mandatory [8,11]. Laser driven atomic media are also known to exhibit significant nonlinear optical properties [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. A particular example is the occurrence of an intensity dependent refractive index, with applications such as beam focussing, pulse compression, selfphase-or cross-phase modulation or optical switching [23,24,25,26,27,28,29,30]. Here, the connection to coherence properties is the following. While an atomic resonance can greatly enhance nonlinear effects in atomic media, the accompanying linear absorption of the same resonance typically renders the medium opaque to the probe field. This can be overcome by tailoring the response via coherence and interference effects. An advantageous situation arises, e.g., if the linear absorption vanishes due to destructive interference while the nonlinear effect is enhanced by constructive interference. Motivated by this, we investigate nonlinear effects in pulse propagation through a closed-loop atomic medium. In particular, we study a four-level atomic system where the four dipole-allowed transitions form a double-Λ type scheme (see Fig. 1). Three of the fields are assumed to be continous-wave coupling laser fields, while the fourth field is a pulsed probe field. We use a time-dependent analysis, as the multiphoton resonance condition cannot be applied due to the finite frequency spectrum of the probe pulses. The medium is modelled as a dilute gas vapor including Doppler and pressure broadening and an additional buffer gas using realistic parameters. Our main observable is the nonlinear index of refraction of the medium. We find that our system exhibits a high nonlinear index of refraction with small linear and nonlinear absorption over a spectral range of several natural linewidths. In this spectral region of interest, the real part of linear and non-linear susceptibility show linear dispersion, such that pulse shape distortions are minimized. For Sodium atoms with Argon buffer gas, we obtain a nonlinear selfphase modulation of π after 2.9 cm of passage through the medium. The paper is organized as follows. In the following Sec. II A we present our model. In Sec. II B, we solve for the time-dependent long-time limit arising from the closed interaction loop in the form of a series. The interpretation of the series coefficients with respect to their physical meaning (Sec. II C) will enable us to identify the quantities necessary to calculate the linear and nonlinear susceptibility for the probe field of our system (Sec. II D). Doppler and pressure broadening are discussed in Secs. II E and II F. Our results are presented in Sec. III, both with and without broadening. Finally, Sec. IV discusses and summarizes our results. II. THEORETICAL ANALYSIS A. The model In this section we present the Hamiltonian for the fourlevel system and the interaction with the coupling fields in a suitable interaction picture. We write the field coupling to transition \|j ↔ \|k (j ∈ {3, 4}, k ∈ {1, 2}) as E jk = E jk 2 ê jk e −iω jk t + c.c. ,(1) with amplitude E jk , unit polarization vectorê jk , and frequency ω jk . For better readability we suppress the spacedependence of the fields. The Hamiltonian in dipole and rotating-wave approximation reads [4,5] H = 4 j=1 ω j A jj − 4 j=3 2 k=1 Ω jk 2 e −i(ω jk t−φ jk ) A jk + H.c. . (2) The energy of level \|j is denoted by ω j and we have introduced Rabi frequencies Ω jk = E jk \|ê jk · d jk \|/ with d jk being the dipole matrix element of transition \|j ↔ \|k (j ∈ {3, 4}, k ∈ {1, 2}). The complex phase of the Rabi frequencies was included into the exponential function where φ jk = arg(ê jk · d jk ). The atomic transition operator is defined as A jk = \|j k\|. The canonical approach with a Hamiltonian of the sort we have just introduced would be to transform it into an interaction picture where the time dependence fully vanishes. Unfortunately, this is not possible in our case. Due to the closed interaction loop, in general a residual time dependence in the Hamiltonian remains. Physically, this means that we cannot expect the system to reach a true stationary state in the long time limit. The best we can do is to use a unitary transformation that gathers all the time dependence in a single exponential factor in front of the probe field Rabi frequency. In this interaction picture we obtain H I = (∆ 32 − ∆ 31 )A 22 − ∆ 31 A 33 + (∆ 32 − ∆ 31 − ∆ 42 )A 44 − 2 (Ω 31 A 31 + Ω 32 A 32 + Ω 42 A 42 +Ω 41 A 41 e −i(∆t−φ) + H.c. ,(3) where the detunings are defined as ∆ jk = ω jk −(ω j −ω k ). We have also defined the so-called multiphoton detuning and an equivalent combination of the dipole phases ∆ =∆ 41 + ∆ 32 − ∆ 31 − ∆ 42 , (4a) φ =φ 41 + φ 32 − φ 31 − φ 42 . (4b) The multiphoton detuning is a typical quantity characterizing a system with a closed interaction loop. Its significance will become more apparent in Sec. II C. We now set up the master equation for the atomic density matrix ̺. We include the unitary evolution due to the Hamiltonian in the interaction picture and relaxation dynamics due to spontaneous decay in Born-Markov approximation. The collision induced dynamics will be considered in Sec. II F. The unitary evolution is given by the Von-Neumann equation and the spontaneous decay can be written in Lindblad form [4]. The master equation in the interaction picture then reads ∂ t ̺ I = 1 i H I , ̺ I − 4 j=3 2 k=1 γ jk 2 ̺ I A jk , A kj + H.c. ,(5) where ̺ I is the density matrix in the interaction picture and γ jk is the radiative decay rate of transition \|j ↔ \|k . For the further analysis we rewrite the master equation in a matrix-vector form. Because the trace of the density matrix is conserved we use the corresponding condition 4 j=1 ̺ I jj = 1 (6) to eliminate the diagonal element ̺ 44 . Here, ̺ I jk = j\|̺ I \|k . Introducing the vector R = (̺ I 11 , ̺ I 12 , ̺ I 13 , . . . , ̺ I 43 ) T containing the remaining fifteen elements of the density matrix we find ∂ t R + Σ = M R ,(7) with an inhomogeneous part Σ that stems from the elimination of ̺ 44 and a coefficient matrix M . Both Σ and M can be directly derived from the master Eq. (5) and contain the explicit time dependence arising from the time dependent Hamiltonian Eq. (3). The explicit form of M and Σ is given in the appendix. B. Time-Dependent Solution To treat the explicit time dependence of the equation of motion we first separate Σ and M into the time independent part and the explicitly time dependent part. For this, we define Σ =Σ 0 + Σ −1 Ω 41 e i(∆t−φ) + Σ 1 Ω 41 e −i(∆t−φ) ,(8a)M =M 0 + M −1 Ω 41 e i(∆t−φ) + M 1 Ω 41 e −i(∆t−φ) , (8b) with time-independent Σ j and M j (j ∈ {0, ±1}). We see that under the condition ∆ = 0 the explicit time dependence vanishes. This is the so-called multiphoton resonance condition. For fixed coupling field frequencies this condition can only be fulfilled for a single probe field detuning ∆ 41 . But we want to investigate probe fields consisting of pulses with finite temporal length, which due to the Fourier relations implies that a whole spectrum of probe field frequencies interacts with the medium at the same time. Thus, we cannot assume the multiphoton resonance condition to be fulfilled [8]. Instead, we have to solve Eq. (7) including the explicit time dependence. To do so, we expand R as a power series in Ω 41 , R = ∞ n=0 R n Ω n 41 .(9) If we assume that the probe field strength is small compared to the control fields this series will converge. Inserting Eqs. (8) and (9) in Eq. (7), we can derive equations of motion for the individual coefficients R n . In order O[Ω n 41 ] we find ∂ t R n =M 0 R n + δ n,1 Σ −1 e i(∆t−φ) + Σ 1 e −i(∆t−φ) + M −1 e i(∆t−φ) + M 1 e −i(∆t−φ) R n−1 . (10) This is an equation for R n where the coefficient matrix M 0 is time independent and only the inhomogeneous part is time dependent. This time dependence is twofold, first again explicitly because of the exponential functions and second because of the dependence on R n−1 . Thus, we make an ansatz for the solution and write R n in a Fourier series, R n = ∞ m=−∞ R (m) n e −im(∆t−φ) .(11) Projecting on the Fourier basis functions we derive a hierarchy of time independent equations for the coefficients R (m) n . Up to order O[Ω 3 41 ] we find R (0) 0 =M −1 0 Σ 0 , (12a) R (±1) 1 = (M 0 ± i∆1) −1 Σ ±1 − M ±1 R (0) 0 ,(12b)R (0) 2 = − M −1 0 M −1 R (1) 1 + M 1 R (−1) 1 , (12c) R (±2) 2 = − (M 0 ± 2i∆1) −1 M ±1 R (±1) 1 , (12d) R (±1) 3 = − (M 0 ± i∆1) −1 × M ±1 R (0) 2 + M ∓1 R (±2) 2 , (12e) R (±3) 3 = − (M 0 ± 3i∆1) −1 M ±1 R (±2) 2 ,(12f) where 1 is the unit matrix and all other R (m) n up to this order vanish. In general we find that R = ∞ n=0 n m=−n, −n+2,... R (m) n Ω n 41 e −im(∆t−φ) .(13) Since Fourier coefficients R n−1 of the next lower order, the full solution can be calculated recursively. C. Physical Interpretation To physically interpret the meaning of the different coefficients we study the influence of the different parts of the solution on the probe field. First, we write down the expansion series for the relevant probe field coherence in the Schrödinger picture ̺ 41 using the explicit transformation relation connecting the Schrödinger picture with our interaction picture. We find ̺ 41 =̺ I 41 e −i(ω41t−φ41) e i(∆t−φ) .(14) With ̺ I 41 given as component of the solution for R we find ̺ 41 = ∞ n=0 n m=−n, −n+2,... R (m) n 13 Ω n 41 × e −i[ω41+(m−1)∆]t e i[φ41+(m−1)φ] ,(15) where [R 13 gives a contribution at the probe field frequency ω 41 plus a frequency shift of (m − 1)∆. The corresponding physical process can be identified as follows. A combination of dipole phases φ = φ 41 −φ 42 +φ 32 −φ 31 indicates a full evolution through a loop which extends from state \|1 to \|4 and via \|2 and \|3 back to state \|1 . The transition direction is given by the sign of the corresponding dipole phase. The evolution around the interaction loop is also the physical reason for the frequency shift ∆ of such a process. Altogether, [R D. Linear and Non-Linear Susceptibility With the above interpretation we can easily identify the parts of the solution leading to the linear and nonlinear susceptibility in the probe field. Because both contributions should oscillate at the probe field frequency we see that m = 1 must be fulfilled in Eq. (15). The order of Ω 41 enables one to identify χ (1) (ω 41 ) ∝ R (1) 1 13 at O Ω 1 41 ,(16a)χ (3) (ω 41 ) ∝ R (1) 3 13 at O Ω 3 41 .(16b) There is no second order contribution to the susceptibility as it should be for an isotropic medium [18]. By comparing the microscopically calculated value for the polarization [4,5] P 41 =N (d 14 ̺ 41 + c.c.) ,(17) with the definition of the susceptibility [18] P 41 =ε 0 E 41 2 χ (1) + 3 4 E 2 41 χ (3) ê 41 e −iω41t + c.c. ,(18) we find χ (1) (ω 41 ) = 3 8π 2 λ 3 41 N γ 41 R (1) 1 13 ,(19)3 4 E 2 41 χ (3) (ω 41 ) = 3 8π 2 λ 3 41 N γ 41 Ω 2 41 R (1) 3 13 ,(20) with ε 0 being the permittivity of free space, λ 41 the wave length of the probe field transition, and N the density of atoms in the gas. We remark that χ (3) (ω 41 ) = χ (3) (ω = ω 41 −ω 41 +ω 41 ) is the lowest order nonlinear contribution at the probe field frequency. It leads to an intensity dependent refractive index that also depends on ω 41 and can be different for each respective frequency of the probe pulse spectrum. This is not the case for other contributions to χ (3) . For example, [R (0) 0 ] 13 oscillates at the frequency ω = ω 41 − ∆ and leads to a contribution χ (3) (ω = ω 31 − ω 32 + ω 42 ) (four-wave mixing). Here, the resulting frequency is independent of ω 41 . Nevertheless, in principle those processes can influence the result for the linear and thirdorder susceptibility at certain probe field frequencies. For example, light can be scattered into the probe field mode via different processes. Whether this or similar contributions change the probe pulse depends on the pulse's frequency width compared to the multiphoton detuning ∆ and more general also on the propagation direction of the probe field relative to the control fields. A definite answer to this question requires an analysis of the full pulse propagation dynamics through the medium which is beyond the scope of this work. E. Doppler Broadening A typical experimental setup to investigate the coherence properties of a laser driven atomic gas would be a gas cell with a dilute alkali-atom vapor. For a dilute atomic gas theoretical predictions for the linear and nonlinear susceptibility can be made on the basis of a single atom analysis. This greatly facilitates the theoretical analysis. However, in a dilute gas at room temperature or above the atoms move at velocities where the frequency shift due to Doppler effect cannot be neglected compared to the natural line width given by the radiative decay rate γ. To calculate the Doppler effect for a single field, we assume a Maxwell-Boltzmann velocity distribution in laser propagation direction with a most probable velocity given by [32] v m = 2k B T m (21) with k B the Boltzmann constant, T the temperature, and m the mass of the atom. The non-relativistic Doppler frequency shift is given by ω eff =ω 1 − v c ,(22) where ω eff is the shifted frequency seen by the moving atom, ω is the lab frame laser frequency, v is the velocity of the atom in laser propagation direction, and c is the speed of light. The Doppler shift effectively leads to an additional detuning ∆ Dop with a Gaussian distribution [32] f (∆ Dop ) d∆ Dop = 1 √ πkv m e − " ∆ Dop kvm " 2 d∆ Dop ,(23) where k is the wave number. The corresponding line width (FWHM) is then given by To actually calculate the linear and nonlinear susceptibility for a Doppler broadened medium, for each propagation direction, we have to add ∆ Dop to the detuning of the fields propagating in this direction and then average the resulting susceptibility over the velocity distribution Eq. (23). δω =k ln(2) 8k B T m .(24) F. Buffer Gas and Pressure Broadening Introducing a buffer gas to the gas cell leads to more frequent collisions between the atoms. This has two main consequences. First of all it causes pressure broadening. For moderate densities, a collision between two atoms disturbs the level energies for a short time which results in the loss of phase coherence. In a simple approach this can be modeled by an additional decay rate γ c for the coherences. This collisional decay rate consists of a contribution due to the studied gas itself and a contribution due to the buffer gas. Both depend linearly on the respective density [18], γ c =C s N s + C b N b ,(25) with gas specific constants C s and C b . A second major effect of a buffer gas is closely connected to Doppler broadening. Due to the higher density the mean free path of a single atom moving in the gas is reduced. If it is reduced below the transition wavelength an averaging over different velocities during a single emission or absorption process can effectively re-narrow a Doppler broadened line. This phenomenon is known as Dicke narrowing [31]. III. RESULTS In principle, Eqs. (12) can be used to calculate analytical results for the desired χ (1) and χ (3) . But in our situation of interest where all four electromagnetic fields, possibly all with different detuning, interact with the atom, these are usually to lengthy to give any physical insight. Therefore, we proceed with a numerical study of the linear and nonlinear susceptibility. A. Without Doppler Broadening Here, our primary goal is to find a set of parameters where the intensity dependent refractive index is large enough to cause an appreciable amount of nonlinear selfphase modulation while the attenuation of a light pulse due to absorption is small. To achieve a high non-linear index of refraction with low linear and non-linear loss all in the same spectral region is challenging because resonances that enhance the nonlinear response typically come with strong absorption. Still, we find such a suitable parameter set by manipulating the linear and nonlinear susceptibility of the probe field as described next. We first split the unperturbed resonance of the probe field transition by a strong coupling field Ω 42 and again about half as much by the second coupling field Ω 31 . This gives rise to four resonance structures in the linear response, see Fig. 2. In this figure, the linear absorption of the resonance at ∆ 41 ≈ −25γ can be lowered by a small detuning ∆ 31 , which modifies the dressed state populations. Finally, optimizing the result with the third coupling Ω 32 , we can tune one half of the resonance to a small linear and nonlinear absorption while still maintaining a substantial nonlinear real part. In Fig. 3 it is shown how gradually introducing a detuning ∆ 31 influences the linear absorption, the nonlinear gain, and the real part of the nonlinear susceptibility. It decreases the linear absorption and the nonlinear gain faster than the real part and thereby improves their ratio. Interestingly, the imaginary parts of the linear and the nonlinear parts of the susceptibility can have opposite signs in this spectral region . The linear response induces absorption, while the nonlinear response leads to gain. Absorption could in this spectral region therefore be reduced even further by a partial cancelling of linear absorption and nonlinear gain. However, these results are preliminary in the sense, that no effects due to Doppler and pressure broadening have been included yet. B. Including Doppler Broadening Using our considerations from Secs. II E and II F we now want to calculate the linear and nonlinear susceptibility in a Doppler broadened atomic gas. As a realistic example we want to assume a Sodium vapor with a To reach a vapor pressure that corresponds to this density the gas cell must be heated to a temperature of T = 547.6 K [33]. At this temperature the Doppler linewidth is δω = 2π×1.78 GHz which is very broad compared to the natural linewidth of the Sodium D 1 transition of γ = 2π×9.76 MHz. In a pure Sodium vapor the spectral features we found in Sec. III A would be averaged out by the Doppler effect. But if we introduce a buffer gas strong pressure broadening can preserve them. For Argon and Sodium, the gas parameters in Eq. (25) are given by C s = 1.50 × 10 −13 m 3 s −1 and C b = 2.53 × 10 −15 m 3 s −1 [18]. We want to assume a collision-induced coherence loss rate of γ c = 1.0 GHz which corresponds to a buffer gas density of N b = 3.95 × 10 23 m −3 . At such a density the mean free path is of order Λ = 10 −5 m. This is much larger than the transition wavelength λ = 589.2 × 10 −9 m such that the limit of Dicke narrowing is not reached. ∆ 41 /γ ∆ 41 /γ ∆ 41 /γ ∆ 41 /γ (a) (b) (c) (d) We now try to recover results similar to the unbroadened case shown in Fig. 3. Because of the strong broadening we have to apply correspondingly stronger control fields. For Ω 42 = 60.0 GHz and Ω 31 = 30.0 GHz, we find the resonance studied in the unbroadened case at around ∆ 41 = −15.0 GHz. The third control field is set to Ω 32 = 25.0 GHz and the detuning to ∆ 31 = 1.6 GHz. For the Doppler averaging we have assumed all fields to be co-propagating. The different subpanels in Fig. 4 correspond to different Doppler linewidths, and thus via Eq. (24) to different temperatures. In Fig. 4(a), the Doppler linewidth is chosen below the natural linewidth of the probe transition, and as expected we finds results that are similar in shape to the unbroadened case (see Fig. 3(d)). Differences are mainly due to pressure broadening. Gradually increasing the Doppler linewidth up to the full Doppler width expected for the gas parameters discussed above in subfigure (d), we find that while the shapes of the different curves change, our main result of high nonlinear index of refraction with small linear and non-linear absorption persists with Doppler broadening. Also in the broadened case, a partial cancelling of linear absorption and nonlinear gain could be possible. Note that since the averaging process affects not only the probe field detuning but all four detunings at the same time the results cannot be explained in terms of a simple smoothing of the curves without Doppler effect. We also considered different laser geometries, such as control fields propagating perpendicular to the probe field, or one or two control field propagating in opposite directions, and found the co-propagating case to be the most advantageous one. This is similar to the case of Doppler broadening in typical electromagnetically induced transparency setups where co-propagating lasers typically are preferable. We finally use our results at probe field detuning ∆ = −17.8 GHz to calculate the required optical length for a nonlinear selfphase modulation of π. This probe field frequency is indicated by the vertical blue dotted line in Fig. 4(d). The nonlinear selfphase modulation is given by [18] ∆Φ Nl =n 2 I k L , with n 2 the intensity dependent refractive index, I the probe field intensity, k the wavevector, and L the optical length. We assume a probe field strength one tenth of the smallest control field and find L π =2.9 cm . From Fig. 4 (d) we see that the magnitude of the imaginary parts of the linear and nonlinear susceptibility are more than one order of magnitude smaller. Therefore, the equivalent characteristic length scale is more than one order of magnitude larger. Furthermore, both parts give rise to small gain rather than absorption. Thus, our results show, that in a certain spectral region a nonlinear selfphase modulation of π can be achieved on a realistic laboratory lengthscale. Since the real part of both the linear and the nonlinear susceptibility display approximately linear dispersion in the spectral region of interest, pulse shape distortions can be expected to be small. Interestingly, the real part of the linear susceptibility has a negative slope in the considered frequency region, in contrast to a positive slope typically found in an electromagnetically induced transparency window. IV. CONCLUSION We have studied nonlinear effects in pulse propagation through a laser-driven medium where the applied fields form a closed interaction loop. Such loop systems in general only allow for a time-independent treatment at a single probe field frequency, where the so-called multiphoton resonance condition is fulfilled. As a probe field pulse has a finite frequency width, this condition which allows for a straightforward theoretical treatment could not be applied. Instead, we treated the time-dependent problem by turning it into a hierarchy of equations that describe the various physical processes occurring in the medium. We have included Doppler and pressure broadening as well as a buffer gas in our analysis and have used realistic parameters for a medium consisting of Sodium vapor. We could show that the studied system can exhibit a high non-linear refractive index with small absorption or gain over a spectral range of several natural line widths. For the chosen parameters, both the linear and the non-linear susceptibilities show near-linear dispersion such that pulse shape distortions are minimized, and the slope of the linear dispersion is negative. A nonlinear selfphase modulation of π is obtained after 2.9 cm propagation through the medium. 13 represents a process with m − 1 loop cycles where the sign of m−1 defines the direction, clockwise for positive or counter-clockwise for negative sign. The remaining n − (m − 1) probe transitions can be interpreted as direct transitions. FIG. 2 : 2(Color online) Real part (solid blue line) and imaginary part (dashed red line) of the linear susceptibility of the probe field. Due to strong control fields Ω42 = 100γ and Ω31 = 50γ the probe field resonance is split into four different resonances. Further, Ω32 = ∆31 = ∆32 = ∆42 = 0, and all spontaneous decay rates γ jk have been set to γ. The susceptibility is plotted in units of 3/8π 2 λ 3 41 N . FIG. 3 : 3(Color online) Real part (dash-dotted blue line) and imaginary part (solid red line) of the nonlinear susceptibility together with the imaginary part of the linear susceptibility (dashed red line). All figures show the resonance around ∆41 = −25γ. The susceptibility is plotted in units of 3/8π 2 λ 3 41 N and for comparability χ (3) has been scaled with 3/4E 2 41 . The parameters are ∆32 = ∆42 = 0, Ω31 = 50γ, Ω32 = 34γ, and Ω42 = 100γ. The probe field strength is assumed to be one tenth of the weakest control field in all cases. The detuning ∆31 is chosen as (a) ∆31 = 0, (b) ∆31 = 0.7γ, (c) ∆31 = 1.5γ, and (d) ∆31 = 1.7γ. Note the different axis scales in the four subpanels. density of N = 1.0 × 10 20 m −3 . FIG. 4 : 4(Color online) Real part (dash-dotted blue line) and imaginary part (solid red line) of the nonlinear susceptibility together with the real part (blue dotted line) and the imaginary part of the linear susceptibility (dashed red line) at the resonance around ∆41 = −15.0 GHz. The control fields have Rabi frequencies Ω42 = 60 GHz, Ω31 = 30 GHz, Ω32 = 25 GHz, and the detunings are ∆31 = 1.6 GHz, ∆32 = ∆42 = 0. The medium parameters described in the main text correspond to Sodium as the active medium with Argon as a buffer gas. The four different plots show Doppler averaged results with a Doppler linewidth of (a) below the natural linewidth, (b) 50%, (c) 90%, and (d) 100% of the full Doppler linewidth of δω = 2π × 1.78 GHz. 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[ "Mass Effect on Axial Charge Dynamics", "Mass Effect on Axial Charge Dynamics" ]	[ "Er-Dong Guo \nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nKavli Institute of Theoretical Physics China\nChinese Academy of Sciences\n100190BeijingChina\n", "Shu Lin [email protected] \nInstitute of Astronomy and Space Sciences\nSun Yat-Sen University\nNo 135 Xingang Xi Rd510275GuangzhouChina\n" ]	[ "Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina", "Kavli Institute of Theoretical Physics China\nChinese Academy of Sciences\n100190BeijingChina", "Institute of Astronomy and Space Sciences\nSun Yat-Sen University\nNo 135 Xingang Xi Rd510275GuangzhouChina" ]	[]	We studied effect of finite quark mass on the dynamics of axial charge using the D3/D7 model in holography. The mass term in axial anomaly equation affects both the fluctuation (generation) and dissipation of axial charge. We studied the dependence of the effect on quark mass and external magnetic field. For axial charge generation, we calculated the mass diffusion rate, which characterizes the helicity flipping rate.The rate is a non-monotonous function of mass and can be significantly enhanced by the magnetic field. The diffusive behavior is also related to a divergent susceptibility of axial charge. For axial charge dissipation, we found that in the long time limit, the mass term dissipates all the charge effectively generated by parallel electric and magnetic fields. The result is consistent with a relaxation time approximation. The rate of dissipation through mass term is a monotonous increasing function of both quark mass and magnetic field. * [email protected] †	10.1103/physrevd.93.105001	[ "https://arxiv.org/pdf/1602.03952v2.pdf" ]	119,297,720	1602.03952	c06e27882f889947a959e6482efe5bc61e888e2c	Mass Effect on Axial Charge Dynamics 4 Jun 2016 June 7, 2016 Er-Dong Guo Institute of Theoretical Physics State Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina Kavli Institute of Theoretical Physics China Chinese Academy of Sciences 100190BeijingChina Shu Lin [email protected] Institute of Astronomy and Space Sciences Sun Yat-Sen University No 135 Xingang Xi Rd510275GuangzhouChina Mass Effect on Axial Charge Dynamics 4 Jun 2016 June 7, 20161 We studied effect of finite quark mass on the dynamics of axial charge using the D3/D7 model in holography. The mass term in axial anomaly equation affects both the fluctuation (generation) and dissipation of axial charge. We studied the dependence of the effect on quark mass and external magnetic field. For axial charge generation, we calculated the mass diffusion rate, which characterizes the helicity flipping rate.The rate is a non-monotonous function of mass and can be significantly enhanced by the magnetic field. The diffusive behavior is also related to a divergent susceptibility of axial charge. For axial charge dissipation, we found that in the long time limit, the mass term dissipates all the charge effectively generated by parallel electric and magnetic fields. The result is consistent with a relaxation time approximation. The rate of dissipation through mass term is a monotonous increasing function of both quark mass and magnetic field. * [email protected] † Introduction It is believed that parity odd domains with chiral imbalance are produced in finite temperature quark-gluon plasma (QGP). Their presence can be detected via axial anomaly as chiral magnetic effect (CME) [1][2][3][4] and chiral magnetic wave (CMW) [5,6] in heavy ion collisions, see [7][8][9] for recent reviews. The former leads to the generation of vector current along the direction of external magnetic field: j V = N c e 2π 2 µ 5 B,(1) where µ 5 is the axial chemical potential characterizing the chiral imbalance. The latter leads to the propagation of axial and vector charges along the direction of external magnetic field. Analogous effects exist when the magnetic field is replaced by vorticity of QGP [10,11]. These effects are being intensively searched for in heavy ion collision experiments in recent years [12][13][14]. Theoretical descriptions of CME and CMW have been developed in different frameworks including hydrodynamics [10,[15][16][17][18] and kinetic theory [19][20][21][22][23][24][25] etc. Most frameworks assume quarks being massless, see exception for example in [26,27]. While it is known that finite quark mass does not modify CME, we do expect quark mass to have imprints on the dynamics of axial charge. Naively, if the mass of one quark flavor is much larger than the temperature of QGP, that quark flavor decouples from axial current. We would like to ask quantitative questions on the mass effect on dynamics of axial charge. This is relevant in reality because the mass of strange quark is comparable to the temperature of QGP created at relativistic heavy ion collider (RHIC) and large hadron collider (LHC). With the inclusion of mass term, the axial anomaly equation reads ∂ µ j µ 5 = 2imψγ 5 ψ − e 2 16π 2 ǫ µνρσ F µν F ρσ − g 2 16π 2 trǫ µνρσ G µν G ρσ ,(2) where the three terms on the right hand side (RHS) corresponds to mass term, QED anomaly term and QCD anomaly term respectively. (2) is written for one flavor of quark with mass m. All three terms lead to modification of axial charge dynamics. The effect of QED anomaly term is extensively studied in the above mentioned references. The effect of QCD anomaly was studied recently [28][29][30][31]. In this work, we will focus on the effect of the mass term. On one hand, finite quark mass explicitly breaks axial symmetry, offering a mechanism of axial charge generation. We find that the mass operator diffuses at low frequency the same way as the Chern-Simon (CS) number. The diffusion of the CS number is known to generate axial charge. The same is true for the mass operator. We calculate the diffusion rate of mass term as a measure of axial charge generation. We also define a dynamical susceptibility by CME, and find it to be divergent in the low frequency limit. We explain the common physical reason for the diffusive mass operator and the divergent susceptibility. On the other hand, finite quark mass also leads to axial charge dissipation. The dissipation effect is studied recently in [32,33] in a relaxation time approximation. We will discuss axial charge dissipation in an indirect way: we set up parallel electric and magnetic field and measure the rate of dissipation through the mass term. The situation is further complicated by the existence of a reservoir of adjoint matter, to which axial charge can dissipate. By taking into account the additional loss rate, we find that the axial charge dissipates entirely in the long time limit, which is consistent with the relaxation time approximation. We will study these effects as a function of both quark mass and external magnetic field using a holographic model. The paper is organized as follows: In Sec II we give a self-contained review of the holographic model. In Sec III we discuss separately mass effect on axial charge generation and dissipation, which we coined mass diffusion rate and mass dissipation effect respectively. We summarize the results in Sec IV. We collect technical details in obtaining phase diagram and hydrodynamic solutions in two appendices. and N = 2 fields are in the adjoint and fundamental representations of the SU (N c ) group respectively. By analogy with QCD, we will loosely refer to the N = 4 and N = 2 fields as gluons and quarks respectively. A detailed account of field content can be found in [34]. The N = 4 theory has a SO(6) R global symmetry, which is broken by the N = 2 theory to SO(4) × U (1) R . As we will see, the U (1) R symmetry is anomalous. We will identify it with axial symmetry. We start with the finite temperature black hole background of D3 branes following the notations of [35]: ds 2 = g tt dt 2 + g xx d x 2 + g ρρ dρ 2 + g θθ dθ 2 + g φφ dφ 2 + g SS dΩ 2 3 , = − r 2 0 2 f 2 H ρ 2 dt 2 + r 2 0 2 Hρ 2 dx 2 + dρ 2 ρ 2 + dθ 2 + sin 2 θdφ 2 + cos 2 θdΩ 2 3 .(3) where f = 1 − 1 ρ 4 , H = 1 + 1 ρ 4 .(4) The temperature is fixed by T = r 0 /π. Note that we have factorized S 5 into S 3 and two additional angular coordinates θ and φ, which makes the breaking of global symmetry SO(6) R → SO(4) × U (1) R manifest. There is a nontrivial background Ramond-Ramond form C 4 = r 2 0 2 ρ 2 H 2 dt∧dx 1 ∧dx 2 ∧dx 3 − cos 4 θdφ∧dΩ 3 .(5) In the probe limit N f /N c ≪ 1, the D7 branes do not backreact on the background of the D3 branes. This corresponds to the quenched limit of QCD. The D3 and D7 branes occupy the following dimensions. x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 D3 × × × × D7 × × × × × × × ×(6) The D3 and D7 branes are separated in the x 8 -x 9 plane. Using translation symmetry, we put D3 branes at the origin of the plane and parameterize the position of D7 branes by radius ρ cos θ and polar angle φ. The D7 branes have rotational symmetry in the x 8 -x 9 plane, corresponding to U (1) R symmetry in the dual field theory. We use the symmetry to choose φ = 0. The embedding function θ(ρ) of D7 branes in D3 background is determined by minimizing the action including a DBI term and WZ term S D7 = S DBI + S W Z , S DBI = −N f T D7 d 8 ξ −det g ab + (2πα ′ )F ab , S W Z = 1 2 N f T D7 (2πα ′ ) 2 P [C 4 ]∧F ∧F .(7) Here T D7 is the D7 brane tension. g ab andF ab are the induced metric and worldvolume field strength respectively. Defining F ab = (2πα ′ )F ab , N = N f T D7 2π 2 = N f N c (2π) 4 ,(8) we simplify the action to S DBI = − N 2π 2 d 8 ξ −det (g ab + F ab ), S W Z = 1 4π 2 N P [C 4 ]∧F ∧F.(9) The mass of the quark is realized as the separation of the D7 branes from the D3 branes at infinity. Explicitly, the mass M is determined from the asymptotic behavior of θ: sin θ = m ρ + c ρ 3 + · · · .(10) with M = r 0 m. We will turn on a constant magnetic field, which amounts to including worldvolume magnetic field in D7 branes. There are two possible embeddings with D7 branes crossing/not crossing the black hole horizon, corresponding to meson melting/mesonic phase respectively [35][36][37][38]. Using t, x, ρ and angular coordinates on S 3 as worldvolume coordinates, the induced metric is given by ds 2 ind = − r 2 0 2 f 2 H ρ 2 dt 2 + r 2 0 2 Hρ 2 d x 2 + 1 ρ 2 + θ ′ (ρ) 2 dρ 2 + cos 2 θdΩ 2 3 .(11) We also turn on a constant magnetic field in z-direction: F xy = B, the action of D7 branes can be written as S DBI = −N dρ r 2 0 2 2 f Hρ 3 1 + ρ 2 θ ′2 1 + 2B 2 r 2 0 Hρ 2 cos 3 θ,(12) with a vanishing WZ term. The phase diagram in the m-B plane has been obtained in [37,38]. We reproduce the result in appendix A and show the result at fixed temperature in Figure. Fluctuations and realization of axial anomaly We consider the fluctuation of embedding function φ and worldvolume gauge field A M . The quadratic action can be written in the following compact form S = N d 5 x − 1 2 √ −GG M N ∂ M φ∂ N φ − 1 4 √ −HF 2 − N κ d 5 xΩǫ M N P QR F M N F P Q ∂ R φ,(13) where M = t, x 1 , x 2 , x 3 , ρ. The EOM of φ is given by δS δφ − ∂ M δS δ∂ M φ = 0.(14) Since φ is a phase, only its derivative enters the action, we have from (14), ∂ µ δS δ∂ µ φ + ∂ ρ δS δ∂ ρ φ = 0,(15)with µ = t, x 1 , x 2 , x 3 . Defining J µ R = dρ δS δ∂µφ , we obtain ∂ µ J µ R + δS δ∂ ρ φ \| ∞ ρ=ρ h = 0.(16) We will identify J R as the axial current. The non-conservation of J R follows from two boundary terms in the integration. The boundary term at the horizon ρ = ρ h indicates axial charge exchange between D7 branes and D3 branes. It is pointed out in [34] that this term represents leakage of R-charge from fundamental sector to adjoint sector as fields in both sectors are charged under the U (1) R symmetry. The other boundary term at ρ = ∞ can be related to axial anomaly: O φ ≡ − δS δ∂ ρ φ \| ρ=∞ = − δS ∂ δφ(ρ → ∞) ,(17) where we have used the defining property of on-shell action S ∂ . For action (13), we have O φ = N √ −GG M ρ ∂ M φ\| ρ=∞ + κN Ωǫ M N P Q F M N F P Q \| ρ=∞ .(18) For our model, √ −GG M N = √ −hg M N g φφ , √ −H = √ −h, Ω = cos 4 θ, κ = 1 8 ,(19) with h to be defined in the next section. Field theory analysis shows that [34] 1 O φ = miψγ 5 ψ + · · · + N E · B,(20) where · · · represents contribution from supersymmetric partners. Noting that θ → 0 as ρ → ∞, we readily identify the second term in (18) with the last term in (20). The remaining term in (18) can then be identified with the mass term in (20). For convenience, we define the remaining term by O η = N √ −GG M ρ ∂ M φ\| ρ=∞ .(21) We have thus holographically split O φ into the mass term O η and anomaly term N E · B, which represent respectively explicit and anomalous breaking of axial symmetry: ∂ µ J µ R = O φ = O η + N E · B.(22) Finite Quark Mass Effect We will study two aspects of finite quark mass effect: i, the mass term, similar to QCD anomaly term, has diffusive behavior at low frequency. This gives rise to fluctuation (random walk behavior) of axial charge. The rate of diffusion, to be referred to as mass diffusion rate, determines the rate of axial charge generation; ii, in the presence of nonvanishing E · B, net axial charge would be produced. However, the axial charge dissipates due to finite quark mass, resulting in a reduced rate of axial charge generation. We will refer to 1 Note that we have φ = 0, thus no axial chemical potential is introduced. this as mass dissipation effect. The above effects are captured by correlators of J z and O η . The mass diffusion rate involves the correlator of O η itself, while the mass dissipation effect involves the correlator between J z and O η . We stress that J z is the vector current coupled to boundary gauge field A z . In holographic formulation, we need to study the fluctuation of bulk fields A z and φ, which are dual to J z and O η (O φ ). For our purpose, it is sufficient to turn on homogeneous (in both x and S 3 ) fluctuation of A z (t, ρ) and φ(t, ρ). The fluctuation leads to the following modification of the following quantities ds 2 ind = g tt + g φφφ 2 dt 2 + g xx dx 2 + g ρρ + g θθ θ ′2 + g φφ φ ′2 dρ 2 + g 2 SS dΩ 2 3 + 2g φφφ φ ′ dtdρ, δF =Ȧ z dt∧dz + A ′ z dρ∧dz, δP [C 4 ] = − cos 4 θ φ dt + φ ′ dρ ∧dΩ 3 .(23) With (23), we can write down the quadratic action of A z (t, ρ) and φ(t, ρ): S DBI + S WZ = −N dtd 3 xdρ 1 2 √ −h g tt g φφφ 2 + g ρρ g φφ φ ′2 + g tt g xxȦ2 z + g ρρ g xx A ′ z 2 + cos 4 θB φ ′Ȧ z −φA ′ z ,(24) where we have defined √ −h = −g tt g xx (g 2 xx + B 2 ) (1 + ρ 2 θ ′2 ) g ρρ g 3 SS .(25) Variation with respect to the fluctuations gives both the EOM and the on-shell action δS = −N dtd 3 xdρ √ −h× −∂ t (g tt g φφφ )δφ − ∂ ρ (g ρρ g φφ φ ′ )δφ − ∂ t (g tt g xxȦ z )δA z − ∂ ρ (g ρρ g xx A ′ z )δA z − ∂ ρ (cos 4 θBȦ z )δφ + ∂ t (cos 4 θBA ′ z )δφ − ∂ t (cos 4 θBφ ′ )δA z + ρ(cos 4 θBφ)δA z − N dtd 3 x √ −h g ρρ g φφ φ ′ δφ + g ρρ g xx A ′ z δA z + cos 4 θB(Ȧ z δφ −φδA z ) .(26) Working with a single Fourier mode e −iωt , we obtain the EOM ω 2 √ −hg tt g φφ φ − ∂ ρ √ −hg ρρ g φφ φ ′ − B∂ ρ (cos 4 θ)A z (−iω) = 0, ω 2 √ −hg tt g xx A z − ∂ ρ √ −hg ρρ g xx A ′ z + B∂ ρ (cos 4 θ)φ(−iω) = 0.(27) The asymptotic expansion of φ and A z can be determined from EOM: φ = f 0 + f 1 ρ 2 + f h ρ 2 ln ρ + · · · , A z = a 0 + a 1 ρ 2 + a h ρ 2 ln ρ + · · · .(28) f 0 and a 0 correspond to sources coupled to O φ and J z . The coefficients of the logarithmic terms correspond to counter terms 2 : f h = ω 2 r 2 0 f 0 , a h = ω 2 r 2 0 a 0 .(29) The vevs of O φ and J z are determined by O φ = δS ∂ δφ(ρ → ∞) = −N √ −hg ρρ g φφ φ ′ − N cos 4 θBȦ z \| ρ→∞ = 2N r 2 0 2 2 m 2 f 1 − N Ba 0 (−iω), J z = δS ∂ δA z (ρ → ∞) = −N √ −hg ρρ g xx A ′ z + N cos 4 θBφ \| ρ→∞ = 2N r 2 0 2 a 1 + N Bf 0 (−iω),(30) where we have used ρ sin θ\| ρ→∞ = m according to (10). Comparing (21) and (30), we arrive at the following dictionary O η = 2N r 2 0 2 2 m 2 f 1 .(31) Mass diffusion rate and susceptibility The mass operator O η can lead to fluctuation of axial charge. It is well known that the origin of axial charge fluctuation from QCD anomaly is topological transitions. The counterpart for O η is helicity flipping from elementary scattering [39,40]. The rate of axial charge generation in case of topological transition is given by CS diffusion rate. Similarly, the corresponding rate in case of helicity flipping is given by the diffusion rate of O η , which we calculate below. The diffusion rate of O η is encoded in the low frequency limit of retarded correlator. To calculate the retarded correlator, we need to turn on source for φ while keeping A z vanish on the boundary. Both φ and A z satisfy infalling wave condition on the horizon. It follows from (31) that the retarded correlator is given by G ηη (ω) = dt [O η (t), O η (0)] Θ(t)e iωt = −2N r 2 0 2 2 m 2 f 1 f 0 .(32) The diffusion rate is defined by For the case B = 0, there is no mixing between φ and A z . We can simply use φ (0) (54) in appendix B: Γ m = lim ω→0 2iT ω G ηη (ω).(33)φ (0) (ρ) = 1 − iω 2r 0 ρ 1 dρ ′    r 2 0 2 2 8 cos 3 θ h sin 2 θ h −h(ρ ′ )g ρρ (ρ ′ )g φφ (ρ ′ ) − 1 ρ ′ − 1    + ln(ρ − 1) .(34) This gives the following retarded correlator of O η : G ηη (ω) = −2N r 2 0 2 2 iω 4r 0 8 cos 3 θ h sin 2 θ h .(35) (35) gives a mass diffusion rate Γ m as analog of CS diffusion rate: Γ m = N π r 2 0 2 2 8 cos 3 θ h sin 2 θ h .(36) The dependence on m is encoded in the combination of trigonometric functions, which clearly indicates an upper bound of the mass diffusion rate. We also extract Γ m using (33) with numerical solutions for general B and m in the meson melting phase. We plot numerical results of Γ m as a function of m 2 for different values of B in Figure. 2. The case B = 0 agrees well with analytic expression (35). We find the mass diffusion rate is a non-monotonous function of m. This is not difficult to understand: in the limit m → 0, Γ m obviously should vanish as O η ∼ m. When m approaches the phase boundary between meson melting phase and mesonic phase, we also expect helicity flipping to freeze due to formation of meson bound states. In between, there must be a maximum for Γ m . Furthermore, the linear behavior of Γ m -m 2 plot in small m region supports the scaling Γ m ∼ m 2 , which is consistent with field theory expectation. The B dependence is more interesting: Γ m shows rapid growth with B. The presence of B enhances the diffusion, which cannot be explained as the increase of effective mass. The enhancement of helicity flipping might provide a way to generate axial charge more efficiently. It is worth mentioning that an enhancement of CS diffusion rate due to magnetic field was also obtained in [41,42]. We would like to comment on the diffusive behavior of O η . On general ground, the mass diffusion effect leads to accumulation of axial charge, which prevents its further generation. It would lead to modification of the long time (low frequency) behavior of G R ηη . However, this does not happen due to the existence of the adjoint reservoir. The generated axial charge entirely dissipates to the adjoint reservoir. To see that, we compare O η and O loss , which are the same quantity below evaluated at ρ = ∞ and ρ = 1 respectively. N √ −hg ρρ ρ φφ φ ′ .(37) It follows from the EOM (27) that the above quantity is constant in the limit ω → 0, ∂ ρ N √ −hg ρρ ρ φφ φ ′ = 0,(38) meaning that the generated charge is entirely balanced by the loss to the reservoir. Consequently, the low frequency behavior of O η correlator is still diffusive. Turing on the source for O η also allows us to study the susceptibility of axial charge. In the presence of finite quark mass, the axial charge is not even approximately conserved, making the susceptibility a subtle concept. Following [31], we can use CME to define a dynamical susceptibility χ. In the present model, it is given by χ = n 5 µ 5 = N Bn 5 J z .(39) We need to calculate both n 5 and J z from response to source for O φ in the hydrodynamic limit. n 5 is essentially known already. Denoting the source by f m , we can express n 5 as −iωn 5 (ω) = O η (ω) = −G R ηη (ω)f m (ω) ∼ O(ω)f m (ω).(40) Therefore we obtain n 5 ∼ O(ω 0 )f m (ω). On the other hand, J z is calculated using the dictionary (30). It is generated through the mixing between φ and A z . J z is also expressible as response to f m J z (ω) = −G R jη (ω)f m (ω).(41) there are two contributions to G R jη , both of which are of order O(ωB). Therefore we have J z ∼ O(ωB)f m (ω). Plugging the above qualitative results into (39), we obtain χ ∼ O(ω −1 ).(42) It simply means that the susceptibility is divergent in the static limit ω → 0. Recalling that the susceptibility is well-defined in the massless limit, we arrive at the non-commutativity of the limits m → 0 and ω → 0. The physical reason for divergent susceptibility is not difficult to understand. On on hand, the mass diffusion effect can spontaneously generate axial charge density at the cost of no energy. On the other hand, as we have seen already, the adjoint reservoir is a perfect sink for axial charge in the flavor sector, preventing accumulation of axial charge. Consequently, the axial charge can be continuously generated in the flavor sector. Note that the situation is different in case of axial charge generation by QCD anomaly. There the breaking of axial symmetry is suppressed by 1/N c (or the quenched limit), resulting in a finite dynamical susceptibility. Mass dissipation effect To study the mass dissipation effect, we turn on an electric field in z-direction by a time dependent A z on the boundary. We do not need to source φ on the boundary. Its profile is entirely generated via mixing of A z and φ in the bulk. The resulting O η from nontrivial profile of φ corresponds to the mass dissipation effect we are after. We also impose infalling wave boundary condition for φ and A z since we are interested in calculating response. We define the dimensionless mass dissipation rate r = O η N E · B .(43) The rate is a function of ω, m and B. In the hydrodynamic limit ω → 0, we can show that r(ω → 0) is a real function of m and B. In fact it can be related to embedding function for given m and B in the meson melting phase. To obtain r(ω → 0) analytically, we need to solve the coupled EOM (27) in the hydrodynamic limit. The hydrodynamic solutions can be found in appendix B. We simply quote the results here. The leading nontrivial order is the zeroth order for A z and the first order for φ: where θ h is the value of θ on the horizon, which needs to be obtained from numerical embedding function for given m and B. For φ (1) , we only retain its asymptotic behavior relevant for extracting O η . (44) leads to the following rate A (0) z = a 0 , φ (1) = 1 − cos 4 θ h Biωa 0 r 2 0 2 2 m 2 (−2) ρ −2 + · · · ,(44)r = 1 − cos 4 θ h .(45) We also study the rate of dissipation by numerical solutions. In practice, we generate two independent infalling numerical solutions at the horizon and use their linear combination to construct the solution with desired boundary condition. We show m-dependence of r in the limit B = 0 and B-dependence of r at different values of m in Figure 3 and Figure 4. We find good agreement with analytic expression (45). On general ground, we expect the rate to be a monotonous increasing function of m. In particular, r → 0 as m → 0. Indeed, this is confirmed in Fig. 3. We further note that the effect of B enhances the dissipation on top of the mass effect in Fig. 4. The physical interpretation of the dissipation rate r turns out to be a subtle question. Recalling the axial anomaly equation (22), we would draw the following conclusion: for every one unit of axial charge generated by parallel electric and magnetic field, r unit of it dissipates through the mass term, with a unit of 1 − r axial charge remaining. The remaining axial charge survives even in the hydrodynamic limit since we have ω → 0. This is not true because we have ignored a third source of axial charge dissipation, i.e. loss to the adjoint reservoir. The anomaly equation (22) should be supplemented by the loss rate with the explicit form of loss rate given by ∂ µ J µ R = O η + N E · B − O loss ,(46)O loss = N √ −hg ρρ r φφ ∂ ρ φ\| ρ→1 + N ΩE · B\| ρ→1 .(47) It is known that the loss rate can be IR unsafe [43]. Indeed, plugging in the hydrodynamic solution A (0) z and φ (1) in appendix B into (46), we find both terms becomes infinitely oscillatory as ρ → 1. Nevertheless we can still extract useful information by taking the hydrodynamic limit ω → 0 before the IR limit ρ → 1. Using this regularization we find the N ΩE · B term becomes N cos 4 θ h E · B, while the other term is higher order in ω. We immediately note that N cos 4 θ h E · B is precisely the 1 − r unit of axial charge. Subtracting the charge loss to the reservoir, we find only r unit of axial charge is effectively generated in the flavor sector by parallel electric and magnetic field. All dissipates by the mass term. This simply means no axial charge survives in the hydrodynamic limit. After clarifying the role of axial charge loss to adjoint reservoir, we should interpret r as a measure of mass dissipation effect compared to dissipation to the adjoint reservoir. The dissipation through mass term is favored at large m and B. The statement on the non-survival of axial charge can receive correction higher order in ω, which quantifies the charge survival rate. We can compare with the relaxation time approximation employed in [32,33], in which the following form of axial anomaly equation is assumed (here we use rN E · B for effectively axial charge generation) ∂ t n 5 = − n 5 τ + rN E · B,(48) with τ being the relaxation time. Physically it means the presence of axial charge n 5 induces O η = − n 5 τ . Plugging it into (48), we can solve for O η in frequency space O η = − rN E · B 1 − iωτ .(49) The leading order result O η = −rN E · B corresponds to our result of full dissipation. In principle, by going to high order in ω, we could calculate the relaxation time τ . We will not attempt it in this paper. Summary We Γ m ∼ m 2 F (B),(50) with F (B) a rapid growing function in the meson melting phase. We also defined a dynamical susceptibility of axial charge using CME. We found the susceptibility to be divergent in the static limit ω → 0. It is due to two reasons: i, spontaneous generation of axial charge by mass diffusion effect; ii, continuous leakage of axial charge from flavor sector to the adjoint sector, preventing the accumulation of axial charge. For axial charge dissipation, we found that a mass term is induced in the presence of parallel electric and magnetic fields, reducing the generation of axial charge. After carefully subtracting the axial charge loss rate to the adjoint sector, we found that the axial charge dissipate entirely through the mass term in the long time limit. To the order we consider, it is consistent with a relaxation time approximation. Acknowledgments We thank K. Landsteiner and Y. Yin for critical comments on an early version of the paper. We also thank D. Kharzeev It is known that D3/D7 brane system as a model for finite temperature QGP in the quenched limit has a first order phase transition [35][36][37][38]. At large quark mass and strong magnetic field (with temperature fixed), the probe D7 brane lie outside of the black hole horizon. In this phase the meson stays in bound state and its spectrum possesses a mass gap. At small quark mass and weak magnetic field, the brane crosses the horizon and this corresponds to the meson melting phase. The case in which the D7 branes touch the horizon corresponds to critical embedding, giving rise to critical mass and condensate. The embeddings close to the critical embedding show oscillatory behavior for the corresponding mass and condensate parameters around their critical values. This implies that the condensate is a multivalued function of mass, corresponding to different states. The true ground state is determined by the embedding that minimizes the free energy. Denoting χ = sin θ, we can rewrite the action (12) as S DBI = −N dρ (1 − ρ 4 )(1 − χ 2 ) (1 − χ 2 + ρ 2 χ ′2 ) (1 + (2 + 4B 2 )ρ 4 + ρ 8 ) 4ρ 5 ,(51) where we have set r 0 = 1, which amounts to fixing the temperature T = 1 π . The EOM following from (51) is solved by numerical integration of the EOM. The black hole embedding and Minkowski embedding satisfy different boundary conditions. For black hole embedding, the boundary condition is χ(ρ = 1) = χ 0 , χ ′ (ρ = 1) = 0, with integration domain from ρ = 1 to ρ = ρ max . For Minkowski embedding, the boundary condition is χ(ρ = ρ min ) = 1, χ ′ (ρ = ρ min ) = (1−ρ 4 )(1+(2+4B 2 )ρ 4 +ρ 8 ) ρ(1+ρ 4 )(1+2B 2 ρ 4 +ρ 8 ) , with integration domain from ρ = ρ min > 1 to ρ = ρ max . The initial condition for the derivative is chosen such that χ ′′ (ρ = ρ min ) can be uniquely determined by EOM. In practice, we start the integration at ρ = 1 + ǫ for black hole embedding and ρ = ρ min + ǫ for Minkowski embedding. We note that the free energy F = T S contains a UV divergence and therefore needs to be renormalized. Following [35], we add to the action the counter term S counter = − N 4 (ρ 2 max − m 2 ) 2 − 4mc + B 2 2 ln ρ max .(52) Note the appearance of a new term due to magnetic field as compared to [35]. The renormalized action S = S DBI + S counter is finite as we take ρ max → ∞. embedding is possible. Below B c , metastable phases of black hole embedding are found as we increases B. We illustrate the structure of metastable phases in Figure 5. B Hydrodynamic solution of fluctuations We wish to solve (27) in the hydrodynamic limit. We reproduce (27) for convenience. ω 2 √ −hg tt g φφ φ − ∂ ρ √ −hg ρρ g φφ φ ′ − B∂ ρ (cos 4 θ)A z (−iω) = 0, ω 2 √ −hg tt g xx A z − ∂ ρ √ −hg ρρ g xx A ′ z + B∂ ρ (cos 4 θ)φ(−iω) = 0.(53) For pedagogical reason, we work in the small B limit and solve (53) order by order in B. Since correction to φ (0) (ρ) = 1 − iω 2r 0 ρ 1 dρ ′    r 2 0 2 2 8 cos 3 θ h sin 2 θ h −h(ρ ′ )g ρρ (ρ ′ )g φφ (ρ ′ ) − 1 ρ ′ − 1    + ln(ρ − 1) , A (0) z (ρ) = 1 − iω 2r 0 ρ 1 dρ ′ r 2 0 2 4 cos 3 θ h −h(ρ ′ )g ρρ (ρ ′ )g xx (ρ ′ ) − 1 ρ ′ − 1 + ln(ρ − 1) .(54) We have chosen a specific normalization for the homogeneous solutions. At order O(B), we need to solve the inhomogeneous equations sourced by the mixing terms. This can be achieved by using Green's function for φ and A z , which are defined by ∂ 2 ρ G φ (ρ, ρ ′ ) + ∂ ρ G φ (ρ, ρ ′ )∂ ρ ln √ −hg ρρ g φφ − ω 2 g tt g ρρ G φ (ρ, ρ ′ ) = δ(ρ − ρ ′ ), ∂ 2 ρ G A (ρ, ρ ′ ) + ∂ ρ G A (ρ, ρ ′ )∂ ρ ln √ −hg ρρ g xx − ω 2 g tt g ρρ G A (ρ, ρ ′ ) = δ(ρ − ρ ′ ).(55) We require that the inhomogeneous solutions satisfy the infalling wave condition on the horizon and vanish on the boundary. It is convenient to construct the Green's function using two independent solutions satisfying the above boundary conditions. We illustrate the procedure using G φ as an example. The two independent solutions are chosen as below: φ h = (ρ − 1) − iω 2r 0 (1 + · · ·) = h 0 + h 1 ρ 2 + · · · , φ b = φ h h * 0 − φ * h h 0 = (h * 0 h 1 − h 0 h * 1 ) ρ −2 + · · · .(56) Here φ h satisfies the infalling wave condition on the horizon. h 0 and h 1 (not to be confused with h) are coefficients of asymptotic expansion of φ h . φ b is constructed from linear combination of φ h and its complex conjugate such that it vanishes on the boundary. G φ can be constructed as follows G φ (ρ, ρ ′ ) = 1 φ ′ b (ρ ′ )φ h (ρ ′ )φ b (ρ ′ )φ ′ h (ρ ′ ) φ h (ρ ′ )φ b (ρ)θ(ρ − ρ ′ ) + φ b (ρ ′ )φ h (ρ)θ(ρ ′ − ρ) . (57) The Wronskian appearing in (57) can be fixed up to normalization from the homogeneous equation: φ ′ b φ h − φ ′ h φ b = # √ −hg ρρ g φφ .(58) We can fixed the normalization by taking the limit ρ → ∞ of (58). Comparing the limit with (56), we obtain # = r 2 0 2 2 m 2 (h * 0 h 1 − h 0 h * 1 ) (−2)h 0 ,(59) where we used the fact ρ sin θ\| ρ→∞ = m. The inhomogeneous solution is given by the convolution of Green's function and corresponding source φ (1) (ρ) = ∞ 1 dρ ′ G φ (ρ, ρ ′ )s(ρ ′ ),(60) with the source s(ρ) = B∂ ρ (cos 4 θ)A (0) z iω √ −hg ρρ g φφ .(61) We are only interested in the limit ρ → ∞ of (60), which is φ (1) (ρ) = ∞ 1 dρ ′ φ h (ρ ′ )B∂ ρ ′ (cos 4 θ(ρ ′ ))A (0) z (ρ ′ )iω r 2 0 2 2 m 2 (−2)h 0 ρ −2 + · · · .(62) Following the same procedure, we obtain the counterpart of A z : A (1) z (ρ) = ∞ 1 dρ ′ −A z,h (ρ ′ )B∂ ρ ′ (cos 4 θ(ρ ′ ))φ (0) (ρ ′ )iω r 2 0 2 (−2)a 0 ρ −2 + · · · ,(63) where A z,h is defined as the solution satisfying infalling wave condition on the horizon, with boundary value a 0 . Before closing this section, we claim that (54), (62) and (63) 1 . 1The two phases are mesonic phase with larger m and B and meson melting phase with smaller m and B. In the former case, R-charge (axial charge) exchange between fundamental matter and adjoint sector is not possible due to the formation of meson bound state, while in the latter case, R-charge (axial charge) can leak from fundamental matter to adjoint sector. The phase diagram implies that large quark mass and magnetic field favors formation of meson bound state. The effect of magnetic field may be understood via an increased effective quark mass. We are interested in the meson melting phase, which is more relevant for application in QGP. Figure 1 : 1m-B phase diagram of D3/D7 background. The axis labels are dimensionless numbers with units set by πT = 1. The region with small m and B corresponds to the meson melting phase, while the region with large m and B corresponds to the mesonic phase. Figure 2 : 2(left) The mass diffusion rate Γ m as a function of m 2 for B = 0 (blue point), B = 2 (purple square), B = 4 (brown diamond). The units are set by πT = 1. The blue line is given by (36) which fits well for B = 0. To guide eyes, we also include linear fittings (red line) in the small mass region. The linear behavior is consistent with field theory expectation. We have used empty symbols for points in metastable phases. (right) Γ m as a function of B at m = 1/20. A rapid growth of Γ m with B is found. Figure 3 : 3The mass dissipation rate r as a function of m from numerics with small B and small ω. It is a monotonous increasing function of m as expected. The analytic function (45) is drawn in blue line and fits the numerical results well. The units are set by πT = 1. Figure 4 : 4The mass dissipation rate r as a function of B for m = 7/20 (blue point), m = 9/20 (purple square), and m = 11/20 (brown diamond). The units are set by πT = 1. The analytic function (45) is drawn in blue line and fits the numerical results well. It is a monotonous increasing function of B. have investigated the effect of finite quark mass and magnetic field in the generation and dissipation of axial charge, using a D3/D7 model. For axial charge generation, we calculated the mass diffusion rate. It is analogous to the Chern-Simon diffusion rate as a measure of axial charge fluctuation. The mass diffusion rate is a bounded non-monotonous function of mass at vanishing magnetic field. The presence of magnetic field enhances the diffusion. At small m, our numerical results are consistent with an approximate scaling for the mass diffusion rate , J. F. Liao, Y. Liu, L. Yaffe, H.-U. Yee and Y. Yin for useful discussions. The work of S.L. is in part supported by Junior Faculty's Fund at Sun Yat-Sen University. Figure 5 : 5Free energy F as a function of 1/m at different B for D3/D7 system. The units are set by πT = 1. The red continuous (blue dashed) curves correspond to the black hole (Minkowski) embedding. √ −h starts from O(B 2 ), we can simply use the B = 0 limit of √ −h for the solution up to O(B). The order O(B 0 ) solution satisfies homogeneous equation. In the hydrodynamic regime, the solution is given by can also be understood as series expansions in ω: we should discard the O(ω) terms in (54) and view the rest as zeroth solution. The first order solutions get contribution from the discarded terms in (54), (62) and (63). Furthermore, we can allow for arbitrary dependence of √ −h on B in (54), (62) and (63) provided that we work with sufficient small ω. We use the D3/D7 model to study the effect of finite quark mass. The background is sourced by N c D3 branes. The worldvolume fields of D3 branes are N = 4 supersymmetric Yang-Mills (SYM) theory. In addition, there are N f D7 branes in the background. The open string stretching between D3 and D7 branes is dual to N = 2 hypermultiplet. 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[ "A Joint Model for IEEE 802.15.4 Physical and Medium Access Control Layers", "A Joint Model for IEEE 802.15.4 Physical and Medium Access Control Layers" ]	[ "Mohamed-Haykel Zayani s:[email protected] \nLab. CNRS SAMOVAR UMR 5157\nTelecom SudParis\nEvryFrance\n", "Vincent Gauthier [email protected] \nLab. CNRS SAMOVAR UMR 5157\nTelecom SudParis\nEvryFrance\n", "Djamal Zeghlache [email protected] \nLab. CNRS SAMOVAR UMR 5157\nTelecom SudParis\nEvryFrance\n" ]	[ "Lab. CNRS SAMOVAR UMR 5157\nTelecom SudParis\nEvryFrance", "Lab. CNRS SAMOVAR UMR 5157\nTelecom SudParis\nEvryFrance", "Lab. CNRS SAMOVAR UMR 5157\nTelecom SudParis\nEvryFrance" ]	[]	Many studies have tried to evaluate wireless networks and especially the IEEE 802.15.4 standard. Hence, several papers have aimed to describe the functionalities of the physical (PHY) and medium access control (MAC) layers. They have highlighted some characteristics with experimental results and/or have attempted to reproduce them using theoretical models. In this paper, we use the first way to better understand IEEE 802.15.4 standard. Indeed, we provide a comprehensive model, able more faithfully to mimic the functionalities of this standard at the PHY and MAC layers. We propose a combination of two relevant models for the two layers. The PHY layer behavior is reproduced by a mathematical framework, which is based on radio and channel models, in order to quantify link reliability. On the other hand, the MAC layer is mimed by an enhanced Markov chain. The results show the pertinence of our approach compared to the model based on a Markov chain for IEEE 802.15.4 MAC layer. This contribution allows us fully and more precisely to estimate the network performance with different network sizes, as well as different metrics such as node reliability and delay. Our contribution enables us to catch possible failures at both layers.	10.1109/iwcmc.2011.5982651	[ "https://arxiv.org/pdf/1205.3305v1.pdf" ]	16,475,149	1205.3305	5ae1de00b7b07e6ae3359b335e788906b39f54ed	A Joint Model for IEEE 802.15.4 Physical and Medium Access Control Layers Mohamed-Haykel Zayani s:[email protected] Lab. CNRS SAMOVAR UMR 5157 Telecom SudParis EvryFrance Vincent Gauthier [email protected] Lab. CNRS SAMOVAR UMR 5157 Telecom SudParis EvryFrance Djamal Zeghlache [email protected] Lab. CNRS SAMOVAR UMR 5157 Telecom SudParis EvryFrance A Joint Model for IEEE 802.15.4 Physical and Medium Access Control Layers Index Terms-IEEE 802154Physical layer modelingMedium Access Control Many studies have tried to evaluate wireless networks and especially the IEEE 802.15.4 standard. Hence, several papers have aimed to describe the functionalities of the physical (PHY) and medium access control (MAC) layers. They have highlighted some characteristics with experimental results and/or have attempted to reproduce them using theoretical models. In this paper, we use the first way to better understand IEEE 802.15.4 standard. Indeed, we provide a comprehensive model, able more faithfully to mimic the functionalities of this standard at the PHY and MAC layers. We propose a combination of two relevant models for the two layers. The PHY layer behavior is reproduced by a mathematical framework, which is based on radio and channel models, in order to quantify link reliability. On the other hand, the MAC layer is mimed by an enhanced Markov chain. The results show the pertinence of our approach compared to the model based on a Markov chain for IEEE 802.15.4 MAC layer. This contribution allows us fully and more precisely to estimate the network performance with different network sizes, as well as different metrics such as node reliability and delay. Our contribution enables us to catch possible failures at both layers. I. INTRODUCTION Wireless sensor networks have been closely studied in recent years. Several studies have investigated behaviors and performances of these networks. Some of them have highlighted such networks properties by relying on empiric results [2]- [6] whereas others have focused on reproducing a standard or mechanism functionalities tied to sensors by proposing analytical models [1], [7]- [9]. Empirical studies have shown that wireless communication networks are radically different from some simulation models (disc-shaped nodes range for example). Analytical studies have attempted to reproduce mechanisms and technical aspects widely used/seen in these networks in order to track network performances. Among these approaches, those of Zuniga and Krishchnamachari [10], [11] stand out. They emphasize the limits of disc-shaped node range models that are used in simulators, and highlight the existence of a transitional region between the connected and disconnected areas. This observation, based on experiments, enables us to understand clearly the reason behind link unreliability in low power wireless networks. Moreover, Zuniga and Krishchnamachari underline the impact of asymmetry in transitional region expansion and its negative effect on reliability [11]. Meanwhile, lot of the performance analysis of MAC layer protocol are derived from the Markov model proposed by Bianchi [7] for the IEEE 802.11 standard [12]. This model consists in a Markov chain that reproduces the functionalities of the IEEE 802.11 standard while assuming saturated traffic and ideal channel conditions. This approach has inspired many others, for instance the Park et al. model [1]. It presents itself as a relevant contribution which aims to measure reliability, delay and energy consumption in a wireless network based on IEEE 802.15.4 standard [13]. In this paper, we develop an IEEE 802. 15 The remainder of this paper is as follows. In Section II, we present the related work which gives an overview of approaches that inspire our model. We focus on our contribution by giving details on the combined PHY and MAC layers models in Section III. In Section IV, we compare our proposition to the enhanced Park et al. approach and estimate nodes performances with different network densities. Finally, Section V concludes the paper and discusses some future research challenges. II. RELATED WORK Many studies have aimed to understand and to evaluate standards and protocols. The works that tried to identify the properties of these networks mechanisms fall into two categories: i.e. simulations-based [2]- [6] relying on empiric observations, or analytical works [1], [7]- [9]. Most of analytical studies are based on the Markov model proposed by Bianchi [7] for the IEEE 802.11 standard. This model consists in a Markov chain that mimics the functionalities of the IEEE 802.11 standard while assuming a saturated traffic and ideal channel conditions. Zhai et al. [14] and Daneshgaran et al. [9] exploit the Bianchi model and extend it through more realistic assumptions. These approaches have inspired Griffith 978-1-4244-9538-2/11/$26.00 c 2011 IEEE and Souryal [15] to develop a model for the IEEE 802.11 MAC layer that adds a frame queue to each node. This contribution enables us to estimate the packet reception rate, the delay, the medium access control layer (MAC layer) service time and the throughput. Similar studies have been developed for the wireless sensor networks, and more especially the IEEE 802.15.4 standard. Hence, we note the models developed by Pollin et al. [16] and Park et al. [1]. The two approaches provide a generalized analysis that allows to measure reliability, delay and energy consumption. In each proposed model, the exponential backoff process is modeled by a Markov chain. Retry limits and acknowledgements in an unsaturated traffic scenario are also taken into consideration. Park et al. propose a generalized analytical model of the slotted CSMA/CA mechanism with beacon enabled mode in IEEE 802.15.4. This model includes retry limits for each packet transmission. The scenario of a star network in which N nodes try to send data to a sink has been considered and defining the state of a single node through a Markov model has been proposed. Each state of the Markov chain is characterized by three stochastic processes: the backoff stage s(t), the state of the backoff counter c(t) and the state of the retransmission counter r(t) at time t. The described modeling allows us to analyze of the link reliability, delay and energy consumption. In another context, numerous works focus on the physical layer modeling. For instance, Zuniga and Krishnamachari [10], [11] have analyzed the major causes behind unreliability [10], [11] and the negative impact of asymmetry on link efficiency [11] in low power wireless links. Instead of the binary discshaped model these models reproduce the called transitional region [3]- [5] in order to model the transmission range. The packet reception rate and the upper-layer protocol reliability are highly instable when a neighbor is located in this region. To understand it, two models have been proposed: a channel model that is based on the log-normal path loss propagation model [17] and a radio reception model closely tied to the determination of packet reception ratio. Through these models, it is possible to derive the expected distribution and the variance of the packet reception ratio according to the distance. III. DEVELOPED IEEE 802.15.4 MODEL FOR SMART GRID Our contribution joins the initiative of Smart Grid [15] to provide tools that evaluate wireless communications standards. The developed model that we propose analyzes an IEEE 802.15.4 PHY and MAC layer channel in which multiple non-saturated stations compete in communicating with a sink. The aim is to combine two relevant models: A PHY model that bypasses the disk shaped node range and takes into consideration the called transitional region and a MAC model that reproduces the CSMA/CA mechanism. The model described enables us to add the impact of PHY layer errors onto the MAC layer and to provide some improvements for the adopted MAC model, in order to obtain more precise output estimations. Our developed model is available at the SGIP NIST Smart Grid Collaboration website [15]. A. IEEE 802.15.4 PHY Model Description To model the PHY layer, we have adopted the Zuniga and Krishnamachari approach [10], [11]. The main objective is to identify the causes of the transitional region and quantify their influence on performance without considering interferences (assumption of a light traffic or static interference). To do this, the expressions of the packet reception rate as function of distance are derived. These expressions take into account radio and channel parameters such as the path loss exponent (log-normal shadowing path loss model [17]), the channel shadowing variance, the modulation, the coding and hardware heterogeneity. They describe how the channel and radio influence transitional region growth. We use mathematical frameworks provided to compute packet delivery rate independently of interferences. For more details of the Zuniga and Krishnamachari models see [10], [11]. We believe that including the PHY model in the MAC model considered (the next subsection describes the MAC model) is an interesting challenge. Indeed, collisions are the major factor behind frame losses. Nonetheless, considering errors that can happen at the PHY layer includes constraints imposed by SNR (signal-to-noise ratio), modulation, encoding and asymmetry (heterogeneous hardware). Therefore, our contribution allows us to have a more realistic estimation by taking into account the causes of failures at both layers. B. Operation details for the IEEE 802.15.4 MAC Model and the interactions with the PHY Model The model of IEEE 802.15.4 MAC layer is inspired from Park et al. Markov chain [1]. It captures the state of the station backoff stage, the backoff counter and the retransmission counter. We insert into Park et al. model an M/M/1/K queuing model that endows a finite buffer to a station. On the one hand, the Markov model determines the steady state probability when a station senses the channel in order to transmit a frame and the probability that a frame experiences a failure (due to a collision or to PHY layer failure). On the other hand, the queuing model gives as output some measurements such as the throughput or the probability that the station is idle. The Park et al. approach, inspired from [16], consists in a generalized analytical model of the slotted CSMA/CA mechanism of beacon enabled IEEE 802.15.4 with retry limits for each packet transmission (the complete description is provided in [1]). The model takes the scenario of N stations that try to communicate with a sink. Park et al. define the probabilities for the following events: a node attempts a first carrier sensing to transmit a frame, a node finds the channel busy during CCA1 or a node finds the channel busy during CCA2. They are denoted by the variables τ , α and β respectively. These three probabilities are related by a system of three non-linear equations that arises from finding the steady state probabilities. Our model, described by the flowchart presented in Fig. 1 (the main PHY and MAC inputs are listed in Table I and Table II respectively), aims to solve the nonlinear system that expresses these probabilities. In addition, it estimates p 0 , the probability of going back to the idle state by considering the offered load per node parameter λ. In this way, our contribution enables the MAC model to determine this probability, in opposition to [1] (p 0 is taken as an input for the performances analysis). We start from equations (16), (17) and (18) in [1] and make changes in some of these expressions to enhance the model. The equations (17) and (18) are expressed with probability τ to mention that a node is transmitting. In our mind, this consideration is insufficient because a transmitting node must not be idle, that is why we substitute τ by (1-p 0 )τ . Thereby, τ is the probability that a node tries to transmit and 1-p 0 is the probability that a station has a frame to send. The system considered is given by equations (1), (2) and (3): τ = 1 − x m+1 1 − x 1 − y n+1 1 − y b 0,0,0(1)α = L + N (1 − p 0 )τ (1 − τ (1 − p 0 )) N −1 1 − (1 − τ (1 − p 0 )) N L ACK 1 − (1 − (1 − p 0 )τ ) N −1 (1 − α)(1 − β) (2) β = 1 − (1 − τ (1 − p 0 )) N −1 DV + N (1 − p 0 )τ (1 − (1 − p 0 )τ ) N −1 DV (3) Where DV = 2 − (1 − (1 − p 0 )τ ) N + N (1 − p 0 )τ (1 − (1 − p 0 )τ ) N −1 , x = α + (1 − α)β and y = P f ail (1 − x m+1 ). The parameter P f ail represents the probability of a failed transmission attempt, m is the maximum number of backoffs the CSMA/CA algorithm will attempt before declaring a channel access failure, n is the maximum number of retries allowed after a transmission failure, L is the length of the data frame in slots (a slot has a length of 80 bits), L ACK is the length of an acknowledgement in slots, N is the number of stations and b 0,0,0 is the state where the state variables of the backoff stage counter, the backoff counter and the retransmission counter are equal to 0 (an approximation is proposed in [1]). The mechanism that computes these probabilities (using the MATLAB f solve function) allows us to determine the probability of failed transmission P f ail , given by: P f ail = 1 − (1 − P col )(1 − P e )(4)Where P col = 1 − (1 − τ (1 − p 0 )) N −1 . In the above expressions, P e is the probability of loss due to channel and radio constraints (computed by the PHY model) and P col is the probability of a collision occurring (modified as done with (17) and (18) in [1]). This mechanism is embedded in a loop that updates p 0 . The developed model solves the system of non-linear equations to determine τ , α, β and therefore P f ail . Then, P f ail , α and β are used to estimate the mean MAC service time, or the mean time to process a frame, expressed also as Expected Time or ET (in [1], Section V.B details how to compute this time. We substitute, of course, P col by P f ail to catch errors that can occur at PHY and MAC layers). So, a new value for p 0 is generated and the updated p 0 is used in the next iteration. It is possible to determine p 0 since each device has a buffering capacity. Every node is modeled as an M/M/1/K queue and each queue receives frames following a Poisson arrival process λ frames/s. The queue utilization ρ is the product of the arrival rate λ and the inverse of the mean MAC service time ET . The steady state probability that there are i frames in the queue is: p i = ρ i / K j=0 ρ j(5) Hence, the value of p 0 is given by: p 0 =   K j=0 ρ j   −1(6) The process continues until the value of p 0 converges to a stable value. Once p 0 converges, all outputs concerning queuing analysis can be computed for each value of λ (the per-node load offered). Four outputs are considered in this study: the average waiting time to receive a frame (Eq. (7)), the failure probability (probability of packet loss due to collisions or link constraints)(Eq. (4)), the reliability of a node (the probability of a good frame reception)(Eq. (8)) and the average throughput per node(Eq. (9)). D = L λ (1 − p k ) (7) R = (1 − p k ) (1 − P cf ) (1 − P cr ) (8) S avg = λRL p(9) Where p k is the probability of having full buffer, P cf is the probability that the frame is discarded due to channel access failure (Eq. (19) in [1]), P cr is the probability that the packet is discarded due to retry limits (Eq. (20) in [1]), L is the payload size and L p is the application data size. Therefore, this contribution enables us to enhance Park and al. model at two levels: • Providing a more precise computation of failure probability by considering possible errors at PHY and MAC layers (link unreliability and collisions). IV. SIMULATION AND RESULTS We propose two scenarios for the simulations. Firstly, we compare the performances of a node obtained in two different ways. On one hand, we use the Park et al. Markov chain (MAC layer) and on the other hand we test our model. Secondly, we expose the same performances, using our developed model, for different densities. Table I presents the main inputs at the MAC layer and Table II enumerates the main ones at the PHY layer. All the simulations test different values for the offered per-station load, measured in units of frame/s. We choose to start from 0.5 frame/s and increase the offered load to 25 frames/s with a step of 0.5 frame/s (or from 400 bits/s to 20000 bits/s). We select four outputs to illustrate node performances: the average waiting time for a frame reception, the failure probability (probability of frame loss due to collisions or link constraints), the reliability of a node (the probability of a good frame reception) and the throughput. A. Comparison between combined PHY and MAC layers and simple MAC layer models As described in Section III, when we include the constraints at the physical layer, delivery failures happen more often. There are many reasons for this: weak SNR and modulation and/or encoding errors. We run simulation for a star network with 10 nodes. The results confirm a notable degradation of node performances. In Fig. 2(a), the average waiting time is for the the combination of the PHY and MAC models. Inserting link constraints increases the number of retransmissions. Thus, the delay increases. The delay difference between the two approaches reaches 40 ms at offered load equal to 11 frames/s. Fig. 2(b) compares the evolution of failure probability for the two approaches. With light offered loads, the impact of the PHY model is conspicuous, especially since the number of collisions is likely to be low. The collisions are more frequent with heavier loads and the probability of occurrences grows quickly, generating network saturation. Meanwhile, the probability of losses due to link conditions remains constant (this probability is determined independently of interferences and computed through an integration over the distance covered by the maximum range and over asymmetry variations). So, the difference between the two approaches is less significant. The same interpretation can be used for reliability evolution, presented in Fig. 2(c). Reliability also undergoes the frame discards due to the reaching of maximum frame retries or maximum CSMA backoffs. The rejected frames due to full node queue represent also a possible interpretation with high offered loads. The throughput evolution, presented in Fig. 2(d), also undergoes the constraints of the PHY layer, and is logically less significant since it follows the same evolution of reliability (throughput is the product of reliability, offered load and data frame size). B. Evolution of node performance with growing densities We use our model to compare node performances with three densities. We propose a network with 5 nodes, another with 10 nodes and a third with 50 nodes. We take into account the same outputs cited in the previous section. The major observation is that the IEEE 802.15.4 networks do not support heavy traffic. The denser the network is, the poorer are the performances are. We note an increasing delay for denser networks, as observed in Fig. 3(a). As the number of nodes increases, and with growing offered loads, collisions are more frequent and so the retransmissions are more recurrent. The switching phase to saturated network shows more significant differences between the three network scenarios. Each node queue begin to experience congestion problems; with more retransmission requirements, the queues are busier and the delays are longer. At saturation, the frame losses are widespread (collisions, link constraints, frames discarded due to retry limits, etc.) for the three scenarios, but the number of nodes still has an impact because it has a negative influence on performances and channel availability (more collisions, more retransmissions, channel congestion,. . . ). The same reasoning explains a higher failure probability, as presented in Fig. 3(b) and a lower reliability as outlined in Fig. 3(c) for denser networks and with increasing offered load. The evolution of throughput, shown in Fig. 3(d), also matches with the interpretations cited above. V. CONCLUSION We have presented, in this paper, a model that mimics the IEEE 802.15.4 functionalities at the PHY and the MAC layers. We aim to combine two relevant propositions. On the one hand, we model constraints that affect link quality using the Zuniga and Krishnamachari mathematical framework: distance, output power, noise, asymmetry and errors related to encoding and modulation. The PHY model bypasses the diskshaped node range and expresses more precisely the degree of link unreliability. The output of this model represents an important outcome for estimating more faithfully the probability of transmitting frame failure. On the other hand, we enhance the Park et al. IEEE 802.15.4 MAC layer model to extract the delay and the reliability. Our contribution seeks to improve the Park et al. approach in determining inherent probabilities (frame transmission, free channel in CCA1/CCA2, failure,...) and combining it with the PHY model to better estimate wireless network parameters. The methodology adopted relies on a Markov chain that follows the flowchart described in Fig.1 and on an M/M/1/K queue that we includes with the Park et al. approach. The joint model is available at the SGIP NIST Smart Grid Collaboration website [15] for use. The simulations show that more precise estimations are provided by our model versus that by the Park et al. MAC model. The comparison between the two considerations highlights a notable performance deterioration using the combined model. This observation is quite logical since this combination joins PHY constraints to collisions. Thus, our contribution improves the Park et al. approach by bypassing the assumption that failures are restricted to collisions. The amelioration of the Park et al. approach is not limited to the above description. We try also to enhance the estimation of inherent probabilities by adjusting some expressions (as for α, β and P f ail ) and modifying the resolution method to gather new parameters (such as p 0 , the probability that a node returns to the idle state, which is considered as an input in Park et al. work). Our contribution proposes to mimic the IEEE 802.15.4 PHY and MAC layers mechanisms. Nonetheless, it is extensible for reproducing more precise wireless networks standards related to IEEE 802.15.4. It is also adjustable to other standards. Indeed, the considered PHY layer model is quite relevant but assumes that interferences are weak and/or stable. Moreover, the probability of an error at the PHY layer is averaged (through integration over maximum range and maximum asymmetry variation). Hence, as future work, we will seek to extend our model in order to consider distance between nodes and thereby topologies. Also, it will be challenging to plan a model that deals with node mobility to appreciate its impact on performances. From there, our model can be used as a tool to verify metrics efficiency in mobile environments. Fig. 1 . 1IEEE 802.15.4 PHY and MAC model flowchart • Enhancing the MAC model of Park et al. by estimating the probability p 0 for the resolution of non-linear equations (this probability is an input in Park et al. model), modifying some expressions to more efficient estimations and determining outputs relative to a precise scenario (number of nodes and per-node load offered). Fig. 2 .Fig. 3 . 23Comparison between IEEE 802.15.4 PHY & MAC Model and IEEE 802.15.Performances evolutions for different densities using IEEE 802.15.4 PHY & MAC Model .4 model that takes into consideration the interactions on PHY and MAC layers. The model, at the PHY layer level, is derived from the Zuniga and Krishnamachari mathematical framework for quantifying link unreliability. The MAC layer model is inspired from an enhanced Park et al. Markov chain. The joint model that combines both PHY and MAC models enables us to consider the causes behind packet losses either at PHY or MAC levels. Indeed, in the Park et al. model, collisions appear to be the only reason for losses, whereas, our model includes constraints posed by SNR (signal-to-noise ratio), modulation, encoding and asymmetry (heterogeneous hardware). TABLE I MAIN IPARAMETERS USED IN MAC LAYERParameter Value Parameter Value Number of Nodes 5, 10, 50 Queue Size 51 Frames Smallest Backoff Win 8 Data Rate 19.2 kbit/s Max Frame Retries 3 ACK Size 88 bits Max CSMA Backoff 4 Shadowing STD 4 Max Backoff Exponent 5 IFS 640 µs Min Backoff Exponent 3 Max TX-RX Time 192 µs MAC Frame Payload 800 bits MAC Overhead 48 bits TABLE II MAIN PARAMETERS USED IN PHY LAYER Parameter Value Parameter Value Noise Figure 23dB Bandwidth 30kHz Pathloss exp 4 STD Tx power 5dBm Noise 15dB Preamb. Length 40 bits Max Tx range 20 m Min Tx Range 1 m A generalized Markov chain model for effective analysis of slotted IEEE 802.15.4. P Park, P Di Marco, P Soldati, C Fischione, K H Johansson, 2009 IEEE 6th International Conference on Mobile Adhoc and Sensor Systems. IEEEP. Park, P. Di Marco, P. Soldati, C. Fischione, and K. H. 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[ "EFFICIENT EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM. II", "EFFICIENT EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM. II", "EFFICIENT EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM. II", "EFFICIENT EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM. II" ]	[ "Svetlana Boyarchenko ", "Sergei Levendorskiȋ ", "Svetlana Boyarchenko ", "Sergei Levendorskiȋ " ]	[]	[]	We prove a simple general formula for the expectation of a function of a Lévy process and its running extremum, which is more convenient for applications than general formulas in the first version of the paper. The derivation of explicit formulas in applications significantly simplifies. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the maximum of a stock price for a power of the price are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 -in a fraction of a second.	10.2139/ssrn.4140462	[ "https://export.arxiv.org/pdf/2207.02793v2.pdf" ]	250,223,424	2207.02793	2d360a760510144fb8cc7a5e67311a1df7bd678d	EFFICIENT EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM. II Svetlana Boyarchenko Sergei Levendorskiȋ EFFICIENT EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM. II Lévy processextrema of a Lévy processlookback optionsbarrier optionsFourier transformHilbert transformFast Fourier transformfast Hilbert transformGaver- Wynn Rho algorithmsinh-acceleration MSC2020 codes: 60-0842A3842B1044A1065R1065G5191G2091G60 We prove a simple general formula for the expectation of a function of a Lévy process and its running extremum, which is more convenient for applications than general formulas in the first version of the paper. The derivation of explicit formulas in applications significantly simplifies. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the maximum of a stock price for a power of the price are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 -in a fraction of a second. Let X be a one-dimensional Lévy process on the filtered probability space (Ω, F, {F t } t≥0 , Q) satisfying the usual conditions, and let E be the expectation operator under Q. There exists a large body of literature devoted to calculation of expectations V (f ; T ; x 1 , x 2 ) of functions of spot value x 1 of X and its running maximum or minimum x 2 and related optimal stopping problems, standard examples being barrier and American options, and lookback options with barrier and/or American features. See, e.g., [41,13,14,15,48,5,6,45,44,9,10,47,46,8,26,42,39,43,53,38,52] and the bibliographies therein. In many papers, in the infinite time horizon case, the Wiener-Hopf factorization technique in various forms is used, and the finite time horizon problems are reduced to the infinite time horizon case using the Laplace transform or its discrete version. The present paper belongs to this strand of the literature. We consider Lévy models, and options with continuous monitoring, without American features; the method admits a straightforward modification to random walk models (hence, to options with discrete monitoring), and can be used as a building block to price American options with lookback features, and options in regime-switching Lévy models. The modification to options with discrete monitoring is straightforward. The modification to stable Lévy processes is less straightforward but can be designed in the same vein as the general approach to the calculation of the probability distributions in Lévy models with exponentially decreasing tails [30] is adjusted to stable Lévy distribtions in [31]. The second contribution of the paper is a simple very efficient algorithm. Once a general exact formula in terms of a sum of 1-3 dimensional integrals is derived, good changes of variables allows one to evaluate the integrals with an almost machine precision and at a much smaller CPU cost than using any previously developed method; the error tolerance of the order of E-7 can be satisfied in milliseconds using Matlab and Mac with moderate characteristics. The algorithm is short and involves a handful of vector operations and multiplication by matrices of a moderate size at 3 places of the algorithm. A secondary contribution is the observation that the choice of an approximately optimal parameters of the numerical scheme simplifies significantly if the process is a Stieltjes-Lévy process (SL-process). This class is defined in [33], where it is shown that all popular classes of Lévy processes bar the Merton model and Meixner processes are SL processes. For the Merton model and Meixner processes, the computational cost is several times larger. LetX t = sup 0≤s≤t X s and X t = inf 0≤s≤t X s be the supremum and infimum processes (defined path-wise, a.s.); X 0 =X 0 = X 0 = 0. For a measurable function f , consider V (f ; T ; x 1 , x 2 ) = E[f (x 1 + X T , max{x 2 , x 1 +X T })]. In Section 3.1, we derive simple explicit formulas for the Laplace transformṼ (f ; q; x 1 , x 2 ) of V (f ; T ; x 1 , x 2 ) using the operator form of the Wiener-Hopf factorization technique [15,14,13,17,20,9]. Basic facts of the Wiener-Hopf factorization technique in the form used in the paper and definitions of general classes of Lévy processes amenable to efficient calculations are collected in Section 2. The formulas are in terms of the (normalized) expected present value operators E + q and E − q defined by E + q u(x) = E[u(x +X Tq )], E − q u(x) = E[u(x + X Tq )], where q > 0 and T q is an exponentially distributed random variable of mean 1/q independent of X. In the case of bounded functions (Theorem 3.1), the formulas are proved for any Lévy process, stable ones including; in the case of functions of exponential growth (Theorem 3.2), the tail(s) of the Lévy density must decay exponentially. A special case x 1 = x 2 = 0 appeared earlier in the working paper [46]; the version formulated and proved in the present paper is more efficient for applications. Theorems 3.1 and 3.2 generalize formulas for E[f (x 1 + X Tq , min{x 2 , X Tq })] and E[f (x 1 + X Tq , max{x 2 ,X Tq })] derived in [15,14,13,17,20,9] for the payoff functions of the form f (x 1 , x 2 ) = g(x 1 )1 (h,+∞) (x 2 ) and f (x 1 , x 2 ) = g(x 1 )1 (−∞,h) (x 2 ), respectively. In Section 3.2, we use the Fourier transform and the equalities E ± q e ixξ = φ ± q (ξ)e ixξ , where φ ± q (ξ) are the Wiener-Hopf factors, to realize the general formula derived in Section 3 as a sum of integrals. As applications of the general theorems, in Section 3.3, we derive explicit formulas for the cumulative distribution function (cpdf) of the Lévy process and its maximum, and for the option to exchange eX T for the power e βX T . In Section 4, we demonstrate how the sinh-acceleration technique used in [30] to price European options and applied in [32,31,34] to pricing barrier options, evaluation of special functions and the coefficients in BPROJ method respectively can be applied to greatly decrease the sizes of grids and the CPU time needed to satisfy the desired error tolerance. This feature makes the method of the paper more efficient than methods that use the fast inverse Fourier transform, fast convolution or fast Hilbert transform. The changes of variables must be in a certain agreement as in [26,51], where a less efficient family of fractional-parabolic deformations was used. Note that Talbot's deformation [57] cannot be applied if the conformal deformations technique is applied to the integrals with respect to the other dual variables. In Section 5, we summarize the results of the paper and outline several extensions of the method of the paper. We relegate to Appendices technical details, and the outline of other methods that are used to price options with barrier/lookback features. Figures and one of the tables are in Appendix B. Preliminaries 2.1. Wiener-Hopf factorization. Let X be a Lévy process on R, X 0 =X 0 = X 0 = 0, and let T q be an exponentially distributed random variable of mean 1/q, independent of X; operators E ± q are defined in Introduction. We will also need the EPV operator E q (normalized resolvent) defined by E q u(x) = E[u(X Tq )]. Lemma 2.1 and equalities (2.1) and (2.2) are three equivalent forms of the Wiener-Hopf factorization for Lévy processes. Eq. (2.2) and (2.1) are special cases of the Wiener-Hopf factorization in complex analysis and the general theory of boundary problems for pseudo-differential operators (pdo), where more general classes of functions and operators appear (see, e.g., [37]). In probability, the version (2.2) was obtained (see [55] for references) before Lemma 2.1; the version (2.1) was proved in [15,14,13,17,20] under additional regularity conditions on the process, and in [9], for any Lévy process. [54, p.81]) Let X and T q be as above. Then (a) the random variablesX Tq and X Tq −X Tq are independent; and (b) the random variables X Tq and X Tq −X Tq are identical in law. (By symmetry, the statements (a), (b) are valid withX and X interchanged). By definition, part (a) amounts to the statement that the probability distribution of the R 2 -valued random variable (X Tq , X Tq −X Tq ) is equal to the product (in the sense of "product measure") of the distribution ofX Tq and the distribution of X Tq −X Tq . The two basic forms of the Wiener-Hopf factorization (both immediate from Lemma 2.1) are (2.1) E q = E + q E − q = E − q E + q , (2.2) q q + ψ(ξ) = φ + q (ξ)φ − q (ξ). Evidently, the EPV-operators are bounded operators in L ∞ (R). In exponential Lévy models, payoff functions may increase exponentially, hence, we consider the action of the EPV op- erators in L ∞ (R; w), L ∞ -spaces with the weights w(x) = e γx , γ ∈ [µ − , µ + ], and w(x) = min{e µ − x , e µ + x }, where µ − ≤ 0 ≤ µ + , µ − < µ + ; the norm is defined by u L∞(R;w) = wu L∞(R) . Recall that a function f is said to be analytic in the closure of an open set U if f is analytic in the interior of U and continuous up to the boundary of U . We need the following straightforward result (see, e.g., [14,20]). Lemma 2.2. Let E[e −γX 1 ] < ∞, ∀ γ ∈ [µ − , µ + ], where µ − ≤ 0 ≤ µ + , µ − < µ + . Then (i) ψ(ξ) admits analytic continuation to the strip S [µ − ,µ + ] := {ξ ∈ C \| Im ξ ∈ [µ − , µ + ]}; (ii) let σ > max{−ψ(iµ − ), −ψ(iµ + )}. Then there exists c > 0 s.t. \|q + ψ(ξ)\| ≥ c for q ≥ σ and ξ ∈ S [µ − ,µ + ] ; (iii) let q ≥ σ. Then φ + q (ξ) (resp., φ − q (ξ)) admit analytic continuation to {Im ξ ≥ µ − } (resp., {Im ξ ≤ µ + }) given by φ + q (ξ) = q (q + ψ(ξ))φ − q (ξ) , Im ξ ∈ [µ − , 0], (2.3) φ − q (ξ) = q (q + ψ(ξ))φ + q (ξ) , Im ξ ∈ [0, µ + ]; (2.4) (iv) φ + q (ξ) (resp., φ − q (ξ)) is uniformly bounded on {Im ξ ≥ µ − } (resp., {Im ξ ≤ µ + }); (v) for any weight function of the form w(x) = e γx , γ ∈ [µ − , µ + ], and w(x) = min{e µ − x , e µ + x }, operators E ± q are bounded in L ∞ (R; w). We have E ± q e ixξ = φ ± q (ξ)e ixξ . Hence, E ± q are pseudo-differential operators with symbols φ ± q , which means that E ± q u(x) = F −1 ξ→x φ ± q (ξ)F x→ξ u(x) for sufficiently regular functions u. 2.2. General classes of Lévy processes amenable to efficient calculations. The conditions of Lemma 2.2 are satisfied for all popular classes of Lévy processes bar stable Lévy processes. See [14,13,15], where the general class of Regular Lévy processes of exponential type (RLPE) is introduced. An additional property useful for development of efficient numerical methods is a regular behavior of the characteristic exponent at infinity. In the definition below, we relax the conditions in [14,13,15] allowing for non-exponential decay of one of the tails of the Lévy density. Indeed, for calculations in the dual space, it does not matter whether the strip of analyticity contains the real line or is adjacent to the real line. For ν = 0+ (resp., ν = 1+), set \|ξ\| ν = ln \|ξ\| (resp., \|ξ\| ν = \|ξ\| ln \|ξ\|), and introduce the following complete ordering in the set {0+, 1+} ∪ (0, 2]: the usual ordering in (0, 2]; ∀ ν > 0, 0+ < ν; ∀ ν > 1, 1 < 1+ < ν. We use cones C γ − ,γ + = {e iϕ ρ \| ρ > 0, ϕ ∈ (γ − , γ + ) ∪ (π − γ + , π − γ − )}, C γ = {e iϕ ρ \| ρ > 0, ϕ ∈ (−γ, γ)}, and the strip S (µ − ,µ + ) = {ξ \| Im ξ ∈ (µ − , µ + )}. Definition 2.3. ([33, Defin 2.1]) We say that X is a SINH-regular Lévy process (on R) of order ν ∈ {0+, 1+} ∪ (0, 2] and type ((µ − , µ + ); C; C + ), iff the following conditions are satisfied: (i) µ − < 0 ≤ µ + or µ − ≤ 0 < µ + ; (ii) C = C γ − ,γ + , C + = C γ − ,γ + , where γ − < 0 < γ + , γ − ≤ γ − ≤ 0 ≤ γ + ≤ γ + , and \|γ − \|+γ + > 0; (iii) the characteristic exponent ψ of X can be represented in the form (2.5) ψ(ξ) = −iµξ + ψ 0 (ξ), where µ ∈ R, and ψ 0 admits analytic continuation to i(µ − , µ + ) + (C ∪ {0}); (iv) for any ϕ ∈ (γ − , γ + ), there exists c ∞ (ϕ) ∈ C \ (−∞, 0] s.t. (2.6) ψ 0 (ρe iϕ ) ∼ c ∞ (ϕ)ρ ν , ρ → +∞; (v) the function (γ − , γ + ) ϕ → c ∞ (ϕ) ∈ C is continuous; (vi) for any ϕ ∈ (γ − , γ + ), Re c ∞ (ϕ) > 0. Example 2.4. A generic process of Koponen's family [11,12] was constructed as a mixture of spectrally negative and positive pure jump processes, with the Lévy measure (2.7) F (dx) = c + e λ − x x −ν + −1 1 (0,+∞) (x)dx + c − e λ + x \|x\| −ν − −1 1 (−∞,0) (x)dx, where c ± > 0, ν ± ∈ [0, 2), λ − < 0 < λ + . Starting with [33], we allow for c + = 0 or c − = 0, λ − = 0 < λ + and λ − < 0 ≤ λ + . This generalization is almost immaterial for evaluation of probability distributions and expectations because for efficient calculations, the first crucial property, namely, the existence of a strip of analyticity of the characteristic exponent, around or adjacent to the real line, holds if λ − < λ + and λ − ≤ 0 ≤ λ + . 1 Furthermore, the Esscher transform allows one to reduce both cases λ − = 0 < λ + and λ − < 0 ≤ λ + to the case λ − < 0 < λ + . If ν ± ∈ (0, 2), ν ± = 1, (2.8) ψ 0 (ξ) = c + Γ(−ν + )((−λ − ) ν + − (−λ − − iξ) ν + ) + c − Γ(−ν − )(λ ν − + − (λ + + iξ) ν − ). Note that a specialization ν ± = ν = 1, c = c ± > 0, of KoBoL used in a series of numerical examples in [11] was named CGMY model in [36] (and the labels were changed: letters C, G, M, Y replace the parameters c, ν, λ − , λ + of KoBoL): (2.9) ψ 0 (ξ) = cΓ(−ν)[(−λ − ) ν − (−λ − − iξ) ν + λ ν + − (λ + + iξ) ν ]. Evidently, ψ 0 given by (2.9) is analytic in C \ iR, and ∀ ϕ ∈ (−π/2, π/2), (2.6) holds with (2.10) c ∞ (ϕ) = −2cΓ(−ν) cos(νπ/2)e iνϕ . In [33], we defined a class of Stieltjes-Lévy processes (SL-processes). In order to save space, we do not reproduce the complete set of definitions. Essentially, X is called a (signed) SLprocess if ψ is of the form (2.11) ψ(ξ) = (a + 2 ξ 2 − ia + 1 ξ)ST (G 0 + )(−iξ) + (a − 2 ξ 2 + ia − 1 ξ)ST (G 0 − )(iξ) + (σ 2 /2)ξ 2 − iµξ, where ST (G) is the Stieltjes transform of the (signed) Stieltjes measure G, a ± j ≥ 0, and σ 2 ≥ 0, µ ∈ R. We call a (signed) SL-process regular if it is SINH-regular. We proved in [33] that if X is a (signed) SL-process then ψ admits analytic continuation to the complex plane with two cuts along the imaginary axis, and if X is SL-process, then, for any q > 0, equation q +ψ(ξ) = 0 has no solution on C \ iR. We also proved that all popular classes of Lévy processes bar the Merton model and Meixner processes are regular SL-processes, with γ ± = ±π/2; the Merton model and Meixner processes are regular signed SL-processes, and γ ± = ±π/4. In [33], the reader can find a list of SINH-processes and SL-processes, with calculations of the order and type. Evaluation of the Wiener-Hopf factors. For numerical realizations, we need the following explicit formulas for φ ± q (see, e.g., [14,9,26,51,32]). Lemma 2.5. Let X and q satisfy the conditions of Lemma 2.2. Then (a) for any ω − ∈ (µ − , µ + ) and ξ ∈ {Im ξ > ω − }, (2.12) φ + q (ξ) = exp 1 2πi Im η=ω − ξ ln(1 + ψ(η)/q) η(ξ − η) dη ; (b) for any ω + ∈ (µ − , µ + ) and ξ ∈ {Im ξ < ω + }, (2.13) φ − q (ξ) = exp − 1 2πi Im η=ω + ξ ln(1 + ψ(η)/q) η(ξ − η) dη . The integrands above decay very slowly at infinity, hence, a fast and accurate numerical realization is impossible unless additional tricks are used. If X is SINH-regular, the rate of decay can be greatly increased using appropriate conformal deformations of the line of integration and the corresponding changes of variables. Assuming that in Definition 2.3, γ ± are not extremely small is absolute value (and, in the case of regular SL-processes, γ ± = ±π/2 are not small), the most efficient change of variables is the sinh-acceleration (2.14) η = χ ω 1 ,b,ω (y) = iω 1 + b sinh(iω + y), where ω ∈ (−π/2, π/2), ω 1 ∈ R, b > 0. Typically, the sinh-acceleration is the best choice even if \|γ ± \| are of the order of 10 −5 . The parameters ω 1 , b, ω are chosen so that the contour L ω 1 ,b,ω := χ ω 1 ,b,ω (R) ⊂ i(µ + , µ + ) + C γ − ,γ + and, in the process of deformation, ln(1 + ψ(η)/q) is a well-defined analytic function (on C or an appropriate Riemann surface). Lemma 2.6. Let X be SINH-regular of type ((µ − , µ + ), C γ − ,γ + , C γ − ,γ + ). Then there exists σ > 0 s.t. for all q > σ, (i) φ + q (ξ) admits analytic continuation to i(µ − , +∞)+i(C π/2−γ − ∪{0}). For any ξ ∈ i(µ − , +∞)+ i(C π/2−γ − ∪ {0}), and any contour L − ω 1 ,b,ω ⊂ i(µ − , µ + ) + (C γ − ,γ + ∪ {0}) lying below ξ, (2.15) φ + q (ξ) = exp 1 2πi L − ω 1 ,b,ω ξ ln(1 + ψ(η)/q) η(ξ − η) dη ; (ii) φ − q (ξ) admits analytic continuation to i(−∞, µ + )−i(C π/2+γ + ∪{0}). For any ξ ∈ i(µ − , +∞)− i(C π/2+γ + ∪ {0}), and any contour L + ω 1 ,ω,b ⊂ i(µ − , µ + ) + (C γ − ,γ + ∪ {0}) lying above ξ, (2.16) φ − q (ξ) = exp − 1 2πi L + ω 1 ,ω,b ξ ln(1 + ψ(η)/q) η(ξ − η) dη . See Fig. 1 for an example of the curves L ± ω 1 ,b,ω . The integrals are efficiently evaluated making the change of variables ξ = χ ω 1 ,b,ω (y) and applying the simplified trapezoid rule. Remark 2.1. In the process of deformation, the expression 1 + ψ(ξ)/q may not assume value zero. In order to avoid complications stemming from analytic continuation to an appropriate Riemann surface, it is advisable to ensure that 1 + ψ(ξ)/q ∈ (−∞, 0]. Thus, if q > 0 (and only positive q's are used in the Gaver-Stehfest method or Gaver-Wynn Rho algorithm) and X is a SL-process, any sinh-deformation is admissible in (2.15) and (2.16). If the sinh-acceleration is applied to the Bromwich integral, then additional conditions on the parameters of the deformations must be imposed. See Sect. 4.3. Remark 2.2. In the remaining part of the paper, we assume that the Wiener-Hopf factors φ ± q (ξ), q > 0, admit the representations φ ± q (ξ) = a ± q + φ ±± q (ξ) and E ± q = a ± q I + E ±± q , where a ± q ≥ 0, and φ ±± (ξ) satisfy the bounds \|φ +,+ q (ξ)\| ≤ C + (q)(1 + \|ξ\|) −ν + , Im ξ ≥ µ − , (2.17) \|φ −,− q (ξ)\| ≤ C + (q)(1 + \|ξ\|) −ν − , Im ξ ≤ µ + , (2.18) where ν ± > 0 and C ± (q) > 0 are independent of ξ. These conditions are satisfied for all popular classes of Lévy processes bar the driftless Variance Gamma model. See Sect. A.1 for details. 3. Expectations of functions of the Lévy process and its extremum 3.1. Main theorems. Let f be measurable and uniformly bounded on U + := {(x 1 , x 2 ) \| x 2 ≥ 0, x 1 ≤ x 2 }. Fix (x 1 , x 2 ) ∈ U + . The function R + T → V (f ; T ; x 1 , x 2 ) is measurable and uniformly bounded, hence,Ṽ (f ; q; x 1 , x 2 ), the Laplace transform of V (f ; T ; x 1 , x 2 ) w.r.t. T , is a well-defined analytic function of q in the right half-plane. Assuming thatṼ (f ; q; x 1 , x 2 ) is sufficiently regular, V (f ; T ; x 1 , x 2 ) can be represented by the Bromwich integral (3.1) V (f ; T ; x 1 , x 2 ) = 1 2πi Re q=σ e qTṼ (f ; q; x 1 , x 2 ) dq, where σ > 0 is arbitrary. We derive an analytical representation for V (f ; q; x 1 , x 2 ) = q −1 E[f (x 1 + X Tq , max{x 2 , x 1 +X Tq })], where q > 0 and T q is an exponentially distributed random variable of mean 1/q, independent of X, and prove that the resulting expression forṼ (f ; q; x 1 , x 2 ) admits analytic continuation to the right half-plane. One can impose additional general conditions on X and f which ensure thatṼ (f ; q; x 1 , x 2 ) is sufficiently regular so that (3.1) holds. Such general conditions are either too messy or exclude some natural examples, for which the regularity can be established on the case-by-case basis. A standard trick which is used in [14,13,8] is as follows. Firstly, (3.1) holds in the sense of generalized functions. One integrates by parts in (3.1), and proves that the derivativeṼ q (f ; q; x 1 , x 2 ) is of class L 1 as a function of q. Hence, V (f ; T ; x 1 , x 2 ) equals the RHS of (3.1) with −T −1Ṽ q (f ; q; x 1 , x 2 ) in place ofṼ (f ; q; x 1 , x 2 ) . After that, one proves that it is possible to integrate by parts back and obtain (3.1) for T > 0. In examples that we consider, the integrands are of essentially the same form as in [14,13,8] for barrier options, and enjoy all properties that are used in [14,13,8] to justify (3.1). In the theorem below, I denotes the identity operator, f + is the extension of f to R 2 by zero, and ∆ is the diagonal map: ∆(x) = (x, x). Theorem 3.1. Let X be a Lévy process on R, q > 0, and let f : U + → R be a measurable and uniformly bounded function s.t. ((E − q ⊗ I)f ) • ∆ : R → R is measurable. Then (1) for any x 1 ≤ x 2 , qṼ (f ; q; x 1 , x 2 ) = ((E q ⊗ I)f + )(x 1 , x 2 ) + (E + q w(f ; q, ·, x 2 ))(x 1 ), (3.2) where (3.3) w(f ; q, y, x 2 ) = 1 [x 2 ,+∞) (y)(((E − q ⊗ I)f + )(y, y) − ((E − q ⊗ I)f + )(y, x 2 )); (2) the RHS' of (3.2) and (3.3) admit analytic continuation w.r.t. q to the right half-plane. Proof. Lemma 2.1 allows us to apply Fubini's theorem. For x 1 ≤ x 2 , we have E[f + (x 1 + X Tq , max{x 2 , x 1 +X Tq })] = E[f + (x 1 + X Tq −X Tq +X Tq , max{x 2 , x 1 +X Tq })] = E[((E − q ⊗ I)f + )(x 1 +X Tq , max{x 2 , x 1 +X Tq })] = E[((E − q ⊗ I)f + )(x 1 +X Tq , x 2 )] +E[1 x 1 +X Tq ≥x 2 (((E − q ⊗ I)f + )(x 1 +X Tq , x 1 +X Tq ) − ((E − q ⊗ I)f + )(x 1 +X Tq , x 2 ))]. Using (2.1), we write the first term on the rightmost side as ((E q ⊗ I)f + )(x 1 , x 2 ), and finish the proof of (i). As operators acting in the space of bounded measurable functions, E ± q admit analytic continuation w.r.t. q to the right half-plane, which proves (ii). Remark 3.1. The inverse Laplace transform of q −1 (E q ⊗I)f + (x 1 , x 2 ) equals E[f (x 1 +X T , x 2 )] , and, therefore, can be easily calculated using the Fourier transform technique. Essentially, we have the price of the European option of maturity T , the riskless rate being 0, depending on x 2 as a parameter. Thus, the new element is the calculation of the second term on the RHS of (3.2). We calculate both terms in the same manner in order to facilitate the explanation of various blocks of our method. Theorem 3.2. Let a Lévy process X on R, function f : U + → R and real q > 0 satisfy the following conditions (a) there exist µ − ≤ 0 ≤ µ + such that ∀ γ ∈ [µ − , µ + ], E[e −γX 1 ] < ∞ and q + ψ(iγ) > 0; (b) f is a measurable function admitting the bound (3.4) \|f (x 1 , x 2 )\| ≤ C(x 2 )e −µ + x 1 , where C(x 2 ) is independent of x 1 ≤ x 2 ; (c) the function ((E − q ⊗ I)f ) • ∆ is measurable and admits the bound (3.5) \|((E − q ⊗ I)f )(x 1 , x 1 )\| ≤ Ce −µ − x 1 , where C is independent of x 1 ≥ 0. Then the statements (i)-(iii) of Theorem 3.1 hold. Proof. It suffices to consider non-negative f . Define f n (x 1 , x 2 ) = min{n, f (x 1 , x 2 )}, n = 1, 2, . . .. By the dominated convergence theorem,Ṽ (f n ; q; x 1 , x 2 ) ↑Ṽ (f ; q; x 1 , x 2 ), a.s., and since f n is bounded, (3.2) holds with f n in place of f . We rewrite (3.2) in the form qṼ (f ; q; x 1 , x 2 ) = E[((E − q ⊗ I)f + )(x 1 +X Tq ,x 2 )1 x 1 +X Tq <x 2 ] (3.6) +E[((E − q ⊗ I)f + )(x 1 +X Tq , x 1 +X Tq )1 x 1 +X Tq ≥x 2 ], and denote byW 1 (f n ; q; x 1 , x 2 ) +W 2 (f n ; q; x 1 , x 2 ) the sum on the RHS of (3.6) with f n in place of f . Fix x 2 . On the strength of (a) and (3.5), W 2 (f n ; q; x 1 , x 2 ) admits a bound via CE + q e −µ − x 1 ≤ C 1 φ + q (iµ − )e −µ − x 1 , where C is independent of n. On the strength of (b) and (3.4),W 2 (f n ; q; x 1 , x 2 ) admits a bound via C(x 2 )E + q E − q e −µ + x 1 ≤ C 1 (x 2 )e −µ + x 1 , where C 1 (x 2 ) is independent of n. Operators E ± q being positive and bounded in L ∞ -spaces with weights e γx , γ ∈ [−µ − , −µ − ] (see Lemma 2.2, (v)), the limit of the RHS' ofW 1 (f n ; q; x 1 , x 2 ) +W 2 (f n ; q; x 1 , x 2 ) is finite and equal to to the RHS of (3.6). Remark 3.2. For functions of a Lévy process and its running minimum, results are mirror reflections of the results for a Lévy process and its supremum: change the direction of the real axis, flip the lower and upper half-plane and operators E ± q . Let V (G; h; T ; x) be the price of the barrier option with the payoff G(X T ) at maturity and no rebate if the barrier h is crossed before or at time n; the rsikless rate is 0. Applying Theorem 3.2, we obtain the formula for the price of the single barrier options, which is equivalent to the formula derived in [12,14,15,13] for wide classes of Lévy processes and generalized for all Lévy processes in [9]. Theorem 3.3. Let the Lévy process X on R and q ∈ (0, 1) satisfy condition (a) of Theorem 3.2, and let G be a measurable function admitting the bound \|G( x)\| ≤ C(e −µ + x + e −µ − x ), where C is independent of x ∈ R. Then, for x < h, (3.7)Ṽ (G; h; q, x) = q −1 (E q G)(x) − q −1 (E + q 1 [h,+∞) E − q G)(x). 3.2. Integral representation of the Laplace transform of the value function. In this Section, we assume that q > 0. The RHS' of the formulas for the Wiener-Hopf factors and formulas that we derive below admit analytic continuation w.r.t. q so that the inverse Laplace transform can be applied. We assume that the representations E ± q = a ± q I + E ±,± q (see Remark 2.2 and Lemma A.1) hold. This excludes the driftless Variance Gamma model which requires a separate treatment. Using the equality w(f ; q, x 1 , x 2 ) = 1 [x 2 ,+∞) (x 1 )(((E − q ⊗ I)f + )(x 1 , x 1 ) − ((E − q ⊗ I)f + )(x 1 , x 2 )) = 0, x 1 ≤ x 2 , we write the second term on the RHS of (3.2) as (3.8) (E + q w(f ; q, ·, x 2 ))(x 1 ) = (E ++ q w(f ; q, ·, x 2 ))(x 1 ), and (3.3) as (3.9) w(f ; q, y, x 2 ) = a − q w 0 (y, x 2 ) + w − (f ; q, y, x 2 ), where w 0 (y, x 2 ) = 1 [x 2 ,+∞) (y)(f + (y, y) − f + (y, x 2 )), and (3.10) w − (f ; q, y, x 2 ) = 1 [x 2 ,+∞) (y)(((E −− q ⊗ I)f + )(y, y) − ((E −− q ⊗ I)f + )(y, x 2 )). Substituting (3.9) into (3.8), we obtain (3.11) (E + q w(f ; q, ·, x 2 ))(x 1 ) = c − q (E ++ q ⊗ I)w 0 )(x 1 , x 2 ) + ((E ++ q ⊗ I)w − )(f ; q, x 1 , x 2 ). In order to derive explicit integral representations for the terms on the RHS of (3.11), we impose the following conditions, which can be relaxed: (a) condition (a) of Theorem 3.2 is satisfied; (b) there exist µ − , µ + ∈ (µ − , µ + ), µ − < µ + such that f admits bounds \|f (x 1 , x 2 )\| ≤ C(x 2 )e −µ + x 1 , x 1 ≤ x 2 , (3.12) \|((E − q ⊗ I)f + )(x 1 , x 1 )\| ≤ Ce −µ − x 1 , x 1 ∈ R, (3.13) where C(x 2 ) and C are independent of x 1 ≤ x 2 , and x 1 ∈ R, respectively; (c) for any x 2 , there exists C(x 2 ) > 0 such that \| (f + ) 1 (ξ 1 , x 2 )\| ≤ C(x 2 )(1 + \|ξ 1 \|) −1 , ξ 1 ∈ S [µ + ,µ + ] , (3.14) \| (w 0 ) 1 (η, x 2 )\| ≤ C(x 2 )(1 + \|η\|) −1 , η ∈ S [µ − ,µ − ] ; (3.15) (d) there exists C > 0 such that for ξ 1 ∈ S [µ + ,µ + ] and ξ 2 ∈ S [µ − ,µ − ] , (3.16) \| (f + )(ξ 1 , ξ 2 )\| ≤ C(1 + \|ξ 1 \|) −1 (1 + \|ξ 2 \|) −1 .(3.17) ω, ω 1 ∈ (µ + , µ + ), ω 2 ∈ (µ − , µ − ), ω − ∈ (µ − , ω 1 + ω 2 ) , and x 1 ≤ x 2 , we havẽ V (f ; q; x 1 , x 2 ) = 1 2π Im ξ=ω e ix 1 ξ 1 q + ψ(ξ 1 ) (f + )(ξ 1 , x 2 )dξ 1 (3.18) + a − q 2πq Im η=ω − e ix 1 η φ ++ q (η) (w 0 ) 1 (η, x 2 )dη + 1 2πq Im η=ω − e i(x 1 −x 2 )η φ ++ q (η) w − 0 (f ; q, η, x 2 )dη, where w − 0 (f ; q, η, x 2 ) is given by w − 0 (f ; q, η, x 2 ) (3.19) = 1 2π Im ξ 1 =ω 1 dξ 1 e ix 2 ξ 1 i(ξ 1 − η) φ −− q (ξ 1 )( f + ) 1 (ξ 1 , x 2 ) + 1 (2π) 2 Im ξ 1 =ω 1 Im ξ 2 =ω 2 dξ 1 dξ 2 e ix 2 (ξ 1 +ξ 2 ) i(η − ξ 1 − ξ 2 ) φ −− q (ξ 1 )( f + )(ξ 1 , ξ 2 ). Proof. The first term on the RHS of (3 .2) is ((E q ⊗ I)f + )(x 1 , x 2 ) , and the first term on the RHS of (3.18) is q −1 ((E q ⊗ I)f + )(x 1 , x 2 ) . Consider the second term. Since (3.15) holds and φ ++ q (η) = O(\|η\| −ν + ) as η → ∞ in the strip S [µ − ,µ + ] , where ν + > 0, the integral (3.20) ((E ++ q ⊗ I)w 0 )(x 1 , x 2 ) = 1 2π Im η=ω − e ix 1 η φ ++ q (η) (w 0 ) 1 (η, x 2 )dη is absolutely convergent. It remains to consider (E ++ q w − (f ; q, ·, x 2 ))(x 1 ). If Im η = ω − , w − (f ; q, η, x 2 ) = − +∞ x 2 dy e −iyη 1 2π Im ξ 1 =ω 1 dξ 1 e iξ 1 y φ −− q (ξ 1 )( f + ) 1 (ξ 1 , x 2 ) + +∞ x 2 dy e −iyη 1 (2π) 2 Im ξ 1 =ω Im ξ 2 =ω 2 dξ 1 dξ 2 e i(ξ 1 +ξ 2 )y φ −− q (ξ 1 )( f + )(ξ 1 , ξ 2 ). We apply Fubini's theorem to the first integral. The integral +∞ x 2 dy e i(−η+ξ 1 )y = e ix 2 (ξ 1 −η) i(η−ξ 1 ) converges absolutely since −ω − + ω 1 > 0, and the repeated integral converges absolutely because φ −− q (ξ) is uniformly bounded on the line of integration and (3.14) holds. Similarly, since −ω − + ω 1 + ω 2 > 0, the integral +∞ x 2 dy e i(−η+ξ 1 +ξ 2 )y = e ix 2 (ξ 1 +ξ 2 −η) /(i(η − ξ 1 − ξ 2 )) converges absolutely. Since φ −− q(ξ) = O(\|ξ 1 \| −ν − ) as ξ 1 → ∞ along the line of integration, where ν − > 0, (3.16) holds, and (3.21) R R dξ 1 dξ 2 (1 + \|ξ 1 + ξ 2 \|) −1 (1 + \|ξ 1 \|) −1−ν − (1 + \|ξ 2 \|) −1 < ∞, the Fubini's theorem is applicable to the second integral as well. Thus, (3.22) w − (f ; q, η, x 2 ) = e −iηx 2 w − 0 (f ; q, η, x 2 ), where w − 0 (f ; q, η, x 2 ) is given by (3.19), and we obtain the triple integral (3.23) (E ++ q w − (·, x 2 ))(x 1 ) = 1 2π Im η=ω − e i(x 1 −x 2 )η φ ++ q (η) w − 0 (f ; q, η, x 2 )dη. The integrand admits a bound via Cg(η, ξ 1 , ξ 2 ), where g(η, ξ 1 , ξ 2 ) = (1 + \|η\|) −ν + (1 + \|η − ξ 1 − ξ 2 \|) −1 (1 + \|ξ 1 \|) −1−ν − (1 + \|ξ 2 \|) −1 . Since (3.24) R 3 g(η, ξ 1 , ξ 2 )dη dξ 1 dξ 2 < ∞() := (E −− q ⊗ I)f + (y, y) − (E −− q ⊗ I)f + (y, x 2 ) is a linear combination of exponential functions (with the coefficients depending on x 2 ). Then w − (q; η, x 2 ) can be calculated directly, the double integral on the RHS of (3.19) can be reduced to 1D integrals, and the condition (3.16) replaced with the condition on h similar to (3.15). Analogous simplifications are possible in more involved cases when h is a piece-wise exponential polynomial in y. Two examples. 3.3.1. Example I. The joint cpdf of X T andX T . The case ν ± > 0. For a 1 ≤ a 2 , and x 1 ≤ x 2 , set f (x 1 , x 2 ) = 1 (−∞,min{a 1 ,x 2 }] (x 1 )1 (−∞,a 2 ] (x 2 ) and consider V (f ; T, x 1 , x 2 ) = Q[x 1 + X T ≤ a 1 , x 2 +X T ≤ a 2 ]. If x 2 > a 2 , then V (f ; T, x 1 , x 2 ) = 0. Hence, we assume that x 2 ≤ a 2 . Theorem 3.5. Let q > 0, a 1 ≤ a 2 , x 1 ≤ x 2 ≤ a 2 , and let X satisfy conditions of Theorem 3.4. Then, for any µ − < ω − < 0 < ω 1 < µ + , V (f ; q, x 1 , x 2 ) (3.25) = 1 2π Im ξ 1 =ω 1 e i(x 1 −a 1 )ξ 1 −iξ 1 (q + ψ(ξ 1 )) dξ 1 + 1 (2π) 2 q Im η=ω − dη e i(x 1 −a 2 )η φ ++ q (η) Im ξ 1 =ω 1 dξ 1 e iξ 1 (a 2 −a 1 ) φ −− q (ξ 1 ) ξ 1 (ξ 1 − η) . Proof. We have f + (x 1 , x 2 ) = 1 (−∞,a 1 ] (x 1 )1 (−∞,a 2 ] (x 2 ), therefore, for x 2 ≤ a 2 , w 0 (y, x 2 ) = 1 [x 2 ,+∞) (y)1 (−∞,a 1 ] (y)(1 (−∞,a 2 ] (y) − 1 (−∞,a 2 ] (x 2 )) = −1 [x 2 ,+∞) (y)1 (−∞,a 1 ] (y)1 (a 2 ,+∞) (y) = 0, hence, the second term on the RHS of (3.18) is 0. Next, (f + ) 1 (ξ 1 , x 2 ) = 1 (−∞,a 2 ] (x 2 ) a 1 −∞ e −ix 1 ξ 1 dξ 1 = 1 (−∞,a 2 ] (x 2 ) e −ia 1 ξ 1 −iξ 1 dξ 1 is well-defined in the upper half-plane, and satisfies the bound (3.14) in any strip S [µ + ,µ + ] , where µ + ∈ (0, µ + ). Hence, the first term on the RHS of (3.18) becomes the first term on the RHS of (3.25). It remains to evaluate the double integral on the RHS of (3.18). As mentioned in Remark 3.3, in the present case, it is simpler to directly evaluate w − and then w − : for any x 2 ≤ a 2 , ω 1 ∈ (0, µ + ) and any η ∈ {Im η ∈ (µ − , ω 1 )}, w − (q, y, x 2 ) = 1 (x 2 ,+∞) (y)(E −− q 1 (−∞,a 1 ] )(y)(1 (−∞,a 2 ] (y) − 1) = −1 [a 2 ,+∞) (y)(E −− q 1 (−∞,a 1 ] )(y) = −1 (a 2 ,+∞) (y) 1 2π Im ξ 1 =ω 1 dξ 1 e i(y−a 1 )ξ 1 φ −− (ξ 1 ) −iξ 1 , w − (q, η, x 2 ) = − +∞ a 2 e −iyη 1 2π Im ξ 1 =ω 1 dξ 1 e i(y−a 1 )ξ 1 φ −− (ξ 1 ) −iξ 1 (3.26) = − e −ia 2 η 2π Im ξ 1 =ω 1 dξ 1 e i(a 2 −a 1 )ξ 1 φ −− (ξ 1 ) i(η − ξ 1 )(−iξ 1 ) . It is easy to see that both integrals are absolutely convergent. Substituting (3.26) into the double integral on the RHS of (3.18), we obtain (3.25). Remark 3.4. If x 1 > a 1 , then it advantageous to move the line of integration in the first integral on the RHS of (3.25) down, and, on crossing the simple pole, apply the residue theorem. In the result, the first term on the RHS turns into 1 q + 1 2π Im η=ω − e i(x 1 −a 1 )η −iη(q + ψ(η)) dη. Remark 3.5. The first step of the proof of Theorem 3.5 implies that we can replace φ −− q in the double integral on the RHS of (3.25) with φ − q . From the computational point of view, if we make the conformal change of variables, this change does not lead to a significant increase in sizes of arrays necessary for accurate calculations, especially if a 2 − a 1 > 0. The advantage is that it becomes unnecessary to evaluate a − q . Recall that the same a − q appears for all ξ 1 in the formula φ −− q (ξ 1 ) = φ − q (ξ 1 ) − a − q , hence, it is necessary to evaluate a − q with a higher precision that φ − q (ξ 1 ). At the same time, the integrand in the formula for a − q decays slower at infinity than the integrand in the formula for φ − q (ξ 1 ). Remark 3.6. Denote by I 2 (q; x 1 , x 2 ) the double integral on the RHS of (3.25) multiplied by q. It follows from (3.8) that we can replace φ ++ q in the double integral with φ + q . If a 1 < a 2 and the conformal deformations are used, then this replacement causes no serious computational problems. If a 1 = a 2 , then the replacement leads to errors typical for the Fourier inversion at points of discontinuity. However, in this case, the RHS of (3.25) can be simplified as follows. We replace φ ±,± q with φ ± q , which is admissible, then push the line of integration in the inner integral down, cross two simple poles at ξ 1 = 0 and ξ 1 = η, and apply the residue theorem. The double integral becomes the following 1D integral: I 2 (q; x 1 , x 2 ) = 1 2π Im η=ω − dη e i(x 1 −a 2 )η φ + q (η)(1 − φ − q (η)) −iη . We push the line of integration to {Im η = ω 1 } and use the equality φ + q (η)φ − q (η) = q/(q + ψ(η)) to obtain the formula for the perpetual no-touch option: qṼ (f, q; x 1 , x 2 ) = 1 2π Im ξ 1 =ω 1 dξ 1 e i(x 1 −a 2 )ξ 1 φ + q (ξ 1 ) −iξ 1 , x 1 ≤ x 2 ≤ a 2 .I 2 (q; x 1 , x 2 ) = 1 4π Im ξ 1 =ω 1 dξ 1 e i(x 1 −a 1 )ξ 1 φ ++ q (ξ 1 )φ −− q (ξ 1 ) −iξ 1 + 1 (2π) 2 v.p. Im η=ω 1 dη e i(x 1 −a 2 )η φ ++ q (η) Im ξ 1 =ω 1 dξ 1 e iξ 1 (a 2 −a 1 ) φ −− q (ξ 1 ) ξ 1 (ξ 1 − η) , where v.p. denotes the Cauchy principal value. After that, one can apply the fast Hilbert transform. The integrand decaying very slowly at infinity, therefore, accurate calculations are possible only if very long grids are used, hence, the CPU cost is very large even for a moderate error tolerance. 3.3.2. Example II. Option to exchange the supremum for a power of the underlying. Let β > 1. Consider the option to exchange the supremumS T = eX T for the power S β T = e βX T . The payoff function f (x 1 , x 2 ) = (e βx 1 − e x 2 ) + 1 (−∞,x 2 ] (x 1 ) satisfies (3.12)-(3.13) with arbitrary µ + > 0, µ − < −β. In Sect. A.3, we prove Proposition 3.6. Let β > 1 and let conditions of Theorem 3.4 hold with µ − < −β, µ + > 0. Then, for x 1 ≤ x 2 , and any 0 < ω 1 < µ + , µ − < ω − < −β, (3.28)Ṽ (f ; q, x 1 , x 2 ) = I 1 (q, x 1 , x 2 ) + q −1 j=2,3 I j (q, x 1 , x 2 ), where I j (q, x 1 , x 2 ), j = 1, 2, 3, are given by I 1 (q, x 1 , x 2 ) = 1 2π Im ξ 1 =ω 1 dξ 1 e i(x 1 −x 2 )ξ 1 q + ψ(ξ 1 ) (3.29) · e x 2 β β − iξ 1 + β e x 2 (1+iξ 1 (1−1/β)) (β − iξ 1 )(−iξ 1 ) − e x 2 −iξ 1 , I 2 (q, x 1 , x 2 ) = a − q e x 2 2π Im η=ω − dη e i(x 1 −x 2 )η φ ++ q (η) iη(1 − iη) , (3.30) and I 3 (q, x 1 , x 2 ) (3.31) = 1 (2π) 2 Im η=ω − dη e i(x 1 −x 2 )η φ ++ q (η) Im ξ 1 =ω 1 dξ 1 e −ix 2 ξ 1 φ −− q (ξ 1 ) i(η − ξ 1 ) · e βx 2 iη − β + βe (1+iξ 1 (1−1/β)x 2 (1 − iξ 1 /β) (β − iξ 1 )(−iξ 1 )(iη − 1 − iξ 1 (1 − 1/β)) − e x 2 (1 − iξ 1 ) (−iξ 1 )(iη − 1) . 4. Numerical evaluation of V (f ; T ; x 1 , x 2 ) in Example I 4.1. Standing assumption. In this section, we assume that X is a SINH-regular process of order ν and type ([µ − , µ + ], C γ − ,γ + , C γ − ,γ + ), where µ − < 0 < µ + and γ − < 0 < γ + . Furthermore, we assume that either ν ≥ 1 or ν < 1 and the "drift" µ in (2.5) is 0. Then (4.1) Re ψ(ξ) ≥ c ∞ \|ξ\| ν − C, ∀ ξ ∈ i(µ − , µ + ) + (C γ − ,γ + ∪ {0}), where C, c ∞ > 0 are independent of ξ. Eq. (4.1) allows us to use one sinh-deformed contour in the lower half-plane and the other one in the upper half-plane for all purposes: the calculation of the Wiener-Hopf factors and evaluation of the integrals on the RHS of (3.25). If either µ − = 0 or µ + = 0, then both contours must cross iR in the same half-plane but the types of contours (two non-intersecting contours, one with the wings deformed upwards, the other one with the wings deformed downwards) remain the same as in the case µ − < 0 < µ + . If (4.1) fails, for instance, if ν < 1 and µ = 0, then the contour of integration in the formulas for the Wiener-Hopf factors can be deformed only upwards (if µ > 0) or downwards (if µ < 0). Hence, we need to use an additional contour to evaluate the Wiener-Hopf factors. Even more importantly, the conformal deformations can be used only if the Gaver-Stephest method or Gaver-Wynn Rho algorithm are used or the line of integration in the Bromwich integral is not deformed; conformal deformations of the contours of integration in the formula for V (f ; q, x 1 , x 2 ) and the Bromwich integral are impossible if we want to preserve the analyticity of the double and triple integrands. To see this, it suffices to consider the degenerate case ψ(ξ) = −iµξ: the conditions q − iµξ 1 ∈ (0, ∞], q − iµη ∈ (0, ∞] are impossible to satisfy if Re q → −∞, and the ξ 1 -and η-contours are deformed upward and downward. Hence, we can either use the Gaver-Wynn Rho algorithm (see Sect. A.7) or acceleration schemes of the Euler type, e.g., the summation by parts formula (see Sect. A.6). See Sect. A.8 for details. Finally, if either γ − = 0 or γ + = 0 (but not both), then additional complications arise, and some of deformations have to be of a less efficient sub-polynomial type. See [31] for examples in the context of calculation of stable probability distributions. 4.2. Sinh-acceleration. Consider the first term on the RHS of (3.25), denote it I 1 (q; x 1 − a 1 ). As ξ → ∞ along the line of integration, the integrand decays not faster than \|ξ\| −3 . The error of the truncation \|j\|≤N of the infinite sum j∈Z in the infinite trapezoid rule is approximately equal to the error of the truncation Λ −Λ , Λ = N ζ, of the integral +∞ −∞ , hence, for a small error tolerance > 0, Λ must be of the order of −1/2 , and the complexity of the numerical scheme of the evaluation of the integral is of the order of −1/2 ln(1/ ). If x 1 − a 1 is not small in the absolute value, acceleration schemes of the Euler type can be employed to decrease the number of terms of the simplified trapezoid rule. If x 1 − a 1 is zero or very close to 0, Euler acceleration schemes are rather inefficient. If X is SINH-regular, the rate of convergence of the simplified trapezoid rule to the infinite trapezoid rule can be greatly increased using appropriate conformal deformations of the line of integration and the corresponding changes of variables. Assuming that in Definition 2.3, γ ± are not extremely small is absolute value, the sinh-acceleration (2.14) is the most efficient change of variables (typically, the sinh-acceleration is the best choice even if \|γ ± \| are of the order of 10 −5 ). Note that in (2.14), ω 1 ∈ R is, generally, different from ω 1 in the formulas of the preceding sections, ω ∈ (−π/2, π/2) and b > 0. The parameters ω 1 , b, ω are chosen so that the contour L ω 1 ,b,ω := χ ω 1 ,b,ω (R) ⊂ i(0, µ + ) + (C γ − ,γ + ∪ {0}). The parameter ω is chosen so that the oscillating factor becomes a fast decaying one. Under the integral sign of the integral I 1 (q; x 1 − a 1 ), the oscillating factor is e i(x 1 −a 1 )ξ 1 . Hence, if x 1 < a 1 , we must choose ω ∈ (γ − , 0) (an approximately optimal choice is ω = γ − /2), if x 1 − a 1 > 0, we must choose ω ∈ (0, γ + ) (an approximately optimal choice is ω = γ + /2), and if x 1 = a 1 , any ω ∈ (γ − , γ + ) is admissible (an approximately optimal choice is ω = (γ − + γ + )/2). If x 1 − a 1 < 0, it is advantageous to push the line of integration in the 1D integral into the lower half-plane, and, on crossing the pole, apply the residue theorem. To evaluate the repeated integral on the RHS of (3.25), we deform both lines of integration. Since a 2 − a 1 > 0, it is advantageous to deform the wings of the contour of integration w.r.t. ξ 1 up; denote this contour L + := L ω + 1 ,b + ,ω + . Since x 1 − a 2 ≤ 0, it is advantageous to deform the wings of the contour of integration w.r.t. η down, denote this contour L − := L ω − 1 ,b − ,ω − . Hence, we choose ω + = γ + /2, ω − = γ − /2; the remaining parameters are chosen so that Fig. 1. The result is µ + > ω + 1 + b + sin ω + > 0 > ω − 1 + b − sin ω − > µ − . SeeV (f ; q, x 1 , x 2 ) = 1 2π L ω 1 ,b,ω qe i(x 1 −a 1 )ξ 1 (q + ψ(ξ 1 ))(−iξ 1 ) dξ 1 (4.2) + 1 (2π) 2 q L − dη e i(x 1 −a 2 )η φ + q (η) L + dξ 1 e iξ 1 (a 2 −a 1 ) φ − q (ξ 1 ) ξ 1 (ξ 1 − η) . We make an appropriate sinh-change of variables in each integral, and apply the simplified trapezoid rule w.r.t. each new variable. 4.3. Calculations using the sinh-acceleration in the Bromwich integral. Define (4.3) χ L;σ ,b ,ω (y) = σ + ib sinh(iω + y), where ω ∈ (0, π/2), b > 0, σ −b sin ω > 0, and deform the line of integration in the Bromwich integral to L (L) = χ L;σ ,b ,ω (R). For q ∈ L (L) , we can calculateṼ (f ; q, x 1 , x 2 ) using the same algorithm as in the case q > 0, if there exist R, q 0 , γ > 0 and γ −− < 0 < γ ++ such that q + ψ(η) = 0 for all q ∈ L (L) and η ∈ C γ −− ,γ ++ , \|η\| ≥ R. In order to avoid the complications of the evaluation of the logarithm on the Riemann surface, it is advisable to ensure that 1 + ψ(η)/q ∈ (−∞, 0] for pairs (q, η) used in the numerical procedure. See Fig. 2 for an illustration. These conditions can be satisfied if (4.1) holds. The sequence of deformations is as follows. First, for q on the line of integration {Im q = σ} in the Bromwich integral, we deform the contours of deformation w.r.t. η and ξ 1 (and contours in the formulas for the Wiener-Hopf factors). Then we deform the line of integration w.r.t. q into the contour L (L) . We choose ω and ω ± sufficiently small in absolute value so that, in the process of deformation, for all 1 + ψ(ξ)/q = 0 and q + ψ(η)/q = 0 for all dual variables q, η, ξ 1 that appear in the formulas forṼ (f ; q; x 1 , x 2 ) and formulas for the Wiener-Hopf factors. To make an appropriate choice, the bound (4.1) must be taken into account. See [26] for details. In [26], fractional-parabolic deformations and changes of variables were used. The modification to the sinh-acceleration is straightforward. 4.4. The main blocks of the algorithm. For the sake of brevity, we omit the block for the evaluation of the 1D integral on the RHS of (3.25); this block is the same as in the European option pricing procedure (see [30]); the type of deformation depends on the sign of x 1 − a 1 . For the 2D integral, the scheme is independent of x 1 −a 1 . We formulate the algorithm assuming that the sinh-acceleration is applied to the Bromwich integral; if the Gaver-Wynn Rho algorithm is used, the modifications of the first step and last step are as described in Sect. A.7. We calculate F (T, a 1 , a 2 ) = V (T, a 1 , a 2 ; 0, 0) (that is, x 1 = x 2 = 0). Step I. Choose the sinh-deformation in the Bromwich integral and grid for the simplified trapezoid rule: y = ζ * (0 : 1 : N ), q = σ + i * b * sinh(i * ω + y). Calculate the derivative der = i * b * cosh(i * ω + y). Step II. Choose the sinh-deformations and grids for the simplified trapezoid rule on L ± : y ± = ζ ± * (−N ± : 1 : N ± ), ξ ± = i * ω ± 1 + b ± * sinh(i * ω ± + i y ± ). Calculate ψ ± = ψ( ξ ± ) and der ± = b ± * cosh(i * ω ± + y ± ). Step III. Calculate the arrays D + = [1/(ξ + j − ξ − k )] and D − = [1/(ξ − k − ξ + j )] (the sizes are (2 * N + + 1) × (2 * N − + 1) and (2 * N − + 1) × (2 * N + + 1), respectively). Step IV. The main block (the same block is used if the Gaver-Wynn Rho algorithm is applied). For given x 1 , x 2 , a 1 , a 2 , in the cycle in q ∈ q, evaluate (1) φ + q at points of the grid L + and φ − q at points of the grid L − using (2.15)-(2.16): φ ± q = exp ((∓ζ ± * i * ζ ∓ /(2 * π)) * ξ ± . * (log(1 + ψ ∓ /q)./ ξ ∓ . * der ∓ ) * D ± ) ; (2) calculate φ ± q at points of the grid L ∓ : φ ± q,∓ = q./(q + ψ ∓ )./ φ ∓ q ; (3) evaluate the 2D integral on the RHS of (3.25) Int2(q) = ((ζ − * ζ + /(2 * π) 2 ) * (exp(−i * a 2 * ξ − ). * φ + q,− . * der − ) * D + ) * conj((exp((i * (a 2 − a 1 )) * ξ + ). * φ − q,+ / ξ + . * der + ) ). (4) depending on the sign of x 1 −a 1 , use either the arrays ξ + , der + , Φ + or ξ − , der − , Φ − to evaluate Int1(q), the 1D integral on the RHS of (3.25). Step V. Laplace inversion. Set Int( q) = Int2( q)./ q + Int2( q), Int(q 1 ) = Int(q 1 )/2, and, using the symmetryṼ (q) =Ṽ (q), calculate V = (ζ /π) * real(sum(exp(T * q). * Int( q). * der )). 4.5. Numerical examples. Numerical results are produced using Matlab R2017b on Mac-Book Pro, 2.8 GHz Intel Core i7, memory 16GB 2133 MHz. The CPU times reported below can be significantly improved because (a) the main block of the program, namely, evaluation ofṼ (q) for a given array of (a 1 , a 2 ), is used both for complex and positive q's. However, if q > 0, we can use the well-known symmetries to decrease the sizes of arrays, hence, the CPU time. Furthermore, the block admits the trivial parallelization; (b) we use the same grids for the calculation of the Wiener-Hopf factors φ ± q and evaluation of integrals on the RHS of (4.2). However, φ ± q need to be evaluated only once and used for all points (a 1 , a 2 ). But if x 1 − a 2 and a 2 − a 1 are not very small in absolute value, then much shorter grids can be used to evaluate the integrals on the RHS of (3.25). See examples in [27,50,30,32]. Therefore, if the arrays (x 1 − a 2 , a 2 − a 1 ) are large, then the CPU time can be decreased using shorter arrays for calculation of the integrals on the RHS of (4.2). (c) If the values F (T, a 1 , a 2 ) are needed for several values of T in the range [T 1 , T 2 ], where T 1 is not too close to 0 and T 2 is not too large, then the CPU time can be significantly decreased if the sinh-acceleration is applied to the Bromwich integral. Indeed, the main step is time independent, and the last step, which is the only step where T appears, admits an easy parallelization. Hence, the CPU time for many values of T is essentially the same as for one value of T . Item (a), and, partially, (b) are motivated by our aim to compare the performance of the algorithm based on the Gaver-Wynn Rho algorithm and the one based on the sinh-acceleration applied to the Bromwich integral. Since the same subprogram for the evaluation ofṼ (q) is used in both cases, and, even in the more complicated second case, we can achieve the precision of the order of E − 14, we can safely say that the errors in the first case are the errors of the Gaver-Wynn Rho algorithm itself 2 ; and these errors are of the order of E − 7 in the cases we considered (sometimes, larger, in other cases, somewhat smaller), which agrees with the general empirical observation E − 0.9M , for all choices of the parameters of the numerical scheme. The errors remain essentially the same even if we use much finer and longer grids in the η-and ξ-spaces than it is necessary. The second motivation for (b) is that we wish to give a relatively short description of the choice of the main parameters of the numerical scheme. In the two examples that we consider, X is KoBoL with the characteristic exponent ψ(ξ) = cΓ(−ν)(λ ν + − (λ + + iξ) ν + (−λ − ) ν − (−λ − − iξ) ν ) , where λ + = 1, λ − = −2 and (I) ν = 0.2, hence, the process is close to Variance Gamma; (II) ν = 1.2, hence, the process is close to NIG. In both cases, c > 0 is chosen so that the second instantaneous moment m 2 = ψ (0) = 0.1. For X 0 =X 0 = 0, we calculate the joint cpdf F (T, a 1 For SL-processes, and KoBoL is an SL-process, any sinh-deformation is admissible provided q 1 +ψ(i(ω 1 −b sin(ω)) > 0 for the smallest q = q 1 > 0 used in the Gaver-Wynn Rho algorithm. If q 1 +ψ(i(ω 1 −b sin(ω)) ≤ 0 then we can reduce the calculations to the case q 1 +ψ(i(ω 1 −b sin(ω)) > 0 using the simple trick (A.13) or take the purely imaginary zero of q + ψ(ξ) into account explicitly as in [51]. In the example that we consider, q 1 + ψ(i(ω 1 − b sin(ω)) > 0. For SLprocesses, the choice of the most important parameters ω ± trivializes: ω ± = ±π/4·min{1, 1/ν}, and the half-width d ± of the strips of analyticity in the new coordinates is d = \|ω ± \|. It can be easily shown that, for the Merton model and Meixner processes, one can choose ω ± = ±π/8 and d ± = \|ω ± \| (see [33] for the analysis of the domain of analyticity and zeros of q + ψ(ξ) for popular Lévy models). Thus, given the error tolerance , we can easily write a universal approximate recommendation for the choice of ζ. The recommendation for an approximately optimal choice of the truncation parameter Λ = N ζ is the same as in [32]; both lead to grids somewhat longer, typically, 1.2-1.5 times longer than necessary. Choosing the parameters by hand, we observe that the results with the errors of the order of E-7, which are inevitable with the Gaver-Wynn Rho algorithm, can be achieved using the sinh-acceleration in the ξ-and ηspaces, with grids of the length 100 or even smaller (depending on ν and T ). It is evident that if the calculations are made using the Hilbert transform or simplified trapezoid rule without the conformal deformations, then much longer arrays will be needed (thousand times longer and more) to satisfy even larger error tolerance, and the increase of the speed due to the use 2 We use the Gaver-Wynn Rho algorithm with M = 8, hence, 16 positive values of q (depending on T ) appear. M = 7 does not work because the error of the Gaver-Wynn Rho algorithm itself is too large, M = 9 does not work because some of the coefficients are so large that qṼ (q) must be calculated with high precision Table 1. Joint cpdf F (T, a 1 , a 2 ) := Q[X T ≤ a 1 ,X T ≤ a 2 \| X 0 =X 0 = 0], and errors (rounded) and CPU time (in msec) of two numerical schemes. KoBoL close to Variance Gamma, with an almost symmetric jump density, and no "drift": m 2 = 0.1, ν = 0.2, λ − = −2, λ + = 1; T = 0.25. of the fast Hilbert transform or fast convolution and fast inverse Fourier transform cannot compensate for the very large increase of the sizes of the arrays. If the sinh-acceleration in the Bromwich integral is used, then we can satisfy the error tolerance of the order E-14 and smaller using the q-grids of the order of 100-150, and the ξand η-grids of the order of 250−450. We use two types of deformations: (I) ω = (π/2)/9, ω ± = ±(π/2)/4.5 · min{1, 1/ν} ("+" for L + , "-" for L − ) and (II) ω = (π/2)/10, ω ± = ±(π/2)/5 · min{1, 1/µ}. Since each of the three curves has changed, the probability of a random agreement between the two results is negligible. The differences being less than e−14, with some exceptions in the case T = 15, we take these values as the benchmark. The errors in Tables 1 and 2 are calculated w.r.t. the benchmark probabilities. The CPU time for the benchmark prices is in the range 5-8 msec, for one pair (a 1 , a 2 ), and 35-60 msec for 44 points (average of 100 runs). Choosing the parameters by hand, we calculated prices with errors somewhat smaller than the errors of the Gaver-Wynn Rho algorithm. The ξ-and η-grids can be chosen shorter than in the case of the Gaver-Wynn Rho algorithm but the length of the q-grid is several times larger than 16 in the Gaver-Wynn Rho algorithm. In the result, the CPU time is several times larger. Remark 4.1. The factor min{1, 1/ν} is needed to ensure that the image of the strip of analyticity S (−d,d) in the y-coordinate under the map y → q + ψ(χ ω 1 ,b,ω (y)), used to satisfy the error tolerance for the infinite trapezoid rule, does not cross the imaginary axis. We give the universal recommendation which can be used for all SL-processes. As the matter of fact, in the case of KoBoL, NIG and Variance Gamma, the analytic continuation across the cuts is admissible. Then larger (in absolute value) ω's can be used and the CPU times documented in Tables 1-2 decreased. See [27], where the analytical extension to the Riemann surface was used in the context of pricing European options using the fractional-parabolic deformations. We will analyze this improvement in a separate publication. Remark 4.2. The reader observes that in the case ν = 0.2 (process is close to Variance Gamma, Table 1), the target precision can be achieved at a smaller computational cost than in the case ν = 1.2 (process is close to NIG, Table 2). For any method that does not explicitly use Table 2. Errors (rounded) and CPU time (in msec) of two numerical schemes for the calculation of the joint cpdf F (T, a 1 , a 2 ) := Q[X T ≤ a 1 ,X T ≤ a 2 \| X 0 =X 0 = 0]; T = 0.25. KoBoL close to NIG, with an almost symmetric jump density, and no "drift": m 2 = 0.1, ν = 1.2, λ − = −2, λ + = 1. The benchmark values (for T = 0.05, 0.25, 1, 5, 15) are in Table 3 in Section B. the conformal deformation technique, one expects that the case ν = 0.2 must be much more time consuming because the integrands decay much slower than in the case ν = 1.2. However, due to the factor min{1, 1/ν}, we can use a larger step in the infinite trapezoid rule in the case ν = 0.2, and the truncation parameter Λ = N ζ is essentially the same for all ν unless ν is very close to 0. Conclusion In the paper, we derive explicit formulas for the Laplace transforms of expectations of functions of a Lévy process on R and its running maximum, in terms of the EPV operators E ± q (factors in the operator form of the Wiener-Hopf factorization). If the explicit formulas can be efficiently realized for q's used in a numerical realization of the Bromwich integral, then the expectations can be efficiently calculated. Standard applications to finance are options with barrier and lookback features, with flat barriers. In the paper, we consider in detail numerical realizations for wide classes of Lévy processes with the characteristic exponents admitting analytic continuation to a strip around or adjacent to the real axis, equivalently, with the Lévy density of either positive or negative jumps decaying exponentially at infinity. Thus, we allow for a stable Lévy component of negative jumps 3 . The numerical part of the paper is a two-step procedure. First, we derive explicit formulas in terms of a sum of 1D-3D integrals; in many cases of interest, the triple integrals are reducible to double integrals over the Cartesian product of two flat contour in the complex plane. As applications, we calculate the cpdf of the Lévy process and its maximumX and the price of the option to exchange eX T for a power e βX T . The repeated integrals can be calculated using the simplified trapezoid rule and the Fast Fourier transform technique (or fast convolution or fast Hilbert transform) if the expectations need to be calculated at many points in the state space. In popular Lévy models, the characteristic exponent admits analytic continuation to a union of a strip and cone around or adjacent to the real line. Then the computational cost can be decreased manifold using the conformal deformation technique. We use the most efficient version: the sinh-acceleration, and explain how the deformations of several contours should be made: two contours for each q > 0 used in the Gaver-Wynn Rho algorithm, and three contours if the sinh-acceleration method is applied to the Bromwich integral. Numerical examples demonstrate the efficiency of the method; the conformal deformation technique applied to the Bromwich integral achieves the precision of the order of E-14 and the Gaver-Wynn Rho algorithm -of the order of E-6-E-8. However, the latter is faster. Note that Talbot's deformation cannot be applied if the conformal deformations technique is applied to the integrals with respect to the other dual variables. As in [14,16,21,20], the results and proofs admit straightforward reformulations for the case of random walks, barrier and lookback options with discrete monitoring in particular. The methodology of the paper can be extended in several directions, and adapted to (1) stable Lévy processes, in the same vein as the sinh-acceleration and several other types of conformal deformations are adapted to evaluation of stable distributions in [31]; (2) barrier and lookback options with time-dependent barriers, similarly to [48,20,24], where American options are priced using multistep maturity randomization (method of lines); (3) regime-switching Lévy models, with different payoff functions in different states, similarly to [22,23,7]; (4) models with stochastic volatility and stochastic interest rates. The first step, namely, approximation by regime-switching models, is the same as in [23,19,35]; (5) models with stochastic interest rates, when the eigenfunction expansion is used to approximate the action of the infinitesimal generator of the process for the interest rates [29]; (6) models with non-standard payoffs arising in applications to real options and Game Theory [25,18,28]. Generalizations of the method of [48,20,24] to American options with lookback features and the method of [7,10,51] to options with two barriers are not so straightforward but possible. [14,15,13] for the class of RLPE (Regular Lévy processes of exponential type); the proof for SINH-regular processes is the same only ξ is allowed to tend to ∞ not only in the strip of analyticity but in the union of a strip and cone. See [8,49,51] for the proof of the statements below for several classes of SINH-regular processes (the definition of the SINH-regular processes formalizing properties used in [8,49,51] was suggested in [30] later.). The contours of analyticity in Lemma A.1 below are in a domain of analyticity s.t. q − iµξ = 0 and 1 + ψ 0 (ξ)/(q − iµξ) ∈ (−∞, 0]. The latter condition is needed when ψ 0 (ξ) = O(\|ξ\| ν ) as ξ → ∞ in the domain of analyticity and ν < 1. Clearly, in this case, for sufficiently large q > 0, the condition holds. In the case of RLPE's, the contours of integration in the lemma below are straight lines in the strip of analyticity Lemma A.1. Let µ − < 0 < µ + , q > 0, let X be SINH-regular of type ((µ − , µ + ), C γ − ,γ + , C γ − ,γ + ), µ − < 0 < µ + , and order ν. Then (1) if ν ∈ [1, 2] ∪ {1+} or ν ∈ (0, 1) and the "drift" is µ = 0, then neitherX Tq nor X Tq has an atom at 0, and φ ± q (ξ) admit the bounds (2.17) and (2.18), where ν ± > 0 and C ± (q) > 0 are independent of ξ; (2) if ν ∈ (0, 1) ∪ {0+} and µ > 0, then (a)X Tq has no atom at 0 and X Tq has an atom a − q δ 0 at zero, where (A.1) a − q = exp 1 2π L + ω 1 ,b,ω ln((1 + ψ 0 (η)/(q − iµη)) η dη , and L + ω 1 ,b,ω is a contour as in Lemma 2.6 (ii), lying above 0; (b) for ξ and L − ω 1 ,b,ω in Lemma 2.6 (i), φ + q (ξ) admits the representation (A.2) φ + q (ξ) = q q − iµξ exp 1 2πi L − ω 1 ,b,ω ξ ln(1 + ψ(η)/(q − iµη)) η(ξ − η) dη , and satisfies the bound (2.17) with ν + = 1; (c) φ − q (ξ) = a − q + φ −− q (ξ), where φ −− q (ξ) satisfies (2.18) with arbitrary ν − ∈ (0, 1 − ν); (d) E − q = a − q I + E −− q , where E −− q is the PDO with the symbol φ −− q (ξ);(3) if ν ∈ (0, 1) ∪ {0+} and µ < 0, then (a) X Tq has no atom at 0 andX Tq has an atom a + q δ 0 at zero, where (A.3) a + q = exp 1 2π L − ω 1 ,b,ω ln((1 + ψ 0 (η)/(q − iµη)) η dη , and L − ω 1 ,b,ω is a contour as in Lemma 2.6 (i), lying below 0; (b) for ξ and L + ω 1 ,b,ω in Lemma 2.6 (ii), φ − q (ξ) admits the representation (A.4) φ − q (ξ) = q q − iµξ exp f rac12πi L + ω 1 ,b,ω ξ ln(1 + ψ(η)/(q − iµη)) η(ξ − η) dη , and satisfies the bound (2.18) with ν − = 1; (c) φ + q (ξ) = a + q + φ ++ q (ξ), where φ ++ q (ξ) satisfies (2.17) with arbitrary ν + ∈ (0, 1 − ν); (d) E + q = a + q I + E ++ q , where E ++ q is the PDO with the symbol φ ++ q (ξ); A.2. Proof of bounds (3.21) and (3.24). First, we prove that if a, b > 0, then g a,b defined by g a,b (ξ 1 , ξ 2 ) = (1 + \|ξ 1 + ξ 2 \|) −a (1 + \|ξ 1 \|) −1−b (1 + \|ξ 2 \|) −1 is of class L 1 (R 2 ) . Consider separately regions U j ⊂ R 2 , j = 1, 2, 3, defined by inequalities \|ξ 2 \| ≤ \|ξ 1 \|/2; \|ξ 2 \| ≥ 2\|ξ 1 \|; \|ξ 1 \|/2 ≤ \|ξ 2 \| ≤ 2\|ξ 1 \|, respectively. On U 1 , g a,b (ξ 1 , ξ 2 ) ≤ C 1 (1 + \|ξ 1 \|) −1−a−b (1 + \|ξ 2 \|) −1 ≤ C 2 (1 + \|ξ 1 \|) −1−b (1 + \|ξ 2 \|) −1−a , and the function on the RHS is of class L 1 (R 2 ). On U 2 , g a,b (ξ 1 , ξ 2 ) admits an upper bound via the same function (and a different constant C 2 ). Finally, U 3 dξ 1 dξ 2 g a,b (ξ 1 , ξ 2 ) ≤ C 3 R dξ 1 ln(2 + \|ξ 1 \|)(1 + \|ξ 1 \|) −1−b < ∞, which proves (3.21). To prove (3.24), we consider the restrictions of g on the regions U j ⊂ R 3 , j = 1, 2, 3, defined by the inequalities \|η\| ≤ \|ξ 1 + ξ 2 \|/2; \|η\| ≥ 2\|ξ 1 + ξ 2 \|; \|ξ 1 + ξ 2 \|/2 ≤ \|η\| ≤ 2\|ξ 1 + ξ 2 \|. On U 1 , \|g(η, ξ 1 , ξ 2 )\| ≤ C 1 (1 + \|η\|) −ν + (1 + \|ξ 1 + ξ 2 \|) −1 (1 + \|ξ 1 \|) −1−ν − (1 + \|ξ 2 \|) −1 ≤ C 2 (1 + \|η\|) −ν + /2−1 (1 + \|ξ 1 + ξ 2 \|) −ν + /2 (1 + \|ξ 1 \|) −1−ν − (1 + \|ξ 2 \|) −1 , on U 2 , \|g(η, ξ 1 , ξ 2 )\| ≤ C 1 (1 + \|η\|) −ν + −1 (1 + \|ξ 1 \|) −1−ν − (1 + \|ξ 2 \|) −1 ≤ C 2 (1 + \|η\|) −ν + /2−1 (1 + \|ξ 1 + ξ 2 \|) −ν + /2 (1 + \|ξ 1 \|) −1−ν − (1 + \|ξ 2 \|) −1 . In each case, the function on the RHS' is of the form C(1 + \|η\|) −1−ν + /2 g ν + /2,ν − (ξ 1 , ξ 2 ), hence, of class L 1 (R 3 ). To prove the integrability of g on U 3 , it suffices to note that \|ξ 1 +ξ 2 \|/2≤\|η\|≤2\|ξ 1 +ξ 2 \| dη \|g(η, ξ 1 , ξ 2 )\| ≤ C 3 ln(2 + \|ξ 1 + ξ 2 \|)g ν + ,ν − (ξ 1 , ξ 2 ), and the RHS admits an upper bound via C 4 g ν + /2,ν − (ξ 1 , ξ 2 ). A.3. Proof of Proposition 3.6. We apply Theorem 3.4 with µ + ∈ (0, µ + ), µ − ∈ (µ − , −β). For x 2 > 0 and ξ ∈ C, ( f + ) 1 (ξ 1 , x 2 ) = x 2 x 2 /β e −ix 1 ξ 1 (e βx 1 − e x 2 )dx 1 = e −ix 2 ξ 1 e x 2 β β − iξ 1 + β e x 2 (1+iξ 1 (1−1/β)) (β − iξ 1 )(−iξ 1 ) − e x 2 −iξ 1 , hence, the first term on the RHS of (3.18) equals the integral on the RHS of (3.29). Then we calculate w 0 (y, x 2 ) = 1 [x 2 ,+∞) (y)((e βy − e y ) − (e βy − e x 2 )) = 1 [x 2 ,+∞) (y)(e x 2 − e y ), w 0 (η, x 2 ) = +∞ x 2 e −iyη (e x 2 − e y )dy = e x 2 −ix 2 η iη(1 − iη) , and obtain that the second term on the RHS of (3.18) equals the RHS of (3.30). Next, we calculateŵ − (q, η, x 2 ): w − (q, η, x 2 ) = +∞ x 2 e −iyη 1 2π Im ξ 1 =ω 1 dξ 1 e iyξ 1 φ −− q (ξ 1 ) e (β−iξ 1 )y − e (β−iξ 1 )x 2 β − iξ 1 +β e (1−iξ 1 /β)y − e (1−iξ 1 /β)x 2 (β − iξ 1 )(−iξ 1 ) − e (1−iξ 1 )y − e (1−iξ 1 )x 2 −iξ 1 = e −ix 2 η 2π Im ξ 1 =ω 1 φ −− q (ξ 1 ) e (β−iξ 1 )x 2 β − iξ 1 1 i(η − ξ 1 ) − (β − iξ 1 ) − 1 i(η − ξ 1 ) + βe (1−iξ 1 /β)x 2 (β − iξ 1 )(−iξ 1 ) 1 i(η − ξ 1 ) − (1 − iξ 1 /β) − 1 i(η − ξ 1 ) − e (1−iξ 1 )x 2 −iξ 1 1 i(η − ξ 1 ) − (1 − iξ 1 ) − 1 i(η − ξ 1 ) = e −ix 2 η 2π Im ξ 1 =ω 1 dξ 1 φ −− q (ξ 1 ) i(η − ξ 1 ) e (β−iξ 1 )x 2 iη − β + βe (1−iξ 1 /β)x 2 (1 − iξ 1 /β) (β − iξ 1 )(−iξ 1 )(iη − 1 − iξ 1 (1 − 1/β)) − e (1−iξ 1 )x 2 (1 − iξ 1 ) (−iξ 1 )(iη − 1) , and, finally, derive the representation (3.31) for the double integral on the RHS of (3.18). A.4. General remarks on numerical Laplace inversion. The final result is obtained applying a chosen numerical Laplace inversion procedure toṼ (q, ·, ·) defined by (3.25). The methods that we construct (main texts: [26,51,29,32]) can be regarded as further steps in a general program of study of the efficiency of combinations of one-dimensional inverse transforms for high-dimensional inversions systematically pursued by Abate-Whitt, Abate-Valko [2, 3, 1, 58, 4] and other authors. Additional methods can be found in [56]. Abate and Valko and Abate and Whitt consider three main different one-dimensional algorithms for the numerical realization of the Bromwich integral: (1) Fourier series expansions with Euler summation (the summation-by-part formula in Sect. A.6 can be regarded as a special case of Euler summation); (2) combinations of Gaver functionals, and (3) deformation of the contour in the Bromwich integral. Talbot's contour deformation q = rθ(cot θ + i), −π < θ < π, is suggested, and various methods of multi-dimensional inversion based on combinations of these three basic blocks are discussed. It is stated that for the popular Gaver-Stehest method, the required system precision is about 2.2 * M , and about 0.9 * M significant digits are produced for f (t) with good transforms. "Good" means that f is of class C ∞ , and the transform's singularities are on the negative real axis. If the transforms are not good, then the number of significant digits may be not so great and may be not proportional to M . In our previous publications [26,51], we develop numerical methods for pricing barrier and lookback options based on the fractional-parabolic deformations, and observed that when we were able to evaluateṼ (q) with the precision E-10 and better, the Gaver-Stehfest method with M = 8, produced fairly accurate results (errors of the order of E-4 or even E-5) although, according to the general remark in [4], V (q)'s had to be calculated with the precision E-15. However, in many cases, the fractionalparabolic acceleration requires too long grids and the accumulation of errors of the calculation of the Wiener-Hopf factors leads to the failure of the Gaver-Stehfest method. If the simplified trapezoid rule, without acceleration, is applied to the integral under the exponential sign on the RHS' of (2.12) and (2.13), then the arrays of the size of the order of 10 9 and more are needed. Hence, sufficiently accurate calculations (nothing to say fast) are impossible. Indeed, the integrands decay slower than \|η\| −2 as η → ∞ in the strip of analyticity. In [7,9,10], it is demonstrated that Carr's randomization (equivalently, the method of lines) allows one to calculate prices of single and double barrier options and barrier options in regimeswitching models with the precision of the order of E-02-E-03 because Carr's randomization procedure works even if the calculations at each time step are with the precision of the order of E-04-E-05 only. In [9,10], calculations are relatively fast because grids of different sizes for the evaluation of the Wiener-Hopf factors and fast convolution at each time step and the refined version of the inverse FFT (iFFT) constructed in [9] are used (standard iFFT and fractional iFFT do not suffice in the majority of cases). In [7], regime-switching hyper-exponential jump diffusion models are considered, hence, the Wiener-Hopf factors are easy to calculate with the precision E-14. In the present paper, as in [32], we use the sinh-acceleration to evaluate the Wiener-Hopf factors. The summation of several hundreds of terms suffices to achieve the precision better than E-15, hence, the effect of accumulation of machine errors is insignificant, and we can calculateṼ (q) with the precision E-14 and better. Thus, the errors of the Gaver-Wynn Rho algorithm which we document are the errors of the algorithm itself. These errors are in the range E-05-E-8, depending on the parameters of the model, T and a 1 , a 2 . For the sake of brevity, we produce the results for x 1 = x 2 = 0; T, a 1 and a 2 vary. More accurate results are obtained when we apply the sinh-acceleration to the Bromwich integral. The CPU cost increases several times because the number of q's used is several times larger; but we can achieve the precision E-14 and better. Note that a more efficient Talbot deformation cannot be applied ifṼ (q) is evaluated using the sinh-acceleration technique. Hence, the best two versions are: the Gaver-Wynn Rho algorithm, if the accuracy of the final result of the order of E-6 is admissible, and the sinh-acceleration applied to the Bromwich integral if a higher precision is needed. In both cases, the Wiener-Hopf factors andṼ (q)'s are calculated using the sinh-acceleration. \|g(ia + y)\|dy < ∞ is finite. We write g ∈ H 1 (S (−d,d) ). The integral I = R g(ξ)dξ can be evaluated using the infinite trapezoid rule (A.6) I ≈ ζ j∈Z g(jζ), where ζ > 0. The following key lemma is proved in [56] using the heavy machinery of sincfunctions. A simple proof can be found in [50]. Once an approximately bound for H(g, d) is derived, it becomes possible to satisfy the desired error tolerance with a good accuracy. A.6. Summation by parts. The rate of decay of the series can be significantly increased if the infinite trapezoid rule is of the form I(a) = ζ j∈Z e −iajζ g(jζ), where a ∈ R \ 0, and g (y) decreases faster than g(y) as y → ±∞. Indeed, then, by the mean value theorem, the finite differences ∆g j = (∆g)(jζ), where (∆g)(ξ) = g(ξ + ζ) − g(ξ), decay faster than g(jζ) as j → ±∞ as well. The summation by parts formula is as follows. Let e iaζ − 1 = 0. Then ζ j∈Z e −iajζ g(jζ) = ζ e iaζ − 1 j∈Z e −iajζ ∆g j . If additional differentiations further increase the rate of decay of the series as j → ±∞, then the summation by part procedure can be iterated: (A.8) ζ j∈Z e −iajζ g j = ζ (e iaζ − 1) n j∈Z e −iajζ ∆ n g j . After the summation by parts, the series on the RHS of (A.8) needs to be truncated. The truncation parameter can be chosen using the following lemma. Lemma A.3. Let n ≥ 1, N > 1 be integers, ζ > 0, a ∈ R and e iaζ − 1 = 0. Let g (n) be continuous and let the function ξ → G n (ξ, ζ) := max η∈[ξ,ξ+nζ] \|g (n) (η)\| be in L 1 (R). Then Table 3. Joint cpdf F (T, a 1 , a 2 ) := Q[X T ≤ a 1 ,X T ≤ a 2 \| X 0 =X 0 = 0]. KoBoL close to NIG, with an almost symmetric jump density, and no "drift": m 2 = 0.1, ν = 1.2, λ − = −2, λ + = 1. EVALUATION OF EXPECTATIONS OF FUNCTIONS OF A LÉVY PROCESS AND ITS EXTREMUM 33 S Numerical evaluation of V (f ; T ; x 1 , x 2 ) in Example .B.: Department of Economics, The University of Texas at Austin, 2225 Speedway Stop C3100, Austin, TX 78712-0301, [email protected] S.L.: Calico Science Consulting. Austin, TX. Email address: [email protected]. Theorem 3. 4 . 4Let conditions (a)-(d) hold and let the representations of the Wiener-Hopf factors in Remark 2.2 be valid. Then, for any ω, ω 1 , ω 2 and ω − satisfying ,(3.27) can be obtained using the main theorem directly. Remark 3. 7 . 7One can push the line of integration in the outer integral in the double integral on the RHS of (3.25) up and obtain , a 2 ) := V (T, a 1 , a 2 ; 0, 0) for T = 0.25 in Case (I) and for T = 0.05, 0.25, 1, 5, 15 in Case (II). In both cases, a 1 is in the range [−0.075, 0.1] and a 2 in the range [0.025, 0.175]; the total number of points (a1 , a 2 ), a 1 ≤ a 2 , is 44. The parameters of the numerical schemes are chosen as follows. the benchmark values: better than e-14. CPU time per 1 point: 118, per 44 points: 1,089. A: Gaver-Wynn Rho algorithm, 2M = 16, N ± = 110. CPU time per 1 point: 6.4; per 44 points: 44.3. B: SINH applied to the Bromwich integral, with N = 65, N ± = 91. CPU time per 1 point 13.3, per 44 points: 175. If in A, N ± = 115 instead of N ± = 110 are used, the rounded errors do not change but the CPU time increases. the benchmark values: better than e-14. CPU time per 1 point: 305, per 44 points: 3,160. A: Gaver-Wynn Rho algorithm, 2M = 16, N ± = 110. CPU time per 1 point: 8.7; per 44 points: 58.1. B: SINH applied to the Bromwich integral, with N = 79, N ± = 115. CPU time per 1 point 22.3, per 44 points: 203. If in A, N ± = 115 are used, the rounded errors do not change. A. 5 . 5Infinite trapezoid rule. Let g be analytic in the strip S (−d,d) := {ξ \| Im ξ ∈ (−d, d)} and decay at infinity sufficiently fast so that lim A→±∞ d −d \|g(ia + A)\|da = 0, and (A.5) H(g, d) := g H 1 (S (−d,d) ) := lim a↓−d R \|g(ia + y)\|dy + lim a↑d R Lemma A. 2 2([56], Thm.3.2.1). The error of the infinite trapezoid rule admits an upper bound (A.7) Err disc ≤ H(g, d) exp[−2πd/ζ] 1 − exp[−2πd/ζ] . Figure 1 . 1e iaζ − 1) n j≤−N e −iajζ ∆ n g j ≤ ζ \|e iaζ − 1\| n −N ζ −∞ G n (ξ, ζ)dξ.(A.10) Example of curves L + (solid line) and L − (dash-dots). Example with λ + = 1, λ − = −2, ν = 1.2. Dots: boundaries of the domain of analyticity around L + used to derive the bound for the discretization error of the infinite trapezoid rule in the y-coordinate: dotted lines become straight lines {Im y = ±d}. Figure 2 . 2Plots of curves η → 1 + ψ(η)/q, for q in the SINH-Laplace inversion and η on the contours L ± (upper and lower panels) in the numerical example with ν = 1.2. Lemma 2.1. ([40, Lemma 2.1], and see Sect. A.2 for the proof), the triple integral on the the RHS of (3.23) is absolutely conver-Remark 3.3. In standard situations such as in the two examples that we consider below, the function y → h(ygent. Substituting (3.11), (3.20) and (3.23) into (3.2), we obtain (3.18). The property does not hold if there is no such a strip (formally, λ− = 0 = λ+). The classical example are stable Lévy processes. The conformal deformation technique can be modified for this case as well[31]. A polynomially decaying stable Lévy tail is important for applications to risk management, however, from the computational point of view, the cases of two exponentially decaying tails and only one exponentially decaying tail are essentially indistinguishable. Proof. Using the mean value theorem, we obtain \|(∆ n g)(ξ)\| ≤ ζ max ξ 1 ∈[ξ,ξ+ζ] \|(∆ n−1 g )(ξ 1 )\| ≤ · · · ≤ ζ n max η∈[ξ,ξ+nζ]\|g (n) (η)\|. A.7. Gaver-Wynn Rho algorithm. The inverse Laplace transform V (T ) ofṼ is approximated bywhere M ∈ N,and a denotes the largest integer that is less than or equal to a. If T is large which in applications to option pricing means options of long maturities, then q = k ln(2)/T is small. In the present paper, efficient calculations ofṼ (f ; q, x 1 , x 2 ) are possible if q ≥ σ, where σ > 0 is determined by the parameters of the process and payoff function. Hence, if T is large, we modify (3.1)where a > 0 is chosen so that ln(2)/T + a > max{−ψ(iµ − ), −ψ(iµ + )}. In the paper, as in[51,32], we apply Gaver-Wynn-Rho (GWR) algorithm, which is more stable than the Gaver-Stehfest method. Given a converging sequence {f 1 , f 2 , . . .}, Wynn's algorithm estimates the limit f = lim n→∞ f n via ρ 1 N −1 , where N is even, and ρ j k , k = −1, 0, 1, . . . , N , j = 1, 2, . . . , N − k + 1, are calculated recursively as follows:We apply Wynn's algorithm with the Gaver functionals(−1) j j f ((j + ) ln 2/T ).A.8. 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J.Inst.Math.Appl., 23:97-120, 1979. ). (I-) If x 1 − a 1 ≤ 0, it is advantageous to deform the line of integration downwards. Hence, the contour L ω 10 ,b 0 ,ω 0 in the new integral is defined by ω 0 < 0, and ω 10 ∈ R, b 0 > 0 such that that σ 0 := Im ψ(χ ω10,b 0 ,ω 0 (0)) = ω 10 + b 0 sin ω 0 ∈ (0, µ + ) and q + ψ(χ ω10,b 0 ,ω 0 (0) = q + µσ 0 + ψ 0 (iσ 0 )) > 0. Alternatively, one can push the line of integration below 0, apply the residue theorem (the additional term 1/q appears), and choose ω 0 < 0, and ω 10 ∈ R. A.8.1. Gaver-WynnRho algorithm is used. Consider 1D integral on the RHS of (3.25. b 0 > 0 so that that σ 0 := Im ψ(χ ω10,b 0 ,ω 0 (0)) = ω 10 + b 0 sin ω 0 ∈ (µ − , 0) and q + ψ(χ ω10,b 0 ,ω 0 (0A.8.1. Gaver-Wynn Rho algorithm is used. Consider 1D integral on the RHS of (3.25). (I-) If x 1 − a 1 ≤ 0, it is advantageous to deform the line of integration downwards. Hence, the contour L ω 10 ,b 0 ,ω 0 in the new integral is defined by ω 0 < 0, and ω 10 ∈ R, b 0 > 0 such that that σ 0 := Im ψ(χ ω10,b 0 ,ω 0 (0)) = ω 10 + b 0 sin ω 0 ∈ (0, µ + ) and q + ψ(χ ω10,b 0 ,ω 0 (0) = q + µσ 0 + ψ 0 (iσ 0 )) > 0. Alternatively, one can push the line of integration below 0, apply the residue theorem (the additional term 1/q appears), and choose ω 0 < 0, and ω 10 ∈ R, b 0 > 0 so that that σ 0 := Im ψ(χ ω10,b 0 ,ω 0 (0)) = ω 10 + b 0 sin ω 0 ∈ (µ − , 0) and q + ψ(χ ω10,b 0 ,ω 0 (0)) = Since x 1 − a 2 < 0, it is advantageous to deform the outer line of integration downwards. Hence, the contour L ω 1− ,b − ,ω − in the new integral is defined by ω − < 0, and ω 1− ∈ R, b − > 0 such that that σ − := Im ψ. II) Now we consider the 2D integral. χ ω 1− ,b − ,ω − (0II) Now we consider the 2D integral. Since x 1 − a 2 < 0, it is advantageous to deform the outer line of integration downwards. Hence, the contour L ω 1− ,b − ,ω − in the new integral is defined by ω − < 0, and ω 1− ∈ R, b − > 0 such that that σ − := Im ψ(χ ω 1− ,b − ,ω − (0)) = 0) and q + ψ(χ ω1−,b − ,ω − (0)) = q + µσ − + ψ 0 (iσ − ) > 0. Both conditions can be satisfied choosing suffciently small (in absolut value) ω 1− and b − . The inner contour is deformed upward. − + B − Sin Ω − ∈, and the same contour as in the case (I+) can be used− + b − sin ω − ∈ (µ − , 0) and q + ψ(χ ω1−,b − ,ω − (0)) = q + µσ − + ψ 0 (iσ − ) > 0. Both conditions can be satisfied choosing suffciently small (in absolut value) ω 1− and b − . The inner contour is deformed upward, and the same contour as in the case (I+) can be used. It can be shown that if γ − < 0 < γ + , the n-the derivative of q/(q + ψ(ξ)), each integrand in the formulas for the Wiener-Hopf factors, hence, the price are of the order of O(\|q\| −n ) as q = σ + iu → ∞ along the line of integration {Re q = σ}. Hence, applying the summation-by-parts procedure 3 times, one can reduce to the series which decays fairly fast, hence, the truncated sum with several hundreds of terms can satisfy a moderately small error tolerance. However, as in the case when the Gaver-Wynn Rho acceleration method is applied, the Wiener-Hopf factors have to be calculated for each q in the truncated sum. Since ψ 0 (η)/(q − iµη) → 0 as (q, η) → ∞ (q along the line of integration. A.8.2. Infinite trapezoid rule applied to the Bromwich integral. After the infinite trapezoid rule is applied, one can use the summation-by-parts procedure (see Sect. A.6. and η in the intersection of the half-plane {µ Im η > 0} and a domain of analyticity), the sinh-deformed contours for an efficient valuation of the Wiener-Hopf factors are easy to constructA.8.2. Infinite trapezoid rule applied to the Bromwich integral. After the infinite trapezoid rule is applied, one can use the summation-by-parts procedure (see Sect. A.6). It can be shown that if γ − < 0 < γ + , the n-the derivative of q/(q + ψ(ξ)), each integrand in the formulas for the Wiener-Hopf factors, hence, the price are of the order of O(\|q\| −n ) as q = σ + iu → ∞ along the line of integration {Re q = σ}. Hence, applying the summation-by-parts procedure 3 times, one can reduce to the series which decays fairly fast, hence, the truncated sum with several hundreds of terms can satisfy a moderately small error tolerance. However, as in the case when the Gaver-Wynn Rho acceleration method is applied, the Wiener-Hopf factors have to be calculated for each q in the truncated sum. Since ψ 0 (η)/(q − iµη) → 0 as (q, η) → ∞ (q along the line of integration, and η in the intersection of the half-plane {µ Im η > 0} and a domain of analyticity), the sinh-deformed contours for an efficient valuation of the Wiener-Hopf factors are easy to construct.	[]
[ "Reminiscence Of An Open Problem: Remarks On Nevanlinna's Four-Value-Theorem", "Reminiscence Of An Open Problem: Remarks On Nevanlinna's Four-Value-Theorem" ]	[ "Norbert Steinmetz [email protected] \nInstitut für Mathematik\nD-44221Dortmund, DortmundTUGermany\n" ]	[ "Institut für Mathematik\nD-44221Dortmund, DortmundTUGermany" ]	[]	The aim of this paper is to describe the origin, first solutions, further progress, the state of art, and a new ansatz in the treatment of a problem dating back to the 1920's, which still has not found a satisfactory solution and deserves to be better known.	null	[ "https://arxiv.org/pdf/1102.3383v1.pdf" ]	119,622,100	1102.3383	d4181439d2329dec78b7e26315aa3910f4d6efc2	Reminiscence Of An Open Problem: Remarks On Nevanlinna's Four-Value-Theorem 16 Feb 2011 Norbert Steinmetz [email protected] Institut für Mathematik D-44221Dortmund, DortmundTUGermany Reminiscence Of An Open Problem: Remarks On Nevanlinna's Four-Value-Theorem 16 Feb 2011AMS Mathematics Subject Classification(2000): 30D35Keywords: Nevanlinna theoryvalue-sharingfour-value-theorem The aim of this paper is to describe the origin, first solutions, further progress, the state of art, and a new ansatz in the treatment of a problem dating back to the 1920's, which still has not found a satisfactory solution and deserves to be better known. Introduction In [15] G. Pólya considered the problem to determine all pairs (f, g) of distinct entire functions of finite order, such that f and g assume each of the values a ν (1 ≤ ν ≤ 3) at the same points and with the same multiplicities, and solved it as follows. Theorem (Pólya [15]). The functions f and g have a common Picard value a 2 = 1 2 (a 1 + a 3 ), and satisfy (f − a 2 )(g − a 2 ) = (a 2 − a 1 )(a 3 − a 2 ). A typical example is f (z) = e z , g(z) = e −z , a 1 = −1, a 2 = 0, and a 3 = 1. In particular, it is not possible that f and g assume each of four finite values at the same points and with the same multiplicities. In modern terminology Pólya's theorem says that distinct non-constant entire functions of finite order cannot share four finite values by counting multiplicities, and may share three finite values only in some very particular case. The background for this theorem was as follows: if two polynomials, P and Q, say, assume integer values at the same points, then the entire functions f = e 2πiP and g = e 2πiQ have the Picard value 0 and share the value 1. Thus the problem arises to determine all entire functions f and g of finite order having Picard value 0 and sharing the value 1 by counting multiplicities. Theorem (Pólya [( 1 )]) Under these hypotheses either f = g or else f = 1/g holds, hence either P − Q or else P + Q is a constant (an integer multiple of 2πi). Since the Picard value ∞ is trivially shared by counting multiplicities, Pólya's first theorem may be looked at as a predecessor of what is nowadays known as Four-Value-Theorem due to R. Nevanlinna, while the proof of his second theorem inspired Nevanlinna to apply his theory of meromorphic functions to so-called Borel 1 We follow Nevanlinna's paper [13]; the reference "G. Pólya, Deutsche Math.-Ver. Bd. 32, S. 16, 1923" given there is incorrect. 1 identities. To describe these results in more detail we need some notation; familiarity with the standard notions and results of Nevanlinna's theory of meromorphic functions is assumed, see the standard references Nevanlinna [14] and Hayman [8]. Given any pair of distinct meromorphic functions f and g sharing the values a ν (1 ≤ ν ≤ q), we set T (r) = max T (r, f ), T (r, g) , and denote by S(r) the usual remainder term satisfying S(r) = O(log(r T (r))) (r → ∞) outside some set E ⊂ (0, ∞) of finite measure. If f and g have finite [lower] order we have S(r) = O(log r) and E is empty [S(r k ) = O(log r k ) on some sequence r k → ∞]. Furthermore, N (r, a ν ) =n(0, a ν ) + log r Five-Value-Theorem (Nevanlinna [13]). Let f and g be distinct meromorphic functions sharing the values( 2 ) a ν (1 ≤ ν ≤ q). Then q ≤ 4, and in case q = 4 the following is true: (N a ) T (r, f ) = T (r) + S(r) and T (r, g) = T (r) + S(r). (N b ) 4 ν=1 N (r, a ν ) = 2 T (r) + S(r). (N c ) N r, 1 f − g = 2 T (r) + S(r).( 3 ) (N d ) N r, 1 f − b = T (r) + S(r) and N r, 1 g − b = T (r) + S(r) (b = a ν ). The typical example is the same as for Pólya's Theorem. Four-Value-Theorem (Nevanlinna [13]). Let f and g be non-constant meromorphic functions sharing values a ν (1 ≤ ν ≤ 4), but now specifically by counting multiplicities. Then (relabelling the values, if necessary) (a 1 , a 2 , a 3 , a 4 ) = (f, g, a 3 , a 4 ) = −1 holds.( 4 ) In particular, a 1 and a 2 are Picard values of f and g, and g is a Möbius transformation of f that fixes a 3 and a 4 and permutes a 1 and a 2 . Both theorems were proved by R. Nevanlinna in 1926. The novelty and importance of this paper stems from the new and powerful methods, nowadays called Nevanlinna Theory, rather than the fact that meromorphic functions of arbitrary order were considered in contrast to entire functions of finite order. 2 Notably without hypothesis about the multiplicities. 3 To be modified if a 4 = ∞: N r, 1 f − g + N (r, ∞) = 2 T (r) + S(r). 4 (a, b, c, d) = (a − c)(b − d) (a − d)(b − c) denotes the cross-ratio of the values a, b, c, d. Progress and Counterexamples In [4] G. Gundersen attributed the question, whether the Four-Value-Theorem also holds without the condition by counting multiplicities, to L. Rubel. This question or problem, however, was already aware to Nevanlinna, who wrote in [13]: Es wäre nun interessant zu wissen, ob dieses Ergebnis auch dann besteht, wenn die Multiplizitäten der betreffenden Stellen nicht berücksichtigt werden. Einige im ersten Paragraphen gewonnene Ergebnisse [here he refers to conditions (N a )-(N d )] sprechen vielleicht für die Vermutung ... ( 5 ) G. Gundersen was the first to contribute to that problem. He proved the (3+1)-Theorem (Gundersen [4]). The conclusion of the Four-Value-Theorem remains true if f and g share four values, at least three of them by counting multiplicities. In the same paper, however, Gundersen also provided the first counterexample to Nevanlinna's conjecture and thus destroyed the hope for a (0+4)-Theorem. It is easily seen (and this is Gundersen's example) that the functions (1) f (z) = e z + 1 (e z − 1) 2 and g(z) = (e z + 1) 2 8(e z − 1) share the values 1, 0, ∞ and −1/8 in the following manner: f has only simple zeros and 1-points, and only double poles and (−1/8)-points, while g has only simple poles and (−1/8)-points, but has only double zeros and 1-points. Four years later, Gundersen was again able to relax the hypothesis on the number of values shared by counting multiplicities by proving the (2+2)-Theorem (Gundersen [5,6]). The conclusion of the Four-Value-Theorem remains true if f and g share four values, at least two of them by counting multiplicities. Gundersen's proof contained a gap, which, however, could be bridged over (see [6]) by considering the auxiliary function which E. Mues discovered in the early 1980's, but did not make use of until 1987. This function Ψ = f ′ g ′ (f − g) 2 4 ν=1 (f − a ν )(g − a ν ) (for a 4 = ∞ the factor for ν = 4 has to be omitted) contains the complete information: if f and g share the values a ν (1 ≤ ν ≤ 4), then Ψ is an entire function, which is small in the sense that T (r, Ψ) = S(r) holds. It is almost trivial to deduce the 5-Theorem from this condition of smallness, and it will soon be seen that the whole progress made afterwards depends almost completely on Mues' function Ψ. 5 It would be interesting to know whether this result remains true regardless multiplicities. Some of the results derived in the first section seem to support the conjecture ... Gundersen's example may also be characterised by additional properties. This was done by M. Reinders (a student of Mues) in several directions, based on the following observations for the functions (1): (a) f (z 0 ) = − 1 2 ⇔ g(z 0 ) = 1 4 . (b) For ν fixed, one of the functions f and g has only a ν -points of order 2. Theorem (Reinders [16,17]). Assume that f and g share mutually distinct values a ν (1 ≤ ν ≤ 4), and that one of the following conditions holds: (a) There exist values a, b = a ν (1 ≤ ν ≤ 4) such that f (z 0 ) = a implies g(z 0 ) = b, and the conclusion of the Four-Value-Theorem does not hold. (b 1 ) For every ν the zeros of (f − a ν )(g − a ν ) have multiplicity 3. (b 2 ) For every ν either f − a ν or else g − a ν has only zeros of order 2. Then up to pre-composition with some non-constant entire function h, and postcomposition with some Möbius transformation, f and g coincide with the functions in Gundersen's example (1). We note that conditions (b 1 ) and (b 2 ) look quite different (although they turn out a posteriori to be equal), since in case (b 1 ), even for fixed ν, each of f and g may have double a ν -points, while this is not the case in (b 2 ). All attempts to prove or disprove a (1+3)-Theorem failed up to now (January 2011). Before describing further progress we mention a counterexample that is quite different from Gundersen's and has a different origin. It arose from a Comptes Rendus note of H. Cartan [2], where, in modern terminology, the following was stated: There do not exist three mutually distinct meromorphic functions sharing four values. The proof indicated in [2], however, contained a serious gap, as Mues pointed out to the author in the early 1980's. This time the gap could not be bridged over, and it took more than three years to find a way leading to the true statement and, by the way, to a new counterexample characterised by that theorem. Triple Theorem (Steinmetz [21]). Suppose that three mutually distinct meromorphic functions f, g, h share four values, 0, 1, ∞ and a, say. Then • a = −1 is a third root of −1, and • w = f (z), g(z), h(z) are solutions to the algebraic equation (2) w 3 + 3[(ā + 1)u 2 (z) + 2u(z)]w 2 − 3[2u 2 (z) + (a + 1)u(z)]w − u 3 (z) = 0. Here u = v • γ, where γ is some non-constant entire function and v some nonconstant solution to the differential equation (3) (v ′ ) 2 = 4v(v + 1)(v − a). Conversely, given a = 0, −1, the solutions to equation (3) are elliptic functions, and for a = 1 2 (1 ± i √ 3) and any non-constant entire function γ, the solutions to equation (2) with u = v • γ provides three meromorphic functions sharing the values 0, 1, ∞, and a. In the most simple case γ(z) = z these functions are elliptic functions of elliptic order six. They share the values 0, 1, ∞, and a in the following manner: every period parallelogram contains three c-points (c ∈ {0, 1, ∞, a}), each being simple for two of these functions, and having multiplicity four for the third one. Thus the sequence of c-points is divided in a natural way into three subsequences having asymptotically equal counting functions. Reinders constructed a counterexample that is quite different from Gundersen's. It is well-known that the non-constant solutions of the differential equation (4) (u ′ ) 2 = 12 u(u + 1)(u + 4), are elliptic functions of elliptic order two (actually u(z) = ℘( √ 3z + c) − 5/3, where ℘ is the specific P-function of Weierstrass that satisfies the differential equation (℘ ′ ) 2 = 4(℘ − 5 3 )(℘ − 2 3 )(℘ + 7 3 )). Then for any such function, (5) f = 1 8 √ 3 uu ′ u + 1 and g = 1 8 √ 3 (u + 4)u ′ (u + 1) 2 share the values −1, 0, 1, ∞ in the following manner: each of these values is assumed in an alternating way with multiplicity 1 by one of the functions f and g, and with multiplicity 3 by the other one (e.g., f has triply zeros at the zeros of u, and simple zeros when u + 4 = 0.) Again this example can be characterised by this particular property. Theorem (Reinders [18]). Let f and g be meromorphic functions sharing finite values a ν (1 ≤ ν ≤ 4), such that (f − a ν )(g − a ν ) has always zeros of multiplicity at least 4. Then up to pre-composition with some non-constant entire function and post-composition with some Möbius transformation, the functions f and g coincide with those in example (5). Remark. Suppose that there exist positive integers p < q, such that every shared value a ν is assumed by f with multiplicity p and by g with multiplicity q, or vice versa. Then considering Mues' function yields p = 1, and from (1 + q)N (r, a ν ) = N r, 1 f − a ν + N r, 1 g − a ν + S(r) ≤ 2T (r) + S(r) and condition (N b ) easily follows 2(1 + q)T (r) ≤ 8T (r) + S(r), hence 2 ≤ q ≤ 3. On combination with the theorems of Reinders this immediately yields: Theorem (Chen, Chen, Ou & Tsai [3]). Suppose f and g share four values a ν , and assume also that there are integers 1 ≤ p < q such that each a ν -point is either a (p, q)-fold point for (f, g) or for (g, f ). Then the conclusion of Reinders' theorems [16,17,18] hold. Proof of the (2+2)-Theorem Any progress till now relies on the fact that "(2+2) implies (4+0)", and was based on Mues' auxiliary function technique [10]. To show the power of this method we will next give independent and short proofs of the Four-Value-, the (3+1)-, and the (2+2)-Theorem. Proof of the (4+0)-Theorem-Mues [10]. Suppose f and g share four values a ν (1 ≤ ν ≤ 4) by counting multiplicities. Then at least two of the values a ν satisfy N (r, a ν ) = S(r). We may assume that a 4 = ∞ does, hence N (r, f ) = N (r, ∞)+S(r) and N (r, g) = N (r, ∞) + S(r) hold, while N (r, 1/f ′ ) + N (r, 1/g ′ ) = S(r) is an easy consequence of the strong assumption "by counting multiplicities". Thus the auxiliary function φ = f ′′ f ′ − g ′′ g ′ satisfies T (r, φ) = S(r) , but vanishes at all poles that are simple for f and g. This yields a contradiction, namely N (r, ∞) = S(r), if φ ≡ 0. If, however, φ vanishes identically, f = g follows at once. Proof of the (3+1)-Theorem-Rudolph [19]. Suppose f and g share the finite values a ν (1 ≤ ν ≤ 3) by counting multiplicities, and a 4 = ∞ without further hypothesis. Then φ = f ′ 3 ν=1 (f − a ν ) − g ′ 3 ν=1 (g − a ν ) satisfies T (r, φ) = S(r) and vanishes at the poles of f and g. Thus we have either φ ≡ 0 and N (r, ∞) = S(r), or else φ vanishes identically, the latter meaning that also a 4 = ∞ is shared by counting multiplicities. On the other hand it is not hard to show that the hypotheses "N (r, ∞) = S(r)" and "a 4 = ∞ is shared by counting multiplicities" are equally strong in the sense that (on combination with the hypotheses about the values a ν ) they lead to the same conclusions. Proof of the (2+2)-Theorem. We may assume that the values 0 and ∞ are shared by counting multiplicities, while the other values are a 1 and a 2 = 1/a 1 (if the latter does not hold a priori, we consider cf and cg instead of f and g, with c satisfying c 2 a 1 a 2 = 1). The auxiliary function φ = f ′′ f ′ + 2 f ′ f − 2 ν=1 f ′ f − a ν − g ′′ g ′ − 2 g ′ g + 2 ν=1 g ′ g − a ν is regular at all a ν -points (ν = 1, 2) of f and g despite their multiplicities, and is also regular at the zeros and poles that are simple for f and g. Thus φ satisfies T (r, φ) = S(r), and φ(z ∞ ) 2 = (a 1 + a 2 ) 2 Ψ(z ∞ ) holds at any pole which is simple for f and g; here Ψ again is Mues' function. Hence either N (r, ∞) = S(r) or else φ 2 = (a 1 + a 2 ) 2 Ψ holds. Repeating this argument with F = 1/f , G = 1/g and corresponding function χ = F ′′ F ′ + 2 F ′ F − 2 ν=1 F ′ F − 1/a ν − G ′′ G ′ − 2 G ′ G + 2 ν=1 G ′ G − 1/a ν = f ′′ f ′ − 2 f ′ f − 2 ν=1 f ′ f − a ν − g ′′ g ′ + 2 g ′ g + 2 ν=1 g ′ g − a ν instead of f , g and φ, and noting that 1/a 1 + 1/a 2 = a 1 + a 2 and Ψ is also Mues' function for F and G, it follows that essentially four possibilities remain to be discussed: (a) N (r, 0) + N (r, ∞) = S(r); (b) φ = χ; (c) φ = −χ; (d) N (r, ∞) = S(r) and χ 2 = (a 1 + a 2 ) 2 Ψ. ad (a)-From (N b ) follows N (r, a 1 ) + N (r, a 2 ) = 2T (r) + S(r), hence the sequence of a ν -points (ν = 1, 2) which have different multiplicities for f and g have counting function S(r). The conclusion of the Four-Value-Theorem follows immediately from Mues' proof of that theorem. ad (b)-From φ − χ = 4 f ′ f − g ′ g = 0 follows g = cf , hence g = f if there exists some common a ν -point. Otherwise a 1 and a 2 are Picard values for f and g, hence the Four-Value-Theorem holds. ad (c)-Here we have f ′′ f ′ − 2 ν=1 f ′ f − a ν − g ′′ g ′ + 2 ν=1 g ′ g − a ν = 0, hence f ′ (f − a 1 )(f − a 2 ) = κ g ′ (g − a 1 )(g − a 2 ) (κ = 0) and f − a 1 f − a 2 = C g − a 1 g − a 2 κ for some κ ∈ Z \ {0} and C = 0. Then κ = ±1 follows from T (r, f ) = T (r, g) + S(r), and two cases remain to be discussed: κ = −1. The values a ν are Picard values for f and g, hence f and g share all values by counting multiplicities. 1. From f − a 1 f − a 2 = C g − a 1 g − a 2 follows that f and g also share the values a 1 and a 2 , hence all values, by counting multiplicities. ad (d)-Following Mues [10] we consider the auxiliary functions η 1 = χ − (a 1 + a 2 ) f ′ (f − g) f (g − a 1 )(f − a 2 ) and θ 1 = χ − (a 1 + a 2 ) g ′ (f − g) g(f − a 1 )(g − a 2 ) and the corresponding functions η 2 and θ 2 (with a 1 and a 2 permuted). It is easily seen that T (r, η ν ) ≤ T (r) − N (r, a ν ) + S(r) and T (r, θ ν ) ≤ T (r) − N (r, a ν ) + S(r) holds (see also [10]). Now each of these functions vanishes at the zeros of f and g. If the functions η 1 and θ 1 as well as η 2 and θ 2 do not simultaneously vanish identically, we obtain (note that N (r, ∞) = S(r)) 2N (r, 0) ≤ 2T (r) − N (r, a 1 ) − N (r, a 2 ) + S(r) = N (r, 0) + S(r), hence N (r, 0) = S(r). So we are back in case (a), and the (2+2)-Theorem is proved completely in that case. On the other hand, η 1 = θ 1 ≡ 0, say, and a 1 + a 2 = 0 yield a 2 ) , hence f and g share the value a 1 by counting multiplicities, so that the (3+1)-Theorem gives the desired result. (f − a 1 )f ′ f (f − a 2 ) = (g − a 1 )g ′ g(g − The case a 1 + a 2 = 0 (hence a 1 = i and a 2 = −i, on combination with a 1 a 2 = 1) has to be treated separately. From χ ≡ 0 follows (6) f ′ f 2 (f − i)(f + i) = κ g ′ g 2 (g − i)(g + i) (κ = 0). Since the values ±i are not Picard values for f and g (otherwise we were already in the (3+1)-case), κ (or 1/κ) is a positive integer, hence f assumes the values ±i "always" with multiplicity κ, while g has "only" simple ±i-points (up to a sequence of points with counting function S(r)). Now (6) is equivalent to f ′ f 2 − κ g ′ g 2 = i 2 f ′ f − i − f ′ f + i − κ g ′ g − i + κ g ′ g + i , and the right hand side, denoted ϕ, is regular at ±i-points and vanishes at simple zeros z 0 of f and g (note that g ′ (z 0 ) = κf ′ (z 0 )). Also from T (r, ϕ) = S(r) follows either N (r, 0) = S(r), which leads back to case (a), or else ϕ ≡ 0, equivalently f ′ f 2 − κ g ′ g 2 = 0 and 1 f = κ g + c with ±1/i = ±κ/i + c. This is only possible if κ = 1 and c = 0. Progress after Gundersen As already mentioned, all attempts to prove or disprove a (1+3)-Theorem failed up to now (January 2011), and so many authors looked for additional conditions or switched to related problems-but the latter is not the subject of the present paper. In [10] E. Mues introduced the quantity τ (a ν ) = lim inf r→∞ (r / ∈E) N s (r, a ν ) N (r, a ν ) if N (r, a ν ) = S(r), and τ (a ν ) = 1 else; here N s (r, a ν ) denotes the counting function of those a ν -points which are simultaneously simple for f and g, and E is the exceptional set for S(r); we note that τ (a ν ) = 1 in particular holds if a ν is shared by counting multiplicities, and also if a ν is a Picard value for f and g. Example. We have τ (a ν ) = 0 in the counterexamples of Gundersen and Reinders, and τ (a ν ) = 1 3 for any pair of the author's triple. In her diploma thesis, E. Rudolph [19] proved some results in terms of the quantities τ (a ν ) or their natural generalisations τ (a ν , a µ ), τ (a ν , a µ , a κ ) and τ (a 1 , a 2 , a 3 , a 4 ). Her proofs were based on the methods developed in the paper [10], which appeared later than [19], but was written earlier. In the sequel we shall derive several results, which become much more apparent when stated as inequalities involving the counting functions N (r, a ν ) and N s (r, a ν ). Nevertheless they may be credited to Rudolph and Mues for the underlying idea. Key Lemma. Suppose that f and g share distinct values a ν (1 ≤ ν ≤ 4). Then either the conclusion of the Four-Value-Theorem holds or else the following is true: (R a ) 3 2 N s (r, a κ ) + N s (r, a ν ) ≤ N (r, a κ ) + N (r, a ν ) + S(r) for κ = ν; for (a 1 , a 2 , a 3 , a 4 ) = −1 the factor 3 2 can be replaced by 2. (R b ) N s (r, a κ ) + µ =λ N s (r, a µ ) ≤ µ =λ N (r, a µ ) + S(r) for κ = λ. (R c ) N (r, a κ ) + µ =κ N s (r, a µ ) ≤ µ =κ N (r, a µ ) + S(r). (R d ) 4 µ=1 N s (r, a µ ) + µ =κ,ν N s (r, a µ ) ≤ 2 µ =κ,ν N (r, a µ ) for κ = ν. (R e ) 4 κ =λ N s (r, a κ ) ≤ 3 κ =λ N (r, a κ ) + S(r) . (R f ) 3 4 µ=1 N s (r, a µ ) ≤ 2 4 µ=1 N (r, a µ ) + S(r) = 4T (r) + S(r). Several more or less recent results obtained by different authors can easily be derived from the previous lemma. Since, however, the thesis [19] has never been published, these results can be looked upon as independent discoveries. The proof of the Key Lemma will be given in the next section. Corollary. Suppose that f and g share the values a ν (1 ≤ ν ≤ 4). Then the conclusion of the Four-Value-Theorem is true, provided one of the following hypotheses is assumed in addition: (A) One value is shared by counting multiplicities, and some other satisfies τ (a κ ) > 2 3 ; for (a 1 , a 2 , a 3 , a 4 ) = −1 the constant 2 3 can be replaced by 1 2 .-(E. Mues [10]).( 6 ) (B) One value is shared by counting multiplicities, while the other values satisfy τ (a ν ) > 1 2 .-(J.P. Wang [24], B. Huang [9]). (C) One value is shared by counting multiplicities and simultaneously satisfies N (r, a κ ) ≥ ( 4 5 + δ) T (r) for some δ > 0 on some set of infinite measure.-(G. Gundersen [7]). (D) Two of the values satisfy τ (a ν ) > 4 5 ; for (a 1 , a 2 , a 3 , a 4 ) = −1 the constant 4 5 can be replaced by Proof. Assuming that the conclusion of Nevanlinna's Four-Value-Theorem does not hold, we will derive a contradiction to the respective hypothesis by applying one of the inequalities (R a ) -(R f ). 6 It is remarkable that Mues did not refer to Gundersen's (2+2)-Theorem, hence, in particular, gave an independent proof for it. The other authors actually proved that the hypotheses of the (2+2)-Theorem follow from their own, thus their theorems generalising Gundersen's (2+2)-result actually depend on it. 7 Ueda has a slightly stronger result involving some additional term, which, however, cannot be controlled. (A) is true since N (r, a ν ) = N s (r, a ν ) + S(r) for some ν and (R a ) for the same ν imply (C) is obtained from (R a ) as follows. We assume that a κ is shared by counting multiplicities, hence N (r, a κ ) = N s (r, a κ ) + S(r) holds. Adding up for ν = κ we obtain 3 2 N s (r, a κ ) ≤ N (r, a κ ) + S(r) (κ = ν),3 2 N (r, a κ ) + ν =κ N s (r, a ν ) ≤ ν =κ N (r, a ν ) + S(r) = 2T (r) − N (r, a κ ) + S(r). (D) again follows from (R a ) by adding up the symmetric inequalities 3 2 N s (r, a ν ) + N s (r, a κ ) ≤ N (r, a κ ) + N (r, a ν ) + S(r) 3 2 N s (r, a κ ) + N s (r, a ν ) ≤ N (r, a κ ) + N (r, a ν ) + S(r). Again we note that 4 5 can be replaced by 2 3 if (a 1 , a 2 , a 3 , a 4 ) = −1. Finally, (E) and (F) follow from (R e ) and (R f ), respectively. Proof of the Key Lemma The proof of the Key Lemma is based on Mues' auxiliary function technique [10]. As long as no particular hypotheses are imposed, the proofs and results are symmetric and Möbius invariant, this meaning that everything proved for a 1 , a 2 , a 3 , a 4 holds for arbitrary permutations, and that three of the four values can be given prescribed numerical values. To prove (R a ) we proceed as in the proof of the (2+2)-Theorem. We set κ = 4, a 4 = ∞, and ν = 3, and consider Mues' function Ψ and the auxiliary function φ = f ′′ f ′ + 2 f ′ f − a 3 − 2 ν=1 f ′ f − a ν − g ′′ g ′ − 2 g ′ g − a 3 + 2 ν=1 g ′ g − a ν . Then φ has only simple poles, exactly at those a 3 -points and poles (a 4 -points) that are not simultaneously simple for f and g, and the usual technique yields T (r, φ) = N (r, a 3 ) − N s (r, a 3 ) + N (r, a 4 ) − N s (r, a 4 ) + S(r). Since φ(z ∞ ) 2 = (2a 3 − a 1 − a 2 ) 2 Ψ(z ∞ ) holds at every pole z ∞ that is simple for f and g, To prove (R b ) we may assume κ = 4, a 4 = ∞ and λ = 3. Then N s (r, a 4 ) ≤ 2T (r, φ) + O(1) ≤ 2 N (r, a 3 ) − N s (r, a 3 ) + N (r,aφ = f ′′ f ′ − 2 f ′ f − a 3 + f ′ (f − a 3 ) (f − a 1 )(f − a 2 ) − g ′′ g ′ − 2 g ′ g − a 3 − g ′ (g − a 3 ) (g − a 1 )(g − a 2 ) is regular at poles of f and g (a 4 = ∞), has simple poles exactly at those a ν -points (1 ≤ ν ≤ 3) of f and g that have different multiplicities, thus (7) T (r, φ) = N (r, φ) + S(r) = 3 ν=1 N (r, a ν ) − N s (r, a ν ) + S(r) holds. Now φ vanishes at a 3 -points which are simultaneously simple for f and g, hence φ ≡ 0 implies To prove (R c ), for κ = 4 and a 4 = ∞, say, we just consider φ = f ′ (f − a 1 )(f − a 2 )(f − a 3 ) − g ′ (g − a 1 )(g − a 2 )(g − a 3 ) ; then φ has poles exactly at those a ν -points (1 ≤ ν ≤ 3) that have different multiplicities for f and g, and thus satisfies a fortiori (7). Since φ vanishes at poles (a 4 -points) of f and g, this yields N (r, a 4 ) ≤ 3 ν=1 N (r, a ν ) − N s (r, a ν ) + S(r) provided φ does not vanish identically. If, however, φ = 0, then f and g share the values a µ (1 ≤ µ ≤ 3) by counting multiplicities, and thus the hypothesis of the (3+1)-Theorem and the conclusion of the Four-Value-Theorem holds. To prove (R d ) we may assume a µ ∈ C (1 ≤ µ ≤ 4) and consider φ = f ′ (f − a 1 )(f − a 2 ) − g ′ (g − a 1 )(g − a 2 ) . Then φ has poles exactly at those a 1 -and a 2 -points that are not simultaneously simple for f and g, and is regular at poles of f and of g, hence T (r, φ) = 2 µ=1 N (r, a µ ) − N s (r, a µ ) + S(r) follows. On the other hand, φ( z ρ ) = (−1) ρ f ′ (z ρ ) − g ′ (z ρ ) µ =ρ (a ρ − a µ ) (a 4 − a 3 ) holds at any a ρ -point z ρ (ρ = 3, 4), and Ψ(z ρ ) = (f ′ (z ρ ) − g ′ (z ρ )) 2 µ =ρ (a ρ − a µ ) 2 holds whenever z ρ is simple for f and g. Thus if φ 2 ≡ (a 4 − a 3 ) 2 Ψ, the assertion (for κ = 3, ν = 4) follows from the First Main Theorem of Nevanlinna: 4 µ=3 N s (r, a µ ) ≤ T r 1 φ 2 − (a 4 − a 3 ) 2 Ψ + O(1) = 2T (r, φ) + S(r) ≤ 2 2 µ=1 N (r, a µ ) − N s (r, a µ ) + S(r). If, however, φ 2 = (a 4 − a 3 ) 2 Ψ, then f and g share the values a 1 and a 2 by counting multiplicities, hence the conclusion of the Four-Value-Theorem holds. Finally, (R e ) and (R f ) follow by adding up inequality (R b ) for κ = λ and (R c ) for κ = 1, . . . , 4, respectively. 6. Towards or way off a (1+3)-Theorem? Assuming that f and g share the values a ν ∈ C (1 ≤ ν ≤ 3) and a 4 = ∞, we set φ f = f ′ (f − a 1 )(f − a 2 )(f − a 3 ) and φ g = g ′ (g − a 1 )(g − a 2 )(g − a 3 ) , Φ f = (f − g)φ f and Φ g = (f − g)φ g , Φ = Φ f /Φ g = φ f /φ g and Ψ = Φ f Φ g (Mues' function). Example. functions f, g Φ f Φ g Ψ Φ e z , e −z e −z e z 1 e −2z Gundersen's 1 − e z 8 1 − e z 8 (1 − e z ) 2 8 Reinders' 12 √ 3 u + 1 12(u + 1) √ 3 144 9 (u + 1) 2 Key Observations. • If the value a 4 = ∞ is shared by counting multiplicities, then Φ f and Φ g are entire functions satisfying N (r, 1/Φ f ) + N (r, 1/Φ g ) = S(r). • If T (r) has finite lower order lim inf r→∞ log T (r) log r , then Φ f = p f e Q , Φ g = p g e −Q , Ψ = p f p g , Φ = p f p g e 2Q , and T (r) ≍ r deg Q hold with polynomials p f , p g , and Q. Proof. Noting that Ψ = f 2 f ′ P (f ) g ′ P (g) + 2 f f ′ P (f ) gg ′ P (g) + f ′ P (f ) g 2 g ′ P (g) (with P (w) = 3 ν=1 (w − a ν ) or P (w) = 4 ν=1 (w − a ν )) holds, the lemma on the logarithmic derivative gives T (r k , Ψ) = m(r k , Ψ) = O(log r k ) on some sequence r k → ∞. Hence Ψ is a polynomial, Φ f and Φ g have only finitely many zeros and satisfy log T (r k , Φ f ) + log T (r k , Φ g ) = O(log r k ). Thus the assertion on Φ f , Φ g and Ψ, holds, since entire functions e h(z) have finite lower order if and only if h is a polynomial (by the Borel-Carathéodory inequality or the lemma on the logarithmic derivative). The assertion on T (r) follows from the subsequent theorem and T (r, e Q ) ≍ r deg Q . Theorem. If f and g share four values, at least one of them by counting multiplicities, then either the conclusion of the Four-Value-Theorem or else Proof. We assume that a 4 = ∞ is shared by counting multiplicities, and first suppose that Φ = Φ f /Φ g = φ f /φ g is non-constant. Then from Now suppose that f (z 0 ) = g(z 0 ) = a ν holds with multiplicities ℓ f and ℓ g , say, hence Φ(z 0 ) = ℓ f /ℓ g . Noting that the sequence of a ν -points with min{ℓ f , ℓ g } > 1 has counting function S(r), and restricting ℓ f and ℓ g to the range {1, . . . , ℓ}, we obtain by Nevanlinna's second main theorem N (r, 1/Φ g ) ≤ N (T (r) ≤ 3 ν=1 N (r, a ν ) + S(r) ≤ (2ℓ + 1)T (r, Φ) + 1 ℓ + 2 3 ν=1 N r, 1 f − a ν + N (r, 1 g − a ν + S(r) ≤ (2ℓ + 1)T (r, Φ) + 6 ℓ + 2 T (r) + S(r), hence (8) T (r) ≤ (2ℓ + 1)(ℓ + 2) ℓ − 4 T (r, Φ) + S(r) (ℓ > 4) holds. The factor 209 5 is obtained for ℓ = 9. If, however, Φ is constant, then we have actually Φ ≡ ℓ or Φ ≡ 1/ℓ for some ℓ ∈ N. This yields, in the first case, say, 3 ν=1 (f − a ν ) = C 3 ν=1 (g − a ν ) ℓ (C = 0 some constant). From T (r, f ) ∼ ℓT (r, g) then follows ℓ = 1, hence f and g share all values a ν by counting multiplicities. Combining the previous result with the second Key Observation, we obtain: [25]). Suppose f and g share four values a ν , at least one of them by counting multiplicities. Then either both functions have infinite lower order or else have equal finite integer order that also equals the lower order. Corollary (Yi & Li Remark. The authors of [25] proved in a way similar to ours the inequality 1 77 T (r) ≤ T (r, Φ) + S(r) ≤ 4T (r) + S(r). Each of the counterexamples (by Gundersen, Reinders and the author) are (can be reduced to) either rational functions of e z or else elliptic functions. Since elliptic functions have no deficient value, we obtain: Corollary. No pair of elliptic functions can share four values, at least one of them by counting multiplicities. Theorem. Let f and g be 2πi-periodic functions of finite order sharing four values, one of them by counting multiplicities. Then f and g are rational functions of e z . Proof. The functions Ψ = Φ f Φ g , Φ f and Φ g are entire of finite order and also 2πiperiodic. But Ψ is a polynomial, hence constant, and thus Φ f and Φ g are zero-free. This yields (9) Φ f (z) = e mz+c f and Φ g (z) = e −mz+cg for some m ∈ Z \ {0} and complex constants c f and c g . From T (r, Φ f /Φ g ) ∼ 2\|m\| π r, hence T (r) ≍ r then follows that f and g are rational functions of e z . Remark. Generally spoken, meromorphic functions h(z) = R(e z ), where R is rational with deg R = d > 1, have Nevanlinna characteristic T (r, h) ∼ d π r and deficient values R(0) and R(∞), which may, of course, coincide. If, e.g., R(u) ∼ a + bu −ρ as u → ∞, then the contribution of the right half plane to m r, 1/(f − a)) is ∼ ρ π r. In Gundersen's example we have δ(a, f ) = 1/2 for a = 0, 1, and δ(a, g) = 1/2 for a = ∞, −1/8. Let f (z) = R(e z ) and g(z) = S(e z ) share four values a ν (∈ C for technical reasons), without assuming anything about multiplicities. Then Mues' function Ψ = Φ f Φ g is a non-zero constant, and from R(u) ∼ a µ + bu −ρ and S(u) ∼ a ν + cu −σ (bc = 0, ρ, σ > 0) as u → ∞ follows Φ f (z) ∼ b 1 (a µ − a ν ) + O(e − min{ρ,σ}Re z ) (b 1 = 0) Φ f (z) ∼ c 1 (a µ − a ν ) + O(e − min{ρ,σ}Re z ) (c 1 = 0) as Re z → +∞. Thus a µ = a ν , and a similar result near u = 0 shows that at least one of the values a ν (but no value b = a ν ) is deficient for f , and the same is true for g (and the same or some other value a µ ). If a 4 , say, is shared by counting multiplicities, then δ(a 4 , f ) = δ(a 4 , g) > 0. More precisely we have R(u) ∼ a 4 + bu −ρ (u → ∞), S(u) ∼ a 4 + cu ρ (u → 0), δ(a 4 , f ) = δ(a 4 , g) = ρ/d, and R(0) = a µ , S(∞) = a ν for some µ, ν < 4. We switch now to values a 1 , a 2 , a 3 and a 4 = ∞ and assume that a 4 is shared by counting multiplicities, and also that R(∞) = S(0) = ∞ holds. From m(r, f ) ∼ m(r, g) ∼ ρ π r for some ρ (1 ≤ ρ < d) we obtain R(u) = P (u) Q(u) , S(u) =P (u) u ρ Q(u) , degP ≤ d and ρ + deg Q = deg P = d; the zeros of Q are simple and = 0. Functions of finite order Under certain circumstances it may happen that a problem for meromorphic function of arbitrary order of growth can be reduced to a problem for functions of finite order by the well-known Zalcman Lemma [26]. A prominent example can be found in [1]. This might also be the case here, although there are some obstacles, as will be seen later. Rescaling Lemma. Let f and g share the values a ν (1 ≤ ν ≤ 4). Then either the spherical derivatives f # and g # are bounded on C, or else there exist sequences (z k ) ⊂ C and (ρ k ) ⊂ (0, ∞) with ρ k → 0, such that the sequences (f k ) and (g k ), defined by f k (z) = f (z k + ρ k z) and g k (z) = g(z k + ρ k z), simultaneously tend to non-constant meromorphic functionsf andĝ, respectively; f andĝ share the values a ν and have bounded spherical derivatives. Proof. We assume that f # is not bounded. Then the existence of the sequences (z k ) (tending to infinity) and (ρ k ) can be taken for granted for the function f by Zalcman's Lemma. If we assume that the sequence (g k ) is not normal on C, then again by Zalcman's Lemma there exist sequences (k ℓ ) ⊂ N,ẑ ℓ → z 0 ∈ C and σ ℓ ↓ 0, such thatĝ ℓ = g k ℓ (ẑ ℓ + σ ℓ z) tends to some non-constant meromorphic functiong. Then on one hand, the corresponding sequence (f ℓ ) tends to a constant, while on the other hand every limit function of (f ℓ ) shares the values a ν withg by Hurwitz' Theorem. Sincẽ f assumes at least two of these values, our assumption on the sequence (g k ) was invalid. We may assume that (g k ) tends toĝ. Thenĝ shares the values a ν withf , hence is non-constant. Finally, if we assume thatĝ # is unbounded on C, then we may apply the first argument to the functionsĝ andf in this order, but now with the a priori knowledge thatf # is bounded. Then some sequenceĝ(z k + τ k z) tends to some non-constant limit, while a sub-sequence of the sequencef (z k + τ k z) tends to a constant (the sequence of spherical derivatives is O(τ k )). This proves the Rescaling Lemma. Remark. Till now, however, it cannot be excluded that (a) a shared value gets lost in the sense that it becomes a Picard value forf andĝ, although it is not for f and g; (b) multiplicities get lost in the sense thatf andĝ share some value a ν by counting multiplicities, although f and g do not; (c)f =ĝ (worst case). In the third case everything is lost. But even iff =ĝ holds and the (3+1)-Conjecture turns out to be true for functions with bounded spherical derivative, we can only deduce (a 1 , a 2 , a 3 , a 4 ) = −1, except if we are able to rule out also the first and second case. Nevertheless we proceed in this direction. Theorem. Suppose that f and g share the values a ν (1 ≤ ν ≤ 4) and have bounded spherical derivative. Then Mues' function Ψ is a constant. Moreover, if one value is shared by counting multiplicities, then (10) Φ f (z) = e Q(z)+c f and Φ g (z) = e −Q(z)+cg holds, with Q some polynomial of degree one or two. Proof. From f # + g # ≤ C follows that f and g have finite order (actually the order is at most two), and by Nevanlinna's lemma on the logarithmic derivative, Ψ is a polynomial. We assume Ψ(z) ∼ cz m as z → ∞ for some m ≥ 1 and c = 0, and consider, for some sequence z n → ∞, the functions f n (z) = f (z n + z −m/2 n z) and g n (z) = g(z n + z −m/2 n z), which also share the values a ν and have Mues' function (11) Ψ n (z) = z −m n Ψ(z n + z −m/2 n z) ∼ c (n → ∞), while obviously f n and g n tend to constants. If the sequence (z n ), which is quite arbitrary, can be chosen in such a way that f (z n ) → b = a ν and g(z n ) → b ′ = a ν (1 ≤ ν ≤ 4), then f n and g n tend to constants b and b ′ , hence Ψ n tends to 0 in contrast to relation (11). Now for any sequence (ẑ n ) the sequencesf n (z) = f (ẑ n + z) and g n (z) = g(ẑ n + z) are normal on C, we may assume that (some sub-sequence off n , again denoted by)f n tends to some non-constant limit functionf , e.g. by choosinĝ z n → ∞ such that \|f ′ (ẑ n )\| = 1. Sinceĝ n shares the values a ν withf n we may (by normality and Picard's theorem) also assume thatĝ n →ĝ ≡ const. Now we choose z 0 such thatf (z 0 ) = b = a ν andĝ(z 0 ) = b ′ = a ν , and set z n =ẑ n + z 0 . Finally, since Ψ = Φ f Φ g is constant, the entire functions Φ f and Φ g are zero-free and have order one or two (as do f and g), hence (10) holds with deg Q = 1 or 2. t, a ν ) −n(0, a ν )] dt t denotes the (integrated) Nevanlinna counting function of the sequence of a ν -points of f and g, each point being counted simply despite of multiplicities. Then Nevanlinna's Theorems may be stated as follows: 2 3 . 3-(E. Rudolph[19], S.P. Wang[23], B. Huang[9]).(E) Three of the values satisfy τ (a ν ) > 3 4 .-(E. Rudolph [19], G. Song & J. Chang [20]). (F) All values satisfy τ (a ν ) > 2 3 .-(E.Rudolph[19], H. Ueda[22]( 7 ), J.P. Wang[24], also conjectured by G. Song & J. Chang[20]). with 3 2 N 2replaced by 2 if (a 1 , a 2 , a 3 , a 4 ) = −1.To prove (B) we may assume that a 4 is shared by counting multiplicities, thus N (r, a 4 ) = N s (r, a 4 ) + S(r)holds. From (R c ) with κ < 4 then follows N (r, a κ ) + (r, a µ ) + S(r). 4 ) − N s (r, a 4 ) + S(r) and (R a ) for κ = 4 and ν = 3 follow, provided φ 2 ≡ (2a 3 − a 1 − a 2 ) 2 Ψ holds. On the other hand, φ 2 = (2a 3 − a 1 − a 2 ) 2 Ψ implies that φ is a small function, thus f and g share the values a 3 and a 4 = ∞ by counting multiplicities, and hence all values a ν by the (2+2)-Theorem. For 2a 3 − a 1 − a 2 = 0 (equivalently (a 1 , a 2 , a 3 , ∞) = −1) we obtain the better inequality 2N s (r, a 4 ) + N s (r, a 3 ) ≤ N (r, a 3 ) + N (r, a 4 ) + S(r), by counting the zeros of φ rather than those of φ 2 − (2a 3 − a 1 − a 2 ) 2 Ψ. N (r, a ν ) − N s (r, a ν ) + S(r), thus (R b ) for κ = 4 and λ = 3. On the other hand, the conclusion of the Four-Value-Theorem holds if φ vanishes identically. N r, 1/Ψ) + S(r) = S(r), N (r, 1/φ g ) = N (r, ∞) + S(r) m(r, φ g ) = S(r) (and the same for g replaced by f ) follows the upper estimate T (r, Φ) = m(r, Φ) + S(r) ≤ m(r, 1/φ g ) + S(r) = N (r, φ g ) − N (r, 1/φ g ) + S(r) (r, a ν ) − N (r, ∞) + S(r) = 2T (r) − 2N (r, ∞) + S(r). 209 T (r) ≤ T (r, Φ) + S(r) ≤ 2T (r) − 2N (r, ∞) + S(r) holds. On the singularities of the inverse to a meromorphic function of finite order. W Bergweiler, A Eremenko, Rev. Mat. Iberoamericana. 11W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), 355-373. Un nouveau théorème d'unicité relatif aux fonctions méromorphes. H Cartan, C. R. Acad. Sci. Paris. 188H. Cartan, Un nouveau théorème d'unicité relatif aux fonctions méromorphes, C. R. Acad. Sci. Paris 188 (1929), 301-303. A remark on meromorphic functions sharing four values. T G Chen, K Y Chen, T C Ou, Y L Tsai, Taiwan. J. Math. 12T. G. Chen, K. Y. Chen, T. C. Ou and Y. L. Tsai, A remark on meromorphic functions sharing four values, Taiwan. J. Math. 12 (2008), 1733-1737. Meromorphic functions that share three or four values. G G Gundersen, J. London Math. Soc. 20G. G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc. 20 (1979), 457-466. Meromorphic functions that share four values. G G Gundersen, Trans. Amer. Math. Soc. 277G. G. Gundersen, Meromorphic functions that share four values, Trans. Amer. Math. Soc. 277 (1983), 545-567. Meromorphic functions that share four values. G G Gundersen, Trans. Amer. Math. Soc. 304Correction toG. G. Gundersen, Correction to "Meromorphic functions that share four values", Trans. Amer. Math. Soc. 304 (1987), 847-850. Meromorphic functions that share three values IM and a fourth value CM. G G Gundersen, Complex Variables. 20G. G. Gundersen, Meromorphic functions that share three values IM and a fourth value CM, Complex Variables 20 (1992), 99-106. W K Hayman, Meromorphic functions. Oxford Clarendon PressW. K. Hayman, Meromorphic functions, Oxford Clarendon Press 1975. On the unicity of meromorphic functions that share four values. B Huang, Indian J. pure appl. Math. 35B. Huang, On the unicity of meromorphic functions that share four values, Indian J. pure appl. Math. 35 (2004), 359-372. Meromorphic functions sharing four values. E Mues, Complex Variables. 12E. Mues, Meromorphic functions sharing four values, Complex Variables 12 (1989), 169-179. Shared value problems for meromorphic functions. E Mues, Value distribution theory and complex differential equations. JoensuuE. Mues, Shared value problems for meromorphic functions, in Value distribution theory and complex differential equations, Joensuu (1994), 17-43. Zur Theorie der meromorphen Funktionen. R Nevanlinna, Acta Math. 46R. Nevanlinna, Zur Theorie der meromorphen Funktionen, Acta Math. 46 (1925), 1-99. Einige Eindeutigkeitssätze in der Theorie der meromorphen Funktionen. R Nevanlinna, Acta Math. 48R. Nevanlinna, Einige Eindeutigkeitssätze in der Theorie der meromorphen Funktionen, Acta Math. 48 (1926), 367-391. Eindeutige analytische Funktionen. R Nevanlinna, SpringerR. Nevanlinna, Eindeutige analytische Funktionen, Springer 1936. Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen. G Pólya, Mat. Tidsskrift B. G. Pólya, Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen, Mat. Tidsskrift B, København (1921), 16-21. A new characterisation of Gundersen's example of two meromorphic functions sharing four values. M Reinders, Results Math. 24M. Reinders, A new characterisation of Gundersen's example of two meromorphic functions sharing four values, Results Math. 24 (1993), 174-179. Eindeutigkeitssätze für meromorphe Funktionen, die vier Werte teilen. M Reinders, Mitt. Math. Sem. Giessen. 200M. Reinders, Eindeutigkeitssätze für meromorphe Funktionen, die vier Werte teilen, Mitt. Math. Sem. Giessen 200 (1991), 15-38. Eindeutigkeitssätze für meromorphe Funktionen, die vier Werte teilen. M Reinders, HannoverPhD thesisM. Reinders, Eindeutigkeitssätze für meromorphe Funktionen, die vier Werte teilen, PhD thesis, Hannover (1990). Über meromorphe Funktionen, die vier Werte teilen. E Rudolph, KarlsruheDiploma ThesisE. Rudolph,Über meromorphe Funktionen, die vier Werte teilen, Diploma Thesis, Karlsruhe (1988). Meromorphic functions sharing four values. G D Song, J M Chang, Southeast Asian Bull. Math. 26G. D. Song and J. M. Chang, Meromorphic functions sharing four values, Southeast Asian Bull. Math. 26 (2002), 629-635. A uniqueness theorem for three meromorphic functions. N Steinmetz, Ann. Acad. Fenn. Sci. 13N. Steinmetz, A uniqueness theorem for three meromorphic functions, Ann. Acad. Fenn. Sci. 13 (1988), 93-110. Some estimates for meromorphic functions sharing four values. H Ueda, Kodai Math. J. 17H. Ueda, Some estimates for meromorphic functions sharing four values, Kodai Math. J. 17 (1994), 329-340. On meromorphic functions that share four values. S P Wang, J. Math. Anal. Appl. 173S. P. Wang, On meromorphic functions that share four values, J. Math. Anal. Appl. 173 (1993), 359-369. Meromorphic functions sharing four values. J P Wang, Indian J. pure appl. Math. 32J. P. Wang, Meromorphic functions sharing four values, Indian J. pure appl. Math. 32 (2001), 37-46. Meromorphic functions sharing four values. H X Yi, X M Li, Proc. Japan Acad. Japan Acad83H. X. Yi and X. M. Li, Meromorphic functions sharing four values, Proc. Japan Acad. 83, Ser. A (2007), 123-128. A heuristic principle in function theory. L Zalcman, Amer. Math. Monthly. 82L. Zalcman, A heuristic principle in function theory, Amer. Math. Monthly 82 (1975), 813- 817. Normal families: new perspectives. L Zalcman, Bull. Amer. Math. Soc. 35L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215-230.	[]
[ "Sensitivity of quantum information to environment perturbations measured with the out-of-time-order correlation function", "Sensitivity of quantum information to environment perturbations measured with the out-of-time-order correlation function" ]	[ "Mohamad Niknam \nInstitute for Quantum Computing\nUniversity of Waterloo\nN2L3G1WaterlooONCanada\n\nDepartment of Physics\nUniversity of Waterloo\nN2L3G1WaterlooONCanada\n", "Lea F Santos \nDepartment of Physics\nYeshiva University\n10016New YorkNew YorkUSA\n", "David G Cory \nInstitute for Quantum Computing\nUniversity of Waterloo\nN2L3G1WaterlooONCanada\n\nDepartment of Chemistry\nUniversity of Waterloo\nN2L3G1WaterlooONCanada\n\nPerimeter Institute\nUniversity of Waterloo\nN2L2Y5WaterlooCanada\n\nCanadian Institute For Advanced Research\nM5G1Z8TorontoONCanada\n" ]	[ "Institute for Quantum Computing\nUniversity of Waterloo\nN2L3G1WaterlooONCanada", "Department of Physics\nUniversity of Waterloo\nN2L3G1WaterlooONCanada", "Department of Physics\nYeshiva University\n10016New YorkNew YorkUSA", "Institute for Quantum Computing\nUniversity of Waterloo\nN2L3G1WaterlooONCanada", "Department of Chemistry\nUniversity of Waterloo\nN2L3G1WaterlooONCanada", "Perimeter Institute\nUniversity of Waterloo\nN2L2Y5WaterlooCanada", "Canadian Institute For Advanced Research\nM5G1Z8TorontoONCanada" ]	[]	In a quantum system coupled with a non-Markovian environment, quantum information may flow out of or in to the system. Measuring quantum information flow and its sensitivity to perturbations is important for a better understanding of open quantum systems and for the implementation of quantum technologies. Information gets shared between a quantum system and its environment by means of system-environment correlations (SECs) that grow during their interaction. We design a nuclear magnetic resonance (NMR) experiment to directly observe the evolution of the SECs and use the second moment of their distribution as a natural metric for quantifying the flow of information. In a second experiment, by accounting for the environment dynamics, we study the sensitivity of the shared quantum information to perturbations in the environment. The metric used in this case is the out-of-time-order correlation function (OTOC). By analyzing the decay of the OTOC as a function of the SEC spread, instead of the evolution time, we are able to demonstrate its exponential behavior.The development of quantum technologies is obstructed by the loss of quantum properties caused by interactions with the environment that lead to decoherence [1][2][3][4][5]. Quantum information is shared with the environment by means of systemenvironment correlations (SECs)[6][7][8]. In the case of interactions with a non-Markovian environment, the SECs may lead to the flow of quantum information back to the system [9, 10]. Interferences arising from this backflow may be catastrophic to quantum information processes taking place in the system. The purpose of this work is to investigate the growth of the SECs as a function of time and to gauge how susceptible they are to perturbations in the environment. An open question with respect to the latter point is whether environment perturbations can be used to reduce information backflow.Our experiment is well-equipped to directly measure correlations between the system and the environment. It builds upon solid-state nuclear magnetic resonance (NMR) methods that have been employed to detect multiple-quantum coherences in homonuclear many-body systems[11][12][13][14][15][16][17][18][19]. These methods have recently been used for the investigation of multiple-quantum coherences in ion traps[20,21]. A change in the encoding basis has allowed for the observation of multispin dynamics of correlation growth during the free induction decay experiment[14,16]. Here, we extend these methods to composite heteronuclear systems to measure the growth of correlations with the environment.We consider a central spin model, which consists of a single spin-1/2 interacting with environment spins of another spin species that may also be coupled[22][23][24][25]. Quantum information initially resides in the central spin and is later shared with the environment in the form of multi-spin SECs. * Email: [email protected] NMR techniques make it possible to separate the systemenvironment evolution from the internal evolution of the environment spins[26][27][28]. This allows us to examine the impact of each process individually.We design two different NMR echo experiments. In the first, the environment dynamics is off. We analyze the evolution of the SECs in time and discuss how the second moment of the distribution of these correlations can be used to quantify the flow of quantum information between the system and the environment. In analogy with recent studies [21], we show that this metric is related with the quantum Fisher information.In the second experiment, the environment dynamics is turned on, causing the scrambling of quantum information.To quantify the sensitivity of information to the environment scrambling, we employ the out-of-time-order correlation function (OTOC). Despite great theoretical interest, very few experiments have had access to this quantity. To measure the OTOC, one needs to reverse the time evolution, which makes NMR echo experiments the method of choice. The evolution of the OTOC was previously studied in closed systems of ion traps [20, 21] and nuclear spins[29,30].The OTOC has become a prominent quantity in the analysis of the scrambling of quantum information in black holes and many-body quantum systems. It has been conjectured that the exponential behavior of this quantity should be an indicator of quantum chaos[31,32], the exponential rate being associated with the classical Lyapunov exponent. This correspondence has so far been confirmed theoretically for one-body chaotic systems[33,34]and for the Dicke model [35], but not yet experimentally.Experimental studies of the Loschmidt echo in many-body quantum systems have been carried out in NMR platform and depending on the interaction Hamiltonian, both Gaussian and exponential decays have been observed[36,37]. Our echo arXiv:1808.04375v2 [quant-ph] 7 Apr 2019 2 experiment shows that in the central spin system, the OTOC presents a Gaussian decay in time. However, an exponential behavior is revealed when it is studied as a function of the spread of the correlations between system and environment (that is, as a function of the Hamming weight spread of SECs). The capability of our experiments to directly measure these correlations is an essential ingredient for describing the flow of quantum information and for uncovering the exponential decay of the OTOC.Sample descriptionThe sample studied is a polycrystalline solid at room temperature composed of an ensemble of triphenylphosphine molecules, as shown inFig. 1a(see Sec. A in Methods). Each molecule has a central 31 P nuclear spin coupled to fifteen 1 H environment spins via the heteronuclear dipolar interactionwhere 'cs' stands for central spin, N = 15 is the number of environment spins and σ j X,Y, Z represent Pauli matrices for the j th spin. The dipolar Hamiltonian is a second rank spherical tensor, where the coupling constant ω j ∝ (3 cos 2 θ j − 1)/r 3 j has radial and angular dependence on the vector r j connecting the central spin to the j th spin in the environment. θ j is the angle between r j and the static field of the NMR magnet, which is along z. An environment spin located on the cone defined by the magic angle θ M , with 3 cos 2 θ M − 1 = 0, does not interact with the central spin. The orientation of the sample molecule illustrated inFig. 1ais such that two of the environment spins lie on the magic-angle cone and consequently, do not interact with the central spin. They belong to the "nonconnected" group, while environment spins coupled to the central spin are part of the "connected" group, as sketched inFig. 1b. The size of the connected group grows in time (see Sec. B in Methods).The central spin is initially in the state ρ cs (0) = [1+ σ X ]/2, where is the strength of the nuclear spin polarization which is of the order of 10 −6 at room temperature. In what follows, we drop the identity operator to simplify the notation, since it does not lead to any observable signal. The N spins in the environment are initially in the maximally mixed state ρ E (0) = (1/2) ⊗N , with no correlations. Thus, the initial state of the composite system is uncorrelated, ρ(0) = ρ cs (0)⊗ρ E (0).Mapping the system-environment correlationsWe design an echo experiment, which we call multispin correlation detection (MCD), to measure the correlation growth between the central spin and the environment spins. The stages of the MCD experiment are sketched inFig. 2a. During the evolution time T , the environment self-interaction is averaged to zero using the MREV-8 pulse sequence[26], while the remaining system-environment interaction correlates the two. The dynamics in the composite Hilbert space is described by the unitary propagator U SE (T ) = e −i H SE T , where H SE represents the system-environment interaction Hamiltonian in the toggling frame of the MREV-8 pulse sequence [27] (see Sec. C in Methods).	10.1103/physrevresearch.2.013200	[ "https://arxiv.org/pdf/1808.04375v2.pdf" ]	102,350,630	1808.04375	12a83c63ef30fcb8d29b63a8a3aeef92e05970ab	Sensitivity of quantum information to environment perturbations measured with the out-of-time-order correlation function Mohamad Niknam Institute for Quantum Computing University of Waterloo N2L3G1WaterlooONCanada Department of Physics University of Waterloo N2L3G1WaterlooONCanada Lea F Santos Department of Physics Yeshiva University 10016New YorkNew YorkUSA David G Cory Institute for Quantum Computing University of Waterloo N2L3G1WaterlooONCanada Department of Chemistry University of Waterloo N2L3G1WaterlooONCanada Perimeter Institute University of Waterloo N2L2Y5WaterlooCanada Canadian Institute For Advanced Research M5G1Z8TorontoONCanada Sensitivity of quantum information to environment perturbations measured with the out-of-time-order correlation function (Dated: April 9, 2019) In a quantum system coupled with a non-Markovian environment, quantum information may flow out of or in to the system. Measuring quantum information flow and its sensitivity to perturbations is important for a better understanding of open quantum systems and for the implementation of quantum technologies. Information gets shared between a quantum system and its environment by means of system-environment correlations (SECs) that grow during their interaction. We design a nuclear magnetic resonance (NMR) experiment to directly observe the evolution of the SECs and use the second moment of their distribution as a natural metric for quantifying the flow of information. In a second experiment, by accounting for the environment dynamics, we study the sensitivity of the shared quantum information to perturbations in the environment. The metric used in this case is the out-of-time-order correlation function (OTOC). By analyzing the decay of the OTOC as a function of the SEC spread, instead of the evolution time, we are able to demonstrate its exponential behavior.The development of quantum technologies is obstructed by the loss of quantum properties caused by interactions with the environment that lead to decoherence [1][2][3][4][5]. Quantum information is shared with the environment by means of systemenvironment correlations (SECs)[6][7][8]. In the case of interactions with a non-Markovian environment, the SECs may lead to the flow of quantum information back to the system [9, 10]. Interferences arising from this backflow may be catastrophic to quantum information processes taking place in the system. The purpose of this work is to investigate the growth of the SECs as a function of time and to gauge how susceptible they are to perturbations in the environment. An open question with respect to the latter point is whether environment perturbations can be used to reduce information backflow.Our experiment is well-equipped to directly measure correlations between the system and the environment. It builds upon solid-state nuclear magnetic resonance (NMR) methods that have been employed to detect multiple-quantum coherences in homonuclear many-body systems[11][12][13][14][15][16][17][18][19]. These methods have recently been used for the investigation of multiple-quantum coherences in ion traps[20,21]. A change in the encoding basis has allowed for the observation of multispin dynamics of correlation growth during the free induction decay experiment[14,16]. Here, we extend these methods to composite heteronuclear systems to measure the growth of correlations with the environment.We consider a central spin model, which consists of a single spin-1/2 interacting with environment spins of another spin species that may also be coupled[22][23][24][25]. Quantum information initially resides in the central spin and is later shared with the environment in the form of multi-spin SECs. * Email: [email protected] NMR techniques make it possible to separate the systemenvironment evolution from the internal evolution of the environment spins[26][27][28]. This allows us to examine the impact of each process individually.We design two different NMR echo experiments. In the first, the environment dynamics is off. We analyze the evolution of the SECs in time and discuss how the second moment of the distribution of these correlations can be used to quantify the flow of quantum information between the system and the environment. In analogy with recent studies [21], we show that this metric is related with the quantum Fisher information.In the second experiment, the environment dynamics is turned on, causing the scrambling of quantum information.To quantify the sensitivity of information to the environment scrambling, we employ the out-of-time-order correlation function (OTOC). Despite great theoretical interest, very few experiments have had access to this quantity. To measure the OTOC, one needs to reverse the time evolution, which makes NMR echo experiments the method of choice. The evolution of the OTOC was previously studied in closed systems of ion traps [20, 21] and nuclear spins[29,30].The OTOC has become a prominent quantity in the analysis of the scrambling of quantum information in black holes and many-body quantum systems. It has been conjectured that the exponential behavior of this quantity should be an indicator of quantum chaos[31,32], the exponential rate being associated with the classical Lyapunov exponent. This correspondence has so far been confirmed theoretically for one-body chaotic systems[33,34]and for the Dicke model [35], but not yet experimentally.Experimental studies of the Loschmidt echo in many-body quantum systems have been carried out in NMR platform and depending on the interaction Hamiltonian, both Gaussian and exponential decays have been observed[36,37]. Our echo arXiv:1808.04375v2 [quant-ph] 7 Apr 2019 2 experiment shows that in the central spin system, the OTOC presents a Gaussian decay in time. However, an exponential behavior is revealed when it is studied as a function of the spread of the correlations between system and environment (that is, as a function of the Hamming weight spread of SECs). The capability of our experiments to directly measure these correlations is an essential ingredient for describing the flow of quantum information and for uncovering the exponential decay of the OTOC.Sample descriptionThe sample studied is a polycrystalline solid at room temperature composed of an ensemble of triphenylphosphine molecules, as shown inFig. 1a(see Sec. A in Methods). Each molecule has a central 31 P nuclear spin coupled to fifteen 1 H environment spins via the heteronuclear dipolar interactionwhere 'cs' stands for central spin, N = 15 is the number of environment spins and σ j X,Y, Z represent Pauli matrices for the j th spin. The dipolar Hamiltonian is a second rank spherical tensor, where the coupling constant ω j ∝ (3 cos 2 θ j − 1)/r 3 j has radial and angular dependence on the vector r j connecting the central spin to the j th spin in the environment. θ j is the angle between r j and the static field of the NMR magnet, which is along z. An environment spin located on the cone defined by the magic angle θ M , with 3 cos 2 θ M − 1 = 0, does not interact with the central spin. The orientation of the sample molecule illustrated inFig. 1ais such that two of the environment spins lie on the magic-angle cone and consequently, do not interact with the central spin. They belong to the "nonconnected" group, while environment spins coupled to the central spin are part of the "connected" group, as sketched inFig. 1b. The size of the connected group grows in time (see Sec. B in Methods).The central spin is initially in the state ρ cs (0) = [1+ σ X ]/2, where is the strength of the nuclear spin polarization which is of the order of 10 −6 at room temperature. In what follows, we drop the identity operator to simplify the notation, since it does not lead to any observable signal. The N spins in the environment are initially in the maximally mixed state ρ E (0) = (1/2) ⊗N , with no correlations. Thus, the initial state of the composite system is uncorrelated, ρ(0) = ρ cs (0)⊗ρ E (0).Mapping the system-environment correlationsWe design an echo experiment, which we call multispin correlation detection (MCD), to measure the correlation growth between the central spin and the environment spins. The stages of the MCD experiment are sketched inFig. 2a. During the evolution time T , the environment self-interaction is averaged to zero using the MREV-8 pulse sequence[26], while the remaining system-environment interaction correlates the two. The dynamics in the composite Hilbert space is described by the unitary propagator U SE (T ) = e −i H SE T , where H SE represents the system-environment interaction Hamiltonian in the toggling frame of the MREV-8 pulse sequence [27] (see Sec. C in Methods). In a quantum system coupled with a non-Markovian environment, quantum information may flow out of or in to the system. Measuring quantum information flow and its sensitivity to perturbations is important for a better understanding of open quantum systems and for the implementation of quantum technologies. Information gets shared between a quantum system and its environment by means of system-environment correlations (SECs) that grow during their interaction. We design a nuclear magnetic resonance (NMR) experiment to directly observe the evolution of the SECs and use the second moment of their distribution as a natural metric for quantifying the flow of information. In a second experiment, by accounting for the environment dynamics, we study the sensitivity of the shared quantum information to perturbations in the environment. The metric used in this case is the out-of-time-order correlation function (OTOC). By analyzing the decay of the OTOC as a function of the SEC spread, instead of the evolution time, we are able to demonstrate its exponential behavior. The development of quantum technologies is obstructed by the loss of quantum properties caused by interactions with the environment that lead to decoherence [1][2][3][4][5]. Quantum information is shared with the environment by means of systemenvironment correlations (SECs) [6][7][8]. In the case of interactions with a non-Markovian environment, the SECs may lead to the flow of quantum information back to the system [9,10]. Interferences arising from this backflow may be catastrophic to quantum information processes taking place in the system. The purpose of this work is to investigate the growth of the SECs as a function of time and to gauge how susceptible they are to perturbations in the environment. An open question with respect to the latter point is whether environment perturbations can be used to reduce information backflow. Our experiment is well-equipped to directly measure correlations between the system and the environment. It builds upon solid-state nuclear magnetic resonance (NMR) methods that have been employed to detect multiple-quantum coherences in homonuclear many-body systems [11][12][13][14][15][16][17][18][19]. These methods have recently been used for the investigation of multiple-quantum coherences in ion traps [20,21]. A change in the encoding basis has allowed for the observation of multispin dynamics of correlation growth during the free induction decay experiment [14,16]. Here, we extend these methods to composite heteronuclear systems to measure the growth of correlations with the environment. We consider a central spin model, which consists of a single spin-1/2 interacting with environment spins of another spin species that may also be coupled [22][23][24][25]. Quantum information initially resides in the central spin and is later shared with the environment in the form of multi-spin SECs. * Email: [email protected] NMR techniques make it possible to separate the systemenvironment evolution from the internal evolution of the environment spins [26][27][28]. This allows us to examine the impact of each process individually. We design two different NMR echo experiments. In the first, the environment dynamics is off. We analyze the evolution of the SECs in time and discuss how the second moment of the distribution of these correlations can be used to quantify the flow of quantum information between the system and the environment. In analogy with recent studies [21], we show that this metric is related with the quantum Fisher information. In the second experiment, the environment dynamics is turned on, causing the scrambling of quantum information. To quantify the sensitivity of information to the environment scrambling, we employ the out-of-time-order correlation function (OTOC). Despite great theoretical interest, very few experiments have had access to this quantity. To measure the OTOC, one needs to reverse the time evolution, which makes NMR echo experiments the method of choice. The evolution of the OTOC was previously studied in closed systems of ion traps [20,21] and nuclear spins [29,30]. The OTOC has become a prominent quantity in the analysis of the scrambling of quantum information in black holes and many-body quantum systems. It has been conjectured that the exponential behavior of this quantity should be an indicator of quantum chaos [31,32], the exponential rate being associated with the classical Lyapunov exponent. This correspondence has so far been confirmed theoretically for one-body chaotic systems [33,34] and for the Dicke model [35], but not yet experimentally. Experimental studies of the Loschmidt echo in many-body quantum systems have been carried out in NMR platform and depending on the interaction Hamiltonian, both Gaussian and exponential decays have been observed [36,37]. Our echo experiment shows that in the central spin system, the OTOC presents a Gaussian decay in time. However, an exponential behavior is revealed when it is studied as a function of the spread of the correlations between system and environment (that is, as a function of the Hamming weight spread of SECs). The capability of our experiments to directly measure these correlations is an essential ingredient for describing the flow of quantum information and for uncovering the exponential decay of the OTOC. Sample description The sample studied is a polycrystalline solid at room temperature composed of an ensemble of triphenylphosphine molecules, as shown in Fig. 1a (see Sec. A in Methods). Each molecule has a central 31 P nuclear spin coupled to fifteen 1 H environment spins via the heteronuclear dipolar interaction H SE = N j ω j σ cs Z ⊗σ j Z ,(1) where 'cs' stands for central spin, N = 15 is the number of environment spins and σ j X,Y, Z represent Pauli matrices for the j th spin. The dipolar Hamiltonian is a second rank spherical tensor, where the coupling constant ω j ∝ (3 cos 2 θ j − 1)/r 3 j has radial and angular dependence on the vector r j connecting the central spin to the j th spin in the environment. θ j is the angle between r j and the static field of the NMR magnet, which is along z. An environment spin located on the cone defined by the magic angle θ M , with 3 cos 2 θ M − 1 = 0, does not interact with the central spin. The orientation of the sample molecule illustrated in Fig. 1a is such that two of the environment spins lie on the magic-angle cone and consequently, do not interact with the central spin. They belong to the "nonconnected" group, while environment spins coupled to the central spin are part of the "connected" group, as sketched in Fig. 1b. The size of the connected group grows in time (see Sec. B in Methods). The central spin is initially in the state ρ cs (0) = [1+ σ X ]/2, where is the strength of the nuclear spin polarization which is of the order of 10 −6 at room temperature. In what follows, we drop the identity operator to simplify the notation, since it does not lead to any observable signal. The N spins in the environment are initially in the maximally mixed state ρ E (0) = (1/2) ⊗N , with no correlations. Thus, the initial state of the composite system is uncorrelated, ρ(0) = ρ cs (0)⊗ρ E (0). Mapping the system-environment correlations We design an echo experiment, which we call multispin correlation detection (MCD), to measure the correlation growth between the central spin and the environment spins. The stages of the MCD experiment are sketched in Fig. 2a. During the evolution time T , the environment self-interaction is averaged to zero using the MREV-8 pulse sequence [26], while the remaining system-environment interaction correlates the two. The dynamics in the composite Hilbert space is described by the unitary propagator U SE (T ) = e −i H SE T , where H SE represents the system-environment interaction Hamiltonian in the toggling frame of the MREV-8 pulse sequence [27] (see Sec. C in Methods). After the evolution with the system-environment Hamiltonian, the resulting density matrix is ρ(T ) = U SE (T )ρ(0)U † SE (T ) (2) = ρ(0) + iT [ρ(0), H SE ] − T 2 2 [[ρ(0), H SE ], H SE ] + . . . . This equation indicates that at short times only the environment spins strongly interacting with the central spin affect the dynamics. The effects of the nested commutators, which are associated with the multi-spin SECs involving weaker interacting spins, become more pronounced as time evolves. Therefore, the evolution of the composite system can be equivalently described using the number of coupled spins and the weight C n (T ) of each cluster as follows (Sec. D in Methods), ρ(T ) = C 0 (T )σ cs X ⊗1 N (3) + C 1 (T ) N j σ cs Y ⊗σ j Z ⊗1 N −1 + C 2 (T ) N j =k σ cs X ⊗σ j Z ⊗σ k Z ⊗1 N −2 + · · · . The observable signal S(T ) from the central spin corresponds to the inner product of the reduced state of the central spin and the measurement operator, S(T ) = Tr[Tr E [ρ(T )].σ CS X ]. Notice that only the first line of Eq. (3) survives the partial trace. Since \|C n (T )\| 2 = 1, as the multi-spin correlations increase, the observable signal from the central spin decays. This is the free induction decay. The measurement operator in the NMR experiment is a transverse single-spin operator, so only the single-spin term in the density matrix induces NMR signal, while the multispin correlated terms are not directly observable. In our case, the reduced state of the environment would not reveal the evolution of SECs, because Tr cs [ρ(T )] = 1. In order to observe the growth of the SECs, we implement the multiple-quantum coherence method for encoding the coherence orders and then detect them through the central spin, which is our probe. In the MCD experiment, by collectively rotating the environment spins along the x-axis, R x (φ) = exp i φ 2 j 1 cs ⊗σ j X , we manage to get the coherence order encoded in a phase factor ρ φ (T ) = R x (φ)ρ(T )R † x (φ) = n e inφ C n (T )ρ n .(4) In the above, ρ n indicates the subset of spin operators in Liouville space with correlation order n with respect to the xbasis. C n (T ) represents the weight of the multi-spin terms with correlation order n. In this basis, the ladder operators are Σ j ± = σ j Y ± iσ j Z , and the correlation order is defined as the absolute value of the number of Σ + minus Σ − operators. This number represents the Hamming weight. Consequently, the relevant description of the density matrix in the x-basis is ρ(T ) = n C n (T )ρ n .(5) In contrast with Eq.(3), where the multi-spin terms are categorized by the number of correlated spins, in this equation they are distinguished by their correlation orders (see details in the Secs. D and E in Methods). After the encoding rotation, the system-environment Hamiltonian is reversed to create an observable echo at time 2T , S φ (2T ) = Tr[ρ φ (2T )ρ(0)],(6) where ρ φ (2T ) = U † SE (T )R x (φ)U SE (T )ρ(0)U † SE (T )R † x (φ)U SE (T ). The Fourier transform of S φ (2T ) gives the correlation amplitudes \|C n (T )\| 2 . They provide snapshots of the SECs at each time T . As seen in Fig. 2b, the distribution of the correlation orders has a Gaussian shape. This form emerges because the signal is an ensemble average over various molecules where each one has a binomial distribution of C n (T )'s (see Sec. E in Methods). A map of the SEC production is obtained by following the correlation amplitudes as a function of T , as depicted in Fig. 2c. There, we show the six largest amplitudes. At short times, the first term in C 0 (T ) coincides with the C 0 (T ) term and dominates the dynamics. At long times, the evolution of the amplitudes saturates and we observe that the values of the C 0 (T ) and C 2 (T ) terms approach each other. This is because the production of the C 2 (T ) term is responsible for the onset of both C 2 (T ) and also the terms Σ j + ⊗Σ k − in C 0 (T ). To quantify the SEC production, one can use the largest correlation order observed in the MCD experiment, which is plotted in Fig. 3. However, it becomes more difficult to detect the correlation orders as they increase, because the amplitude of the largest correlation order drops exponentially (see Sec. E in Methods). Alternatively, the second moment of the distribution of the correlation orders, n \|C n (T )\| 2 n 2 , is a more reliable experimental measure for quantifying the extent of the correlations. The second moment (variance) of a binomial distribution centered at 0 is equal to n. This value is the square of the width of the SEC distribution in Fig. 2b. Here, we refer to the second moment as the "Hamming weight spread". The second moment of the coherence distribution is also used in [12,15] for quantifying the number of spins involved in the clusters of linked spins in homonuclear solid-state systems. \|C0(T) 2 \|C1(T) 2 \|C2(T) 2 \|C3(T) 2 \|C4(T) 2 \|C5(T) 2 In Fig. 3, we show the Hamming weight spread as a function of T , which initially grows slowly and later linearly before saturating. The point of saturation depends on the size of the connected group of environment spins (Sec. B in Methods). In Ref. [21], it was shown that the second moment of the multiple quantum coherences for many-body systems with mixed states is a lower bound on the quantum Fisher information (QFI). In our experiment, the QFI is associated with the information shared between the system and the environment and the rate of its change measures the information flow, as discussed in [38]. By extension, the slope of the curve for the Hamming weight spread can be used as a metric to quantify the flow of information between the system and the environment. Scrambling of information in the environment In the second experiment, we explore the resistance of the quantum information shared between the system and environment against perturbations in the environment. The latter refer to changes that take place when we turn on the dynamics in the environment. The homonuclear dipolar Hamiltonian is given by H E = 1 cs ⊗ N j<k Ω jk σ j Z σ k Z − 1 4 (σ j + σ k − + σ j − σ k + ) ,(7) where Ω jk ∝ (3 cos 2 θ jk − 1)/r 3 jk is the coupling strength between the environment spins j and k. The operators in parenthesis in Eq. (7) represent the flip-flop term that swaps the states of pairs of environment spins and scrambles quantum information. The eigenvalues of this Hamiltonian satisfy level statistics given by random matrix theory (Sec. G in Methods), as in quantum systems with chaotic classical counterparts. Consequently, the scrambling of information within the environment is expected to take place very fast. To analyze the sensitivity of quantum information to scrambling in the environment, we use the echo experiment outlined in Fig. 4a. In this experiment, the evolution interval T , where only the coupling between the system and the environment is effective, is followed by a scrambling window of length τ , where only the environment spins interact and the propagator is U E (τ ) = e −iH E τ . Information shared with the environment in the course of time T gets scrambled during τ . The length of the scrambling window τ determines the strength of the environment perturbation. The observable echo signal amplitude at 2T + τ is given by S(2T + τ ) = Tr[ρ(2T + τ ) · ρ(0)],(8) where ρ(2T +τ ) = U † SE (T )U E (τ )U SE (T )ρ(0)U † SE (T )U † E (τ )U SE (T ). For a fixed value of τ , as the evolution time T increases, the overlap between the density matrix ρ(2T + τ ) and the initial density matrix decreases, resulting in the decay of the echo signal. The signal S(2T + τ ) for a fixed scrambling window τ can be written in the form of the OTOC function. The latter is defined as F (T ) ≡ W † (T )V (0) † W (T )V (0) ,(9) where V (0) and W (T ) are two unitary operators that commute at T = 0. We choose V (0) to be equal to the initial density matrix ρ(0) = ρ cs (0)⊗( 1 2 ) ⊗N and consider the environment operator in the Heisenberg picture as the second operator W τ (T ) = U † SE (T )U † E (τ )U SE (T ).(10) Therefore, with the assumption of infinite temperature the expectation value in Eq.(9) results in The OTOC function is related to the commutator between V (0) and W τ (T ) as Re[F τ (T )] = {1 − [W τ (T ), V (0)] 2 /2}. As T becomes larger and W τ (T ) gets more distant from W τ (0), the commutator [W τ (T ), V (0)] increases and the OTOC decreases. Physically, what happens is that as the SECs grow, the environment interaction H E has access to a larger subset of correlated spins and the number of swaps that effectively scramble information increases. Therefore, the OTOC decay quantifies the level of sensitivity of quantum information to perturbations in the environment. F τ (T ) ≡ Tr[W † τ (T )ρ(0)W τ (T )ρ(0)] = S(2T + τ )· (11) In Fig. 4b, the result of the OTOC decay is presented for various evolution and perturbation times. When τ = 0, the state of the central spin is completely refocused (revived) and F τ =0 (T ) = 1 for all T . This happens because the information that is initially encoded in the central spin is not lost during the evolution time. It is simply stored in the form of multi-spin correlations between the system and the environment spins from the connected group. By reversing the evolution, the information can be recovered in the system. The refocusing degrades and the echo amplitude decays as τ increases. During the scrambling window, the flip-flop term of H E swaps the states of coupled spin pairs in the environment. As a result, the subsequent evolution under the inverse of the system-environment Hamiltonian can only partially revive the initial state. This situation is aggravated by the existence of the non-connected group of environment spins, which do not develop correlations with the central spin during T , but may have their states swapped with those from the connected group during the scrambling window. Information that is shared with the non-connected group cannot be recovered, which ultimately leads to the loss of quantum information in the environment. Consequently, the sensitivity of shared quantum information to perturbations in the environment, depends on the scrambling window length as well as the size of connected and non-connected spin groups in the environment. Exponential decay of the OTOC For a more detailed analysis of the results for the OTOC presented in Fig.4b, we now show in Fig. 5a, the OTOC as a function of the evolution time T for different perturbation strengths and compare it with Fig. 5b, where the OTOC is presented as a function of the Hamming weight spread. As seen in Fig. 5a, the OTOC decay in time is well described by a Gaussian function. This is understandable because the evolution of environment spins under the homonuclear dipolar interaction H E , examined with a free induction decay experiment, is known to give a signal decay with Gaussian shape. This behavior is typical of solid-state spin systems [39,40]. In the case of our composite system, the scrambling of quantum information that happens only in the environment, is subject to the same homonuclear dipolar Hamiltonian. The effectiveness of quantum information scrambling, probed with the OTOC decay, increases with the extent of quantum information shared with the environment. As seen in Fig. 3, the extent of shared information, which is measured with the Hamming weight spread, does not always grow linearly. So in Fig. 5b, The OTOC decays exponentially as a function of systemenvironment correlation spread. Panel a depicts the OTOC as a function of the evolution time T and b as a function of the Hamming weight spread. Each curve in the panels corresponds to a fixed value of the scrambling window τ (perturbation strength). The data are normalized with respect to the τ = 0 data set. The dashed lines in a and b are Gaussian and exponential fits, respectively. The exponential behavior is uncovered by analyzing the OTOC decay as a function of the Hamming weight spread. Because of the environment's finite size and the memory effects, the late decay data is better described by a Gaussian function. Error bars in panel a and b indicate the inverse of signal-to-noise ratio. In panel c, the scrambling immunity factor indicates the capability of environment interactions in disrupting the system-environment correlations. This plot indicates that with increasing τ , even smaller SECs become sensitive to the environment perturbations. The data is fitted with an exponential decay curve. The error bars correspond to the errors for the exponential fits in the panel b. Similar dynamics is observed for swapping coins in a classical coin game (Sec. F in Methods). periment as the variable for the OTOC instead of time. The resulting behavior of the OTOC is exponential, as corroborated by the exponential fits in Fig. 5b. The search for the quantum counterpart of the exponential instability observed in chaotic classical systems has been a subject of discussion for many years [41][42][43][44][45][46]. The subject is now under intense investigation in part due to a conjecture that associates the exponential rate of change of the OTOC with the classical Lyapunov exponent. In the case of manybody spin systems, the OTOC exponential behavior has not yet been observed [47]. Here, we show that by removing the non-linear rate of the correlation growth, that is by investigating the OTOC against SEC sizes, the exponential instability can be uncovered. This observation indicates that the amount of quantum information shared between the system and environment determines the capability of the flip-flop Hamiltonian in scrambling the shared quantum information. We developed a classical coin game to illustrate the dynamics of spin swaps between connected and non-connected spin groups and to justify the exponential decay of the OTOC (Sec. E in Methods). The idea goes as follows. Among N coins, we randomly flip k. If these same k coins are flipped a second time, the initial state is recovered. However, if we swap some of the N coins before the second flip, the final state may be different from the initial one. This happens when some of the N − k coins get swapped with some of the k coins. In this game, k coins represent the connected group of the environment spins, N −k coins portray the non-connected group, and the number of coin swaps is analogous to the perturbation strength in the environment. We find that similar to Fig. 5b, the probability of recovering the initial coin array decreases exponentially as a function of the initial number of flipped coins k. The inverse rate of this exponential decay characterizes the capability of a fixed number of coin swaps to disrupt the coin array recovery, and is called swap immunity factor. As expected, the swap immunity factor is smaller when a larger number of swaps are performed. This classical game provides a simplified picture of the mechanism underlying the exponential instability of the composite quantum system. Motivated by the coin game analysis, we plot in Fig. 5c the "scrambling immunity factors" obtained from the inverse of the decay rates of the exponential fits in Fig. 5b. Similar to the concept of the swap immunity factor for a coin array, the scrambling immunity factor characterizes the capability of the flip-flop Hamiltonian to disrupt the SECs for a given perturbation window τ , resulting in the incomplete revival of the central spin state. In other words, this factor characterizes the sensitivity of SECs to the scrambling of quantum information in the environment for various perturbation strengths. For the small scrambling windows τ = 6, 8 µs, the scrambling immunity factor is unreasonably large, as the perturbation is too small and leaves the environment effectively unscrambled for most orientations of the spins. For larger perturbation windows, the scrambling immunity factor decays exponentially with τ . This shows that, the perturbation strength needed for the effective information scrambling is much smaller for larger SECs. The scrambling of quantum information is most effective when it involves the non-connected spin group of the environment. Quantum information transferred to the non-connected group can be considered lost. As it cannot produce any echo signal, it does not contribute to the backflow of quantum information to the system. Thus, the scrambling immunity factor provides an upper bound for the effectiveness of environment perturbation in removing quantum information backflow. Conclusion We introduced the scrambling immunity factor to characterize the capability of environment perturbations to disrupt the system-environment correlations. Our experiments also enabled us to quantify the flow of quantum information between the system and environment. We proposed an alternative way to analyze nonequilibrium quantum dynamics, where instead of time, quantities of interest are studied as a function of the SEC spread. This approach uncovered the exponential instability of our many-body spin system by showing that the OTOC decays exponentially as a function of Hamming weight spread. The heart of these experiments is our correlation detection method and this technique is not restricted to the system considered here. It can be used for any quantum system if global control of the environment is available. This is relevant for many-body quantum systems where the measurements are performed on a subsystem and the rest acts as an environment. Methods A. Sample Triphenylphosphine is a common organophosphorous compound and was obtained from SIGMA-ALDRICH with 99% purity. To reduce the T 1 relaxation time of the protons we used Chromium(III) acetylacetonate as a relaxation agent. 1 mmol of the sample and 0.13 mmol of the relaxation agent were resolved in 300 ml of Chloroform-d and left for crystallization over night. The resulting powder was compressed into a NMR-sphere sample tube which was flame-sealed to best preserve the contents. Using the relaxation agent resulted in the reduction of the proton T 1 relaxation time from 630 ± 30s to 2.5 ± 0.2s. B. Spin groups in the environment The multi-spin correlation growth and the information scrambling in the environment, both depend on the strength of the dipolar interactions. The dipolar interaction strength depends on the relative orientation of the spins with respect to the static field of the NMR magnet. The idea of distinguishing connected and non-connected spin groups in the environment can be explored by considering the number of spins in the environment with a high probability of being correlated to the central spin. The total number of these spins in the environment, increase with the evolution time. Fig. 6 plots the average number of environment spins that have a probability larger than 1 2 for being correlated with the central spin as a function of the evolution time in the MCD experiment. Heteronuclear dipolar coupling constants are evaluated for 2000 randomly orientated Triphenylphosphine molecules. For the longest evolution time in the MCD experiment, T = 532 µs on average 8.2 environment spins are found to be more likely to correlate with the central spin. C. NMR experiments The MCD experiment captures snapshots of the multi-spin SECs at specific evolution times. This experiment is designed to initiate the growth of SECs from the central spin, and also to use the central spin itself as a probe for the detection of SECs. Figure 7a shows the two channel NMR pulse program used for simultaneous control of the central spin ( 31 P) and the environment spins ( 1 H) in the MCD experiment. The Cross Polarization (CP) step is employed to remove any initial environment correlations, and to increase the sensitivity of the experiment by enhancing the initial polarization of the central spin, in addition to reducing the necessary repetition delay time. Evolution under the heteronuclear dipolar interaction for time T results in the growth of the SECs, while the homonuclear dipolar interaction in the environment is averaged out with the MREV-8 pulse sequence. Under the MREV8 cycle, the σ Z operator for the environment spins is transformed to a vector pointing at the (1, 0, 1) direction with the scaling factor of [26]: α = √ 2(1 + 2 3tp τc ( 4 π − 1)) 3 ,(12) where t p is the pulse length and τ c is the length of the MREV-8 sequence. Consequently, the zeroth order of the average Hamiltonian for the heteronuclear dipolar interaction in Eq. (1), in the toggling frame of the MREV-8 pulse sequence, is: H SE = 0.36 j ω j (σ cs Z ⊗σ j X + σ cs Z ⊗σ j Z ).(13) Note that because of the symmetry in this Hamiltonian, the σ X and σ Z operators in the environment are produced with the same weight. Hence, all the equations below Eq.(2) are written for H SE instead of H SE . This is allowed because this experiment uses x as the quantization axis, and therefore it is insensitive to σ X operators that appear in H SE and not in H SE . However, the reader should keep in mind that the environment σ X operators have an equal rate of production as the σ Z operators. The only observable difference between these two Hamiltonian in the MCD experiment is that for the H SE , the rate of production of the σ Z operators in the environment is scaled down with the scaling factor α. As shown in Fig. 7a, after T , we apply a collective rotation φ x on the environment spins to encode the correlation order as a phase factor e inφ , which is observed at the end of the experiment. Next, the sign of the heteronuclear dipolar interaction is virtually changed by sandwiching the evolution period with π rotations on the central spin, in order to create an echo signal at time 2T . Finally, a decoupling sequence is applied to remove interactions with the environment during the detection and to achieve the maximum signal-to-noise ratio. Figure 7b sketches the pulse sequence used for measuring the OTOC decay. In this experiment, the collective rotation of the environment spins is removed and a scrambling window is introduced. During this window, the environment spins evolve under the homonuclear dipolar interaction, while the central spin is decoupled from them. The echo signal at the end of this experiment provides the ratio of the multi-spin correlated terms that were not affected by the homonuclear dipolar interaction during the scrambling window. D. Growth of the correlated multi-spin terms For a closed environment with N spins, the unitary evolution of the system-environment under the heteronuclear dipolar Hamiltonian, Eq.(1), can be written as ρ(t) = U (t).ρ(0).U † (t) = 1 2 N +1 {σ CS X ⊗1 ⊗N N i=1 cos(ω i t) + N j=1 σ CS Y ⊗σ j Z ⊗1 ⊗N −1 sin(ω j t) N i =j cos(ω i t) − N j,k σ CS X ⊗σ j Z ⊗σ k Z ⊗1 ⊗N −2 sin(ω j t) sin(ω k t) N i =j,k cos(ω i t) − N j,k,l σ CS Y ⊗σ j Z ⊗σ k Z ⊗σ l Z ⊗1 ⊗N −3 sin(ω j t) sin(ω k t) sin(ω l t) N i =j,k,l cos(ω i t) + N j,k,l,m . . . }.(14) The equation above can be put in the form of Eq. (3) if all of the coupling constants are known for the molecules in the ensemble, one can calculate the weight of each correlation order C n (T ) in Eq. (3). The equation above shows that higher orders of SECs become non-negligible only at longer evolution times. In the x-basis the environment part of the total density matrix leads to off-diagonal elements (coherences) along x axis, that can be accessed experimentally. In this basis, the ladder operators are Σ j ± = σ j Y ± iσ j Z and the density matrix is written as ρ(T ) = C 0 (T ) N j =k σ cs X ⊗[1 N − (Σ j + ⊗Σ k − ⊗1 N −2 ) + · · · ] +C 1 (T ) N j σ cs Y ⊗[(Σ j + ⊗1 N −1 ) − (Σ j − ⊗1 N −1 ) + · · · ] +C 2 (T ) N j =k σ cs X ⊗[(Σ j + ⊗Σ k + ⊗1 N −2 ) + (Σ j − ⊗Σ k − ⊗1 N −2 ) + · · · ] + · · · (15) = n C n (T )ρ n ρ n 's are vectors from the Liouville space describing the measurement basis of the MCD experiment, and they include all permutations of Σ ± operators with correlation order n. E. Correlation orders vs number of correlated spins To understand the spin physics of the MCD experiment, an example of a central spin model with two spins in the environment, N = 2, is explored in this section. After the cross polarization step, the density matrix for the central spin and the environment is ρ(0) = 1 2 3 σ CS X ⊗1⊗1.(16) Since pre-existing correlations between the central spin and the environment spins disappear during the spin locking pulse, the initial system-environment state is uncorrelated. In order to maintain our SEC terms as simple as possible, we assume that the homonuclear dipolar interaction in the environment is completely turned off during the evolution step and the central spin evolves under the heteronuclear dipolar interaction: H 1,2 SE = ω 1 2 {σ CS Z ⊗σ 1 Z ⊗1} + ω 2 2 {σ CS Z ⊗1⊗σ 2 Z }.(17) After the evolution time T , the density matrix evolves to ρ(T ) = 1 8 cos(ω 1 T ) cos(ω 2 T ) σ CS X ⊗1⊗1 (18) + sin(ω 1 T ) cos(ω 2 T ) σ CS Y ⊗σ Z ⊗1 + cos(ω 1 T ) sin(ω 2 T ) σ CS Y ⊗1⊗σ Z − sin(ω 1 T ) sin(ω 2 T ) σ CS X ⊗σ Z ⊗σ Z . This is equivalent to the description of the density matrix using the number of coupled spins, Eq.(3). The coefficients of the various spin terms above correspond to the C n (T )'s in Eq.(3). Notice that in the NMR experiment, spins are not distinguishable and only the sum of all single spin correlation terms in Eq. (18) are observed. Collective rotation of the environment spins by φ about the x axis, R X (φ) = exp(i φ 2 i 1 CS ⊗σ i X ), transforms the density matrix to: ρ φ (T ) = 1 8 σ CS X ⊗1⊗1 cos(ω 1 T ) cos(ω 2 T )(19)+ cos(φ){σ CS Y ⊗σ Z ⊗1 sin(ω 1 T ) cos(ω 2 T ) + σ CS Y ⊗1⊗σ Z cos(ω 1 T ) sin(ω 2 T )} + sin(φ){σ CS Y ⊗σ Y ⊗1 sin(ω 1 T ) cos(ω 2 T ) + σ CS Y ⊗1⊗σ Y cos(ω 1 T ) sin(ω 2 T )} − sin(ω 1 T ) sin(ω 2 T ){cos(φ) 2 σ CS X ⊗σ Z ⊗σ Z + sin(φ) 2 σ CS X ⊗σ Y ⊗σ Y } − sin(ω 1 T ) sin(ω 2 T ) cos(φ) sin(φ){σ CS X ⊗σ Z ⊗σ Y + σ CS X ⊗σ Y ⊗σ Z } . Consequently, the density matrix terms gain a cos(φ) n factor where n corresponds to the number of σ Z operators in the multi-spin correlated terms. The next step is another evolution interval T with the inverse of Eq. (17). The resulting density matrix ρ φ (2T ) is given by a long equation shown in Ref [48]. But from this equation, the only observable terms are the following ones          cos(ω 1 T ) 2 cos(ω 2 T ) 2 σ CS X ⊗1⊗1 cos(φ) sin(ω 1 T ) 2 cos(ω 2 T ) 2 σ CS X ⊗1⊗1 cos(φ) cos(ω 1 T ) 2 sin(ω 2 T ) 2 σ CS X ⊗1⊗1 cos(φ) 2 sin(ω 1 T ) 2 sin(ω 2 T ) 2 σ CS X ⊗1⊗1 The signal amplitude is evaluated with the inner product of the reduced state of the central spin and the measurement operator, σ CS X , at 2T : S φ (2T ) = Tr[Tr E [ρ(2T )].σ CS X ](20) = cos(ω 1 T ) 2 cos(ω 2 T ) 2 + cos(φ){cos(ω 1 T ) 2 sin(ω 2 T ) 2 + sin(ω 1 T ) 2 cos(ω 2 T ) 2 } + cos(φ) 2 sin(ω 1 T ) 2 sin(ω 2 T ) 2 . The data set containing amplitudes of S φ (2T ) for various encoding angles φ is Fourier transformed to evaluate the weight of each correlation order \|C n (T )\| 2 : F[S φ (2T )] = cos(ω 1 T ) 2 cos(ω 2 T ) 2 δ(n)(21)+ {cos(ω 1 T ) 2 sin(ω 2 T ) 2 + sin(ω 1 T ) 2 cos(ω 2 T ) 2 }[ 1 2 δ(n − 1) + 1 2 δ(n + 1)] + sin(ω 1 T ) 2 sin(ω 2 T ) 2 [ 1 4 δ(n − 2) + 1 2 δ(n) + 1 4 δ(n + 2)]. We can compare these coefficients with the density matrix weights C n (T ) expressed in Eq. (18). First, notice that the amplitude of the order n of the Fourier transformed signal is given by the squared coefficients \|C n (T )\| 2 of ρ(T ). The first line in the equation above is the signal resulted from the uncorrelated spin term in the first line of Eq.(18), the C 0 (T ) term. In the SEC spectrum, this is the amplitude for n = 0. The second line is produced by spin terms with one spin correlated to the central spin, that is the C 1 (T ) terms. In the SEC spectrum they show up at n = ±1. The third line is produced by the term which has two environment spins correlated to the central spin, that is the C 2 (T ) term. In the SEC spectrum it shows up at n = 0, ±2. Thus, the C 2 (T ) term contributes to the production of both C 0 (T ) and C 2 (T ) terms. This explains why in Fig. 2c, \|C 0 (T )\| 2 and \|C 2 (T )\| 2 saturate to approximately the same value. It is easy to show that all even(odd) powers of cos(φ) n in the C n (T ) terms produce Fourier components at even(odd) orders of C n (T ) terms, where the amplitude of each peak is evaluated with the coefficients of the binomial distribution [48]. Therefore, the amplitude of the largest observed correlation order for each molecule scales down with a factor of 1 2 n . Consequently, the second moment of the correlation order spectrum is more suitable as a measure for the extent of SECs, than the largest observed order. F. Classical coin game We have designed a classical game to make a parallel with the swap dynamics of the environment spins that take place in our second experiment. This game simulates the loss of echo signal resulting only from spin swaps between the connected and non-connected spin groups. It does not address the decay resulting from spin swaps in the connected spin group. Consider an array of N coins initially set to heads. We randomly flip k of these coins to represent spins in the connected group at time T , with the constraint that each coin may be flipped only once. The remaining N − k coins represent the non-connected group. Subsequently, if the coins are not swapped, flipping the same random k coins for a second time results in the complete return to the initial state. This is equivalent to a perfect echo of the spin signal at time 2T . However, when we add random swap actions between the two flipping stages, the final state of the coin array may be different from its initial state. The distance between the initial and the final state of the coin array depends on the number of swaps performed between flipped coins and un-flipped coins. The probability of having this sort of "successful swap" (ssw) for each spin pair, that is swaps that increase the distance between the two states of the coin array, is given by P ssw = 2 kN − k 2 N 2 − N ·(22) We ignore the cases where the same two coins swap more than once, and we assume that after each swap, the probability of having a successful swap remains unchanged. Then the probability of success for m random coin swaps is m times the probability of success for one coin swap. Consequently, the overlap amplitude A OL between the initial state and the final state of the coin chain after two rounds of flips with a round of coin swap in the middle is A OL (m, k, N ) := 1 − 2m N P ssw 2 = 1 − 4m N kN − k 2 N 2 − N 2 .(23) A OL (m, k, N ) is plotted in Fig. 8 for an array of N = 15 coins, which is the number of environment spins, while k is set according to the Hamming weight spread for various evolution time steps T in the MCD experiment, and m is varied from 0 to 10. We have fitted the data in Fig.8 with a series of decaying exponential functions to characterize the capability of various numbers of coin swaps to disrupt the overlap between the initial and final coin state [48]. The overlap probability becomes 1 e , when k is the inverse of the exponential decay rate. We call this inverse rate the "swap immunity factor" and plot it as a function of m in Fig. 9. The swap immunity factor indicates the effectiveness of the number of coin swaps in obstructing the coin array recovery. Similar to the OTOC experimental results in Fig. 5c, the swap immunity factor decays exponentially with m, for m above a threshold. Fig.5c. The dashed line indicates an exponential fit for this part of the data. For m < 4 the swap immunity factor is larger than the size of the coin array, which means that coin swap cannot effectively disturb the coin array recovery. This final remark is also similar to the discussion about Fig.5c, when τ is small. G. Chaotic environment Quantum chaos refers to properties of the spectrum that indicate whether the classical counterpart of the quantum system is chaotic. One of the main signatures of chaos is the strong repulsion of the eigenvalues [49]. The energy levels of quantum systems that are classically chaotic are correlated and prohibited from crossing. This is detected, for example, with the distribution P (s) of the unfolded spacings s between neighboring levels. In the case of real and symmetric Hamiltonian matrices, as in our case, the level spacing distribution follows closely the Wigner surmise, P (s) = πs 2 exp − πs 2 4 .(24) We verified that P (s) for the Hamiltonian in Eq. (7) is well described with this equation. The spread of information in chaotic systems far from equilibrium happens very fast [50]. This is the scenario of our composite system, where information is initially confined to a single spin and the environment is chaotic. Data availability. The data sets generated during the current study are available from the corresponding author on reasonable request. FIG. 1 : 1Sample structure matching the central spin model. The Triphenylphosphine molecule shown in panel a has a 31 P nucleus at the central spin position and fifteen 1 H nuclei as the environment spins. The NMR experiment is performed on an ensemble of these molecules in random orientations. Due to the angular dependence of the dipolar interaction, environment spins located near the two magic-angle cones (shaded area) are very weakly coupled to the central spin. Therefore, environment spins may be divided into a connected and a non-connected group, as sketched in panel b. FIG. 2 : 2The multi-spin correlation detection (MCD) experiment. The stages of the MCD experiment are sketched in panel a. The plot for the correlation amplitudes in panel b displays the distribution of the system-environment correlations (SECs) at a chosen time T . The dashed line is a Gaussian fit and the arrow indicates the Hamming weight spread. The time evolution of the correlation amplitudes \|Cn(T )\| 2 are shown in panel c. The error bars, corresponding to the inverse of the signal-to-noise ratio, are very small and not visible in panel c. FIG. 3 : 3Hamming weight spread. The second moment of the distribution of multi-spin system-environment correlations (SECs) and the largest correlation order as a function of time are indicators for the growth of the system-environment correlations. The Hamming weight spread is used to quantify the extent of quantum information shared with the environment. FIG. 4 : 4The OTOC Fτ (T ) measures the sensitivity of quantum information to environment perturbations. Sketch of the steps involved in the OTOC experiment is depicted in a. The decay of the echo amplitude as a function of the scrambling window length τ for different values of the evolution time T is shown in b. For small perturbations (short τ ), the central spin for most molecules gets refocused and the OTOC is nearly independent of T , while for large perturbations, the OTOC decays significantly with T . we shift the perspective of the analysis and use the Hamming weight spread obtained in the first ex- Evolution time T (μs)Connected group size FIG. 6: A simulation of 10000 random orientations of the Triphenylphosphine molecule is used to estimate the size of the connected spin group for various evolution times in the MCD experiment. FIG. 7 : 7The NMR pulse sequence for the multi-spin Correlation Detection (MCD) experiment is shown in panel a. Panel b shows the pulse sequence for the OTOC measurement with quantum information scrambling implementation in the environment. FIG. 8 : 8The classical coin game is similar to the scrambling of quantum information in the spin environment. 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[ "Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians", "Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians" ]	[ "K Zelaya \nCentre de Recherches Mathématiques\nUniversité de Montréal\nH3C 3J7MontréalQCCanada\n", "I Marquette \nSchool of Mathematics and Physics\nThe University of Queensland\n4072BrisbaneQLDAustralia\n", "V Hussin \nCentre de Recherches Mathématiques\nUniversité de Montréal\nH3C 3J7MontréalQCCanada\n\nDépartement de Mathématiques et de Statistique\nUniversité de Montréal\nH3C 3J7MontréalQCCanada\n" ]	[ "Centre de Recherches Mathématiques\nUniversité de Montréal\nH3C 3J7MontréalQCCanada", "School of Mathematics and Physics\nThe University of Queensland\n4072BrisbaneQLDAustralia", "Centre de Recherches Mathématiques\nUniversité de Montréal\nH3C 3J7MontréalQCCanada", "Département de Mathématiques et de Statistique\nUniversité de Montréal\nH3C 3J7MontréalQCCanada" ]	[]	In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. The new quantum invariant is constructed by adding a deformation term to the well-known parametric oscillator invariant. Such a deformation depends explicitly on time through the solutions of the Ermakov equation, which ensures the regularity of the new time-dependent potential of the Hamiltonian at each time. On the other hand, with the aid of the proper reparametrization, the fourth Painlevé equation appears, the parameters of which dictate the spectral behavior of the quantum invariant. In particular, the eigenfunctions of the third-order ladder operators lead to several sequences of solutions to the Schrödinger equation, determined in terms of the solutions of a Riccati equation, Okamoto polynomials, or nonlinear bound states of the derivative nonlinear Schrödinger equation. Remarkably, it is noticed that the solutions in terms of the nonlinear bound states lead to a quantum invariant with equidistant eigenvalues, which contains both an (N+1)-dimensional and an infinite sequence of eigenfunctions. The resulting family of time-dependent Hamiltonians is such that, to the authors' knowledge, have been unnoticed in the literature of stationary and nonstationary systems. arXiv:2006.00207v1 [quant-ph] 30 May 2020	10.1088/1751-8121/abcab8	[ "https://arxiv.org/pdf/2006.00207v1.pdf" ]	219,176,830	2006.00207	29be90bdf0c59ab6e5cc239729761579eaf18561	Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians K Zelaya Centre de Recherches Mathématiques Université de Montréal H3C 3J7MontréalQCCanada I Marquette School of Mathematics and Physics The University of Queensland 4072BrisbaneQLDAustralia V Hussin Centre de Recherches Mathématiques Université de Montréal H3C 3J7MontréalQCCanada Département de Mathématiques et de Statistique Université de Montréal H3C 3J7MontréalQCCanada Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. The new quantum invariant is constructed by adding a deformation term to the well-known parametric oscillator invariant. Such a deformation depends explicitly on time through the solutions of the Ermakov equation, which ensures the regularity of the new time-dependent potential of the Hamiltonian at each time. On the other hand, with the aid of the proper reparametrization, the fourth Painlevé equation appears, the parameters of which dictate the spectral behavior of the quantum invariant. In particular, the eigenfunctions of the third-order ladder operators lead to several sequences of solutions to the Schrödinger equation, determined in terms of the solutions of a Riccati equation, Okamoto polynomials, or nonlinear bound states of the derivative nonlinear Schrödinger equation. Remarkably, it is noticed that the solutions in terms of the nonlinear bound states lead to a quantum invariant with equidistant eigenvalues, which contains both an (N+1)-dimensional and an infinite sequence of eigenfunctions. The resulting family of time-dependent Hamiltonians is such that, to the authors' knowledge, have been unnoticed in the literature of stationary and nonstationary systems. arXiv:2006.00207v1 [quant-ph] 30 May 2020 Introduction During the last several decades, physicists have realized the importance of nonlinear equations in the study of physical systems, even in cases where linear equations govern the dynamical laws. For instance, in quantum mechanics, the nonlinear Riccati equation [1,2] has played a fundamental role in the construction and study of new exactlysolvable models, for it relates the factorization method [3][4][5][6][7][8], with the Darboux transformation [9]. The latter can be formulated in the so-called supersymmetric quantum mechanics (SUSYQM) [10][11][12][13]. In this regard, systems with spectrum on-demand are obtained either by adding energy level not present in the original model, or by removing levels through the Darboux-Crum transformation [14]. This formalism represents an outstanding progress in the study of quantum systems; it has allowed extending the families of exactly-solvable models [15,16] beyond the conventional harmonic oscillator, hydrogen atom, the interaction between diatomic molecules, and some few others. Moreover, quantum mechanics in the non-Hermitian regime has been explored by generalizing the factorization method and allowing the Riccati equation to be a complex-valued function. In this form, the realitiy of the spectrum is preserved [17][18][19] in systems with either broken and unbroken P T -symmetry, extending the conventional systems with P T -symmetry and real spectrum [20][21][22]. From the latter, the nonlinear Ermakov equation [23][24][25] emerges naturally from the complexified Riccati one, the solutions of which ensures the regularity of the new complex-valued potentials. For more details on the applications of the Riccati and Ermakov equations in physics, see [2]. Among other nonlinear equations, we have the Painlevé transcendentals, a family of six nonlinear equations P I -P VI with complex parameters, whose solutions are in general transcendental, that is, theyr are not be expressed in terms of classical functions [26,27]. Nevertheless, for some specific values of the parameters, a seed function can be used to generate a complete hierarchy of solutions through the Bäcklund transformation [28], which can be thought as a nonlinear counterpart of the recurrence relations. In particular, the fourth Painlevé equation can be taken into a Riccati equation with the appropriate choice of the parameters. we thus solve a "simpler" nonlinear equation instead. Also, the fourth Painlevé equation has also brought new result in the trend of orthogonal polynomials, where new families were discovered though the hierarchies of rational solutions in terms of the generalized Okamoto, generalized Hermite and Yablonskii-Vorob'ev polynomials [29]. The Painlevé transcendental have also found interesting applications in the study of physical models in nonlinear optics [30], quantum gravity [31], and SUSYQM [32][33][34], to mention some. Interestingly, the fourth Painlevé equation arises quite naturally in third-order shape-invariant SUSYQM [33], where the parameters of the Painlevé transcendental define the eigenvalues of the new Hamiltonians. The respective intertwining operators serve at the same time as ladder operators, from where the eigenfunctions are determined. A striking feature of this approach is that, in general, the so-constructed intertwining operators are not in general factorizale in terms of first-order operators. Thus, the results obtained in this way generalize those of [35]. It is worth to notice that higher-order ladder operators have been also studied, in a different way, in the context of supersymmetric (SUSY) partners for the stationary oscillator in both the Hermitian [16,36] and non-Hermitian regimes [18]. Although a vast literature on families of solvable stationary systems is available, the time-dependent counterparts have not been widely explored. The difficulty lies in the dy-namical law, the Schrödinger equation, which is defined in terms of a partial differential equation that, in general, can not be reduced to an ordinary differential equation. Under some circumstances, we can extract information of the system through approximation techniques such as the sudden and the adiabatic approximations [37]. The latter restricts the range of applicability of the so-obtained solutions. Despite all these difficulties, time-dependent phenomena find exciting applications in physical systems such as electromagnetic traps of charged particles [38][39][40][41], plasma physics [42], and in optical-analogs under the paraxial approximation [43][44][45]. In contradistinction to the stationary cases, the lack of an eigenvalue equation in time-dependent systems prevents us from implementing the Darboux-transformation directly, and some workarounds are in order. The latter has been addressed by Bargov-Samsonov [46,47], where some intertwining operators allow us to relate an exactly solvable Schrödinger equation with another unknown one. In analogy to the stationary Darboux transformation, the solutions of the new models are inherited from the former one. Let us mention that orthogonality is no longer a property that can be taken for granted [48], since the method by itself does not provide essential information about the system such as the constants of motion, which have to be determined separately. Despite such a difficulty, several new families of exactly-solvable time-dependent potentials have been reported in the literature [48][49][50][51]. Among nonstationary quantum systems, the parametric oscillator [52][53][54][55] is perhaps the most well-known model that admits a set of exact solutions. Lewis and Riesenfeld [52] addressed the problem by noticing the existence of a nonstationary eigenvalue equation associated with the appropriate constant of motion (quantum invariant) of the system in which the time dependence appears in the coefficients of the related ordinary differential equation. The latter eigenvalue equation can indeed be factorized in such a way that the Darboux transformation 1 is applied with ease [56,57], resulting in a new quantum invariant rather than a Hamiltonian. Then, the appropriate ansatz allows to determine the respective Hamiltonian and time-dependent potentials with ease [56]. The solutions, and the complex-phases introduced by Lewis-Riesenfeld, are inherited from the former system, ensuring an orthogonal set of solutions for the new system. In this work, we combine the solutions of the Ermakov and fourth Painlevé equation to address the construction of new time-dependent Hamiltonians. This is achieved by considering third-order intertwining operators in the spatial variable with time-dependent coefficients. Those operators generate a third-order shape-invariant condition with respect to an unknown quantum invariant, which is introduced as a deformation of the one associated with the parametric oscillator. In this form, by working with a quantum invariant rather than a Hamiltonian, we generalize the construction presented in [33,34], and the time dependence is introduced into the intertwining operators through the solutions of the Ermakov equation, which ensure the regularity of the resulting quantum invariant and its eigenfunctions at each time. On the other hand, with aid of the appro-priate reparametrization, the fourth Painlevé equation is achieved, the parameters and solutions of which determine the spectral information and the exact form of the quantum invariant. Then, we modify the transitionless tracking algorithm [58] to construct the time-dependent Hamiltonians from the quantum invariant, from where the respective solutions of the Schrödinger equation are determined with the addition of a time-dependent complex-phase. The text is structured as follows. In Sec. 2, we introduce the basic notions of shapeinvariance for time-dependent Hamiltonians and their respective quantum invariants. Then, a couple of differential ladder operators of third-order are introduced such that an initial, and unknown, quantum invariant satisfies a higher-order shape-invariant relationship. From the latter, the explicit form of the ladder operators and the quantum invariants are determined. In Sec. 3, with the aid of the third-order ladder operators, we determine the respective spectral information of the quantum invariant. In Sec. 4, the time-dependent Hamiltonians associated with the quantum invariants are identified, together with the solutions to the Schrödinger equation. In Sec. 5, we discuss the solutions of the Ermakov equation for some specific time-dependent frequency profiles. In particular, it is shown that the constant-frequency case leads to periodic potentials, whose solutions are in agreement with the Floquet theorem. Also, the appropriate limit to recover the well-known stationary results is presented. In turn, in Sec. 6, we consider some particular solutions of the Painlevé equation obtained through solutions of the Riccati equation, in the form of rational solutions, or by solutions of another nonlinear models such as the derivative nonlinear Schrödinger equation. For completeness, in App. A, we briefly revisit the parametric oscillator and its solutions through the approach of Lewis-Riesenfeld. 2 Time-dependent quantum invariants, third-order ladder operators and the fourth Painlevé equation In quantum mechanics, the quantum invariants play a fundamental role in determining the exact solutions of the quantum models. For stationary systems (time-independent Hamiltonians), it is straightforward to realize that the Hamiltonian is a constant of motion, such that it leads to an eigenvalue equation in the form of a Sturm-Liouville problem. For time-dependent Hamiltonians, the determination of such quantum invariants becomes a challenging task in most of the cases. A prime example is given by the parametric oscillator, where the respective quantum invariants are determined with relative ease. As pointed out in [52], such an invariant admits a nonstationary eigenvalue equation, where its spectral information leads to the solutions of the Schrödinger equation. For the sake of self-consistency, in App. A we provide a brief discussion on the matter. Thoughout this manuscript, we construct new time-dependent models and their respective solutions using a new approach based on the ladder operator structure associated with the quantum invariant. The method relies on the existence of an unknown quantum invariantÎ 1 (t), which admits a set of ladder operators {Â(t),Â † (t)} defined in coordi-nate representation as N -order differential operators in the spatial variable with timedependent coefficients. In particular, to reduce the possible family of quantum invariants, we considerÎ 1 (t) as a deformation of the invariant associated with the parametric oscillator,Î 0 (t). Thus, for a fixed order N , we determine the exact form ofÎ 1 (t). It is worth to mention that ladder operators of first and second-order have been reported for the parametric oscillator [54,56] and the nonstationary singular oscillator [57,59]. Nevertheless, in those cases, the quantum invariants are already known, and the ladder operators are constructed following the polynomial structure of the eigenfunctions. Throughout the rest of the manuscript, we focus in third-order differential ladder operators and the related quantum invariants. In this case, we determine several families of time-dependent HamiltoniansĤ 1 (t), together with the respective solutions to the Schrödinger equation. Before proceeding, we would like to stress the meaning of shape-invariance in supersymmetric quantum mehcnics (SUSYQM) for both stationary and nonstationary systems. It is said that two time-independent HamiltoniansĤ ± are shape-invariant [13] if their respective potentials V + (x; {c n }) and V − (x; {d n }), with {c n } and {d n } two sets of constant parameters, are related by the condition V + (x; {c n }) = V − ({d n })+S({c n }), where S({c n }) is a function of the set of parameters {c n } and independent of x. In turn, for nonstationary systems, we define the shape-invariance considering two quantum invariantsÎ ± (t) by means of the relationshipÎ + (t) =Î − (t)+f 0 , with f 0 a real constant. The latter implies that the respective time-dependent HamiltomniansĤ ± (t) are related asĤ + (t) =Ĥ − (t) + f (t), with f (t) an arbitrary real-valued function of time. As a consequence, the solutions of the respective Schrödinger equations differ only from a global time-dependent complex-phase e i t dt f (t ) , see [56] for details. From the previous considerations, let us introduce two unknown quantum invariantŝ I 1,2 (t), whereÎ 1 (t) is the invariant operator under consideration, andÎ 2 (t) serves as an auxiliary operators to determinedÎ 1 (t). From the latter, there are two time-dependent HamiltoniansĤ 1,2 (t) such that dÎ j (t) dt = i Ĥ j (t),Î j (t) + ∂Î j (t) ∂t = 0 , j = 1, 2 .(1) We focus on self-adjoint time-dependent Hamiltonians and quantum invariants. It is thus guaranteed that the nonstationary eigenvalue equationŝ I j (t)φ (j) n (x, t) = Λ (j) n φ (j) n (x, t) , j = 1, 2,(2) lead to real and time-independent eigenvalues Λ (j) n , and orthogonal nonstationary eigenfunctions φ n (x, t)} ∞ n=0 , with j = 1, 2. As mentioned above, we construct the quantum invariantÎ 1 (t) from the third-order SUSYQM shape-invariant condition, defined in term of the intertwining relationshipŝ I 1 (t)Â † (t) =Â † (t) Î 1 (t) + 2λ ,Î 1 (t)Â(t) =Â(t) Î 1 (t) − 2λ ,(3) whereÂ(t) andÂ † (t) are also the annihilation and creation operators, respectively, for φ (1) n (x, t). Indeed, the action ofÂ † (t) (Â(t)) on φ (1) n (x, t) increases (decreases) the eigenvalue Λ (1) n by 2λ units. Now, to provide a specific form to the quantum invariantsÎ 1,2 (t), we consider them as deformations of the quantum invariant associated with the parametric oscillatorÎ 0 (t), see Eq. (A-5), that is, I j (t) =Î 0 (t) + R j (x, t) ≡ −σ 2 ∂ 2 ∂x 2 + ixσσ ∂ ∂x + R(x, t) + R j (x, t) , j = 1, 2,(4) where σ ≡ σ(t) given in (A-7) is solution to the Ermakov equation associated with the parametric oscillator (A-6), andσ ≡ dσ/dt. The function R(x, t) is given as R(x, t) = σ 2 4 + 1 σ 2 x 2 + iσ σ 2 ,(5) and R 1,2 (x, t) are real-valued functions, where R 1 (x, t) is determined from the shapeinvariant condition (3), and R 2 (x, t) defines the auxiliary invariantÎ 2 (t). As discussed in [33], higher-order intertwining operators can be factorized as products of first-order operators for a reducible factorization, or as combination of first and second-order operators for irreducible factorizations [60]. In the sequel, we focus on the reducible factorization of the intertwining relationships (3); however, for the sake of completeness, we discuss the more general irreducible factorization in this section. We thus decompose the set of intertwining operators {Â(t),Â † (t)} as the product of first and second-order differential operators {Q † (t),Q(t)} and {M † (t),M (t)}, respectively, as follows: A † (t) =Q † (t)M (t) ,Â =M † (t)Q(t) .(6) The new operators give rise the additional set of intertwining relationships of the form I 1 (t)Q † (t) =Q † (t) Î 2 (t) + 2λ ,Î 2 (t)Q(t) =Q(t) Î 1 (t) − 2λ ,(7)I 2 (t)M (t) =M (t)Î 1 (t) ,Î 1 (t)M † (t) =M † (t)Î 2 (t) ,(8) where in the latter we have introduced the auxiliary quantum invariantÎ 2 (t) as an intermediate. In contradistinction to (3), the Eqs. (7)-(8) by themselves do not define a shape-invariant relation. Nevertheless, their combined action take us back to the shapeinvariant condition (3), see Fig. 1 for details. Now, given thatQ(t) andQ † (t) are considered as first-order differential operators, we use the general form introduced in [57], that is, Q † = σ ∂ ∂x + w(x, t) ,Q = −σ ∂ ∂x + w * (x, t) ,(9) with w(x, t) a complex-valued function, σ ≡ σ(t) given in (A-7), and f * stands for the complex-conjugate of f . In turn, the second-order differential operatorsM (t) andM † (t) Figure 1: Third-order shape-invariant SUSYQM for the quantum invariantÎ 1 (t). The arrow indicates intertwining relationship between the quantum invariants; for instance, the arrow on top implieŝ are constructed as a generalization of those reported in [60] by introducing the timedependent coefficients of the form Ĥ 1 (t)Ĥ 1 (t) + 2λ/σ 2 I 1 (t)Î 1 (t) + 2λ I 2 (t) + 2λÎ(t) + 2λ H 2 (t) + 2λ A † (t) Q † (t)M (t)M 2 (t)M 1 (t)I 1 (t)Â † (t) =Â † [Î 2 (t) + 2λ].M † = σ 2 ∂ 2 ∂x 2 − 2g(x, t) ∂ ∂x + b(x, t) , M = σ 2 ∂ 2 ∂x 2 + 2g * (x, t) ∂ ∂x + b * (x, t) − 2 [g (x, t)] * ,(10) where b(x, t) and g(x, t) are complex-valued functions, and g (x, t) stands for the partial derivative of g(x, t) with respect to x. The complex-valued functions w(x, t), g(x, t) and b(x, t) are determined from the respective intertwining relationships. For instance, w(x, t) is obtained after substituting (9) in (7), leading to w(x, t) = −iσ 2 x + W (z(x, t)) , z(x, t) := x σ .(11) Given that the solution to the Ermakov equation σ(t) is a nodeless function, we can guarantee that the reparametrizated variable z(x, t) is non-singular for t ∈ R. In turn, W (z(x, t)) is a real-valued function that solves the Riccati equations z 2 + R 1 (z) = ∂ z W + W 2 , z 2 + R 2 (z) = −∂ z W + W 2 − 2λ , ∂ z ≡ ∂ ∂z .(12) From (12) we also get R 2 (z) − R 1 (z) = −2∂ z W − 2λ, which resembles the conventional relationship between the potential and the super-potential of the conventional stationary SUSYQM construction [33,34]. On the other hand, the functions g(x, t) and b(x, t) are computed after inserting (10) in (8), leading to g(x, t) = i σσ 2 x + σG(z(x, t)) , b(x, t) = iσxG(z(x, t)) − iσ σ 2 −σ 2 4 x 2 + B(z(x, t)) ,(13) with the real-valued functions G(z(x, t)) and B(z(x, t)) determined from the nonlinear relationships B = 2G 2 + ∂ z G − z 2 + R 2 (z) + γ , z 2 + R 1 (z) = −2∂ z G + G 2 + ∂ 2 z G 2G − (∂ z G) 2 4G 2 − d 4G 2 + γ , z 2 + R 2 (z) = 2∂ z G + G 2 + ∂ 2 z G 2G − (∂ z G) 2 4G 2 − d 4G 2 + γ ,(14) with γ and d constants of integration with respect to z(x, t), that is, those constants do not depend on x or t. From (14) we obtain a complementary relationship of the form R 2 (z) − R 1 (z) = 4∂ z G that, together with the one obtained from (12), gives W (z) = −2G(z) − λz .(15) A differential equation for G(z) can be found after substituting (15) into any of the Riccati equations in (12) and comparing with (14). The straightforward calculation leads to ∂ 2 z G = (∂ z G) 2 2G + 6G 3 + 8λzG 2 + 2 λ 2 z 2 − (γ + λ) G + d 2G ,(16) where the following reparametrizations: y = √ λz , G = √ λ 2 w(y) , α = γ λ + 1 , β = 2d λ 2 ,(17) allow us to rewrite (16) as the fourth Painlevé differential equation [61] w yy = (w y ) 2 2w + 3 2 w 3 + 4yw 2 + 2(y 2 − α)w + β w .(18) Solutions for the fourth Painlevé equation have been extensively studied in the literature, in particular it is known that w(y) can be determined in terms of elementary functions [26,27]. Before finishing this section, we would like to recall that the factorization (6) allowed us to find the functions R 1,2 (x, t), which define uniquely the respective quantum invariantŝ I 1,2 (t) in terms of the solutions of the fourth Painlevé equation (18). A summary of the steps followed so far is presented in the diagram of Fig. 1, where the time-dependent HamiltoniansĤ 1,2 (t) are discussed in Sec. 4. Spectral information ofÎ 1 (t) As pointed out in the previous section, the shape-invariant condition (3) implies thatÂ(t) andÂ † (t) are the ladder operators for the nonstationary eigenfunctions φ (1) n (x, t) ofÎ 1 (t). The latter indeed allows determining the spectral information (2), for j = 1. To this end, we first determine the zero-mode eigenfunction, which is an element in the kernel of the annihilation operator K A ≡ Ker(Â(t)) = {φ (1) }, withÂ(t)φ (1) = 0. However, in our case, the annihilation operator under consideration is a differential third-order one, and thus K A is composed of three linearly independent zero-mode solutions, K A = {φ (1) 0;1 , φ (1) 0;2 , φ (1) 0;3 }. Nevertheless, we must verify whether the elements in K A fulfill the finite-norm condition. With the zero-modes already identified, the remaining eigenfunctions are computed from the iterated action of the creation operatorÂ † (t) on the zero-mode eigenfunctions, and the respective eigenvalues increase by 2λ at each iteration. For convenience, in this section we consider the case for whichM (t) † factorizes as the product of two first-order operators, that is, a reducible case. Let us consider the factorizationM † (t) ≡M † 1 (t)M † 2 (t) ,M (t) =M 2 (t)M 1 (t) ,(19) whereM 1,2 (t) are first-order operators constructed in analogy to (9) aŝ M † 1 (t) := σ ∂ ∂x − iσ 2 x + W 1 (z) ,M † 2 (t) := σ ∂ ∂x − iσ 2 x + W 2 (z) .(20) The straightforward calculations show that the real-valued functions W 1 (z) and W 2 (z) are given by W 1 = −G + G z − √ −d 2G , W 2 = −G − G z − √ −d 2G . From the latter result, it is clear that the factorization ofM (t) requires d < 0. Recall thatM † (t) intertwines the quantum invariantÎ 1 (t) withÎ 2 (t). Thus, to inspect the respective intertwining relationships fulfilled byM 1,2 (t) we introduce a new auxiliary quantum invariantÎφ n (x, t) =Λ nφn (x, t)(22) withH = Span{φ n } ∞ n=0 the respective vector space composed with the finite-norm solutions. The spectral information ofÎ(t) is not relevant, for it just serves as an aid to solve the eigenvalue problem associated withÎ 1 (t). For this reason, the respective Hamiltonian associated withÎ(t) is not considered throughout the rest of the text. The new auxiliary invariant satisfies the intertwining relationshipŝ I 1 (t)M † 1 (t) =M † 1 (t)Î(t) ,Î(t)M † 2 (t) =M † 2 (t)Î 2 (t) .(23) Given that both operatorsM 1,2 (t) are of first-order, the relationships (23) are then equivalent toÎ 1 (t) =M † 1 (t)M 1 (t) + 1 ,Î(t) =M 1 (t)M † 1 (t) + 1 , I(t) =M † 2 (t)M 2 (t) + 2 ,Î 2 (t) =M 2 (t)M † 2 (t) + 2 ,(24)H 1 (t) H 2 (t)H(t) φ (1) 0;3 φ (1) 0;2 φ (1) 0;1Q φ (2) 1 φ (2) 0M † 2φ 0M † 1 Q Q † M † 2 M 2 Figure 2: Zero-mode eigenfunctions {φ (1) 0;1 , φ (1) 0;2 , φ(1) 0;3 } and their relationship with the respective eigenfunctions of the auxiliary quantum invariantsÎ 2 (t) andÎ(t). Dotted-black lines denote the null-vector in vector space, and the dashed-red arrows indicate the annihilation operation of the intertwining operators. where the substitution of (20)-(21) into (24) leads to 1 = γ − √ −d , 2 = γ + √ −d .(25) From (7) and (23), we can see that the first-order operators define mappings among the vector spaces H 1 (t), H 2 (t) andH(t) in the following form: M † 2 (t) : H 2 (t) →H(t) ,M 2 (t) :H(t) → H 2 (t) , M † 1 (t) :H(t) → H 1 (t) , M 1 (t) : H 1 (t) →H(t) , Q † (t) : H 2 (t) → H 1 (t) , Q(t) : H 1 (t) → H 2 (t) .(26) From the latter mappings, we construct the elements of K A (zero-modes), that is, the eigenfunctions annihilated byÂ(t). We thus havê A(t)φ (1) 0;k =M † (t)Q(t)φ (1) 0;k =M † 1M † 2 (t)Q(t)φ (1) 0;k = 0 , k = 1, 2, 3,(27) where it is worth discussing three different cases. •Q(t)φ(1) 0;1 = 0. Here, a first solution is determined, up to a normalization constant, by solving a trivial first-order differential equation. •M † 2 (t)Q(t)φ (1) 0;2 = 0 withQ(t)φ(1) 0;2 = 0. From the mappings defined by the intertwining relationship (7), it is clear thatQ(t)φ (1) 0;2 = φ (2) 0 ∈ H 2 (t), with the latter being annihilated byM † 2 (t). Thus, in order to determine the zero-mode φ (1) 0;2 , we should solve the first-order differential equationM † 2 (t)φ (2) 0 = 0, and from it we determine φ 0;2 , together with the respective eigenvalue, after mapping φ (2) 0 throughQ † (t) (see Fig. 2). •M † 1 (t)M † 2 (t)Q(t)φ(1)0;3 = 0 withM † 2 (t)Q(t)φ(1) 0;3 = 0. In this case, the non-null element M † 2 (t)Q(t)φ(1)0;3 =φ 0 ∈H(t) is annihilated byM † 1 (t). To extract the zero-mode φ 0;3 , we solve the first-order differential equationM † 1 (t)φ 0 = 0. Then, we takeφ 0 to H 1 (t) by consecutively performing the mappingsM 2 (t) andQ † (t) (see Fig. 2). The complete procedure is summarized in the scheme depicted in Fig. 2. The straightforward calculations lead to φ (1) 0;1 (x, t) := N (1) 0;1 e i 4σ σ x 2 √ σ e z dz W (z ) , φ (1) 0;2 (x, t) = N (1) 0;1 [W (z) − W 2 (z)] e i 4σ σ x 2 √ σ e − z dz W 2 (z ) , φ (1) 0;3 (x, t) = N (1) 0;3 −2 √ −d + (W (z) − W 2 (z))(W 1 (z) + W 2 (z)) e i 4σ σ x 2 √ σ e − z dz W 1 (z ) ,(28) with the respective eigenvalues Λ 0;1 = 0, Λ(1)0;2 = 2 + 2λ = γ + √ −d + 2λ and Λ(1)0;3 = 1 + 2λ = γ − √ −d + 2λ. The terms N(1) 0;j stand for the normalization factors that might depend on time. From (28), the rest of the eigenfunctions are determined from the action Â † (t) n , for n = 0, 1, · · · , on each element φ (1) 0;j . By doing so, we generate at most three sequences of eigenfunctions, where the eigenvalues Λ (1) 0;j , for j = 1, 2, 3, increase by 2nλ. For the conventional stationary oscillator, it is well-known that the creation operator does not lead to finite-norm eigenfunctions. On the other hand, the one-step SUSY partner Hamiltonians admit a creation operator for which a finite-norm eigenfunction is achived. In the context of SUSYQM, such an eigenfunction is the so-called missing state [16]. Thus, it is natural to look for the solutions that are annihilated by the creation operator. In the case under consideration, we have constructedÂ † (t) as a third-order differential operator, which admits three linearly independent eigenfunctions, and at least one finitenorm solution is possible. The existence of the latter implies a truncation of the sequences generated from the zero-modes φ 0;2 , Φ(1) 0;3 } as the set containing the finite-norm eigenfunctions ofÂ † (t). If the set K A † is empty, three infinite sequences are generated (see Sec. 6.2). In turn, if K A † contains one single element, we generate at most two infinite sequences, together with one finite-dimensional sequence (see Sec. 6.3), which in particular could be a singlet (see Sec. 6.1). Following the same steps as in (28), it is straightforward to show that the eigenfunc- tions of A † are Φ (1) 0;1 (x, t) = N (1) 0;1 e i 4σ σ x 2 √ σ e z dz W 1 (z ) , Φ(1)0;2 (x, t) := N (1) 0;2 [W 1 (z) + W 2 (z)] e i 4σ σ x 2 √ σ e z dz W 2 (z ) Φ (1) 0;3 (x, t) := N(1)0;3 [ 2 + 2λ + (W 1 (z) + W 2 (z))(W 2 (z) − W (z))] e i 4σ σ x 2 √ σ e − z dz W (z ) ,(29) where the respective eigenvalues are given by Λ 0,1 = 1 = γ − √ −d, Λ(1)0,2 = 2 = γ + √ −d and Λ(1)0,3 = −2λ.(1) In general, we can not say which solutions in (28) and (29) fulfill the finite-norm condition, since it depends on the specific solutions of the fourth Painlevé equation. However, we may get more insight by considering the possible behavior of the asymptotics. To this end, let us suppose that the real-valued functions W 1 (z), W 2 (z) and W (z) are smooth, and such that they converge to a finite value for \|z\| → ∞. Then, finite-norm solutions are achieved depending on the convergence of the exponential functions in (28)- (29). For instance, if e z dz W (z ) → 0 for \|z\| → ∞, then φ 0;1 (x, t) becomes a finite-norm solution, whereas Φ (1) 0;3 (x, t) do not. The same analysis can be extended to the rest of solutions in (28)- (29). In such a case, we may conclude that at most three out of the six solutions have a finite-norm. This indeed corresponds to the solutions discussed in Sec. 6.1 and Sec. 6.2. New families of time-dependent Hamiltonians So far, we have determined the families of exactly solvable quantum invariantÎ 1 (t), related to the fourth Painleé transcendents, which fulfill a third-order SUSYQM shape-invariant condition. Nevertheless, the respective Hamiltonian of the systemĤ 1 (t) has not been identified yet. The latter is required to properly define the Schrödinger equation that characterize the quantum system under consideration. Such a task have been addressed in previous works using the factorization method for time-dependent Hamiltonians [56,57] by imposing the appropriate ansatz. In this work, we consider an alternative approach based on the transitionless tracking algorithm [58,62]. In this form, additional information is obtained about the classes of time-dependent Hamiltonians that can be constructed in term of the nonstationary eigenfunctions. In App. A we have discussed the nonstationary eigenvalue equation of the quantum invariant associated with the parametric oscillator. Remarkably, the results from Lews-Riesenfeld [52] hold for any quantum invariant 2 . Therefore, for the quantum invariantŝ I 1,2 (t) constructed in Sec. 2 we can determine the respective time-dependent Hamiltonianŝ H 1,2 (t) such that the Schrödinger equation, in coordinate-free representation, i∂ t \|ψ (j) n (t) =Ĥ j (t)\|ψ (j) n (t) , \|ψ (j) n (t) = e iθ (j) n (t) \|φ (j) n (t) , j = 1, 2,(30) is fulfilled, where the wavefunctions and nonstationary eigenfunctions are recovered from the coordinate-representation ψ (j) n (x, t) = x\|ψ (j) n (t) and φ (j) n (x, t) = x\|φ (j) n (t) , respectively. Notice that the wavefunctions and eigenfunctions differ by just a time-dependent complex-phase, which is computed from (30) through the expectation value d dt θ (j) n (t) = φ (j) n (t)\| i∂ t −Ĥ j (t) \|φ (j) n (t) , j = 1, 2.(31) provided that the Hamiltonian is already known. However, in our case, both the Hamiltonian and the complex-phase are unknown, and a workaround should be implemented. To this end, we consider the time-evolution operatorÛ j (t; t 0 ), that is, an operator that maps a solution defined at a time t 0 into one defined at a time t, \|ψ (j) n (t) =Û j (t; t 0 )\|ψ (j) n (t 0 ) . Given that both the HamiltoniansĤ j (t) and the quantum invariantsÎ j (t) are self-adjoint, it follows that the time-evolution operator is unitary and it takes the diagonal form U j (t; t 0 ) := ∞ n=0 \|ψ (j) n (t) ψ (j) n (t 0 )\| = ∞ n=0 e i θ (j) n (t)−θ (j) n (t 0 ) \|φ (j) n (t) φ (j) n (t 0 )\| ,(32) where it is worth to recall that φ n (t) =Û j (t; t 0 )\|ψ (j) n (t 0 ) into (30) lead us to an expression for the HamiltonianĤ j (t) in terms ofÛ j (t; t 0 ) aŝ H j (t) = i∂ tÛj (t; t 0 ) Û † j (t; t 0 ) = − ∞ n=0θ (j) n (t)\|φ (j) n (t) φ (j) n \| + i ∞ n=0 ∂ t \|φ (j) n (t) φ (j) n (t)\| . (33) Therefore, from (33), the HamiltonianĤ j (t) is determined once the complex-phase θ (j) n (t) has been specified. Moreover, it is straightforward to show that the Hamiltonian obtained from (33) is such thatÎ j (t) is its respective quantum invariant. Such a conclusion holds true regardless of the choice of θ (j) n (t). From (33) we have to point out thatĤ j (t) is composed by the sum of a diagonal and a non-diagonal operator. Thus, in general, thê H j (t) is not diagonizable in H j (t). Moreover, sinceĤ j (t) andÎ j (t) do not commute, a common basis that simultaneously diagonalizes both operators does not exist. Now, the time-dependent Hamiltonians related to the quantum invariantsÎ j of Sec. 2 are determined by proposingĤ j (t) as the sum of a kinetic energy term and a timedependent potential energy term V j (x, t). Given that the complex-phase is arbitrary, we introduce it in the convenient forṁ θ (j) n (t) ≡ d dt θ (j) n (t) = − Λ (j) n σ 2 (t) ,(34) leading to the time-dependent Hamiltonianŝ H j (t) = 1 σ 2Î j (t) +F (t) ,F (t) := i ∞ n=0 ∂ t \|φ (j) n (t) φ (j) n (t)\| ,(35) where the first part of the Hamiltonian becomes proportional to the invariant operator I j (t), and the factor σ −2 (t) has been introduced such that we recover the kinetic energy termp 2 . The operatorF (t) can be simplified by using the coordinate representation x\|∂ t \|φ n (x, t) = e iσx 2 /4σ σ −1/2 K(z(x, t)), with z(x, t) = x/σ, and K(z(x, t)) a function that depends explicitly on z, and implicitly on x and t. After some calculations we obtain ∂ t \|φ (j) n (t) = i 4 σ σ +σ 2 σ 2 x 2 − i 2σ σ {x,p} \|φ (j) n .(36) From the latter result, together with the Ermakov equation (A-6), the time-dependent Hamiltonians take the final form H j (t) =p 2 + V j (x, t) , V j (x, t) = Ω 2 (t)x 2 + 1 σ 2 R j (x, t) , j = 1, 2,(37) wherex ≡ x andp ≡ −i∂ x stand for the position and momentum operators, respectively, and the time-dependent potentials V j (x, t) are written in terms of the functions R j (x, t) given in (12). Notice that a different choice of θ n (x, t) given in (30) are simplified, by using the solutions of the Ermakov equation [19,54,56], as ψ (j) n (x, t) = e iθ (j) n (t) φ (j) n (x, t) , θ (j) n (t) = −Λ (j) n t dt σ 2 (t ) = − Λ (j) n 2 arctan W 0 2 ac − 4 W 2 0 + c q 1 (t) q 2 (t) ,(38) with a, c some arbitrary positive constants given in (A-7), and W 0 the Wronskian of two linearly independent solutions q 1,2 (t) of the linear equation (A-8). We thus have properly identified the time-dependent Hamiltonians whose potential energy term is related to the solutions of the fourth Painlevé transcendent. Frequency profiles and solutions of the Ermakov equation In this section, we discuss the specific form of σ(t) by considering some particular forms of the time-dependent frequency term Ω(t) that appears in the new time-dependent potentials V j (x, t) given in (37), and in the parametric oscillator Hamiltonian (A-1). With the solutions to the Ermakov equation properly identified, we determine the reparametrization z(x, t) = x/σ(t), and equivalently y(x, t) = √ λz(x, t), which are singular-free at each time (see Sec. A). We have to remark that the form of V j (x, t) depend on the solutions of the Ermakov and fourth Painlevé equations. However, the latter are independent one of the other, and thus the solutions constructed in this section are valid for any solution of the fourth Painlevé equation discussed in Sec. 6. To exemplify our results, we consider two different time-dependent frequency profiles. First, we consider the simplest constant frequency case, for which time-dependent potentials are achieved in the most general case, and the stationary results are determined as a particular limit. On the other hand, we consider a frequency profile that changes smoothly from a constant value at t → −∞ to another different constant at t → ∞. Such a profile can be seen as a regularization of the Heaviside distribution [61]. Frequency Ω 2 (t) = 1 In this case, two linearly independent solutions of (A-8) are given by q 1 (t) = cos 2(t − t 0 ) and q 2 (t) = sin 2(t − t 0 ), with t 0 ∈ R an arbitrary initial time and the Wronskian W 0 = 2. After some calculations we obtain σ(t) = a + c 2 + √ ac − 1 sin 4(t − t 0 ) + a − c 2 cos 4(t − t 0 ) 1/2(39) with a, c > 0 such that ac ≥ 1. Notice that, even if the frequency Ω(t) is a constant, the resulting potentials V 1,2 (x, t) are in general time-dependent, and periodic functions in time. This class of systems are usually studied under the Floquet theory, and already discussed for the parametric oscillator in [40]. For a = c = 1, it follows that σ(t) = 1 and z(x, t) = x. Thus, the conventional stationary results reported in [33,34] are recovered. Frequency 4Ω 2 (t) = Ω 1 + Ω 2 tanh(kt) In this case, we introduce the constraint Ω 1 > Ω 2 to ensure that Ω(t) is a positive function at each time. Exact solutions can be determined for any value of the parameters k, Ω 1 and Ω 2 by taking the linear differential equation (A-8) into the hypergeometric form [63]. After some calculations we obtain two linearly independent solutions q 1 (t) = (1 − T(t)) − i 2 r + (1 + T(t)) − i 2 r − 2 F 1 −iµ , 1 − iµ 1 − ir + 1 − T(t) 2 , q 2 (t) = [q 1 (t)] * , µ = 1 k Ω 1 + Ω 2 1 − Ω 2 2 2 , r ± = µ ± Ω 2 2k 2 µ , T(t) = tanh(kt) ,(40) with 2 F 1 (a, b; c; z) the hypergeometric function [61]. Given that q 2 (t) = [q 1 (t)] * , it is trivial to realize that the respective Wronskian becomes W 0 = q 1q2 −q 1 q 2 = −2ikr + , that is, a pure imaginary constant. Thus, a real-valued solution σ(t) is determined if a = c in (A-7), leading to σ 2 (t) = 2a Re[q 2 1 (t)] + 2 a 2 + 1 k 2 r 2 + \|q 1 (t)\| 2 .(41) The behavior of σ(t) and q 1,2 (t) is depicted in Fig. 3. For asymptotic times \|t\| >> 1, the frequency function Ω(t) converges to a constant value, and the respective linear solutions (a) Figure 3: Functions σ(t) (solid-blue), q 1 (t) (dashed-red), and q 2 (t) (dotted-green) for the frequency profile 4Ω 2 (t) = Ω 1 + Ω 2 tanh(kt). The parameters has been fixed to k = 1/2, Ω 1 = 15, Ω 2 = 10 and a = 1/2. q 1,2 (t) approximate to sinusoidal functions. Thus, the resulting time-dependent potential V 1 (x, t) behaves as a periodic function in the asymptotic limit. Solutions of the Painlevé equation As discussed in Sec. 2, the solutions to the fourth Painlevé equation w(y) allow us to construct the functions R j (x, t) required to determine the time-dependent potentials V j (x, t) given in (37). Also, the finite-norm condition of the zero-mode solutions discussed Sec. 3 depends strongly on the asymptotic behavior and regularity of w(y). The fourth Painlevé equation has been widely studied in the literature [26,27,64], and in this section we discuss some hierarchies of solutions that can be implemented in the construction of timedependent systems. To this end, let us recall the fourth Painlevé equation, w yy = (w y ) 2 2w + 3 2 w 3 + 4yw 2 + 2(y 2 − α)w + β w ,(42) whose solutions w ≡ w(y; α, β) are determined according to the values of the parameters α and β. From the latter, the time-dependent HamiltonianĤ 1 (t) given in (12) is defined in terms of the time-dependent potential V 1 (x, t) = Ω 2 (t) + λ 2 − 1 σ 4 x 2 − λ σ 2 ∂ y w − w 2 − 2yw + 1 , y = √ λz = √ λ x σ(t) , (43) and the respective zero-modes are given in Sec. 3. Throughout the rest of this section, we consider three different hierarchies of solutions w(y; α, β), namely the Riccati-like, rational, nonlinear solutions. Those hierarchies have been considered because the spectral information obtained in each case reveals the existence of sequences of solutions, which can be either truncated, infinite, or a combination of both. In addition, we obtain eigenvalues that are equidistant, or equidistant with intermediate gaps. Solutions in terms of the Riccati equation It is well-known that the solutions of the fourth Painlevé equation can be determined through a Riccati equation [26] of the form w y = µw 2 + 2µyw − 2(1 + αµ) , µ 2 = 1 ,(44) provided that β = −2(1 + αµ) 2 , with α ∈ C. In this form, we have to solve (44), which can be linearized with ease [1] though the use of a logarithmic derivative as w = − 1 µ u y u , u yy − 2µyu y − 2µ(1 + αµ)u = 0 .(45) For the physical case under consideration, the set of Painlevé parameters {α, β} are related to the set of physical parameters {λ, γ, d} through the relationships given in (17). In this form, for the Riccati-like solutions of (42), we obtain the constraint d = −[λ(1+µ)+µγ] 2 , where the reducible condition d ≤ 0 of Sec. 3 is automatically fulfilled. In general, the linearized equation (45) has two linearly independent solutions of the form u 1 (y) = 1 F 1 1 2 + µ λ + γ 2λ , 1 2 ; µy 2 , u 2 (y) = µy 2 1 F 1 1 + µ λ + γ 2λ , 3 2 ; µy 2 ,(46) with 1 F 1 (a, b; z) the confluent hypergeometric function [61]. Interestingly, from (44), one realizes that the time-dependent potential (43) reduces, for µ = 1, to V 1 (x, t) = Ω 2 (t) + λ 2 − 1 σ 4 (t) x 2 + λ σ 2 (t) 3 + γ λ ,(47) which in the context of time-dependent systems corresponds to a shape-invariant potential of the parametric oscillator (see discussion in Sec. 2). The latter holds for any linear combination of the solutions (46). We thus discard the case µ = 1 for the rest of the text. In turn, for µ = −1, the new time-dependent potential takes the form V 1 (x, t) = Ω 2 (t) + λ 2 − 1 σ 4 (t) x 2 − λ σ 2 (t) 2∂ y w − 2 γ λ + 1 ,(48) where now we have, in general, a potential different from the class of shape-invariants. From (46), we can either choose u 1 (y) = 1 F 1 − γ 2λ , 1 2 ; −y 2 , u 2 (y) = iy 1 F 1 λ − γ 2λ , 3 2 ; −y 2 ,(49) or the equivalent Kummer transformations [61] u 1 (y) = e −y 2 1 F 1 γ + λ 2λ , 1 2 ; y 2 , u 2 (y) = y e −y 2 1 F 1 2λ + γ 2λ , 3 2 ; y 2 .(50) as the set of linearly independent solutions. For (49), the general solution is constructed as the linear combination u(y) = k a u 1 (y) + k b u 2 (y), k a k b > Γ 3 2 Γ λ+γ 2λ Γ 1 2 Γ 2λ+γ 2λ ,(51) where the imaginary number i has been absorbed in the constant k b , and constraint between the real constants k a and k b is determined from the asymptotic behavior of the confluent hypergeometric function to ensure the existence of a nodeless solution u(y) for y ∈ R. The latter is required to avoid singularities in the solution of the fourth Painlevé equation given in (45), and consequently the potential V 1 (x, t) in (48). Similar results are obtained by using the solutions (50) instead. Additionally, the asymptotic behavior of the confluent hypergeometric function reveals that u(y) → ∞ and w(y) → 0 at y → ±∞. We thus determine the finite-norm elements in K A and K A † from (28) and (29), respectively, leading to the following spectral information φ (1) 0 (x, t) ≡ φ (1) 0;1 (x, t) = Φ (1) 0;1 (x, t) = N (1) 0 e i 4σ σ x 2 √ σ e −y 2 /2 u(y) , Λ(1)0 = 0 , φ(1)1 (x, t) ≡ φ (2) 0;2 (x, t) = N (1) 1 e i 4σ σ x 2 √ σ u y u + 2y e −y 2 /2 , Λ(1)1 = 2(γ + λ) ,(52) with y(x, t) = √ λx/σ(t). From the latter, the zero-mode solution φ (1) 0 belongs to both K A and K A † , that is, φ (1) 0 is annihilated by both the creation and annihilation operators. In turn, φ (1) 1 is annihilated only byÂ(t), and from it we generate a single sequence of states {φ (1) n+1 } ∞ n=0 through the iterated operation φ (1) n+1 ∝ [Â † (t)] n φ(1) 1 , up to a normalization constant, with n = 0, 1, · · · . On the other hand, the respective eigenvalues are determined by increasing Λ Several special cases can be discussed from the general solution (51), leading to specific hierarchies of solutions of the Painlevé equation. • For γ = 0 together with 2 √ wk a = 1 − √ 2πk 2 and k b = k 2 , we obtain the set of parameters {α, β} = {1, 0}. In such a case we recover the complementary error function hierarchy solutions of the form w(y; 1, 0) = 2 √ 2k 2 e −y 2 1 − √ 2πk 2 Erfc(y) ,(53) leading to the equidistant eigenvalues Λ (1) n = 2nλ. The respective potential V 1 (x, t), determined from (48), reduces to a time-dependent variation of the stationary deformed oscillator potentials reported [8]. Such a time-dependent potential has been obtained previously in [48] through the Bagrov-Samsonov approach [46,47]. • Another interesting case is recovered for k b = 0 and γ = 2N λ, with N = 0, 1, · · · , where we obtain the rational solutions (52). The dotted-black line represents the null state, the dashed-red arrow depict the solutions annihilated by either the creation or annihilation operators, and the solid-black arrow represents the transition to higher modes due to the action ofÂ † . Figure 5: (Color online) Ladder structure of the finite-norm zero-modes for the "−2y/3" hierarchy φ (1) n obtained in (58). The dotted-black line represents the null state, the dashed-red arrow depict the solutions annihilated by either the creation or annihilation operators, and the solid-black arrow represents the transition to higher modes due to the action ofÂ † . u(y) = H 2N (y) , w(y; 2N + 1, −2(2N ) 2 ) = 2N H 2N −1 (y) H 2N (y) ,(54)Sp Î 1 φ (1) 2 A † φ (1) 1 φ (1) 0 A † A infinite sequence singlet (a)Sp Î 1 Â φ (1) 0 φ (1) 1 φ (1) 2 φ (1) 5Â † φ (1) 3 φ (1) 6 φ (1) 4 φ (1) 7 (a) with H N (y) = (−i) N H N (iy) and H n (z) the pseudo-Hermite and Hermite polynomials [61], respectively. Contrary to the previous case, the eigenvalues are non-equidistant and given by Λ (1) 0 = 0 and Λ (1) n+1 = 2λ(2N + n + 1). It is worth to remark that the even pseudo-Hermite polynomials are nodeless, whereas the odd ones have one zero at the origin. Thus, the Painlevé solution in (54) is well defined for every y ∈ R. Such a property is essential since it leads to a rational, nonsingular, and time-dependent potential V 1 (x, t), where y ≡ y(x, t) = √ λx/σ(t). This particular case leads to eigenfunctions written in terms of the exceptional Hermite polynomials, previously discussed for stationary [65,66] and time-dependent systems [56]. Hierarchies of rational solutions In general, it is well-known that the fourth Painlevé equation admits hierarchies of rational solutions if and only if [26] the set of parameters {α, β} take either the values {m, −2(2n − m + 1) 2 } or {m, −2(2n − m + 1/3) 2 }. The class of all the rational solutions are classified as "−1/y", "−2y", and "−2y/3" hierarchies. For instance, it has been shown that solutions in terms of the generalized Hermite polynomials H n,m (z) contain all solutions in the "−1/y" and "−2y" hierarchies, whereas the generalized Okamoto polynomials Q n,m (y) determine the "−2y/3" hierarchy. For a complete discussion see [26]. The relation of such rational solutions with SUSYQM in the stationary regime has been already discussed in [67]. For the sake of simplicity, and to illustrate our general set-up, let us consider the simplest family of solutions in the "−2y/3" hierarchy, that is, w M ≡ w(y; 2M, −2/9) = − 2y 3 + ∂ ∂y ln Q M +1 Q M ,(55) where Q M ≡ Q M (y) stands for the Okamoto polynomials [68]. The latter are determined from the nonlinear recurrence relationship Q M +1 = − 9 2 Q M Q M − [Q M ] 2 Q M −1 + 2y 2 + 3(2M − 1) Q 2 M Q M −1 , M = 1, 2, · · · ,(56) with Q 0 = Q 1 = 1 and Q M ≡ ∂Q M /∂y. Given that the Okamoto polynomials do not contain zeros for real y, one conclude that w M given in (55) is a singular-free solution. To simplify our calculations, let us consider M = 2, in such a case we obtain w 2 = − 2y 3 + 16y 3 (4y 4 + 24y 2 + 45) (2y 2 + 3) (8y 6 + 60y 4 + 90y 2 + 135) , together with the finite-norm zero modes φ (1) 0 (x, t) ≡ φ (1) 0;1 = N (1) 0 e i 4σ σ x 2 √ σ e − y 2 6 (2y 2 + 3) 8y 6 + 60y 4 + 90y 2 + 135 , φ(1)3 (x, t) ≡ φ (1) 0;2 = N (1) 3 e i 4σ σ x 2 √ σ e − y 2 6 y (8y 4 (2y 4 + 24y 2 + 63) − 2835) (8y 6 + 60y 4 + 90y 2 + 135) , φ (1) 4 (x, t) ≡ φ(1)0;3 = N (1) 4 e i 4σ σ x 2 √ σ e − y 2 6 (2 (4y 2 (2y 2 + 15) (2y 4 − 45) − 6075) y 2 + 6075) (8y 6 + 60y 4 + 90y 2 + 135) , where N (1) n stands for the respective normalization constants. From (58) one can see that φ (1) 0 , φ(1) 3 , and φ (1) 4 = 16λ/3. Every zero-mode in (58) is an element of KÂ, and consequently, each mode generates an infinite sequence of solutions, that is, we have three infinite sequences. The behavior of the respective potential and the probability densities associated with the zero-modes is depicted in Figs. 6a-6d, where the frequency profile has been fixed as a constant, Ω 2 (t) = 1, with σ(t) given in (39). Notice that, even in such a case, the resulting potential depends explicitly on time. 3 \| 2 (c), and \|φ (1) 4 \| 2 (d) of the zero-modes given in (58). The frequency profile is Ω 2 (t) = 1, and the parameters have been fixed to λ = 2.5, a = 2, and c = 1. (Lower row) Time-dependent potential V 1 (x, t) (e) computed through the hierarchy of nonlinear bound states w = 2 √ 2η 2 k (χ, N ) of Sec. 6.3, together with the probability densities \|φ (1) 0 \| 2 (f), \|φ (1) N \| 2 (g), and \|φ (1) N +1 \| 2 (h) of the zero-modes given in (64). The frequency profile is Ω 2 (t) = 1, and the parameters have been fixed to N = 3, k = 0.44/ √ 3!, λ = 1, a = 2, and c = 1. Solutions in terms of nonlinear bound states Another special class of solutions to (42) is determined by considering the set of parameters {α, β} = {2ν + 1, 0}, together with the reparametrization w(y; 2ν + 1, 0) = 2 √ 2η 2 k (ξ; ν) , y = ξ √ 2 .(59) The latter leads to a nonlinear differential equation for η k (ξ; ν) of the form [64] d 2 η k dξ 2 = 3η 5 k + 2ξη 3 k + 1 4 ξ 2 − ν − 1 2 η k , η ≡ η k (ξ; ν) , ν k ∈ R ,(60) which arises in the study of the derivative nonlinear Schrödinger equation [69]. A striking feature of η k is provided by the asymptotic behavior η k (ξ; ν) ∼ kD ν (ξ) as ξ → +∞ for ν ∈ R and D ν (ξ) the parabolic cylinder functions [61]. In turn, determining the asymptotic behavior for ξ → −∞ becomes a challenging task, where the asymptotic value depends on ν and it is computed from a connection formulae, see [70] for details. In this section, we restrict ourselves to the special case ν = N , with N = 0, 1, · · · . In such a case, there are solutions η k (ξ; N ) for ξ ∈ R with asymptotic behavior η k (ξ; N ) ∼    kξ N e −ξ 2 /4 x → +∞ kξ N e −ξ 2 /4 √ 1−2 √ 2πN !k 2 x → −∞ , k 2 < 1 2 √ 2πn! .(61) That is, the solutions decay exponentially to zero at both ξ → ±∞. The exact form of η k (ξ; N ) is determined in a recursive way through the combination of several Bäcklund transformations such that β = 0 is preserved in each iteration [64]. We thus have η k √ n+1 (ξ; N + 1) = ξη k (ξ; N ) + 2η 3 k (ξ; N ) − 2η k (ξ; N ) 2 [N + 1 + 2η k (ξ; N )η k (ξ; N ) − ξη 2 k (ξ; N ) − 2η 4 k (ξ; N )] 1/2 ,(62) with η k the partial derivative of η k with respect to ξ. Thus, the N -th solution is determined by iterating N times the solution associated with N = 0 in the recursion formula (62). The N = 0 solution is related to the complementary error function hierarchy (53) as η k (ξ; 0) ≡ 1 2 3/2 w(ξ/ √ 2; 1, 0) 1/2 = ke −ξ 2 /4 1 − √ 2πk 2 Erfc(ξ/ √ 2) 1/2 .(63) The behavior of the solutions η k (ξ; N ) are depicted in Fig. 7a for several nalues of N . In such a figure it can be seen that, indeed, the solutions contain exactly N zeroes while they converge to zero at the boundary points of the domain. Now, with the above solutions and their asymptotic behavior, it is straightforward to n obtained in (64). The dotted-black line represents the null state, the dashed-red arrow depict the solutions annihilated by either the creation or annihilation operators, and the solid-black arrow represents the transition to higher modes due to the action ofÂ † . (a) Sp Î 1 φ (1) N +2 A † φ (1) N +1 A φ (1) N A † φ (1) 1 φ (1) 0 A † A (N + 1)−dim. determine the set of finite-norm zero-mode eigenfunctions. From (28)-(29) we obtain φ (1) 0 (x, t) ≡ φ (1) 0;1 = N (1) 0 e i 4σ σ x 2 √ σ e −λz 2 /2 e −2 z dz G(z ) , φ (1) N (x, t) ≡ Φ (1) 0;1 = Φ (1) 0;2 = N (1) N e i 4σ σ x 2 √ σ e − z dz G(z ) η k (ξ; N ) , φ (1) N +1 (x, t) ≡ φ (1) 0;2 = φ (1) 0;3 = N (1) N +1 e i 4σ σ x 2 √ σ e z dz G(z ) F k (ξ; N )η k √ N +1 (ξ; N + 1), F k (ξ; N ) = N + 1 + 2η k (ξ; N )η k (ξ; N ) − ξη 2 k (ξ; N ) − 2η 4 k (ξ; N ) 1/2 ,(64) with the respective eigenvalues Λ 0 = 0, Λ(1)N = 2λN , and Λ(1) N +1 = 2λ(N + 1). From the asymptotic behavior (61), one realizes that the term exp z dz G(z ) converges to a finite value for z → ±∞, since the integral approximates to the error function at the asymptotic value. Thus, every zero-mode eigenfunction in (64) converges to zero at z → ±∞ and, indeed, we have finite-norm solutions. The remaining elements of the spectrum are determined from the action of the creation operatorÂ † (t) on the zero modes (64), as usual. Notice that φ N ∈ K A † , and thus φ N is annihilated by the creation operator A † (t). Therefore, the creation operator generates a (N +1)-dimensional sequence of eigenfunctions {φ (1) n } N n=0 through the iteration Â † (t) n φ(1) 0 , for n = 0, 1, · · · , N . In turn, an additional infinite sequence {φ (1) n } ∞ n=N +1 is generated from the operation Â † (t) n φ (1) N +1 , for n = 0, 1, · · · . In this form, the case discussed in this section generalizes the complementary error function hierarchy, where the latter is obtained as the special case N = 0. This spectral information is summarized in the diagram depicted in Fig. 7b. Finally, from the zero modes (64), together with the properties of the nonlinear bound states η k (ξ; N ), it follows that φ (1) 0 , φ(1) N , and φ (1) N +1 are solutions with exactly zero, N , and N + 1 nodes, respectively. Therefore, the oscillation theorem for the Sturm-Liouville associated with the quantum invariantÎ 1 (t) is verified. Additionally, given that the action of the creation operator increases the eigenvalue by 2λ units, we determine that in general the eigenvalues ofÎ 1 (t) are equidistant, Λ (1) n = 2λn, for n = 0, 1, · · · . The behavior of the respective time-dependent potential V 1 (x, t) and the probability densities of the zeromodes is depicted in Figs. 6e-6h, where we have chosen Ω 2 (t) = 1, with σ(t) given in (39). Conclusions The results of this manuscript can be seen from two different perspectives. On the one hand our approach represents a time-dependent generalization of the families of potentials reported previously in the stationary regime [33], on the other hand we also introduce some new quantum potentials unnoticed in the literature of stationary models. To this end, it was essential to address the shape-invariant problem from the more general perspective of the quantum invariants rather than the Hamiltonians. In this form, the time-dependence is introduced to both quantities, where the conventional spectral analysis is now carried on for the quantum invariant. Regardless of its time-dependence, the eigenvalues associated with the quantum invariant are time-independent, as it was first proved by Lewis-Reisenfeld [52]. Interestingly, after introducing the time parameter in the construction, a second nonlinear equation appears, namely the Ermakov equation, in such a way that the resulting time-dependent potentials and solutions to the Schrödnger equation are free of singularities at each time. In turn, the fourth Painlevé equation emerges after using a convenient reparametrization, where the parameters of the Painelvé equation dictate the distribution of eigenvalues of the quantum invariant, provided that the respective zero-modes are physically acceptable. It is worth to mention that both nonlinear equations, Ermakov and Painlevé, are not interlaced to each other; that is, the solutions of one equation do not modify the outcome of the solutions of the other equation. We can thus study each equation independently. Regarding the fourth Painlveé equation, a first family of solutions is determined through the related Riccati equation. This does not only allows us to recover the one-step rational extensions of the parametric oscillator, reported previously in [56], but also leads to a family of one-parameter solutions in terms of the error function. The respective potential corresponds to a time-dependent generalization of the deformed oscillator reported by Mielnik [8]. On the other hand, the hierarchy of rational solutions "−2y/3" in terms of the Okamoto polynomials allows constructing a quantum invariant with several gaps in its spectrum, which is generated by three infinite sequences of independent eigenfunctions, that is, the respective eigenvalues in each sequence do not overlap. As a particular example, we have shown that the Okamoto polynomial Q 2 (y) generates two gaps. Nevertheless, our results can be separated for an arbitrary polynomial Q N (y), with N = 0, 1, · · · , where the spectrum acquires precisely N gaps. The latter is indeed a property that could reveal an intrinsic structure in terms of exceptional polynomials. Further analysis is required, and results on the matter will be reported elsewhere. Although a closed expression for the N -th nonlinear bound state is not available, a nonlinear recurrence relation in the form of a Bäcklund transformation allows computing any solution by iterative means from the seed solution given by the error function. Remarkably, the Bäcklund transformation [64] is such that preserves β = 0 for any N -th nonlinear bound state. The latter implies that the spectrum of the quantum invariant is equidistant, for any N . Furthermore, the eigenfunctions are classified by two sequences, one that is (N + 1)-dimensional and one infinite-dimensional. The finite-dimensional sequence has two zero-modes, constructed as eigenfunctions of the annihilation (nodeless function) and creation (N nodes function) operators. The zero-mode related to the infinite sequence is also an eigenfunction of the annihilation operator such that it has exactly N +1 nodes. Interestingly, this case also brings new results in the stationary regime, for it generalizes the singlet and doublet structure introduced in [33]. Thus, it is clear that the families of nonlinear bound states can be explored even further in the context of stationary Hamiltonians, and a detailed analysis will be discussed in an upcoming contribution. and it takes the form [52,54] I 0 (t) = σ 2p2 + σ 2 4 + 1 σ 2 x 2 − σσ 2 {x,p} ,σ ≡ dσ(t) dt , (A-5) with σ = σ(t) a solution of the nonlinear equation σ + 4Ω 2 (t)σ = 4 σ 3 . (A-6) The latter equation is known as the Ermakov equation [23][24][25], and a solution is found through the nonlinear combination [17,18] σ(t) = aq 2 1 (t) + bq 1 (t)q 2 (t) + cq 2 2 (t) 1/2 , b 2 − 4ac = − 16 W 2 0 , (A-7) with W 0 = W (q 1 , q 2 ) the Wronskian of two linearly independent solutions q 1,2 (t) of the linear homogeneous equationq 1,2 + 4Ω 2 (t)q 1,2 = 0 . (A-8) From the form of the differential equation (A-8), it is straightforward to realize that the Wronskian W 0 is time-independent, regardless of the structure of Ω 2 (t). The constraint in the constants given in (A-7) guarantees that σ(t) is a nodeless function at any time. Such a feature is essential to construct regular solutions ψ (0) n (x, t), and also in determining new nonsigular time-dependent potentials, as discussed in Sec. 2. The spectral problem (A-3) has been already determined in the literature through several techniques, such as solving the differential equation directly [52], using a particular complex reparametrization [40], with aid of the Fourier transform [55], and performing geometrical transformations [54]. Thus, the spectral information ofÎ 0 (t) is given by φ (0) n (x, t) = e i 2σ σ x 2 2 n n!σ √ π e − x 2 2σ 2 H n x σ , Λ (0) n = 2n + 1 , n = 0, 1, · · · , (A-9) with H n (z) the Hermite polynomials [61]. Clearly, the elements of the set {φ (0) n } ∞ n=0 do not fulfill (A-2), but it can be easily shown that ψ (0) n (x, t) = e iθ (0) n (t) φ (0) n (x, t) , (A-10) is indeed a solution, where θ (0) n (t) is determined after substituting (A-10) in (A-2). It takes the following expression [52,54] θ (0) n (x, t) = −(2n + 1) t dt σ 2 (t ) = − n + 1 2 arctan W 0 2 ac − 4 W 2 0 + c q 2 (t) q 1 (t) . (A-11) Contrary to the stationary case, the time-dependent complex phase does not represent the time-evolution of the system. In summary, to completely determine the solutions of the parametric oscillator, we only need to find two linearly independent solutions of (A-8), provided that Ω(t) has been already specified. n (x, t) that satisfy the finite-norm condition \| φ n (t) \| < ∞. The orthogonality condition can be used as a base to construct the vector spaces H j (t) = Span{φ j . We thus define K A † := Ker(Â † (t)) = {Φ m (t) = δ n,m , for t = t. Therefore, the set {\|ψ n (t) } ∞ n=0 inherits the orthogonality from the set {\|φ n (t) } ∞ n=0 . In this form, we can be built-up the vector spaces H j (t) = Span{\|φ n (t) } ∞ n=0 , which under the definition of inner-product are equivalent to the vector spaces H j (t) introduced in Sec. 2. Now, substituting (32) and \|ψ n , where the nonstationary eigenfunctions obtained in Sec. 3 are all of the form φ (j) n (t) lead to a Hamiltonian that, in general, can not be written in terms of the position and momentum operators. The physical meaning of such Hamiltonians is not clear, and will not be considered in the rest of this work.Finally, the solutions to the Schrödinger equation ψ the spectral information of the quantum invariant, see Fig. 4. Figure 4 : 4(Color online) Ladder structure of the finite-norm zero-modes for the Riccati hierarchy φ (1) n obtained in Figure 6 : 6(Upper row) Time-dependent potential V 1 (x, t) (a) computed through the hierarchy of rational solutions w 2 in terms of the Okamoto polynomials (57), together with the probability densities \|φ Figure 7 : 7(a) (Color online) Solutions η k (ξ; N ) computed through (62)-(63) with the parameters {N = 1, k = 0.446} (solid-black), {N = 2, k = 0.446/ √ 2!} (dashed-blue), and {N = 3, k = 0.446/ √ 3!} (dottedred). (b) (Color online) Ladder structure of the finite-norm zero-modes for the non-linear bound state hierarchy φ (1) An eigenvalue equation for the time-dependent Hamiltonian is still attainable in the context of the adiabatic approximation[37]. Although, an orthogonal set of eigenfunctions can not be taken for granted for any general quantum invariant. A Parametric oscillatorIn this appendix, we briefly introduce the basic notions of the parametric oscillator, also known as nonstationary oscillator. The latter system is characterized by a time-dependent quadratic Hamiltonian of the formwithx ≡ x andp ≡ −i∂/∂x the canonical position and momentum operators, respectively, and Ω(t) > 0 the time-dependent frequency of oscillation. In contradistinction to the stationary oscillator, the HamiltonianĤ 0 (t) does not admit an eigenvalue equation. However, from the approach of Lewis-Riesenfeld[52], it is known that solutions of the Schrödinger equationare determined from the eigenvalue problemn the time-independent eigenvalues and φn (x, t) the nonstationary eigenfunctions of the quantum invariantÎ 0 (t) of the system. Such an invariant is computed from the Ordinary differential equations. E L Ince, Dover PublicationsNew YorkE.L. Ince, Ordinary differential equations, Dover Publications, New York, 1956. 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[ "Gradient Descent Learns Linear Dynamical Systems", "Gradient Descent Learns Linear Dynamical Systems" ]	[ "Moritz Hardt ", "Tengyu Ma ", "Benjamin Recht " ]	[]	[]	We prove that gradient descent efficiently converges to the global optimizer of the maximum likelihood objective of an unknown linear time-invariant dynamical system from a sequence of noisy observations generated by the system. Even though the objective function is non-convex, we provide polynomial running time and sample complexity bounds under strong but natural assumptions. Linear systems identification has been studied for many decades, yet, to the best of our knowledge, these are the first polynomial guarantees for the problem we consider. * Google.	null	[ "https://arxiv.org/pdf/1609.05191v1.pdf" ]	7,597,719	1609.05191	81429d54395ebe1747fe32e1e99125760f00ef43	Gradient Descent Learns Linear Dynamical Systems September 19, 2016 Moritz Hardt Tengyu Ma Benjamin Recht Gradient Descent Learns Linear Dynamical Systems September 19, 2016 We prove that gradient descent efficiently converges to the global optimizer of the maximum likelihood objective of an unknown linear time-invariant dynamical system from a sequence of noisy observations generated by the system. Even though the objective function is non-convex, we provide polynomial running time and sample complexity bounds under strong but natural assumptions. Linear systems identification has been studied for many decades, yet, to the best of our knowledge, these are the first polynomial guarantees for the problem we consider. * Google. Introduction Many learning problems are by their nature sequence problems where the goal is to fit a model that maps a sequence of input words x 1 , . . . , x T to a corresponding sequence of observations y 1 , . . . , y T . Text translation, speech recognition, time series prediction, video captioning and question answering systems, to name a few, are all sequence to sequence learning problems. For a sequence model to be both expressive and parsimonious in its parameterization, it is crucial to equip the model with memory thus allowing its prediction at time t to depend on previously seen inputs. Recurrent neural networks form an expressive class of non-linear sequence models. Through their many variants, such as long-short-term-memory [HS97], recurrent neural networks have seen remarkable empirical success in a broad range of domains. At the core, neural networks are typically trained using some form of (stochastic) gradient descent. Even though the training objective is non-convex, it is widely observed in practice that gradient descent quickly approaches a good set of model parameters. Understanding the effectiveness of gradient descent for non-convex objectives on theoretical grounds is a major open problem in this area. If we remove all non-linear state transitions from a recurrent neural network, we are left with the state transition representation of a linear dynamical system. Notwithstanding, the natural training objective for linear systems remains non-convex due to the composition of multiple linear operators in the system. If there is any hope of eventually understanding recurrent neural networks, it will be inevitable to develop a solid understanding of this special case first. To be sure, linear dynamical systems are important in their own right and have been studied for many decades independently of machine learning within the control theory community. Control theory provides a rich set techniques for identifying and manipulating linear systems. The learning problem in this context corresponds to "linear dynamical system identification". Maximum likelihood estimation with gradient descent is a popular heuristic for dynamical system identification [Lju98]. In the context of machine learning, linear systems play an important role in numerous tasks. For example, their estimation arises as subroutines of reinforcement learning in robotics [LK13], location and mapping estimation in robotic systems [DWB06], and estimation of pose from video [RRD05]. In this work, we show that gradient descent efficiently minimizes the maximum likelihood objective of an unknown linear system given noisy observations generated by the system. More formally, we receive noisy observations generated by the following time-invariant linear system: h t+1 = Ah t + Bx t (1.1) y t = Ch t + Dx t + ξ t Here, A, B, C, D are linear transformations with compatible dimensions and we denote by Θ = (A, B, C, D) the parameters of the system. The vector h t represents the hidden state of the model at time t. Its dimension n is called the order of the system. The stochastic noise variables {ξ t } perturb the output of the system which is why the model is called an output error model in control theory. We assume the variables are drawn i.i.d. from a distribution with mean 0 and variance σ 2 . Throughout the paper we focus on controllable and externally stable systems. A linear system is externally stable (or equivalently bounded-input bounded-output stable) if and only if the spectral radius of A, denoted ρ(A), is strictly bounded by 1. Controllability is a mild non-degeneracy assumption that we formally define later. Without loss of generality, we further assume that the transformations B, C and D have bounded Frobenius norm. This can be achieved by a rescaling of the output variables. We assume we have N pairs of sequences (x, y) as training examples, S = (x (1) , y (1) ), . . . , (x (N ) , y (N ) ) . Each input sequence x ∈ R T of length T is drawn from a distribution and y is the corresponding output of the system above generated from an unknown initial state h. We allow the unknown initial state to vary from one input sequence to the next. This only makes the learning problem more challenging. Our goal is to fit a linear system to the observations. We parameterize our model in exactly the same way as (1.1). That is, for linear mappings (Â,B,Ĉ,D), the trained model is defined as: h t+1 =Âĥ t +Bx t ,ŷ t =Ĉĥ t +Dx t (1.2) The (population) risk of the model is obtained by feeding the learned system with the correct initial states and comparing its predictions with the ground truth in expectation over inputs and errors. Denoting byŷ t the t-th prediction of the trained model starting from the correct initial state that generated y t , and using Θ as a short hand for (Â,B,Ĉ,D), we formally define population risk as: f ( Θ) = E {xt},{ξt} 1 T T t=1 ŷ t − y t 2 (1.3) Note that even though the predictionŷ t is generated from the correct initial state, the learning algorithm does not have access to the correct initial state for its training sequences. While the squared loss objective turns out to be non-convex, it has many appealing properties. Assuming the inputs x t and errors ξ t are drawn independently from a Gaussian distribution, the corresponding population objective corresponds to maximum likelihood estimation. In this work, we make the weaker assumption that the inputs and errors are drawn independently from possibly different distributions. The independence assumption is certainly idealized for some learning applications. However, in control applications the inputs can often be chosen by the controller rather than by nature. Moreover, the outputs of the system at various time steps are correlated through the unknown hidden state and therefore not independent even if the inputs are. Our results We show that we can efficiently minimize the population risk using projected stochastic gradient descent. The bulk of our work applies to single-input single-output (SISO) systems meaning that inputs and outputs are scalars x t , y t ∈ R. However, the hidden state can have arbitrary dimension n. Every controllable SISO admits a convenient canonical form called controllable canonical form that we formally introduce later. In this canonical form, the transition matrix A is governed by n parameters a 1 , . . . , a n which coincide with the coefficients of the characteristic polynomial of A. The minimal assumption under which we might hope to learn the system is that the spectral radius of A is smaller than 1. However, the set of such matrices is non-convex and does not have enough structure for our analysis. We will therefore make additional assumptions. The assumptions we need differ between the case where we are trying to learn A with n parameter system, and the case where we allow ourselves to over-specify the trained model with n > n parameters. The former is sometimes called proper learning, while the latter is called improper learning. In the improper case, we are essentially able to learn any system with spectral radius less than 1 under a mild separation condition on the roots of the characteristic polynomial. Our assumption in the proper case is stronger and we introduce it next. Proper learning Suppose that the state transition matrix A is given by parameters a 1 , . . . , a n and consider the polynomial q(z) = 1 + a 1 z + a 2 z 2 + · · · + a n z n over the complex numbers C. Complex plane We will require that the image of the unit circle on the complex plane under the polynomial q is contained in the cone of complex numbers whose real part is larger than their absolute imaginary part. Formally, for all z ∈ C such that \|z\| = 1, we require that (q(z)) > \| (q(z))\|. Here, (z) and (z) denote the real and imaginary part of z, respectively. We illustrate this condition in the figure on the right for a degree 4 system. Our assumption has three important implications. First, it implies (via Rouché's theorem) that the spectral radius of A is smaller than 1 and therefore ensures the stability of the system. Second, the vectors satisfying our assumption form a convex set in R n . Finally, it ensures that the objective function is weakly quasi-convex, a condition we introduce later when we show that it enables stochastic gradient descent to make sufficient progress. We note in passing that our assumption can be satisfied via the 1 -norm constraint a 1 ≤ √ 2/2. Moreover, if we pick random Gaussian coefficients with expected norm bounded by o(1/ √ log n), then the resulting vector will satisfy our assumption with probability 1 − o(1). Roughly speaking, the assumption requires the roots of the characteristic polynomial p(z) = z n + a 1 z n−1 + · · · + a n are relatively dispersed inside the unit circle. (For comparison, on the other end of the spectrum, the polynomial p(z) = (z − 0.99) n have all its roots colliding at the same point and doesn't satisfy the assumption.) Theorem 1.1 (Informal). Under our assumption, projected stochastic gradient descent, when given N sample sequence of length T , returns parameters Θ with population risk f ( Θ) ≤ f (Θ) + O n 5 + σ 2 n 3 T N . Recall that f (Θ) is the population risk of the optimal system, and σ 2 refers to the variance of the noise variables. We also assume that the inputs x t are drawn from a pairwise independent distribution with mean 0 and variance 1. Note, however, that this does not imply independence of the outputs as these are correlated by a common hidden state. The stated version of our result glosses over the fact that we need our assumption to hold with a small amount of slack; a precise version follows in Section 4. Our theorem establishes a polynomial convergence rate for stochastic gradient descent. Since each iteration of the algorithm only requires a sequence of matrix operations and an efficient projection step, the running time is polynomial, as well. Likewise, the sample requirements are polynomial since each iteration requires only a single fresh example. An important feature of this result is that the error decreases with both the length T and the number of samples N . Computationally, the projection step can be a bottleneck, although it is unlikely to be required in practice and may be an artifact of our analysis. The power of over-parameterization Endowing the model with additional parameters compared to the ground truth turns out to be surprisingly powerful. We show that we can essentially remove the assumption we previously made in proper learning. The idea is simple. If p is the characteristic polynomial of A of degree n. We can find a system of order n > n such that the characteristic polynomial of its transition matrix becomes p · p for some polynomial p of order n − n. This means that to apply our result we only need the polynomial p · p to satisfy our assumption. At this point, we can choose p to be an approximation of the inverse p −1 . For sufficiently good approximation, the resulting polynomial p · p is close to 1 and therefore satisfies our assumption. Such an approximation exists generically for n = O(n) under mild non-degeneracy assumptions on the roots of p. In particular, any small random perturbation of the roots would suffice. Theorem 1.2 (Informal). Under a mild non-degeneracy assumption, stochastic gradient descent returns parameters Θ corresponding to a system of order n = O(n) with population risk f ( Θ) ≤ f (Θ) + O n 5 + σ 2 n 3 T N , when given N sample sequences of length T . We remark that the idea we sketched also shows that, in the extreme, improper learning of linear dynamical systems becomes easy in the sense that the problem can be solved using linear regression against the outputs of the system. However, our general result interpolates between the proper case and the regime where linear regression works. We discuss in more details in Section 6.3. Multi-input multi-output systems Both results we saw immediately extend to single-input multi-output (SIMO) systems as the dimensionality of C and D are irrelevant for us. The case of multi-input multi-output (MIMO) systems is more delicate. Specifically, our results carry over to a broad family of multi-input multi-output systems. However, in general MIMO systems no longer enjoy canonical forms like SISO systems. In Section 7, we introduce a natural generalization of controllable canonical form for MIMO systems and extend our results to this case. Related work System identification is a core problem in dynamical systems and has been studied in depth for many years. The most popular reference on this topic is the text by Ljung [Lju98]. Nonetheless, the list of non-asymptotic results on identifying linear systems from noisy data is surprisingly short. Several authors have recently tried to estimate the sample complexity of dynamical system identification using machine learning tools [VK08, CW02, WC99]. All of these result are rather pessimistic with sample complexity bounds that are exponential in the degree of the linear system and other relevant quantities. Contrastingly, we prove that gradient descent has an associated polynomial sample complexity in all of these parameters. Moreover, all of these papers only focus on how well empirical risk approximates the true population risk and do not provide guarantees about any algorithmic schemes for minimizing the empirical risk. The only result to our knowledge which provides polynomial sample complexity for identifying linear dynamical systems is in Shah et al [SBTR12]. Here, the authors show that if certain frequency domain information about the linear dynamical system is observed, then the linear system can be identified by solving a second-order cone programming problem. This result is about improper learning only, and the size of the resulting system may be quite large, scaling as the (1 − ρ(A)) −2 . As we describe in this work, very simple algorithms work in the improper setting when the system degree is allowed to be polynomial in (1 − ρ(A)) −1 . Moreover, it is not immediately clear how to translate the frequency-domain results to the time-domain identification problem discussed above. Our main assumption about the image of the polynomial q(z) is an appeal to the theory of passive systems. A system is passive if the dot product between the input sequence u t and output sequence y t are strictly positive. Physically, this notion corresponds to systems that cannot create energy. For example, a circuit made solely of resistors, capacitors, and inductors would be a passive electrical system. If one added an amplifier to the internals of the system, then it would no longer be passive. The set of passive systems is a subset of the set of stable systems, and the subclass is somewhat easier to work with mathematically. Indeed, Megretski used tools from passive systems to provide a relaxation technique for a family of identification problems in dynamical systems [Meg08]. His approach is to lower bound a nonlinear least-squares cost with a convex functional. However, he does not prove that his technique can identify any of the systems, even asymptotically. Bazanella et al use a passivity condition to prove quasi-convexity of a cost function that arises in control design [BGMA08]. Building on this work, Eckhard and Bazanella prove a weaker version of Lemma 3.3 in the context of system identification [EB11], but no sample complexity or global convergence proofs are provided. Proof overview The first important step in our proof is to develop population risk in Fourier domain where it is closely related to what we call idealized risk. Idealized risk essentially captures the 2 -difference between the transfer function of the learned system and the ground truth. The transfer function is a fundamental object in control theory. Any linear system is completely characterized by its transfer function G(z) = C(zI − A) −1 B. In the case of a SISO, the transfer function is a rational function of degree n over the complex numbers and can be written as G(z) = s(z)/p(z). In the canonical form introduced in Section 1.7, the coefficients of p(z) are precisely the parameters that specify A. Moreover, z n p(1/z) = 1 + a 1 z + a 2 z 2 + · · · + a n z n is the polynomial we encountered in the introduction. Under the assumption illustrated earlier, we show in Section 3 that the idealized risk is weakly quasi-convex (Lemma 3.3). Quasi-convexity implies that gradients cannot vanish except at the optimum of the objective function; we review this (mostly known) material in Section 2. In particular, this lemma implies that in principle we can hope to show that gradient descent converges to a global optimum. However, there are several important issues that we need to address. First, the result only applies to idealized risk, not our actual population risk objective. Therefore it is not clear how to obtain unbiased gradients of the idealized risk objective. Second, there is a subtlety in even defining a suitable empirical risk objective. The reason is that risk is defined with respect to the correct initial state of the system which we do not have access to during training. We overcome both of these problems. In particular, we design an almost unbiased estimator of the gradient of the idealized risk in Lemma 5.4 and give variance bounds of the gradient estimator (Lemma 5.5). Our results on improper learning in Section 6 rely on a surprisingly simple but powerful insight. We can extend the degree of the transfer function G(z) by extending both numerator and denominator with a polynomial u(z) such that G(z) = s(z)u(z)/p(z)u(z). While this results in an equivalent system in terms of input-output behavior, it can dramatically change the geometry of the optimization landscape. In particular, we can see that only p(z)u(z) has to satisfy the assumption of our proper learning algorithm. This allows us, for example, to put u(z) ≈ p(z) −1 so that p(z)u(z) ≈ 1, hence trivially satisfying our assumption. A suitable inverse approximation exists under light assumptions and requires degree no more than d = O(n). Algorithmically, there is almost no change. We simply run stochastic gradient descent with n + d model parameters rather than n model parameters. Preliminaries For complex matrix (or vector, number) C, we use (C) to denote the real part and (C) the imaginary part, andC the conjugate and C * =C its conjugate transpose . We use \| · \| to denote the absolute value of a complex number c. For complex vector u and v, we use u, v = u * v to denote the inner product and u = √ u * u is the norm of u. For complex matrix A and B with same dimension, A, B = tr(A * B) defines an inner product, and A F = tr(A * A) is the Frobenius norm. For a square matrix A, we use ρ(A) to denote the spectral radius of A, that is, the largest absolute value of the elements in the spectrum of A. We use I n to denote the identity matrix with dimension n × n, and we drop the subscript when it's clear from the context.We let e i denote the i-th standard basis vector. A SISO of order n is in controllable canonical form if A and B have the following form A =        0 1 0 · · · 0 0 0 1 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · 1 −a n −a n−1 −a n−2 · · · −a 1        B =        0 0 . . . 0 1        (1.4) We will parameterizeÂ,B,Ĉ,D accordingly. We will write A = CC(a) for brevity, where a is used to denote the unknown last row [−a n , . . . , −a 1 ] of matrix A. We will useâ to denote the corresponding training variables for a. Since here B is known, soB is no longer a trainable parameter, and is forced to be equal to B. Moreover, C is a row vector and we use [c 1 , · · · , c n ] to denote its coordinates (and similarly forĈ). A SISO is controllable if and only if the matrix [B \| AB \| A 2 B \| · · · \| A n−1 B] has rank n. This statement corresponds to the condition that all hidden states should be reachable from some initial condition and input trajectory. Any controllable system admits a controllable canonical form [HRS07]. For vector a = [a n , . . . , a 1 ], let p a (z) denote the polynomial p a (z) = z n + a 1 z n−1 + · · · + a n . (1.5) When a defines the matrix A that appears in controllable canonical form, then p a is precisely the characteristic polynomial of A. That is, p a (z) = det(zI − A). Gradient descent and quasi-convexity It is known that under certain mild conditions (stochastic) gradient descent converges even on nonconvex functions to local minimum [GHJY15,LSJR16]. Though usually for concrete problems the challenge is to prove that there is no spurious local minimum other than the target solution. Here we introduce a condition similar to the quasi-convexity notion in [HLS15], which ensures that any point with vanishing gradient is the optimal solution . Roughly speaking, the condition says that at any point θ the negative of the gradient −∇f (θ) should be positively correlated with direction θ * −θ pointing towards the optimum. Our condition is slightly weaker than that in [HLS15] since we only require quasi-convexity and smoothness with respect to the optimum, and this (simple) extension will be necessary for our analysis. Definition 2.1 (Weak quasi-convexity). We say an objective function f is τ -weakly-quasi-convex (τ -WQC) over a domain B with respect to global minimum θ * if there is a positive constant τ > 0 such that for all θ ∈ B, ∇f (θ) (θ − θ * ) ≥ τ (f (θ) − f (θ * )) . (2.1) We further say f is Γ-weakly-smooth if for for any point θ, ∇f (θ) 2 ≤ Γ(f (θ) − f (θ * )). Note that indeed any Γ-smooth convex function in the usual sense is O(Γ)-weakly-smooth. For a random vector X ∈ R n , we define it's variance to be Var [X] = E [ X − EX 2 ]. Definition 2.2. We call r(θ) an unbiased estimator of ∇f (θ) with variance V if it satisfies E [r(θ)] = ∇f (θ) and Var[r(θ)] ≤ V . Projected stochastic gradient descent over some closed convex set B with learning rate η > 0 refers to the following algorithm in which Π B denotes the Euclidean projection onto B: for k = 0 to K − 1 : w k+1 = θ k − ηr(θ k ) θ k+1 = Π B (w k+1 ) return θ j with j uniformly picked from 1, . . . , K (2. 2) The following Proposition is well known for convex objective functions (corresponding to 1weakly-quasi-convex functions). We extend it (straightforwardly) to the case when τ -WQC holds with any positive constant τ . Proposition 2.3. Suppose the objective function f is τ -weakly-quasi-convex and Γ-weakly-smooth, and r(·) is an unbiased estimator for ∇f (θ) with variance V . Moreover, suppose the global minimum θ * belongs to B, and the initial point θ 0 satisfies θ 0 −θ * ≤ R. Then projected gradient descent (2.2) with a proper learning rate returns θ K in K iterations with expected error f (θ K ) ≤ O max ΓR 2 τ 2 K , R √ V τ √ K . Remark 1. It's straightforward to see (from the proof) that the algorithm tolerates inverse exponential bias in the gradient estimator. Technically, suppose E [r(θ)] = ∇f (θ) ± ζ then f (θ K ) ≤ O max ΓR 2 τ 2 K , R √ V τ √ K + poly(K) · ζ. Throughout the paper, we assume that the error that we are shooting for is inverse polynomial and therefore the effect of inverse exponential bias is negligible. We defer the proof of Proposition 2.4 to Appendix A which is a simple variation of the standard convergence analysis of stochastic gradient descent (see, for example, [Bot98]). Finally, we note that the sum of two quasi-convex functions may no longer be quasi-convex. However, if a sequence functions is τ -WQC with respect to a common point θ * , then their sum is also τ -WQC. This follows from the linearity of gradient operation. Proposition 2.4. Suppose functions f 1 , . . . , f n are individually τ -weakly-quasi-convex in B with respect to a common global minimum θ * , then for non-negative w 1 , . . . , w n the linear combination f = n i=1 w i f i is also τ -weakly-quasi-convex with respect to θ * in B. Population risk in frequency domain We next establish conditions under which risk is weakly-quasi-convex. Our strategy is to first approximate the risk functional f ( Θ) by what we call idealized risk. This approximation of the objective function is fairly straightforward; we justify it toward the end of the section. We can show that f ( Θ) ≈ D −D 2 + ∞ k=0 ĈÂ k B − CA k B 2 . (3.1) The leading term D −D 2 is convex inD which appears nowhere else in the objective. It therefore doesn't affect the convergence of the algorithm (up to lower order terms) by virtue of Proposition 2.4, and we restrict our attention to the remaining terms. Definition 3.1 (Idealized risk). We define the idealized risk as g(Â,Ĉ) = ∞ k=0 ĈÂ k B − CA k B 2 . (3.2) We now use basic concepts from control theory (see [HRS07,Hes09] for more background) to express the idealized risk (3.2) in Fourier domain. The transfer function of the linear system is G(z) = C(zI − A) −1 B . (3.3) Note that G(z) is a rational function over the complex numbers of degree n and hence we can find polynomials s(z) and p(z) such that G(z) = s(z) p(z) , with the convention that the leading coefficient of p(z) is 1. In controllable canonical form (1.4), the coefficients of p will correspond to the last row of the A, while the coefficients of s correspond to the entries of C. Also note that G(z) = ∞ t=1 z −t CA t−1 B = ∞ t=1 z −t r t−1 The sequence r = (r 0 , r 1 , . . . , r t , . . .) = (CB, CAB, . . . , CA t B, . . .) is called the impulse response of the linear system. The behavior of a linear system is uniquely determined by the impulse response and therefore by G(z). Analogously, we denote the transfer function of the learned system by G(z) =Ĉ(zI −Â) −1 B =ŝ(z)/p(z) . The idealized risk (3.2) is only a function of the impulse responser of the learned system, and therefore it is only a function of G(z). For future reference, we note that by some elementary calculation (see Lemma B.1), we have G(z) = C(zI − A) −1 B = c 1 + · · · + c n z n−1 z n + a 1 z n−1 + · · · + a n , (3.4) which implies that s(z) = c 1 + · · · + c n z n−1 and p(z) = z n + a 1 z n−1 + · · · + a n . With these definitions in mind, we are ready to express idealized risk in Fourier domain. Proposition 3.2. Suppose pâ(z) has all its roots inside unit circle, then the idealized risk g(â,Ĉ) can be written in the Fourier domain as g(Â,Ĉ) = 2π 0 G(e iθ ) − G(e iθ ) 2 dθ . Proof. Note that G(e iθ ) is the Fourier transform of the sequence {r k } and so is G(e iθ ) the Fourier transform 1 ofr k . Therefore by Parseval' Theorem, we have that g(Â,Ĉ) = ∞ k=0 r k − r k 2 = 2π 0 \| G(e iθ ) − G(e iθ )\| 2 dθ . Quasi-convexity of the idealized risk Now that we have a convenient expression for risk in Fourier domain, we can prove that the idealized risk g(Â,Ĉ) is weakly-quasi-convex whenâ is not so far from the true a in the sense that p a (z) and p a (z) have an angle less than π/2 for every z on the unit circle. We will use the convention that a andâ refer to the parameters that specify A andÂ, respectively. Lemma 3.3. For τ > 0 and everyĈ, the idealized risk g(Â,Ĉ) is τ -weakly-quasi-convex over the domain N τ (a) = â ∈ R n : p a (z) pâ(z) ≥ τ /2, ∀ z ∈ C, s.t. \|z\| = 1 . (3.5) Proof. We first analyze a single term h = \| G(z) − G(z)\| 2 . Recall that G(z) =ŝ(z)/p(z) wherê p(z) = pâ(z) = z n +â 1 z n−1 + · · · +â n . Note that z is fixed and h is a function ofâ andĈ. Then it is straightforward to see that ∂h ∂ŝ(z) = 2 1 p(z) ŝ(z) p(z) − s(z) p(z) * . (3.6) 1 The Fourier transform exists since r k 2 = ĈÂ kB 2 ≤ Ĉ Â k B ≤ cρ(Â) k where c doesn't depend on k and ρ(Â) < 1. and ∂h ∂p(z) = −2 ŝ(z) p(z) 2 ŝ(z) p(z) − s(z) p(z) * . (3.7) Sinceŝ(z) andp(z) are linear inĈ andâ respectively, by chain rule we have that ∂h ∂â ,â − a + ∂h ∂Ĉ ,Ĉ − C = ∂h ∂p(z) ∂p(z) ∂â ,â − a + ∂h ∂ŝ(z) ∂ŝ(z) ∂Ĉ ,Ĉ − C = ∂h ∂p(z) (p(z) − p(z)) + ∂h ∂ŝ(z) (ŝ(z) − s(z)) . Plugging the formulas (3.6) and (3.7) for ∂h ∂ŝ(z) and ∂h ∂p(z) into the equation above, we obtain that ∂h ∂â ,â − a + ∂h ∂Ĉ ,Ĉ − C = 2 −ŝ(z)(p(z) − p(z)) +p(z)(ŝ(z) − s(z)) p(z) 2 ŝ(z) p(z) − s(z) p(z) * = 2 ŝ(z)p(z) − s(z)p(z) p(z) 2 ŝ(z) p(z) − s(z) p(z) * = 2 p(z) p(z) ŝ(z) p(z) − s(z) p(z) 2 = 2 p(z) p(z) G(z) − G(z) 2 . Hence h = \| G(z) − G(z)\| 2 is τ -weakly-quasi-convex with τ = 2 min \|z\|=γ p(z) p(z) . This implies our claim, since by Proposition 3.2, the idealized risk g is convex combination of functions of the form \| G(z) − G(z)\| 2 for \|z\| = 1. Moreover, Proposition 2.4 shows convex combination preserves quasi-convexity. For future reference, we also prove that the idealized risk is O(n 2 /τ 4 1 )-weakly smooth. Lemma 3.4. The idealized risk g(Â,Ĉ) is Γ-weakly smooth with Γ = O(n 2 /τ 4 1 ). Proof. By equation (3.7) and the chain rule we get that ∂g ∂Ĉ = T ∂\| G(z) − G(z)\| 2 ∂p(z) · ∂p(z) ∂Ĉ dz = T 2 1 p(z) ŝ(z) p(z) − s(z) p(z) * · [1, . . . , z n−1 ]dz . therefore we can bound the norm of the gradient by ∂g ∂Ĉ 2 ≤ T ŝ(z) p(z) − s(z) p(z) 2 dz · T 4 [1, . . . , z n−1 ] 2 · \| 1 p(z) \| 2 dz ≤ O(n/τ 2 1 ) · g(Â,Ĉ) . Similarly, we could show that ∂g ∂â 2 ≤ O(n 2 /τ 2 1 ) · g(Â,Ĉ). Justifying idealized risk We need to justify the approximation we made in Equation (3.1). Lemma 3.5. Assume that ξ t and x t are drawn i.i.d. from an arbitrary distribution with mean 0 and variance 1. Then the population risk f ( Θ) can be written as, f ( Θ) = (D − D) 2 + T −1 k=1 1 − k T ĈÂ k−1 B − CA k−1 B 2 + σ 2 . (3.8) Proof of Lemma 3.5. Under the distributional assumptions on ξ t and x t , we can calculate the objective functions above analytically. We write out y t ,ŷ t in terms of the inputs, y t = Dx t + t−1 k=1 CA t−k−1 Bx k + CA t−1 h 0 + ξ t ,ŷ t =Dx t + t−1 k=1ĈÂ t−k−1B x k + CA t−1 h 0 . Therefore, using the fact that x t 's are independent and with mean 0 and covariance I, the expectation of the error can be calculated (formally by Claim B.2), E ŷ t − y t 2 = D − D 2 F + t−1 k=1 ĈÂ t−k−1B − CA t−k−1 B 2 F + E [ ξ t 2 ] . (3.9) Using E [ ξ t 2 ] = σ 2 , it follows that f ( Θ) = D − D 2 F + T −1 k=1 1 − k T ĈÂ k−1B − CA k−1 B 2 F + σ 2 . (3.10) Recall that under the controllable canonical form (1.4), B = e n is known and thereforeB = B is no longer a variable. We useâ for the training variable corresponding to a. Then the expected objective function (3.10) simplifies to f ( Θ) = (D − D) 2 + T −1 k=1 1 − k T ĈÂ k−1 B − CA k−1 B 2 . The previous lemma does not yet control higher order contributions present in the idealized risk. This requires additional structure that we introduce in the next section. Effective relaxations of spectral radius The previous section showed quasi-convexity of the idealized risk. However, several steps are missing towards showing finite sample guarantees for stochastic gradient descent. In particular, we will need to control the variance of the stochastic gradient at any system that we encounter in the training. For this purpose we formally introduce our main assumption now and show that it serves as an effective relaxation of spectral radius. This results below will be used for proving convergence of stochastic gradient descent in Section 5. Consider the following convex region C in the complex plane, C = {z : z ≥ (1 + τ 0 )\| z\|} ∩ {z : τ 1 < z < τ 2 } . (4.1) where τ 0 , τ 1 , τ 2 > 0 are constants that are considered as fixed constant throughout the paper. Our bounds will have polynomial dependency on these parameters. Definition 4.1. We say a polynomial p(z) is α-acquiescent if {p(z)/z n : \|z\| = α} ⊆ C. A linear system with transfer function G(z) = s(z)/p(z) is α-acquiescent if the denominator p(z) is. The set of coefficients a ∈ R n defining acquiescent systems form a convex set. Formally, for a positive α > 0, define the convex set B α ⊆ R n as B α = a ∈ R n : {p a (z)/z n : \|z\| = α} ⊆ C . (4.2) We note that definition (4.2) is equivalent to the definition B α = a ∈ R n : {z n p(1/z) : \|z\| = 1/α} ⊆ C , which is the version that we used in introduction for simplicity. Indeed, we can verify the convexity of B α by definition and the convexity of C: a, b ∈ B α implies that p a (z)/z n , p b (z)/z n ∈ C and therefore, p (a+b)/2 (z)/z n = 1 2 (p a (z)/z n + p b (z)/z n ) ∈ C. We also note that the parameter α in the definition of acquiescence corresponds to the spectral radius of the companion matrix. In particular, an acquiescent system is stable for α < 1. Lemma 4.2. Suppose a ∈ B α , then the roots of polynomial p a (z) have magnitudes bounded by α. Therefore the controllable canonical form A = CC(a) defined by a has spectral radius ρ(A) < α. Proof. Define holomorphic function f (z) = z n and g(z) = p a (z) = z n + a 1 z n−1 + · · · + a n . We apply the symmetric form of Rouche's theorem [Est62] on the circle K = {z : \|z\| = α}. For any point z on K, we have that \|f (z)\| = α n , and that \|f (z) − g(z)\| = α n · \|1 − p a (z)/z n \|. Since a ∈ B α , we have that p a (z)/z n ∈ C for any z with \|z\| = α. Observe that for any c ∈ C we have that \|1 − c\| < 1 + \|c\|, therefore we have that \|f (z) − g(z)\| = α n \|1 − p a (z)/z n \| < α n (1 + \|p a (z)\|/\|z n \|) = \|f (z)\| + \|p a (z)\| = \|f (z)\| + \|g(z)\| . Hence, using Rouche's Theorem, we conclude that f and g have same number of roots inside circle K. Note that function f = z n has exactly n roots in K and therefore g have all its n roots inside circle K. The following lemma establishes the fact that B α is a monotone family of sets in α. The proof follows from the maximum modulo principle of the harmonic functions (z n p(1/z)) and (z n p(1/z)). We defer the short proof to Section C.1. We remark that there are larger convex sets than B α that ensure bounded spectral radius. However, in order to also guarantee monotonicity and the no blow-up property below, we have to restrict our attention to B α . Lemma 4.3 (Monotonicity of B α ). For any 0 < α < β, we have that B α ⊂ B β . Our next lemma entails that acquiescent systems have well behaved impulse responses. Lemma 4.4 (No blow-up property). Suppose a ∈ B α for some α ≤ 1. Then the companion matrix A = CC(a) satisfies ∞ k=0 α −k A k B 2 ≤ 2πnα −2n /τ 2 1 . (4.3) Moreover, it holds that for any k ≥ 0, A k B 2 ≤ min{2πn/τ 2 1 , 2πnα 2k−2n /τ 2 1 } . Proof of Lemma 4.4. Let f λ = ∞ k=0 e iλk α −k A k B be the Fourier transform of the series α −k A k B. Then using Parseval's Theorem, we have ∞ k=0 α −k A k B 2 = 2π 0 \|f λ \| 2 dλ = 2π 0 \|(I − α −1 e iλ A) −1 B\| 2 dλ = 2π 0 n j=1 α 2j \|p a (αe −iλ )\| 2 dλ ≤ 2π 0 n \|p a (αe −iλ )\| 2 dλ. (4.4) where at the last step we used the fact that ( I − wA) −1 B = 1 pa(w −1 ) [w −1 , w −2 . . . , z −n ] (see Lemma B.1), and that α ≤ 1. Since a ∈ B α , we have that \|q a (α −1 e iλ )\| ≥ τ 1 , and therefore p a (αe −iλ ) = α n e −inλ q(e iλ /α) has magnitude at least τ 1 α n . Plugging in this into equation (4.4), we conclude that ∞ k=0 α −k A k B 2 ≤ 2π 0 n \|p a (αe −iλ )\| 2 dλ ≤ 2πnα −2n /τ 2 1 . Finally we establish the bound for A k B 2 . By Lemma 4.3, we have B α ⊂ B 1 for α ≤ 1. Therefore we can pick α = 1 in equation (4.3) and it still holds. That is, we have that ∞ k=0 A k B 2 ≤ 2πn/τ 2 1 . This also implies that A k B 2 ≤ 2πn/τ 2 1 . Efficiently computing the projection In our algorithm, we require a projection onto B α . However, the only requirement of the projection step is that it projects onto a set contained inside B α that also contains the true linear system. So a variety of subroutines can be used to compute this projection or an approximation. First, the explicit projection onto B α is representable by a semidefinite program. This is because each of the three constrains can be checked by testing if a trigonometric polynomial is non-negative. A simple inner approximation can be constructed by requiring the constraints to hold on an a finite grid of size O(n). One can check that this provides a tight, polyhedral approximation to the set B α , following an argument similar to Appendix C of Bhaskar et al [BTR13]. See Section F for more detailed discussion on why projection on a polytope suffices. Furthermore, sometimes we can replace the constraint by an 1 or 2 -constraint if we know that the system satisfies the corresponding assumption. Removing the projection step entirely is an interesting open problem. Learning acquiescent systems Next we show that we can learn acquiescent systems. Theorem 5.1. Suppose the true system Θ is α-acquiescent and satisfies C ≤ 1. Then with N samples of length T ≥ Ω(n + 1/(1 − α)), stochastic gradient descent (Algorithm 1) with projection set B α returns parameters Θ = (Â,B,Ĉ,D) with population risk f ( Θ) ≤ f (Θ) + O n 2 N + n 5 + σ 2 n 3 T N , (5.1) where O(·)-notation hides polynomial dependencies on 1/(1 − α), 1/τ 0 , 1/τ 1 , τ 2 , and R = a . Recall that T is the length of the sequence and N is the number of samples. The first term in the bound (5.1) comes from the smoothness of the population risk and the second comes from the variance of the gradient estimator of population risk (which will be described in detail below). Algorithm 1 Projected stochastic gradient descent with partial loss For i = 0 to N : 1. Take a fresh sample ((x 1 , . . . , x T ), (y 1 , . . . , y T )). Letỹ t be the simulated outputs 2 of system Θ on inputs x and initial states h 0 = 0. 2. Let T 1 = T /4. Run stochastic gradient descent 3 on loss function ((x, y), Θ) = 1 T −T 1 t>T 1 ỹ t − y t 2 . Concretely, let G A = ∂ ∂â , G C = ∂ ∂Ĉ , and , G D = ∂ ∂D , we update [â,Ĉ,D] → [â,Ĉ,D] − η[G A , G C , G D ] . 3. Project Θ = (â,Ĉ,D) to the set B α ⊗ R n ⊗ R. An important (but not surprising) feature here is the variance scale in 1/T and therefore for long sequence actually we got 1/N convergence instead of 1/ √ N (for relatively small N ). We can further balance the variance of the estimator with the number of samples by breaking each long sequence of length T into Θ(T /n) short sequences of length Θ(n), and then run backpropagation (1) on these T N/n shorter sequences. This leads us to the following bound which gives the right dependency in T and N as we expected: T N should be counted as the true number of samples for the sequence-to-sequence model. where O(·)-notation hides polynomial dependencies on 1/(1 − α), 1/τ 0 , 1/τ 1 , τ 2 , and R = a . Algorithm 2 Projected stochastic gradient descent for long sequences Input: N samples sequences of length T Output: Learned system Θ 1. Divide each sample of length T into T /(βn) samples of length βn where β is a large enough constant. Then run algorithm 1 with the new samples and obtain Θ. We remark the the gradient computation procedure takes time linear in T n since one can use chain-rule (also called back-propagation) to compute the gradient efficiently . For completeness, Algorithm 3 gives a detailed implementation. Finally and importantly, we remark that although we defined the population risk as the expected error with respected to sequence of length T , actually our error bound generalizes to any longer (or shorter) sequences of length T max{n, 1/(1 − α)}. By the explicit formula for f ( Θ) (Lemma 3.5) and the fact that CA k B decays exponentially for k n (Lemma 4.4), we can bound the population risk on sequences of different lengths. Concretely, let f T ( Θ) denote the population risk on sequence of length T , we have for all T max{n, 1/(1 − α)}, f T ( Θ) ≤ 1.1f ( Θ) + exp(−(1 − α) min{T, T }) ≤ O n 5 + σ 2 n 3 T N . We note that generalization to longer sequence does deserve attention. Indeed in practice, it's usually difficult to train non-linear recurrent networks that generalize to longer sequences than the training data. Our proof of Theorem 5.1 simply consists of three parts: a) showing the idealized risk is quasiconvex in the convex set B α (Lemma 5.3); b) designing an (almost) unbiased estimator of the gradient of the idealized risk (Lemma 5.4); c) variance bounds of the gradient estimator (Lemma 5.5). First of all, using the theory developed in Section 3 (Lemma 3.3 and Lemma 3.4), it is straightforward to verify that in the convex set B α ⊗ R n , the idealized risk is both weakly-quasi-convex and weakly-smooth. Lemma 5.3. Under the condition of Theorem 5.1, the idealized risk (3.2) is τ -weakly-quasi-convex in the convex set B α ⊗ R n and Γ-weakly smooth, where τ = Ω(τ 0 τ 1 /τ 2 ) and Γ = O(n 2 /τ 4 1 ). Proof of Lemma 5.3. It suffices to show that for allâ, a ∈ B α , it satisfiesâ ∈ N τ (a) for τ = Ω(τ 0 τ 1 /τ 2 ). Indeed, by the monotonicity of the family of sets B α (Lemma 4.3), we have that a, a ∈ B 1 , which by definition means for every z on unit circle, p a (z)/z n , pâ(z)/z n ∈ C. By definition of C, for any point w,ŵ ∈ C, the angle φ between w andŵ is at most π − Ω(τ 0 ) and ratio of the magnitude is at least τ 1 /τ 2 , which implies that (w/ŵ) = \|w\|/\|ŵ\| · cos(φ) ≥ Ω(τ 0 τ 1 /τ 2 ). Therefore (p a (z)/pâ(z)) ≥ Ω(τ 0 τ 1 /τ 2 ), and we conclude thatâ ∈ N τ (a). The smoothness bound was established in Lemma 3.4. Towards designing an unbiased estimator of the gradient, we note that there is a small caveat here that prevents us to just use the gradient of the empirical risk, as commonly done for other (static) problems. Recall that the population risk is defined as the expected risk with known initial state h 0 , while in the training we don't have access to the initial states and therefore using the naive approach we couldn't even estimate population risk from samples without knowing the initial states. We argue that being able to handle the missing initial states is indeed desired: in most of the interesting applications h 0 is unknown (or even to be learned). Moreover, the ability of handling unknown h 0 allows us to break a very long sequence into shorter sequences, which helps us to obtain Corollary 5.2. Here the difficulty is essentially that we have a supervised learning problem with missing data h 0 . We get around it by simply ignoring first T 1 = Ω(T ) outputs of the system and setting the corresponding errors to 0. Since the influence of h 0 to any outputs later than time k ≥ T 1 max{n, 1/(1 − α)} is inverse exponentially small, we could safely assume h 0 = 0 when the error earlier than time T 1 is not taken into account. This small trick also makes our algorithm suitable to the cases when these early outputs are actually not observed. This is indeed an interesting setting, since in many sequence-to-sequence model [SVL14], there is no output in the first half fraction of iterations (of course these models have non-linear operation that we cannot handle). The proof of the correctness of the estimator is almost trivial and deferred to Section C. (5.2) Finally, we control the variance of the gradient estimator. Lemma 5.5. The (almost) unbiased estimator (G A , G C ) of the gradient of g(Â,Ĉ) has variance bounded by Var [G A ] + Var [G C ] ≤ O n 3 Λ 2 /τ 6 1 + σ 2 n 2 Λ/τ 4 1 T . where Λ = O(max{n, 1/(1 − α) log 1/(1 − α)}). Note that Lemma 5.5 does not directly follow from the Γ-weakly-smoothness of the population risk, since it's not clear whether the loss function ((x, y), Θ) is also Γ-smooth for every sample. Moreover, even if it could work out, from smoothness the variance bound can be only as small as Γ 2 , while the true variance scales linearly in 1/T . Here the discrepancy comes from that smoothness implies an upper bound of the expected squared norm of the gradient, which is equal to the variance plus the expected squared mean. Though typically for many other problems variance is on the same order as the squared mean, here for our sequence-to-sequence model, actually the variance decreases in length of the data, and therefore the bound of variance from smoothness is very pessimistic. We bound directly the variance instead. It's tedious but very simple in spirit. We mainly need Lemma 4.4 to control various difference sums that shows up from calculating the expectation. The only tricky part is to obtain the 1/T dependency which corresponds to the cancellation of the contribution from the cross terms. In the proof we will basically write out the variance as a (complicated) function ofÂ,Ĉ which consists of sums of terms involving (ĈÂ k B − CA k B) and A k B. We control these sums using Lemma 4.4. The proof is deferred to Section C. Finally we are ready to prove Theorem 5.1. We essentially just combine Lemma 5.3, Lemma 5.4 and Lemma 5.5 with the generic convergence Proposition 2.3. This will give us low error in idealized risk and then we relate the idealized risk to the population risk. The power of improper learning We observe an interesting and important fact about the theory in Section 5: it solely requires a condition on the characteristic function p(z). This suggests that the geometry of the training objective function depends mostly on the denominator of the transfer function, even though the system is uniquely determined by the transfer function G(z) = s(z)/p(z). This might seem to be an undesirable discrepancy between the behavior of the system and our analysis of the optimization problem. However, we can actually exploit this discrepancy to design improper learning algorithms that succeed under much weaker assumptions. We rely on the following simple observation about the invariance of a system G(z) = s(z) p(z) . For an arbitrary polynomial u(z) of leading coefficient 1, we can write G(z) as G(z) = s(z)u(z) p(z)u(z) =s (z) p(z) , wheres = su andp = pu. Therefore the systems(z)/p(z) has identical behavior as G. Although this is a redundant representation of G(z), it should counted as an acceptable solution. After all, learning the minimum representation 4 of linear system is impossible in general. In fact, we will encounter an example in Section 6.1. While not changing the behavior of the system, the extension from p(z) top(z), does affect the geometry of the optimization problem. In particular, ifp(z) is now an α-acquiescent characteristic polynomial as defined in Definition 4.1, then we could find it simply using stochastic gradient descent as shown in Section 5. Observe that we don't require knowledge of u(z) but only its existence. Denoting by d the degree of u, the algorithm itself is simply stochastic gradient descent with n + d model parameters instead of n. Our discussion motivates the following definition. Definition 6.1. A polynomial p(z) of degree n is α-acquiescent by extension of degree d if there exists a polynomial u(z) of degree d and leading coefficient 1 such that p(z)u(z) is α-acquiescent. For a transfer function G(z), we define it's H 2 norm as G 2 H 2 = 1 2π 2π 0 \|G(e iθ )\| 2 dθ . We assume (with loss of generality) that the true transfer function G(z) has bounded H 2 norm, that is, G H 2 ≤ 1. This can be achieve by a rescaling 5 of the matrix C. Theorem 6.2. Suppose the true system has transfer function G(z) = s(z)/p(z) with a characteristic function p(z) that is α-acquiescent by extension of degree d, and G H 2 ≤ 1, then projected stochastic gradient descent with m = n + d states (that is, Algorithm 2 with m states) returns a system Θ with population risk f ( Θ) ≤ O m 5 + σ 2 m 3 T K . where the O(·) notation hides polynomial dependencies on τ 0 , τ 1 , τ 2 , 1/(1 − α). The theorem follows directly from Corollary 5.2 (with some additional care about the scaling. Proof of Theorem 6.2. Letp(z) = p(z)u(z) be the acquiescent extension of p(z). Since τ 2 ≥ \|u(z)p(z)\| = \|p(z)\| ≥ τ 0 on the unit circle, we have that \|s(z)\| = \|s(z)\|\|u(z)\| = s(z) · O τ (1/p(z)). Therefore we have thats(z) satisfies that s H 2 = O τ ( s(z)/p(z) H 2 ) = O τ ( G(z) H 2 ) ≤ O τ (1). That means that the vector C that determines the coefficients ofs satisfies that C ≤ O τ (1), since for a polynomial h(z) = b 0 + · · · + b n−1 z n−1 , we have h H 2 = b . Therefore we can apply Corollary 5.2 to complete the proof. In the rest of this section, we discuss in subsection 6.1 the instability of the minimum representation in subsection, and in subsection 6.2 we show several examples where the characteristic function p(z) is not α-acquiescent but is α-acquiescent by extension with small degree d. As a final remark, the examples illustrated in the following sub-sections may be far from optimally analyzed. It is beyond the scope of this paper to understand the optimal condition under which p(z) is acquiescent by extension. Instability of the minimum representation We begin by constructing a contrived example where the minimum representation of G(z) is not stable at all and as a consequence one can't hope to recover the minimum representation of G(z). Consider G(z) = s(z) p(z) := z n −0.8 −n (z−0.1)(z n −0.9 −n ) and G (z) = s (z) p (z) := 1 z−0.1 . Clearly these are the minimum representations of the G(z) and G (z), which also both satisfy acquiescence. On the one hand, the characteristic polynomial p(z) and p (z) are very different. On the other hand, the transfer functions G(z) and G (z) have almost the same values on unit circle up to exponentially small error, \|G(z) − G (z)\| ≤ 0.8 −n − 0.9 −n (z − 0.1)(z − 0.9 −n ) ≤ exp(−Ω(n)) . Moreover, the transfer functions G(z) andĜ(z) are on the order of Θ(1) on unit circle. These suggest that from an (inverse polynomially accurate) approximation of the transfer function G(z), we cannot hope to recover the minimum representation in any sense, even if the minimum representation satisfies acquiescence. Power of improper learning in various cases We illustrate the use of improper learning through various examples below. Example: artificial construction We consider a simple contrived example where improper learning can help us learn the transfer function dramatically. We will show an example of characteristic function which is not 1-acquiescent but (α + 1)/2-(α + 1)/2-acquiescent by extension of degree 3. Let n be a large enough integer and α be a constant. Let J = {1, n − 1, n} and ω = e 2πi/n , and then define p(z) = z 3 j∈[n],j / ∈J (z − αω j ). Therefore we have that p(z)/z n = z 3 j∈[n],j∈J (1 − αω j /z) = 1 − α n /z n (1 − ω/z)(1 − ω −1 /z)(1 − 1/z) (6.1) Taking z = e −iπ/2 we have that p(z)/z n has argument (phase) roughly −3π/4, and therefore it's not in C, which implies that p(z) is not 1-acquiescent. On the other hand, picking u(z) = (z−ω)(z−1)(z−ω −1 ) as the helper function, from equation (6.1) we have p(z)u(z)/z n+3 = 1−α n /z n takes values inverse exponentially close to 1 on the circle with radius (α + 1)/2. Therefore p(z)u(z) is (α + 1)/2-acquiescent. Example: characteristic function with separated roots A characteristic polynomial with well separated roots will be acquiescent by extension. Our bound will depend on the following quantity of p that characterizes the separateness of the roots. Definition 6.3. For a polynomial h(z) of degree n with roots λ 1 , . . . , λ n inside unit circle, define the quantity Γ(·) of the polynomial h as: Γ(h) := j∈[n] λ n j i =j (λ i − λ j ) . Lemma 6.4. Suppose p(z) is a polynomial of degree n with distinct roots inside circle with radius α. Let Γ = Γ(p), then p(z) is α-acquiescent by extension of degree d = O(max{(1 − α) −1 log( √ nΓ · p H 2 ), 0}). Our main idea to extend p(z) by multiplying some polynomial u that approximates p −1 (in a relatively weak sense) and therefore pu will always take values in the set C. We believe the following lemma should be known though for completeness we provide the proof in Section D. z n+d p(z) − h(z) ≤ ζ . Proof of Lemma 6.4. Let γ = 1 − α. Using Lemma 6.5 with ζ = 0.5 p −1 H∞ , we have that there exists polynomial u of degree d = O(max{ 1 1−α log(Γ p H∞ ), 0}) such that z n+d p(z) − u(z) ≤ ζ . Then we have that p(z)u(z)/z n+d − 1 ≤ ζ\|p(z)\| < 0.5 . Therefore p(z)u(z)/z n+d ∈ C τ 0 ,τ 1 ,τ 2 for constant τ 0 , τ 1 , τ 2 . Finally noting that for degree n polynomial we have h H∞ ≤ √ n · h H 2 , which completes the proof. Example: Characteristic polynomial with random roots We consider the following generative model for characteristic polynomial of degree 2n. We generate n complex numbers λ 1 , . . . , λ n uniformly randomly on circle with radius α < 1, and take λ i ,λ i for i = 1, . . . , n as the roots of p(z). That is, p(z) = (z − λ 1 )(z −λ 1 ) . . . (z − λ n )(z −λ n ). We show that with good probability (over the randomness of λ i 's), polynomial p(z) will satisfy the condition in subsection 6.2.2 so that it can be learned efficiently by our improper learning algorithm. Theorem 6.6. Suppose p(z) with random roots inside circle of radius α is generated from the process described above. Then with high probability over the choice of p, we have that Γ(p) ≤ exp( O( √ n)) and p H 2 ≤ exp(Õ( √ n)). As a corollary, p(z) is α-acquiescent by extension of degree O((1 − α) −1 n). Towards proving Theorem 6.6, we need the following lemma about the expected distance of two random points with radius ρ and r in log-space. Lemma 6.7. Let x ∈ C be a fixed point with \|x\| = ρ, and λ uniformly drawn on the circle with radius r. Then E [ln \|x − λ\|] = ln max{ρ, r} . Proof. When r = ρ, let N be an integer and ω = e 2iπ/N . Then we have that E [ln \|x − λ\| \| r] = lim N →∞ 1 N N k=1 ln \|x − rω k \| (6.2) The right hand of equation (6.2) can be computed easily by observing that 1 N N k=1 ln \|x − rω k \| = 1 N ln N k=1 (x − rω k ) = 1 N ln \|x N − r N \|. Therefore, when ρ > r, we have lim N →∞ 1 N N k=1 ln \|x − rω k \| = lim N →∞ ρ + 1 N ln \|(x/ρ) N − (r/ρ) N \| = ln ρ. On the other hand, when ρ < r, we have that lim N →∞ 1 N N k=1 ln \|x − rω k \| = ln r. Therefore we have that E [ln \|x − λ\| \| r] = ln(max ρ, r). For ρ = r, similarly proof (with more careful concern of regularity condition) we can show that E [ln \|x − λ\| \| r] = ln r. Now we are ready to prove Theorem 6.6. Proof of Theorem 6.6. Fixing index i, and the choice of λ i , we consider the random variable Y i = ln( \|λ i \| 2n j =i \|λ i −λ j \| j =i \|λ i −λ j \| )n ln \|λ i \|− j =i ln \|λ i −λ j \|. By Lemma 6.7, we have that E [Y i ] = n ln \|λ i \|− j =i E [ln \|λ i − λ j \|] = ln(1 − δ). Let Z j = ln \|λ i − λ j \|. Then we have that Z j are random variable with mean 0 and ψ 1 -Orlicz norm bounded by 1 since E [e ln \|λ i −λ j \| − 1] ≤ 1. Therefore by Bernstein inequality for sub-exponential tail random variable (for example, [LT13, Theorem 6.21]), we have that with high probability (1 − n −10 ), it holds that j =i Z j ≤ O( √ n) where O hides logarithmic factors. Therefore, with high probability, we have \|Y i \| ≤ O( √ n). Finally we take union bound over all i ∈ [n], and obtain that with high probability, for ∀i ∈ [n], \|Y i \| ≤ O( √ n), which implies that n i=1 exp(Y i ) ≤ exp( O( √ n)) . With similar technique, we can prove that p H 2 ≤ exp(Õ( √ n). Example: Passive systems We will show that with improper learning we can learn almost all passive systems, an important class of stable linear dynamical system as we discussed earlier. We start off with the definition of a strict-input passive system. In order to learn the passive system, we need to add assumptions in the definition of strict passivity. To make it precise, we define the following subsets of complex plane: For positive constant τ 0 , τ 1 , τ 2 , define C + τ 0 ,τ 1 ,τ 2 = {z ∈ C : \|z\| ≤ τ 2 , (z) ≥ τ 1 , (z) ≥ τ 0 \| (z)\| } . (6.3) We say a transfer function G(z) = s(z)/p(z) is (τ 0 , τ 1 , τ 2 )-strict input passive if for any z on unit circle we have G(z) ∈ C + τ 0 ,τ 1 ,τ 2 . Note that for small constant τ 0 , τ 1 and large constant τ 2 , this basically means the system is strict-input passive. Now we are ready to state our main theorem in this subsection. We will prove that passive systems could be learned improperly with a constant factor more states (dimensions), assuming s(z) has all its roots strictly inside unit circles and Γ(s) ≤ exp(O(n)). Theorem 6.9. Suppose G(z) = s(z)/p(z) is (τ 0 , τ 1 , τ 2 )-strict-input passive. Moreover, suppose the roots of s(z) have magnitudes inside circle with radius α and Γ = Γ(s) ≤ exp(O(n)) and p H 2 ≤ exp(O(n)). Then p(z) is α-acquiescent by extension of degree d = O τ,α (n), and as a consequence we can learn G(z) with n + d states in polynomial time. Moreover, suppose in addition we assume that G(z) ∈ C τ 0 ,τ 1 ,τ 2 for every z on unit circle. Then p(z) is α-acquiescent by extension of degree d = O τ,α (n). The proof of Theorem 6.9 is similar in spirit to that of Lemma 6.4, and is deferred to Section D. Improper learning using linear regression In this subsection, we show that under stronger assumption than α-acquiescent by extension, we can improperly learn a linear dynamical system with linear regression, up to some fixed bias. The basic idea is to fit a linear function that maps [x k− , . . . , x k ] to y k . This is equivalent to a dynamical system with hidden states and with the companion matrix A in (1.4) being chosen as a = 1 and a −1 = · · · = a 1 = 0. In this case, the hidden states exactly memorize all the previous inputs, and the output is a linear combination of the hidden states. Equivalently, in the frequency space, this corresponds to fitting the transfer function G(z) = s(z)/p(z) with a rational function of the form c 1 z −1 +···+c 1 z −1 = c 1 z −( −1) + · · · + c n . The following is a sufficient condition on the characteristic polynomial p(x) that guarantees the existence of such fitting, Definition 6.10. A polynomial p(z) of degree n is extremely-acquiescent by extension of degree d with bias ε if there exists a polynomial u(z) of degree d and leading coefficient 1 such that for all z on unit circle, p(z)u(z)/z n+d − 1 ≤ ε (6.4) We remark that if p(z) is 1-acquiescent by extension of degree d, then there exists u(z) such that p(z)u(z)/z n+d ∈ C. Therefore, equation (6.4) above is a much stronger requirement than acquiescence by extension. 6 When p(z) is extremely-acquiescent, we see that the transfer function G(z) = s(z)/p(z) can be approximated by s(z)u(z)/z n+d up to bias ε. Let = n + d + 1 and s(z)u(z) = c 1 z −1 + · · · + c . Then we have that G(z) can be approximated by the following dynamical system of hidden states with ε bias: we choose A = CC(a) with a = 1 and a −1 = · · · = a 1 = 0, and C = [c 1 , . . . , c ]. As we have argued previously, such a dynamical system simply memorizes all the previous inputs, and therefore it is equivalent to linear regression from the feature [x k− , . . . , x k ] to output y k . Proposition 6.11 (Informal). If the true system G(z) = s(z)/p(z) satisfies that p(z) is extremelyacquiescent by extension of degree d. Then using linear regression we can learn mapping from [x k− , . . . , x k ] to y k with bias ε and polynomial sampling complexity. We remark that with linear regression the bias ε will only go to zero as we increase the length of the feature, but not as we increase the number of samples. Moreover, linear regression requires a stronger assumption than the improper learning results in previous subsections do. The latter can be viewed as an interpolation between the proper case and the regime where linear regression works. Learning multi-input multi-output (MIMO) systems We consider multi-input multi-output systems with the transfer functions that have a common denominator p(z), G(z) = 1 p(z) · S(z) (7.1) where S(z) is an in × out matrix with each entry being a polynomial with real coefficients of degree at most n and p(z) = z n + a 1 z n−1 + · · · + a n . Note that here we use in to denote the dimension of the inputs of the system and out the dimension of the outputs. Although a special case of a general MIMO system, this class of systems still contains many interesting cases, such as the transfer functions studied in [FHB01,FHB04], where G(z) is assumed to take the form G(z) for λ 1 , . . . , λ n ∈ C with conjugate symmetry and R i ∈ C out× in satisfies that R i =R j whenever λ i =λ j . = R 0 + n i=1 R i z−λ i , In order to learn the system G(z), we parametrize p(z) by its coefficients a 1 , . . . , a n and S(z) by the coefficients of its entries. Note that each entry of S(z) depends on n + 1 real coefficients and therefore the collection of coefficients forms a third order tensor of dimension out × in × (n + 1). It will be convenient to collect the leading coefficients of the entries of S(z) into a matrix of dimension out × in , named D, and the rest of the coefficients into a matrix of dimension out × in n, denoted by C. This will be particularly intuitive when a state-space representation is used to learn the system with samples as discussed later. We parameterize the training transfer functionĜ(z) byâ, C andD using the same way. Let's define the risk function in the frequency domain as, g(Â,Ĉ,D) = 2π 0 G(e iθ ) −Ĝ(e iθ ) 2 F dθ . (7.2) The following lemma is an analog of Lemma 3.3 for the MIMO case. Itss proof actually follows from a straightforward extension of the proof of Lemma 3.3 by observing that matrix S(z) (orŜ(z)) commute with scalar p(z) andp(z), and thatŜ(z),p(z) are linear inâ,Ĉ. Lemma 7.1. The risk function g(â,Ĉ) defined in (7.2) is τ -weakly-quasi-convex in the domain N τ (a) = â ∈ R n : p a (z) pâ(z) ≥ τ /2, ∀ z ∈ C, s.t. \|z\| = 1 ⊗ R in × out×n Finally, as alluded before, we use a particular state space representation for learning the system in time domain with example sequences. It is known that any transfer function of the form (7.1) can be realized uniquely by the state space system of the following special case of Brunovsky normal form [Bru70], A =        0 I in 0 · · · 0 0 0 I in · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · I in −a n I in −a n−1 I in −a n−2 I in · · · −a 1 I in        , B =      0 . . . 0 I in      , (7.3) and, C ∈ R out×n in , D ∈ R out× in . The following Theorem is a straightforward extension of Corollary 5.2 and Theorem 6.2 to the MIMO case. Theorem 7.2. Suppose transfer function G(z) of a MIMO system takes form (7.1), and has norm G H 2 ≤ 1. If the common denominator p(z) is α-acquiescent by extension of degree d then projected stochastic gradient descent over the state space representation (7.3) will return Θ with risk f ( Θ) ≤ poly(n + d, σ, τ, (1 − α) −1 ) T N . We note that since A and B are simply the tensor product of I in with CC(a) and e n , the no blow-up property (Lemma 4.4) for A k B still remains true. Therefore to prove Theorem 7.2, we essentially only need to run the proof of Lemma 5.5 with matrix notation and matrix norm. We defer the proof to the full version. [WC99] Erik Weyer and M. C. Campi. Finite sample properties of system identification methods. In Proceedings of the 38th Conference on Decision and Control, 1999. A Background on optimization The proof below uses the standard analysis of gradient descent for non-smooth objectives and demonstrates that the argument still works for weakly-quasi-convex functions. Proof of Proposition 2.3. We start by using the weakly-quasi-convex condition and then the rest follows a variant of the standard analysis of non-smooth projected sub-gradient descent 7 . We conditioned on θ k , and have that τ (f (θ k ) − f (θ * )) ≤ ∇f (θ k ) (θ k − θ * ) = E [r(θ k ) (θ k − θ * ) \| θ k ] = E 1 η (θ k − w k+1 )(θ k − θ * ) \| θ k = 1 η E θ k − w k+1 2 \| θ k + θ k − θ * 2 − E w k+1 − θ * 2 \| θ k = η E r(θ k ) 2 + 1 η θ k − θ * 2 − E w k+1 − θ * 2 \| θ k (A.1) where the first inequality uses weakly-quasi-convex and the rest of lines are simply algebraic manipulations. Since θ k+1 is the projection of w k+1 to B and θ * belongs to B, we have w k+1 − θ * ≥ θ k+1 − θ * . Together with (A.1), and E r(θ k ) 2 = ∇f (θ k ) 2 + Var[r(θ k )] ≤ Γ(f (θ k ) − f (θ * )) + V, we obtain that τ (f (θ k ) − f (θ * )) ≤ ηΓ(f (θ k ) − f (θ * )) + ηV + 1 η θ k − θ * 2 − E θ k+1 − θ * 2 \| θ k . Taking expectation over all the randomness and summing over k we obtain that K−1 k=0 E [f (θ k ) − f (θ * )] ≤ 1 τ − ηΓ ηKV + 1 η θ 0 − θ * 2 ≤ 1 τ − ηΓ ηKV + 1 η R 2 . where we use the assumption that θ 0 − θ * ≤ R. Suppose K ≥ 4R 2 Γ 2 V τ 2 , then we take η = R √ V K . Therefore we have that τ − ηΓ ≥ τ /2 and therefore K−1 k=0 E [f (θ k ) − f (θ * )] ≤ 4R √ V √ K τ . (A.2) On the other hand, if K ≤ 4R 2 Γ 2 V τ 2 , we pick η = τ 2Γ and obtain that K−1 k=0 E [f (θ k ) − f (θ * )] ≤ 2 τ τ KV 2Γ + 2ΓR 2 τ ≤ 8ΓR 2 τ 2 . (A.3) Therefore using equation (A.3) and (A.2) we obtain that when choosing η properly according to K as above, E k∈[K] [f (θ k ) − f (θ * )] ≤ max 8ΓR 2 τ 2 K , 4R √ V τ √ K . 7 Although we used weak smoothness to get a slightly better bound B Toolbox Lemma B.1. Let B = e n ∈ R n×1 and λ ∈ [0, 2π], w ∈ C. Suppose A with ρ(A) · \|w\| < 1 has the controllable canonical form A = CC(a). Then (I − wA) −1 B = 1 p a (w −1 )      w −1 w −2 . . . w −n      where p a (x) = x n + a 1 x n−1 + · · · + a n is the characteristic polynomial of A. Proof. let v = (I − wA) −1 B then we have (I − wA)v = B. Note that B = e n , and I − wA is of the form I − wA =        1 −w 0 · · · 0 0 1 −w · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · −w a n w a n−1 w a n−2 w · · · 1 + a 1 w        (B.1) Therefore we obtain that v k = wv k+1 for 1 ≤ k ≤ n − 1. That is, v k = v 0 w −k for v 0 = v 1 w 1 . Using the fact that ((I − wA)v) n = 1, we obtain that v 0 = p a (w −1 ) −1 where p a (·) is the polynomial p a (x) = x n + a 1 x n−1 + · · · + a n . Then we have that u(I − wA) −1 B = u 1 w −1 +···+unw −n pa(w −1 ) Claim B.2. Suppose x 1 , . . . , x n are independent variables with mean 0 and covariance matrices and I, U 1 , . . . , U d are fixed matrices, then E n k=1 U k x k 2 = n k=1 U k 2 F . Proof. We have that Proof. Let q a (z) = 1 + a 1 z + · · · + a n z n . Note that q(z −1 ) = p a (z)/z n . Therefore we note that B α = {a : q a (z) ∈ C, ∀\|z\| = 1/α}. Suppose a ∈ B α , then (q a (z)) ≥ τ 1 for any z with \|z\| = 1/α. Since (q a (z)) is the real part of the holomorphic function q a (z), its a harmonic function. By maximum (minimum) principle of the harmonic functions, we have that for any \|z\| ≤ 1/α, (q a (z)) ≥ inf \|z\|=1/α (q a (z)) ≥ τ 1 . In particular, it holds that for \|z\| = 1/β < 1/α, (q a (z)) ≥ τ 1 . Similarly we can prove that for z with \|z\| = 1/β, (q a (z)) ≥ (1 + τ 0 ) (q a (z)), and other conditions for a being in B β . E n k=1 U k x k 2 F = n k, tr(U k x k x U ) = n k tr(U k x k x k U k ) = n k=1 U k 2 F C C.2 Proof of Lemma 5.4 Lemma 5.4 follows directly from the following general Lemma which also handles the multi-input multi-output case. It can be seen simply from calculation similar to the proof of Lemma 3.5. We mainly need to control the tail of the series using the no-blow up property (Lemma 4.4) and argue that the wrong value of the initial states h 0 won't cause any trouble to the partial loss function ((x, y), Θ) (defined in Algorithm 1). This is simply because after time T 1 = T /4, the influence of the initial state is already washed out. Lemma C.2. In algorithm 3 the values of G A , G C , G D are equal to the gradients of g(Â,Ĉ) + (D − D) 2 with respect toÂ,Ĉ andD up to inverse exponentially small error. Proof of Lemma C.2. We first show that the partial empirical loss function ((x, y), Θ) has expectation almost equal to the idealized risk (up to the term forD and exponential small error), E [ ((x, y), Θ)] = g(Â,Ĉ) + (D − D) 2 ± exp(−Ω((1 − α)T )). This can be seen simply from similar calculation to the proof of Lemma 3.5. Note that y t = Dx t + t−1 k=1 CA t−k−1 Bx k + CA t−1 h 0 + ξ t andỹ t =Dx t + t−1 k=1ĈÂ t−k−1B x k . (C.1) Therefore noting that when t ≥ T 1 ≥ Ω(T ), we have that CA t−1 h 0 ≤ exp(−Ω((1 − α)T ) and therefore the effect of h 0 is negligible. Then we have that E [ ((x, y), Θ)] = 1 T − T 1 E   T t>T 1 y t − y t 2   ± exp(−Ω((1 − α)T )) = D − D 2 + 1 T − T 1 T ≥t>T 1 0≤j≤t−1 ĈÂ j B − CA j B 2 ± exp(−Ω((1 − α)T )) = D − D 2 + T 1 j=0 ĈÂ j B − CA j B 2 + T ≥j≥T 1 T − j T − T 1 ĈÂ j B − CA j B 2 ± exp(−Ω((1 − α)T )) = D − D 2 + ∞ j=0 ĈÂ j B − CA j B 2 ± exp(−Ω((1 − α)T )) . where the first line use the fact that CA t−1 h 0 ≤ exp(−Ω((1−α)T ), the second uses equation (3.9) and the last line uses the no-blowing up property of A k B (Lemma 4.4). Similarly, we can prove that the gradient of E [ ((x, y), Θ)] is also close to the gradient of g(Â,Ĉ) + (D − D) 2 up to inverse exponential error. C.3 Proof of Lemma 5.5 Proof of Lemma 5.5. Both G A and G C can be written in the form of a quadratic form (with vector coefficients) of x 1 , . . . , x T and ξ 1 , . . . , ξ T . That is, we will write G A = s,t x s x t u st + s,t x s ξ t u st and G C = s,t x s x t v st + s,t x s ξ t v st . where u st and v st are vectors that will be calculated later. By Claim C.3, we have that Var s,t x s x s u st + s,t x s ξ t u st ≤ O(1) s,t u st 2 + O(σ 2 ) s,t u st 2 . (C.2) Therefore in order to bound from above Var [G A ], it suffices to bound u st 2 and u st 2 , and similarly for G C . We begin by writing out u st for fixed s, t ∈ [T ] and bounding its norm. We use the same set of notations as int the proof of Lemma 5.4. Recall that we set r k = CA k B andr k =ĈÂ k B, and ∆r k =r k − r k . Moreover, let z k =Â k B. We note that the sums of z k 2 and r 2 k can be controlled. By the assumption of the Lemma, we have that ∞ k=t z k 2 ≤ 2πnτ −2 1 , z k 2 ≤ 2πnα 2k−2n τ −2 1 . (C.3) ∞ k=t ∆r 2 k ≤ 4πnτ −2 1 , ∆r k 2 ≤ 4πnα 2k−2n τ −2 1 . (C.4) which will be used many times in the proof that follows. We calculate the explicit form of G A using the explicit back-propagation Algorithm 3. We have that in Algorithm 3,ĥ k = k j=1Â k−j Bx j = k j=1 z k−j x j (C.5) and ∆h k = T j=k (Â ) j−kĈ ∆y j = T j=k α j−k (Â ) j−kĈ 1 j>T 1 ξ j + j =1 ∆r j− x (C.6) Then using G A = k≥2 B ∆h k h k−1 and equation (C.5) and equation (C.6) above, we have that u st = T k=2 j≥max{k,s,T 1 +1} ∆r j−sĈÂ j−k B 1 k≥t+1 ·Â k−t−1 B = T k=2 j≥max{k,s,T 1 +1} ∆r j−srj−k 1 k≥t+1 · z k−t−1 . (C.7) and that, u st = T k=2 z k−1−s · 1 k≥s+1 ·r t−k · 1 t>max{T 1 ,k} = s+1≤k≤t z k−1−s ·r t−k · 1 t>max{T 1 } (C.8) Towards bounding u st , we consider four different cases. Let Λ = Ω {max{n, (1 − α) −1 log( 1 1−α )} be a threshold. where at the second line, we did the change of variables = j − s. Then by Cauchy-Schartz inequality, we have, u st 2 ≤ 2   ≥0, ≥T 1 +1−s ∆r 2     ≥0, ≥T 1 +1−s s≤k≤l+s,k≤Tr +s−k z k−t−1 2   T 1 + 2   ≥0, ≥T 1 +1−s ∆r 2     ≥0, ≥T 1 +1−s s>k>tr +s−k z k−t−1 2   T 2 . (C.9) We could bound the contribution from ∆r 2 k ssing equation (C.4), and it remains to bound terms T 1 and T 2 . Using the tail bounds for z k (equation (C.3)) and the fact that \|r k \| = \|ĈÂ k B\| ≤ Â k B = z k , we have that T 1 = ≥0, ≥T 1 +1−s s≤k≤l+s,k≤Tr +s−k z k−t−1 2 ≤ ≥0   s≤k≤ +s \|r +s−k \| z k−t−1   2 . (C.10) We bound the inner sum of RHS of (C.10) using the fact that z k 2 ≤ O(nα 2k−2n /τ 2 1 ) and obtain that, s≤k≤ +s \|r +s−k \| z k−t−1 ≤ s≤k≤ +s O(nα ( +s−t−1)−2n /τ 2 1 ) ≤ O( nα ( +s−t−1)−2n /τ 2 1 ) . (C.11) Note that equation (C.11) is particular effective when > Λ. When ≤ Λ, we can refine the bound using equation (C.3) and obtain that s≤k≤ +s \|r +s−k \| z k−t−1 ≤   s≤k≤ +s \|r +s−k \| 2   1/2   s≤k≤ +s z k−t−1 2   1/2 ≤ O( √ n/τ 1 ) · O( √ n/τ 1 ) = O(n/τ 2 1 ) . (C.12) Plugging equation (C.12) and (C.11) into equation (C.10), we have that ≥0   s≤k≤ +s \|r +s−k \| z k−t−1   2 ≤ Λ≥ ≥0 O(n 2 /τ 4 1 ) + >Λ O( 2 n 2 α 2( +s−t−1)−4n /τ 4 1 ) ≤ O(n 2 Λ/τ 4 1 ) + O(n 2 /τ 4 1 ) = O(n 2 Λ/τ 4 1 ) . (C.13) For the second term in equation (C.9), we bound similarly, T 2 ≤ ≥0, ≥T 1 +1−s s>k>tr +s−k z k−t−1 2 ≤ O(n 2 Λ/τ 4 1 ) . (C.14) Therefore using the bounds for T 1 and T 2 we obtain that, u st 2 ≤ O(n 3 Λ/τ 6 1 ) (C.15) Case 2: When s − t > Λ, we tighten equation (C.13) by observing that, T 1 ≤ ≥0   s≤k≤ +s \|r +s−k \| z k−t−1   2 ≤ α 2(s−t−1)−4n ≥0 O( 2 n 2 α 2 /τ 4 1 ) ≤ α s−t−1 · O(n 2 /(τ 4 1 (1 − α) 3 )) . (C.16) where we used equation (C.11). Similarly we can prove that T 2 ≤ α s−t−1 · O(n 2 /(τ 4 1 (1 − α) 3 )) . Therefore, we have when s − t ≥ Λ, u st 2 ≤ O(n 3 /((1 − α) 3 τ 6 1 )) · α s−t−1 . (C.17) Case 3: When −Λ ≤ s − t ≤ 0, we can rewrite u s t and use the Cauchy-Schwartz inequality and obtain that u st = T ≥k≥t+1 z k−t−1 j≥max{k,T 1 +1} ∆r j−srj−k = ≥0, ≥T 1 +1−s ∆r t+1≤k≤l+s,k≤Tr +s−k z k−t−1 . and, u st 2 ≤   ≥0, ≥T 1 +1−s ∆r 2     ≥0, ≥T 1 +1−s t+1≤k≤l+s,k≤Tr +s−k z k−t−1 2   . Using almost the same arguments as in equation (C.11) and (C.12), we that t+1≤k≤ +s \|r +s−k \| · z k−t−1 ≤ O( nα ( +s−t−1)−2n /τ 2 1 ) and t+1≤k≤ +s \|r +s−k \| · z k−t−1 ≤ O( √ n/τ 1 ) · O( √ n/τ 1 ) = O(n/τ 2 1 ) . Then using a same type of argument as equation (C.13), we can have that It follows that in this case u st can be bounded with the same bound in (C.15). Case 4: When s − t ≤ −Λ, we use a different simplification of u st from above. First of all, it follows (C.7) that u st ≤ T k=2   j≥max{k,s,T 1 +1} ∆r j−sr j−k z k−t−1 1 k≥t+1   (C.18) ≤ k≥t+1 z k−t−1 j≥max{k,T 1 +1} \|∆r j−sr j−k \| . Since j − s ≥ k − s > 4n and it follows that j≥max{k,T 1 +1} \|∆r j−sr j−k \| ≤ j≥max{k,T 1 +1} O( √ n/τ 1 · α j−s−n ) · O( √ n/τ 1 · α j−k−n ) ≤ O(n/(τ 2 1 (1 − α)) · α k−s−n ) Then we have that u st 2 ≤ k≥t+1 z k−t−1 j≥max{k,T 1 +1} \|∆r j−sr j−k \| ≤   k≥t+1 z k−t−1 2     k≥t+1   j≥max{k,T 1 +1} \|∆r j−sr j−k \|   2   ≤ O(n/τ 2 1 ) · O(n 2 /(τ 4 1 (1 − α) 3 )α t−s ) = O(n 3 /(τ 6 1 δ 3 )α t−s ) Therefore, using the bound for u st 2 obtained in the four cases above, taking sum over s, t, we obtain that O(n 3 /(τ 6 1 (1 − α) 3 )α \|t−s\|−1 ) ≤ O(T n 3 Λ 2 /τ 6 1 ) + O(n 3 /τ 6 1 ) = O(T n 3 Λ 2 /τ 6 1 ) . (C.19) We finished the bounds for u st and now we turn to bound u st 2 . Using the formula for u st (equation C.8), we have that for t ≤ s + 1, u st = 0. For s + Λ ≥ t ≥ s + 2, we have that by Cauchy-Schwartz inequality, u st ≤   s+1≤k≤t z k−1−s 2   1/2   s+1≤k≤t \|r t−k \| 2   1/2 ≤ O(n/τ 2 1 ) ≤ O(n/τ 2 1 ) . On the other hand, for t > s + Λ, by the bound that \|r k \| 2 ≤ z k 2 ≤ O(nα 2k−2n /τ 2 1 ), we have, Hence, it follows that u st ≤ T s+1≤k≤t−1 z k−1−s · \|r t−k \| ≤ T s+1≤k≤t−1 nα t−s−1 /τ 2 1 ≤ O(n(t − s)α t−s−1 /τ 2 1 ) .Var[G A ] ≤ 1 (T − T 1 ) 2 Var[G A ] ≤ O n 3 Λ 2 /τ 6 1 + σ 2 n 2 Λ/τ 4 1 T . We can prove the bound for G C similarly. Claim C.3. Let x 1 , . . . , x T be independent random variables with mean 0 and variance 1 and 4-th moment bounded by O(1), and u ij be vectors for i, j ∈ [T ]. Moreover, let ξ 1 , . . . , ξ T be independent random variables with mean 0 and variance σ 2 and u ij be vectors for i, j ∈ [T ]. Then, Var i,j x i x j u ij + i,j x i ξ j u ij ≤ O(1) i,j u ij 2 + O(σ 2 ) i,j u ij 2 . Proof. Note that the two sums in the target are independent with mean 0, therefore we only need to bound the variance of both sums individually. The proof follows the linearity of expectation and the independence of x i 's: E i,j x i x j u ij 2 = i,j k, E x i x j x k x u ij u k = i E [u ii u ii x 4 i ] + i =j E [u ii u jj x 2 i x 2 j ] + i,j E x 2 i x 2 j (u ij u ij + u ij u ji ) ≤ i,j u ii u jj + O(1) i,j u ij + u ji 2 = i u ii 2 + O(1) i,j u ij 2 where at second line we used the fact that for any monomial x α with an odd degree on one of the x i 's, E [x α ] = 0. Note that E [ i,j x i x j u ij ] = i u ii . Therefore, Var i,j x i x j u ij = E i,j x i x j u ij 2 − E [ i,j x i x j u ij ] 2 ≤ O(1) i,j u ij 2 (C.21) Similarly, we can control Var i,j x i ξ j u ij by O(σ 2 ) i,j u ij 2 . D Missing proofs in Section 6 D.1 Proof of Lemma 6.5 Towards proving Lemma 6.5, we use the following lemma to express the inverse of a polynomial as a sum of inverses of degree-1 polynomials. Lemma D.1. Let p(z) = (z − λ 1 ) . . . (z − λ n ) where λ j 's are distinct. Then we have that 1 p(z) = n j=1 t j z − λ j , where t j = i =j (λ j − λ i ) −1 . (D.1) Proof of Lemma D.1. By interpolating constant function at points λ 1 , . . . , λ n using Lagrange interpolating formula, we have that 1 = n j=1 i =j (x − λ i ) i =j (λ j − λ i ) · 1 (D.2) Dividing p(z) on both sides we obtain equation (D.1). The following lemma computes the Fourier transform of function 1/(z − λ). τ 1 < \|p(z)\| < τ 2 , we have that \| p(z) τ 2 z n − 1\| < 1 − τ 1 /τ 2 . Therefore truncating the Taylor series at k = O τ (1) we obtain a polynomial a rational function h(z) of the form h(z) = k j≥0 ( p(z) τ 2 z n − 1) j , which approximates 1 √ p(z)/z n with precision ζ min{τ 0 , τ 1 }/τ 2 , that is, 1 √ p(z)/z n − h(z) ≤ ζ . Therefore, we obtain that p(z)h(z) z n − p(z)/z n ≤ ζ\|p(z)/z n \| ≤ ζτ 2 . Note that since p(z)/z n ∈ C + τ 0 ,τ 1 ,τ 2 , we have that p(z)/z n ∈ C τ 0 ,τ 1 ,τ 2 for some constants τ 0 , τ 1 , τ 2 . Therefore p(z)h(z) z n ∈ C τ 0 ,τ 1 ,τ 2 . Note that h(z) is not a polynomial yet. Let u(z) = z nk h(z) and then u(z) is polynomial of degree at most nk and p(z)u(z)/z (n+1)k ∈ C τ 0 ,τ 1 ,τ 2 for any z on unit circle. E Back-propagation implementation In this section we give a detailed implementation of using back-propagation to compute the gradient of the loss function. The algorithm is for general MIMO case with the parameterization (7.3). To obtain the SISO sub-case, simply take in = out = 1. Algorithm 3 Back-propagation Parameters:â ∈ R n ,Ĉ ∈ R in ×n out , andD ∈ R in × out . LetÂ = MCC(â) = CC(â) ⊗ I in and B = e n ⊗ I in . Input: samples ((x (1) , y 1 ), . . . , x (N ) , y (N ) ) and projection set B α . for each sample (x (j) , y j ) = ((x 1 , . . . , x T ), (y 1 , . . . , y T )) do Feed-forward pass: h 0 = 0 ∈ R n in . for k = 1 to T h k ←Âh k−1 +Bx k ,ŷ t ←Ĉh k +Dx k andĥ k ←Âh k−1 +Bx k . end for Back-propagation: ∆h T +1 ← 0, G A ← 0, G C ← 0. G D ← 0 T 1 ← T /4 for k = T to 1 if k > T 1 , ∆y k ←ŷ k − y k , o.w. ∆y k ← 0. Let ∆h k ←Ĉ ∆y k +Â ∆h k+1 . update G C ← G C + 1 T −T 1 ∆y kĥk , G A ← G A − 1 T −T 1 B ∆h kĥ k−1 , and G D ← G D + 1 T −T 1 ∆y k x k . end for Gradient update:Â ←Â − η · G A ,Ĉ ←Ĉ − η · G C ,D ←D − η · G D . Projection step: Obtainâ fromÂ and setâ ← Π B (â), andÂ = MCC(â) end for F Projection to the set B α In order to have a fast projection algorithm to the convex set B α , we consider a grid G M of size M over the circle with radius α. We will show that M = O τ (n) will be enough to approximate the set B α in the sense that projecting to the approximating set suffices for the convergence. Let B α,τ 0 ,τ 1 ,τ 2 = {a : p a (z)/z n ∈ C τ 0 ,τ 1 ,τ 2 , ∀z ∈ G M } and B α,τ 0 ,τ 1 ,τ 2 = {a : p a (z)/z n ∈ C τ 0 ,τ 1 ,τ 2 , ∀\|z\| = α}. Here C τ 0 ,τ 1 ,τ 2 is defined the same as before though we used the subscript to emphasize the dependency on τ i 's, C τ 0 ,τ 1 ,τ 2 = {z : z ≥ (1 + τ 0 )\| z\|} ∩ {z : τ 1 < z < τ 2 } . (F.1) We will first show that with M = O τ (n), we can make B α,τ 1 ,τ 2 ,τ 3 to be sandwiched within to two sets B α,τ 0 ,τ 1 ,τ 2 and B α,τ 0 ,τ 1 ,τ 2 . Lemma F.1. For any τ 0 > τ 0 , τ 1 > τ 1 , τ 2 < τ 2 , we have that for M = O τ (n), there exists κ 0 , κ 1 , κ 2 that polynomially depend on τ i , τ i 's such that B α,τ 0 ,τ 1 ,τ 2 ⊂ B α,κ 0 ,κ 1 ,κ 2 ⊂ B α,τ 0 ,τ 1 ,τ 2 Before proving the lemma, we demonstrate how to use the lemma in our algorithm: We will pick τ 0 = τ 0 /2, τ 1 = τ 1 /2 and τ 2 = 2τ 2 , and find κ i 's guaranteed in the lemma above. Then we use B α,κ 0 ,κ 1 ,κ 2 as the projection set in the algorithm (instead of B α,τ 0 ,τ 1 ,τ 2 )). First of all, the ground-truth solution Θ is in the set B α,κ 0 ,κ 1 ,κ 2 . Moreover, since B α,κ 0 ,κ 1 ,κ 2 ⊂ B α,τ 0 ,τ 1 ,τ 2 , we will guarantee that the iterates Θ will remain in the set B α,τ 0 ,τ 1 ,τ 2 and therefore the quasi-convexity of the objective function still holds 8 . Note that the set B α,κ 0 ,κ 1 ,κ 2 contains O(n) linear constraints and therefore we can use linear programming to solve the projection problem. Moreover, since the points on the grid forms a Fourier basis and therefore Fast Fourier transform can be potentially used to speed up the projection. Finally, we will prove Lemma F.1. We need S. Bernstein's inequality for polynomials. Theorem F.2 (Bernstein's inequality, see, for example, [Sch41]). Let p(z) be any polynomial of degree n with complex coefficients. Then, sup \|z\|≤1 \|p (z)\| ≤ n sup \|z\|≤1 \|p(z)\|. We will use the following corollary of Bernstein's inequality. \|p(e 2ikπ/m )\|. Proof. For simplicity let τ = sup k∈[m] \|p(e 2ikπ/m )\|, and let τ = sup k∈[m] \|p(e 2ikπ/m )\|. If τ ≤ 2τ then we are done by Bernstein's inequality. Now let's assume that τ > 2τ . Suppose p(z) = τ . Then there exists k such that \|z − e 2πik/m \| ≤ 4/m and \|p(e 2πik/m )\| ≤ τ . Therefore by Cauchy mean-value theorem we have that there exists ξ that lies between z and e 2πik/m such that p (ξ) ≥ m(τ − τ )/4 ≥ 1.1nτ , which contradicts Bernstein's inequality. Lemma F.4. Suppose a polynomial of degree n satisfies that \|p(w)\| ≤ τ for every w = αe 2iπk/m for some m ≥ 20n. Then for every z with \|z\| = α there exists w = αe 2iπk/m such that \|p(z) − p(w)\| ≤ O(nατ /m). Proof. Let g(z) = p(αz) by a polynomial of degree at most n. Therefore we have g (z) = αp(z). Corollary 5. 2 . 2Under the assumption of Theorem 5.1, Algorithm 2 returns parameters Θ with population riskf ( Θ) ≤ f (Θ) + O n 5 + σ 2 n 3 T N , Lemma 5. 4 . 4Under the assumption of Theorem 5.1, supposeâ, a ∈ B α . Then in Algorithm 1, at each iteration, G A , G C are unbiased estimators of the gradient of the idealized risk (3.2) in the sense that:E [G A , G C ] = ∂g ∂â, ∂g ∂Ĉ ± exp(−Ω((1 − α)T )) . Proof of Theorem 5.1. We consider g (Â,Ĉ,D) = (D − D) 2 + g(Â,Ĉ), an extended version of the idealized risk which takes the contribution ofD into account. By Lemma 5.4 we have that Algorithm 1 computes G A , G C which are almost unbiased estimators of the gradients of g up to negligible error exp(−Ω((1 − α)T )), and by Lemma C.2 we have G D is an unbiased estimator of g with respect toD. Moreover by Lemma 5.5, these unbiased estimator has total variance V = O(n 5 +σ 2 n 3 ) T where O(·) hides dependency on τ 1 and (1 − α). Applying Proposition 2.3 (which only requires an unbiased estimator of the gradient of g ), we obtain that after T iterations, we converge to a point with g (â,Ĉ,D) ≤ O n 2 N + n 5 +σ 2 n 3 T N . Then, by Lemma 3.5 we have f ( Θ) ≤ g (â,Ĉ,D) + σ 2 = g (â,Ĉ,D) + f (Θ) ≤ O n 2 N + n 5 +σ 2 n 3 T N + f (Θ) which completes the proof. Lemma 6 . 5 ( 65Approximation of inverse of a polynomial). Suppose p(z) is a polynomial of degree n and leading coefficient 1 with distinct roots inside circle with radius α, and Γ = Γ(p). Then for d = O(max{( 1 1−α log Γ (1−α)ζ , 0}), there exists a polynomial h(z) of degree d and leading coefficient 1 such that for all z on unit circle, Definition 6. 8 ( 8Passive System, c.f[KA10]). A SISO linear system is strict-input passive if and only if for some τ 0 > 0 and any z on unit circle, (G(z)) ≥ τ 0 . Lemma C. 1 ( 1Lemma 4.3 restated). For any 0 < α < β, we have that B α ⊂ B β . Case 1 : 1When 0 ≤ s − t ≤ Λ, we rewrite u st by rearranging equation (C.7), u st = T ≥k≥s z k−t−1 j≥max{k,T 1 +1} ∆r j−srj−k + t<k<s z k−t−1 j≥max{s,T 1 +1} ∆r j−srj−k = ≥0, ≥T 1 +1−s ∆r s≤k≤l+s,k≤Tr +s−k z k−t−1 + ≥0, ≥T 1 +1−s ∆r s>k>tr +s−k z k−t−1 Corollary F. 3 . 3Let p(z) be any polynomial of degree n with complex coefficients. Then, for m = 20n, sup \|z\|≤1 \|p (z)\| ≤ 2n sup k∈[m] Let w = αe 2iπk/m such that \|z − w\| ≤ O(α/m). Then we have \|p(z) − p(w)\| = \|g(z/α) − p(w/α)\| ≤ sup \|x\|≤1 \|g (x)\| · 1 α \|z − w\| (By Cauchy's mean-value Theorem) ≤ sup \|x\|≤1 \|p (x)\| · \|z − w\| ≤ nτ \|z − w\| . (Corallary F.3) ≤ O(αnτ /m) . 8 with a slightly worse parameter up to constant factor since τi's are different from τi's up to constant factors Therefore taking sum over s, t, similarly to equation (C.19),s,t∈[T ] u st 2 ≤ O(T n 2 Λ/τ 4 1 ) . (C.20) Then using equation (C.2) and equation (C.19) and (C.20), we obtain that Var[ G A 2 ] ≤ O T n 3 Λ 2 /τ 6 1 + σ 2 T n 2 Λ/τ 4 1 . Note thatỹt is different fromŷt defined in equation (1.2) which is used to define the population risk: hereŷt is obtained from the (wrong) initial state h0 = 0 whileŷt is obtained from the correct initial state.3 See Algorithm Box 3 for a detailed back-propagation algorithm that computes the gradient. The minimum representation of a transfer function G(z) is defined as the representation G(z) = s(z)/p(z) with p(z) having minimum degree.5 In fact, this is a natural scaling that makes comparing error easier. Recall that the population risk is essentially Ĝ − G H 2 , therefore rescaling C so that G H 2 = 1 implies that when error 1 we achieve non-trivial performance. We need (1−δ)-acquiescence by extension in previous subsections for small δ > 0, though this is merely additional technicality needed for the sample complexity. We ignore this difference between 1−δ-acquiescence and 1-acquiescence and for the purpose of this subsection AcknowledgmentsWe thank Amir Globerson, Alexandre Megretski, Pablo A. Parrilo, Yoram Singer and Ruixiang Zhang for helpful discussions. We are indebted to Mark Tobenkin for pointers to relevant prior work. We also thank Alexandre Megretski for helpful feedback, insights into passive systems and suggestions on how to organize Section 3.Lemma D.2. Let m ∈ Z, and K be the unit circle in complex plane, and λ ∈ C inside the K. Then we have thatProof of Lemma D.2. For m ≥ 0, since z m is a holomorphic function, by Cauchy's integral formula, we have thatFor m < 0, by changing of variable y = z −1 we have thatsince \|λy\| = \|λ\| < 1, then we by Taylor expansion we have,Since the series λy is dominated by \|λ\| k which converges, we can switch the integral with the sum. Note that y −m−1 is holomorphic for m < 0, and therefore we conclude thatNow we are ready to prove Lemma 6.5.Proof of Lemma 6.5. Let m = n + d. We compute the Fourier transform of z m /p(z). That is, wewhereThen it follows thatUsing Lemma D.2, we obtain thatWe claim that n j=1 t j λ n−1 j = 1 , and n j=1 t j λ s j = 0 , 0 ≤ s < n − 1 .Indeed these can be obtained by writing out the lagrange interpolation for polynomial f (x) = x s with s ≤ n − 1 and compare the leading coefficient. Therefore, we further simplify β k toLet h(z) = k≥0 β k z k . Then we have that h(z) is a polynomial with degree d = m − n and leading term 1. Moreover, for our choice of d,D.2 Proof of Theorem 6.9Theorem 6.9 follows directly from a combination of Lemma D.3 and Lemma D.4 below. Lemma D.3 shows that the denominator of a function (under the stated assumptions) can be extended to a polynomial that takes values in C + on unit circle. Lemma D.4 shows that it can be further extended to another polynomial that takes values in C.Lemma D.3. Suppose the roots of s are inside circle with radius α < 1, and Γ = Γ(s). If transfer function G(z) = s(z)/p(z) satisfies that G(z) ∈ C τ 0 ,τ 1 ,τ 2 (or G(z) ∈ C + τ 0 ,τ 1 ,τ 2 ) for any z on unit circle,, 0}) such that p(z)u(z)/z n+d ∈ C τ 0 ,τ 1 ,τ 2 (or p(z)u(z)/z n+d ∈ C + τ 0 ,τ 1 ,τ 2 respectively) for τ = Θ τ (1) , where O τ (·), Θ τ (·) hide the polynomial dependencies on τ 0 , τ 1 , τ 2 .Proof of Lemma D.3. By the fact that G(z) = s(z)/p(z) ∈ C τ 0 ,τ 1 ,τ 2 , we have that p(z)/s(z) ∈ C τ 0 ,τ 1 ,τ 2 for some τ that polynomially depend on τ . Using Lemma 6.5, there exists u(z) of degree d such thatwhere we set ζ min{τ 0 , τ 1 }/τ 2 · p −1 H∞ . Then we have thatIt follows from equation (D.5) implies that that p(z)u(z)/z n+d ∈ C τ 0 ,τ 1 ,τ 2 , where τ polynomially depends on τ . The same proof still works when we replace C by C + .Lemma D.4. Suppose p(z) of degree n and leading coefficient 1 satisfies that p(z) ∈ C + τ 0 ,τ 1 ,τ 2 for any z on unit circle. Then there exists u(z) of degree d such that p(z)u(z)/z n+d ∈ C τ 0 ,τ 1 ,τ 2 for any z on unit circle with d = O τ (n) and τ 0 , τ 1 , τ 2 = Θ τ (1), where O τ (·), Θ τ (·) hide the dependencies on τ 0 , τ 1 , τ 2 .Proof of Lemma D.4. We fix z on unit circle first. Let's defined p(z)/z n be the square root of p(z)/z n with principle value. 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Inequalities of a. markoff and s. bernstein for polynomials and related functions. A C Schaeffer, Bull. Amer. Math. Soc. 4781941A. C. Schaeffer. Inequalities of a. markoff and s. bernstein for polynomials and related functions. Bull. Amer. Math. Soc., 47(8):565-579, 08 1941. Sequence to sequence learning with neural networks. Ilya Sutskever, Oriol Vinyals, Quoc V Le, Proc. 27th NIPS. 27th NIPSIlya Sutskever, Oriol Vinyals, and Quoc V. Le. Sequence to sequence learning with neural networks. In Proc. 27th NIPS, pages 3104-3112, 2014. + τ i ).The first inequality is trivial. We prove the second one. Consider a such that a ∈ B α,κ 0 ,κ 1 ,κ 2 . We wil show that a ∈ B α,τ 0 ,τ 1 ,τ 2. M Vidyasagar, Rajeeva L Karandikar, Journal of Process Control. 183A learning theory approach to system identification and stochastic adaptive control. Let q a (z) = p(z −1 )z nM. Vidyasagar and Rajeeva L. Karandikar. A learning theory approach to system identification and stochastic adaptive control. Journal of Process Control, 18(3):421- 430, 2008. i + τ i ).The first inequality is trivial. We prove the second one. Consider a such that a ∈ B α,κ 0 ,κ 1 ,κ 2 . We wil show that a ∈ B α,τ 0 ,τ 1 ,τ 2 . Let q a (z) = p(z −1 )z n . 2πik/M for some integer k such that \|q a (z) − q a (w)\| ≤ O(τ 2 n/(αM )). Therefore let M = cn for sufficiently large c (which depends on τ i 's), we have that for every z with \|z\| = 1/α. By Lemma F.4, for every z with \|z\| = 1/α, we have that there exists w = α −1 e. C τ 0 ,τ 1 ,τ 2 . This completes the proofBy Lemma F.4, for every z with \|z\| = 1/α, we have that there exists w = α −1 e 2πik/M for some integer k such that \|q a (z) − q a (w)\| ≤ O(τ 2 n/(αM )). Therefore let M = cn for sufficiently large c (which depends on τ i 's), we have that for every z with \|z\| = 1/α, q a (z) ∈ C τ 0 ,τ 1 ,τ 2 . This completes the proof.	[]
[ "A Monopole Mining Method for High Throughput Screening Weyl Semimetals", "A Monopole Mining Method for High Throughput Screening Weyl Semimetals" ]	[ "Vsevolod Ivanov \nDepartment of Physics\nUniversity of California\n95616DavisCAUSA\n", "Sergey Y Savrasov \nDepartment of Physics\nUniversity of California\n95616DavisCAUSA\n" ]	[ "Department of Physics\nUniversity of California\n95616DavisCAUSA", "Department of Physics\nUniversity of California\n95616DavisCAUSA" ]	[]	Although topological invariants have been introduced to classify the appearance of protected electronic states at surfaces of insulators, there are no corresponding indexes for Weyl semimetals whose nodal points may appear randomly in the bulk Brillouin Zone (BZ). Here we use a well-known result that every Weyl point acts as a Dirac monopole and generates integer Berry flux to search for the monopoles on rectangular BZ grids that are commonly employed in self-consistent electronic structure calculations. The method resembles data mining technology of computer science and is demonstrated on locating the Weyl points in known Weyl semimetals. It is subsequently used in high throughput screening several hundreds of compounds and predicting a dozen new materials hosting nodal Weyl points and/or lines.PACS numbers:There has been recent surge of interest in topological quantum materials caused by the existence in these systems of robust electronic states insensitive to perturbations [1, 2]. Z 2 invariants have been proposed to detect the protected (quantum Hall-like) surface states in topological insulators (TIs) [3], and, for centrosymmetric crystals, this reduces to finding band parities of electronic wave functions at time-reversal invariant points in the Brillouin zone (BZ) [4]. For a general case, the calculation involves an integration of Berry fields [5], and has been implemented in numerical electronic structure calculations[6] with density functional theory. These methods have allowed for exhaustive searches to identify candidate materials hosting topological insulator phases [7-9].Weyl semimetals (WSMs) are closely related systems characterized by a bulk band structure which is fully gapped except at isolated points described by the 2x2 Weyl Hamiltonian [2]. Sometimes these Weyl points extend into lines in the BZ giving rise to nodal line semimetals (NLSMs) [10]. Due to their intriguing properties such as Fermi arc surface states [11], chiral anomaly induced negative magnetoresistance [12], and a semiquantized anomalous Hall effect [13,14], the search for new WSM materials is currently very active. Unfortunately, their identification in infinite space of chemically allowed compounds represents a challenge: there is no corresponding topological index characterizing WSM phase, and the Weyl points may appear randomly in the bulk BZ. General principles, such as broken time reversal or inversion symmetry, or emergence of the WSM phase between topologically trivial and non-trivial insulating phases [11] are too vague to guide their high throughput screening, and recent group theoretical arguments[15,16]to connect crystal symmetry with topological properties still await their practical realization. The progress in this field was mainly serendipitous, although the ideas based on band inversion mechanism[17]or analyzing mirror Chern numbers[18,19]were proven to be useful inFIG. 1:a. A typical cone dispersion relationship E(k)=±v\|k − k WP \| for the Weyl point plotted within a rectangular area in k-space set by divisions of reciprocal lattice translations G1 and G2 for a fixed value along the third translation G3. b. The Weyl point located within a microcell set by the grid vectors q1, q2, q3 generates a Berry flux through each plaquette as given by the (right handed) circulation of the Berry connection with sign convention defined in text. many recent discoveries[20][21][22][23], and computer oriented searches of topological semimetals are beginning to appear[24,25].In this work, we propose a straightforward method to identify Weyl semimetals by using a well-known result that every Weyl point acts as a Dirac monopole [26] producing a non-zero Berry flux when it is completely enclosed by a surface in the BZ. The enclosed charge is given by the chirality of the Weyl point similar to the Gauss theorem in the Coulomb law. Rectangular grids of k-points that are widely employed in self-consistent electronic structure calculations for the BZ integration either via special points (Monkhorst-Pack) technique[27]or a tetrahedron method[28], are ideally suited for this purpose since they divide the volume of the BZ onto microcells and the electronic wave functions are automatically available at the corners of each microcell. It is thus a matter of rearranging the data to extract Berry phases of these wave functions in order to recover the Dirac monopoles inside the BZ. While there are some uncertainties connected to energy bands cutoffs used while arXiv:1810.09402v1 [cond-mat.mtrl-sci]	10.1103/physrevb.99.125124	[ "https://arxiv.org/pdf/1810.09402v1.pdf" ]	55,364,146	1810.09402	6ab87b78b5b174119464ff430995090a09dc463e	A Monopole Mining Method for High Throughput Screening Weyl Semimetals 22 Oct 2018 Vsevolod Ivanov Department of Physics University of California 95616DavisCAUSA Sergey Y Savrasov Department of Physics University of California 95616DavisCAUSA A Monopole Mining Method for High Throughput Screening Weyl Semimetals 22 Oct 2018(Dated: October 23, 2018) Although topological invariants have been introduced to classify the appearance of protected electronic states at surfaces of insulators, there are no corresponding indexes for Weyl semimetals whose nodal points may appear randomly in the bulk Brillouin Zone (BZ). Here we use a well-known result that every Weyl point acts as a Dirac monopole and generates integer Berry flux to search for the monopoles on rectangular BZ grids that are commonly employed in self-consistent electronic structure calculations. The method resembles data mining technology of computer science and is demonstrated on locating the Weyl points in known Weyl semimetals. It is subsequently used in high throughput screening several hundreds of compounds and predicting a dozen new materials hosting nodal Weyl points and/or lines.PACS numbers:There has been recent surge of interest in topological quantum materials caused by the existence in these systems of robust electronic states insensitive to perturbations [1, 2]. Z 2 invariants have been proposed to detect the protected (quantum Hall-like) surface states in topological insulators (TIs) [3], and, for centrosymmetric crystals, this reduces to finding band parities of electronic wave functions at time-reversal invariant points in the Brillouin zone (BZ) [4]. For a general case, the calculation involves an integration of Berry fields [5], and has been implemented in numerical electronic structure calculations[6] with density functional theory. These methods have allowed for exhaustive searches to identify candidate materials hosting topological insulator phases [7-9].Weyl semimetals (WSMs) are closely related systems characterized by a bulk band structure which is fully gapped except at isolated points described by the 2x2 Weyl Hamiltonian [2]. Sometimes these Weyl points extend into lines in the BZ giving rise to nodal line semimetals (NLSMs) [10]. Due to their intriguing properties such as Fermi arc surface states [11], chiral anomaly induced negative magnetoresistance [12], and a semiquantized anomalous Hall effect [13,14], the search for new WSM materials is currently very active. Unfortunately, their identification in infinite space of chemically allowed compounds represents a challenge: there is no corresponding topological index characterizing WSM phase, and the Weyl points may appear randomly in the bulk BZ. General principles, such as broken time reversal or inversion symmetry, or emergence of the WSM phase between topologically trivial and non-trivial insulating phases [11] are too vague to guide their high throughput screening, and recent group theoretical arguments[15,16]to connect crystal symmetry with topological properties still await their practical realization. The progress in this field was mainly serendipitous, although the ideas based on band inversion mechanism[17]or analyzing mirror Chern numbers[18,19]were proven to be useful inFIG. 1:a. A typical cone dispersion relationship E(k)=±v\|k − k WP \| for the Weyl point plotted within a rectangular area in k-space set by divisions of reciprocal lattice translations G1 and G2 for a fixed value along the third translation G3. b. The Weyl point located within a microcell set by the grid vectors q1, q2, q3 generates a Berry flux through each plaquette as given by the (right handed) circulation of the Berry connection with sign convention defined in text. many recent discoveries[20][21][22][23], and computer oriented searches of topological semimetals are beginning to appear[24,25].In this work, we propose a straightforward method to identify Weyl semimetals by using a well-known result that every Weyl point acts as a Dirac monopole [26] producing a non-zero Berry flux when it is completely enclosed by a surface in the BZ. The enclosed charge is given by the chirality of the Weyl point similar to the Gauss theorem in the Coulomb law. Rectangular grids of k-points that are widely employed in self-consistent electronic structure calculations for the BZ integration either via special points (Monkhorst-Pack) technique[27]or a tetrahedron method[28], are ideally suited for this purpose since they divide the volume of the BZ onto microcells and the electronic wave functions are automatically available at the corners of each microcell. It is thus a matter of rearranging the data to extract Berry phases of these wave functions in order to recover the Dirac monopoles inside the BZ. While there are some uncertainties connected to energy bands cutoffs used while arXiv:1810.09402v1 [cond-mat.mtrl-sci] Although topological invariants have been introduced to classify the appearance of protected electronic states at surfaces of insulators, there are no corresponding indexes for Weyl semimetals whose nodal points may appear randomly in the bulk Brillouin Zone (BZ). Here we use a well-known result that every Weyl point acts as a Dirac monopole and generates integer Berry flux to search for the monopoles on rectangular BZ grids that are commonly employed in self-consistent electronic structure calculations. The method resembles data mining technology of computer science and is demonstrated on locating the Weyl points in known Weyl semimetals. It is subsequently used in high throughput screening several hundreds of compounds and predicting a dozen new materials hosting nodal Weyl points and/or lines. PACS numbers: There has been recent surge of interest in topological quantum materials caused by the existence in these systems of robust electronic states insensitive to perturbations [1,2]. Z 2 invariants have been proposed to detect the protected (quantum Hall-like) surface states in topological insulators (TIs) [3], and, for centrosymmetric crystals, this reduces to finding band parities of electronic wave functions at time-reversal invariant points in the Brillouin zone (BZ) [4]. For a general case, the calculation involves an integration of Berry fields [5], and has been implemented in numerical electronic structure calculations [6] with density functional theory. These methods have allowed for exhaustive searches to identify candidate materials hosting topological insulator phases [7][8][9]. Weyl semimetals (WSMs) are closely related systems characterized by a bulk band structure which is fully gapped except at isolated points described by the 2x2 Weyl Hamiltonian [2]. Sometimes these Weyl points extend into lines in the BZ giving rise to nodal line semimetals (NLSMs) [10]. Due to their intriguing properties such as Fermi arc surface states [11], chiral anomaly induced negative magnetoresistance [12], and a semiquantized anomalous Hall effect [13,14], the search for new WSM materials is currently very active. Unfortunately, their identification in infinite space of chemically allowed compounds represents a challenge: there is no corresponding topological index characterizing WSM phase, and the Weyl points may appear randomly in the bulk BZ. General principles, such as broken time reversal or inversion symmetry, or emergence of the WSM phase between topologically trivial and non-trivial insulating phases [11] are too vague to guide their high throughput screening, and recent group theoretical arguments [15,16] to connect crystal symmetry with topological properties still await their practical realization. The progress in this field was mainly serendipitous, although the ideas based on band inversion mechanism [17] or analyzing mirror Chern numbers [18,19] were proven to be useful in FIG. 1: a. A typical cone dispersion relationship E(k)=±v\|k − k WP \| for the Weyl point plotted within a rectangular area in k-space set by divisions of reciprocal lattice translations G1 and G2 for a fixed value along the third translation G3. b. The Weyl point located within a microcell set by the grid vectors q1, q2, q3 generates a Berry flux through each plaquette as given by the (right handed) circulation of the Berry connection with sign convention defined in text. many recent discoveries [20][21][22][23], and computer oriented searches of topological semimetals are beginning to appear [24,25]. In this work, we propose a straightforward method to identify Weyl semimetals by using a well-known result that every Weyl point acts as a Dirac monopole [26] producing a non-zero Berry flux when it is completely enclosed by a surface in the BZ. The enclosed charge is given by the chirality of the Weyl point similar to the Gauss theorem in the Coulomb law. Rectangular grids of k-points that are widely employed in self-consistent electronic structure calculations for the BZ integration either via special points (Monkhorst-Pack) technique [27] or a tetrahedron method [28], are ideally suited for this purpose since they divide the volume of the BZ onto microcells and the electronic wave functions are automatically available at the corners of each microcell. It is thus a matter of rearranging the data to extract Berry phases of these wave functions in order to recover the Dirac monopoles inside the BZ. While there are some uncertainties connected to energy bands cutoffs used while defining non-Abelian Berry fields for metallic systems, our method allows a subsequent refinement provided a signal from a monopole is detected. The entire procedure resembles data mining technology in computer science as an intelligent method to discover patterns from large data sets in a (semi-) automatic way so that the extracted data can subsequently be used in further analysis. Since we are dealing with grids, there is a chance that the grid microcell will enclose both chiral positive and negative charges whose Berry fluxes cancel each other. Although resolution here is obviously adjustable by changing the grid size, and modern computers allow handlings of thousands and even millions of k-points in parallel, going for Weyl points that are too close makes no sense from both practical and fundamental reasons. Practically, properties such as anomalous Hall effect [13,14] are proportional to the distance between the Weyl points and so does the density of Fermi arc surface states [11]. Disorder, electronic interactions, thermal broadening and Heisenberg uncertainty principle provide fundamental limitations. Therefore, distances between the Weyl points need not be smaller than a few percent of the reciprocal lattice spacing, and this does not require dealing with very dense grids. Here we implement this monopole mining method and test it by verifying locations of the Weyl points in several known systems, such as recently proposed TaAs [20] and CuF [23] Weyl semimetals. Next, we demonstrate how it can be used for high throughput screening of WSMs by scanning several hundreds of compounds in the p62m(#189) space group with the ZrNiAl structure. We predict a dozen new materials hosting WSM/NLSM behavior. We first outline the method to evaluate the Berry flux due to a single Weyl point that appears somewhere in the bulk BZ with its typical dispersion relationship E(k) = ±v\|k − k WP \| as illustrated in Fig.1a. We represent the BZ by reciprocal lattice translations G ν=1,2,3 and divide it onto N 1 ×N 2 ×N 3 microcells. Each microcell is spanned by primitive vectors q ν=1,23 = G ν /N ν with its origin given by the grid of k-points represented by three integers n ν = 0, N ν −1 as k = n 1 q 1 +n 2 q 2 +n 3 q 3 . The problem of finding the wave vector k WP is reduced to recovering the microcell that contains the monopole. We define a non-Abelian link field that appears while evaluating the Berry phase using the finite difference method [6] U q (k) = det k + qj \|e iqr \|kj \|det [ k + qj \|e iqr \|kj ]\|(1) Here the matrix elements between the periodic parts of the wave functions are cast into the form k+qj \|e iqr \|kj , which frequently appear in density functional linear response calculations [29] and thus are straightforward to evaluate. The set of energy bands j is spanned over occupied states and includes those that cross the Fermi level. However, some uncertainty exists in this enumeration procedure because the Berry flux from the negative and positive branches of the monopole (bands 1 and 2 for the example shown in Fig.1a)will cancel each other. For the example being discussed, this means that either band 1 or 2 (but not both) needs to be taken into account while evaluating Eq.1. In real materials, this may result in contribution for some monopoles cancelling, but since we are mostly interested in the Weyl points in the immediate vicinity of the Fermi level, varying the upper cutoff value for j by one or two will resolve this problem. We also note that the link field U q (k) needs to be computed for the entire grid of k-points, where the group symmetry operations help to generate the wave functions that are normally available within only irreducible portion of the BZ. We now evaluate the Berry flux through faces of each microcell of the N 1 × N 2 × N 3 grid. This is illustrated in Fig.1b, where the flux Φ i=1..6 through each plaquette with the origin at particular k and spanned by a pair of vectors q µ q ν is conveniently encoded into the following formula 2πΦ ≡ Im ln U qµ (k)U qν (k + q µ ) U qν (k)U qµ (k + q ν )(2) This procedure is similar to one employed while evaluating Z 2 invariants [6] on six two-dimensional tori introduced in Ref. [30] but now the roles of the tori are played by the slices of the BZ spanned by each pair of the reciprocal vectors G µ G ν with a fixed value along the third vector G ξ . We only need to take care of the fact that the flux as given by Eq. 2 produces right (alternatively left) handed circulation of the Berry connection but inner (or outer) normal should be chosen consistently for the total flux through each surface of the microcell. Thus, the total Berry flux is given by c = Φ 1 + Φ 2 + Φ 3 − Φ 4 − Φ 5 − Φ 6(3) Although the flux through each plaquette is generally non-integer, the total flux is guaranteed to be an integer since individual contributions (2) from adjacent plaquettes cancel each other in Eq.(3), up to an addition of 2πn. Therefore c returns ether the chiral charge of the monopole or zero. The entire algorithm is now viewed as an automated procedure that is either done following the self-consistent band structure calculation or "on the fly". We illustrate it on the example of TaAs Weyl semimetal whose electronic properties are well documented in recent literature [20]. We use a full potential linear muffin-tin orbital method (FP LMTO) developed by one of us [31] and perform a self-consistent density functional calculation with spin-orbit coupling using the Generalized Gradient Approximation [32]. We subsequently set up a kgrid using 20 × 20 × 20 divisions of the reciprocal lattice unit cell. These types of grids were previously shown to be sufficient in calculating Z 2 invariants in topological insulators [33]. For evaluating the link field, Eq. (1), the energy window is chosen to span the entire valence band with the cutoff value corresponding to the band number that crosses the Fermi level. It appears this is sufficient to recover all monopoles. The net result is 24 out 8000 microcells produce non-zero Berry flux and give their approximate positions. We take the coordinates of the corresponding microcells (only non-equivalent by symmetry are needed; two for TaAs) and mine these areas of k-space by introducing similar rectangular grids inside each microcell in order to refine the locations of the Weyl points to the positions: (0.009, 0.506, 0), (0.019, 0.281, 0.579) in units 2π/a, 2π/a, 2π/c. This is in agreement with the previous calculation [20]. We also considered CuF, recently predicted to be a Weyl semimetal by one of us [23]. The exact same setup (20 × 20 × 20 divisions with the energy panel spanned till the band that crosses the Fermi level) returns 24 microcells that are all related by symmetry. Zooming into one microcell returns the following location of the Weyl point: (0.281, 0.119, 0)2π/a, consistent with our previous result [23]. To demonstrate the predictive power of the method, we scanned several hundreds noncentrosymmetric hexagonal compounds in the p62m (# 189) space group with the ZrNiAl structure. A complete list of these materials is given in Supplementary Infortmation. Topological electronic structures in few of these systems have already drawn a recent attention. CaAgP was predicted to be a line-node Dirac semimetal while CaAgAs was found to be a strong topological insulator [34]. Similar properties have been discussed for NaBaBi under pressure [35]. The unit cell of these crystals consists of a rhomboid prism with side a, internal angle 2π/3, and height c; the Nitype atoms are located on the vertical edges and in the centers of the two equilateral triangles forming the rhombus base, with the Zr-type and Al-type atoms located on the edges of these triangles, 1/3 ± 1/4(c 2 /a 2 ) away from the corners in the middle and bottom layers respectively [see Fig. 2]. We perform self-consistent band structure calculations and subsequent monopole mining procedure in exactly the same manner as we illustrated for TaAs and CuF. The lattice parameters can be found in Ref. [36]. Out of the compounds that we studied, we clearly identify 11 materials which show WSM behavior, 1 NLSM and 1 hosting both Weyl points and nodal lines. The two NLSMs also host topologically distinct triple fermion points [37]. Table I summarizes our results for each compound , giving the locations of the non-equvalent lowenergy Weyl and/or triple points, their number and energies relative E F in eV. The Weyl points are generally viewed as type II according to classification introduced in Ref. [38]. (Complete crystallographic and electronic structure data for these compounds is given in the supplementary information.) Many of the Weyl semimetals that we predict in our work display remarkably similar locations of their Weyl points. LaInMg, LuGeAg, YGeLi,YPbAg, and YSiAg, exhibit 6 pairs (chiral positive and negative) of points, that are all symmetry related and are only slightly displaced from the k z = 0 plane. They are located along the ΓM direction in the BZ. We illustrate their precise Fig. 3a and refer to them in Table I as Weyl points of sort A. We find that HfPRu, and ZrPRu show another sort (referred to as sort B) of Weyl points, namely 12 pairs that are shifted symmetrically away from the ΓK line (see Fig. 3b). Interestingly, a similar behavior is seen for LaTlMg, and YTlMg which show both sorts (A and B) of Weyl points. LuAsPd shows two kinds of sort A Weyl points (24 total), while ZrAsOs shows two kinds of sort B Weyl points (48 total). Their displacement from k z = 0 plane is much larger than the one found in previous cases. For each reported Weyl point, we also provide independent verification by calculating the band structures along k x , k y and k z directions with the boundary vectors confining the Weyl point. An example of such plot is shown in Fig. 3c for the Weyl point in LaInMg, where one clearly recognizes the band crossings along all three directions that are characteristic of the Weyl cone dispersion. Another interesting outcome of our high-throughput screening is the materials exhibiting nodal lines and triple-point fermions. TiGePd and VAsFe both host 12 pairs (chiral positive and negative) of nodal lines that are located very close to the ΓA direction in the BZ. We illustrate this behavior for TiGePd in Fig. 4a by zooming into the area of the BZ bounded by 0.15 ≤ 2πk z /c ≤ 0.22 and −0.03 ≤ 2πk x,y /a ≤ +0.03. Interestingly, the nodal lines start and end at triple degenerate points that have recently enriched our classification of the topological objects [37]. These triple points are located at the ΓA line of the BZ. We provide their coordinates for TiGePd and VAsFe in Table I. The corresponding band structure plot for one of the triple points in TiGePd is shown in Fig. 4b along the k x and k y directions of the BZ with the boundary vectors confining the triple point. (A complete set of plots for each compound is provided in the supplemen-tary information.) One of the most striking features of Weyl semimetals is the presence of the Fermi arcs in their one-electron surface spectra [11]. Although computations of their shapes are possible via a self-consistent supercell (slab) calculation of the surface energy bands, given the number of compounds that we deal in this work, it is a computationally demanding study. Nevertheless, since the arcs connect the Weyl points of different chirality, one can expect that most of the materials that we list in Table I would have rather short arcs since the distances between positive and negative chiral charges are quite small. One notable exception is VAsFe which, as we list in Table I, exhibits not only nodal lines and triple points, but also a set of Weyl points which are well separated from each other. These are expected to produce very long Fermi arcs for the (100) or (110) crystallographic types of surfaces. One can also expect that their contribution to the anomalous Hall coefficient should be large since the latter is known to be directly proportional to the distance between the Weyl points [13]. We have recently shown [39] that long and straight Fermi arcs are generally capable of supporting nearly dissipationless surface currents, therefore it could be interesting to explore such possibility in VAsFe. In conclusion, using the well-known property that Weyl points act as Dirac monopoles in k-space, we presented an automated monopole mining method to identify Weyl and nodal line semimetals. We tested the method by recovering the Weyl points in several known systems as well as demonstrating its predictive power by high throughput screening hundreds noncentrosymmetric hexagonal compounds in the p62m (# 189) space group and finding 13 new materials whose electronic structures as well as the locations of the topological nodal points and lines have been reported. As we judge from our calculated energy bands, the WSMs identified in this work exhibit regular Fermi surface states while the Weyl points are not exactly pinned at the Fermi level. This is similar to other recently discovered WSMs, such as TaAs [20] whose experimental studies of large negative magnetoresistance have been recently performed [40]. Despite the latter representing a signature of the much celebrated chiral anomaly feature in Weyl semimetals, there exists an obvious problem of distinguishing contributions from the Weyl points and regular Fermi states. In this regard our automated approach should be helpful for scanning vast material databases in identifying an ideal WSM with only nodal points at the Fermi level as it was originally envisioned in pyrochlore iridates [11]. The work was supported by NSF DMR Grant No. 1411336. SUPPLEMENTARY INFORMATION List of Compounds Here we list noncentrosymmetric hexagonal compounds in the p62m (# 189) space group with the ZrNiAl structure studied in this work. Their complete crystallographic data can be found in Ref. [1]. As many of the compounds in this structure include rare earth elements with their f electron states appearing in the vicinity of the Fermi level, we first provide a list of only those compounds that do not explicitly include Lanthanides (see Table I). These are the systems for which density functional based calculations can be trusted in general. We can also comment on the compounds that include Lanthanide elements. They can be separated onto two large groups. The first group includes the materials where the narrow f-band appears crossing the Fermi level in the calculated band structures. This would be an indication that a many-body renormalization of the single particle spectra (such, e.g., as band narrowing, multiplet transitions, etc) is expected. Although modern electronic structure approaches based on combinations of density functional and dynamical mean field theories [2] allow handling such cases, those are outside the scope of the present study, and we do not study topological properties of these compounds. The second group includes the materials with either fully empty or fully occupied f band, namely f 0 : LaAuCd, LaAuIn, LaAuMg, LaCuIn, LaCuMg, LaInMg, LaIrSn, LaNiIn, LaNiZn, LaPdCd, LaPdHg, LaPdIn, LaPdMg, LaPdPb, LaPdSn, LaPdTl, LaPtIn, LaPtPb, LaPtSn, LaRhIn, LaRhSn, LaTlMg; f 14 : LuAsPd, LuAuIn, Lu-AuZn, LuCuIn, LuGaMg LuGeAg, LuGeLi, LuInMg, LuIrSn, LuNiAl LuNiIn, LuNiPb, LuPbAg, LuPdIn, LuPdSn LuPdZn, LuPtIn, LuPtSn, LuRhSn, LuSiAg, LuTlMg. These are the cases where static mean field description can in principle capture single particle excitations (apart from the question whether the position of the f-band is correctly predicted by such theory). There are a few materials that include Sm ion with its non-magnetic configuration f 6 : SmAgMg, SmAuCd, SmAuIn, SmAuMg, SmCuAl, SmCuIn, SmIrIn, SmIrSn, SmNiAl, SmNiIn, SmNiSn, SmNiZn, SmPdCd, Sm-PdHg, SmPdIn, SmPdMg, SmPdPb, SmPdTl, SmPtIn, SmPtMg, SmPtPb, SmPtSn, SmRhIn, SmRhSn, Sm-SiAg, SmTlMg. Here j = 5/2 and j = 7/2 subbands appear below and above the Fermi level, respectively. The Coulomb renormalzation in these compounds has a predictable effect by renormalizing the spin-orbit coupling by the Hubard-type interaction, and the states in the immidiate vicinity of the Fermi level are not affected. [3]. FIG. 2 : 2ZrNiAl-type crystal structure (# 189 space group p62m) of noncentrosymmetric hexagonal compounds compounds studied in this work. FIG. 3 3: a. Positions of 6 pairs (cyan for chiral positive and magenta for chiral negative) of low-energy Weyl points seen along the ΓM direction in the BZ for LaInMg and referenced inTable Ias sort A; b. Positions of 12 pairs of Weyl points that are shifted symmetrically away from the ΓK line for HfPRu and referenced in Table I as sort B; c. Energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.36868, 0.01123) for LaInMg. Point notations are as follows: K1x = (−0.10000, 0.36868, 0.01123), K2x = (0.10000, 0.36868, 0.01123), K1y = (0.0000, 0.26868, 0.01123), K2y = (0.0000, 0.46868, 0.011230), K1z = (0.0000, 0.36868, −0.056150), K2z = (0.0000, 0.36868, 0.056150) in units 2π/a, 2π/a, 2π/c. positions for LaInMg in FIG. 4 4: a. A set of nodal lines for TiGePd that is recovered by the monopole mining method presented in this work. The color (cyan and magenta) distinguishes chiral positive and negative lines, respectively. The zoomed area of the BZ is bounded by 0.15 ≤ 2πkz/c ≤ 0.22 and −0.03 ≤ 2πkx,y/a ≤ +0.03. Also shown in yellow are the triple degenerate topological points[37]. b. Energy band dispersions in the vicinity of the triple point (0, 0, 0.20775) for TiGePd. Point notations are as follows: K1x = (−0.10000, 0.0000, 0.20775), K2x = (0.10000, 0.0000, 0.20775), K1y = (0.0000, −0.10000, 0.20775) in units 2π/a, 2π/a, 2π/c. Figures 1 - 113 provide complete data for for the topological materials predicted in this work: the band structures near the Fermi level, energy panels used for defining non-Abelian Berry connection, positions of low-energy topological nodal points in the Brillouin Zone as well as energy band dispersions in the vicinity of the nodal points. near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.46280, 0.06931, 0.02210). Point notations are as follows: K1x = (0.36280, 0.069310, 0.022100), K2x = (0.56280, 0.069310, 0.022100), K1y = (0.46280, −0.17328, 0.02210), K2y = (0.46280, 0.17328, 0.022100), K1z = (0.46280, 0.06931, −0.11050), K2z = (0.46280, 0.06931, 0.11050) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=12.1207 a.u., c/a=0.58513 [3]. FIG. 2 : 2Results for LaInMg: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.36868, 0.01123). Point notations are as follows: K1x = (−0.10000, 0.36868, 0.01123), K2x = (0.10000, 0.36868, 0.01123), K1y = (0.0000, 0.26868, 0.01123), K2y = (0.0000, 0.46868, 0.011230), K1z = (0.0000, 0.36868, −0.056150), K2z = (0.0000, 0.36868, 0.056150) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=14.789 a.u., c/a=0.61472 [4]. FIG. 3: Results for LaTlMg: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points; d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.38916, 0.03236). Points notations are as follows: K1x = (−0.10000, 0.38916, 0.032360), K2x = (0.10000, 0.38916, 0.032360), K1y = (0.0000, 0.28916, 0.03236), K2y = (0.0000, 0.48916, 0.03236), K1z = (0.0000, 0.38916, −0.16180), K2z = (0.0000, 0.38916, 0.16180) in units 2π/a, 2π/a, 2π/c as well as e. energy band dispersions in the vicinity of the Weyl point kwp = (0.41450, 0.02567, 0.00724). Point notations are as follows: K1x = (0.3145, 0, 0.02567, 0.00724), K2x = (0.51450, 0.02567, 0.00724), K1y = (0.41450, −0.12835, 0.00724), K2y = (0.41450, 0.12835, 0.00724), K1z = (0.41450, 0.02567, −0.0362), K2z = (0.41450, 0.02567, 0.0362) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=14.7644 a.u., c/a=0.61160[5]. FIG. 4 : 4Results for LuAsPd: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.11481, 0.14140). Point notations are as follows: K1x = (−0.10000, 0.11481, 0.14140), K2x = (0.10000, 0.11481, 0.14140), K1y = (0.0000, 0.01481, 0.14140), K2y = (0.0000, 0.21481, 0.14140), K1z = (0.0000, 0.11481, 0.0414), K2z = (0.0000, 0.11481, 0.24140) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=13.1733 a.u., c/a=0.55817 [6].FIG. 5: Results for LuGeAg: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.42190, 0.00098). Point notations are as follows: K1x = (−0.10000, 0.42190, 0.00098), K2x = (0.10000, 0.42190, 0.00098), K1y = (0.0000, 0.32190, 0.00098), K2y = (0.0000, 0.52190, 0.00098), K1z = (0.0000, 0.42190, −0.0049), K2z = (0.0000, 0.42190, 0.0049) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=13.2517 a.u., c/a=0.58948 [7].FIG. 6: Results for TiGePd: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. nodal lines and positions of triple degenerate points. The zoomed area of the BZ is bounded by 0.15 ≤ 2πkz/c ≤ 0.22 and −0.03 ≤ 2πkx,y/a ≤ +0.03; d. energy band dispersions in the vicinity of the triple point ktp = (0.00000, 0.00000, 0.16495). Points notations are as follows: K1x = (−0.10000, 0.0000, 0.16495), K2x = (0.10000, 0.0000, 0.16495), K1y = (0.0000, −0.10000, 0.16495), K2y = (0.0000, 0.10000, 0.16495) in units 2π/a, 2π/a, 2π/c; e. energy band dispersions in the vicinity of the triple point ktp = (0.00000, 0.00000, 0.20775). Point notations are as follows: K1x = (−0.10000, 0.0000, 0.20775), K2x = (0.10000, 0.0000, 0.20775), K1y = (0.0000, −0.10000, 0.20775) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=12.4779 a.u., c/a=0.56032[8]. FIG near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points; d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.38339, 0.17269). Points notations are as follows: K1x = (−0.10000, 0.38339, 0.17269), K2x = (0.10000, 0.38339, 0.17269), K1y = (0.0000, 0.28339, 0.17269), K2y = (0.0000, 0.48339, 0.17269), K1z = (0.0000, 0.38339, 0.07269), K2z = (0.0000, 0.38339, 0.27269) in units 2π/a, 2π/a, 2π/c.; e. nodal lines with triple degenerate points. The zoomed area of the BZ is bounded by 0.3 ≤ 2πkz/c ≤ 0.5 and −0.01 ≤ 2πkx,y/a ≤ +0.01.; f. energy band dispersions in the vicinity of the triple point ktp = (0.00000, 0.00000, 0.32279). Points notations are as follows: K1x = (−0.10000, 0.0000, 0.32279), K2x = (0.10000, 0.0000, 0.32279), K1y = (0.0000, −0.1000, 0.32279), K2y = (0.00000, 0.1000, 0.32279). in units 2π/a, 2π/a, 2π/c; g. energy band dispersions in the vicinity of the triple point ktp = (0.00000, 0.00000, 0.47625). Point notations are as follows: K1x = (−0.10000, 0.0000, 0.47625), K2x = (0.10000, 0.0000, 0.47625), K1y = (0.0000, −0.1000, 0.47625), K2y = (0.00000, 0.1000, 0.47625).in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=11.7352 a.u., c/a=0.56892[9]. FIG near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.27793, 0.00817). Point notations are as follows: K1x = (−0.10000, 0.27793, 0.00817), K2x = (0.10000, 0.27793, 0.00817), K1y = (0.0000, 0.17793, 0.00817), K2y = (0.0000, 0.37793, 0.00817), K1z = (0.0000, 0.27793, −0.040850), K2z = (0.0000, 0.27793, 0.04085) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=13.3509 a.u., c/a=0.59915 [10]. FIG. 9 : 9Results for YPbAg: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.40335, 0.03142). Point notations are as follows: K1x = (−0.10000, 0.40335, 0.03142), K2x = (0.10000, 0.40335, 0.03142), K1y = (0.0000, 0.30335, 0.03142), K2y = (0.0000, 0.50335, 0.03142), K1z = (0.0000, 0.40335, −0.15710), K2z = (0.0000, 0.40335, 0.15710) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=14.140 a.u., c/a=0.59133[11]. FIG. 10 : 10Results for YSiAg: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.37866, 0.00385). Point notations are as follows: K1x = (−0.10000, 0.37864, 0.00385), K2x = (0.10000, 0.37864, 0.00385), K1y = (0.0000, 0.27864, 0.00385), K2y = (0.0000, 0.47864, 0.00385), K1z = (0.0000, 0.37864, −0.0192), K2z = (0.0000, 0.37864, 0.0192) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=13.2623 a.u., c/a=0.59364 [12].FIG. 11: Results for YTlMg: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points; d. energy band dispersions in the vicinity of the Weyl point kwp = (0.00000, 0.43303, 0.02319). Points notations are as follows: K1x = (−0.10000, 0.43303, 0.02319), K2x = (0.10000, 0.43303, 0.02319), K1y = (0.0000, 0.33303, 0.02319), K2y = (0.0000, 0.53303, 0.02319), K1z = (0.0000, 0.43303, −0.11595), K2z = (0.0000, 0.43303, 0.11595) in units 2π/a, 2π/a, 2π/c. e. energy band dispersions in the vicinity of the Weyl point kwp = (0.44076, 0.02908, 0.00441). Point notations are as follows: K1x = (0.34076, 0.02908, 0.00441), K2x = (0.54076, 0.02908, 0.00441), K1y = (0.44076, −0.14540, 0.00441), K2y = (0.44076, 0.14540, 0.00441), K1z = (0.44076, 0.02908, −0.02205), K2z = (0.44076, 0.02908, 0.02205) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=14.1824 a.u., c/a=0.61272[5]. FIG. 12 : 12Results for ZrAsOs: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points; d. energy band dispersions in the vicinity of the Weyl point kwp = (0.47365, 0.02591, 0.04792). Points notations are as follows: K1x = (0.37365, 0.02591, 0.04792), K2x = (0.57365, 0.02591, 0.04792), K1y = (0.47365, −0.12955, 0.04792), K2y = (0.47365, 0.12955, 0.04792), K1z = (0.47365, 0.02591, −0.11980), K2z = (0.47365, 0.02591, 0.11980) in units 2π/a, 2π/a, 2π/c; e. energy band dispersions in the vicinity of the Weyl point kwp = (0.474060.012150.047890). Point notations are as follows: K1x = (0.37406, −0.01215, 0.04789), K2x = (0.57406, −0.01215, 0.04789), K1y = (0.47406, −0.06075, 0.04789), K2y = (0.47406, 0.06075, 0.04789), K1z = (0.47406, 0.01215, −0.11973), K2z = (0.47406, 0.01215, 0.11973) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=12.476 a.u., c/a=0.57467[13]. FIG. 13 : 13Results for ZrPRu: a. band structure near the Fermi level; b. energy panel used for defining non-Abelian Berry connection; c. positions of low-energy Weyl points as well as d. energy band dispersions in the vicinity of the Weyl point kwp = (0.45982, 0.07532, 0.01698). Point notations are as follows: K1x = (0.35982, 0.07532, 0.01698), K2x = (0.55982, 0.07532, 0.01698), K1y = (0.45982, −0.18830, 0.01698), K2y = (0.45982, 0.18830, 0.01698), K1z = (0.45982, 0.07532, −0.0849), K2z = (0.45982, 0.07532, 0.0849) in units 2π/a, 2π/a, 2π/c. Lattice parameters used: a=12.2057 a.u., c/a=0.58492 TABLE I : IList of non-equivalent Weyl and triple points (in units 2π/a, 2π/a, 2π/c), their number and energies relative to the Fermi level (in eV) recovered using the monopole mining method for noncentrosymmetric hexagonal compounds in the p62m (# 189) space group with the ZrNiAl structure that are predicted to exhibit Weyl/nodal line semimetal behavior. The typical appearance of the Weyl points in the Brillouin Zone is cited by referencing to either sort A or B as illustrated inFig. 3ab.Comp. Topological Points Type # E (eV) LaInMg (0.00000, 0.36868, 0.01123) Weyl-A 12 −0.06 LuGeAg (0.00000, 0.42190, 0.00098) Weyl-A 12 −0.23 YGeLi (0.00000, 0.27793, 0.00817) Weyl-A 12 −0.13 YPbAg (0.00000, 0.40335, 0.03142) Weyl-A 12 −0.09 YSiAg (0.00000, 0.37864, 0.00384) Weyl-A 12 −0.09 HfPRu (0.46280, 0.06931, 0.0221) Weyl-B 24 +0.06 ZrPRu (0.45982, 0.07532, 0.01698) Weyl-B 24 +0.06 LaTlMg (0.00000, 0.38916, 0.03236) (0.41450, 0.02567, 0.00724) Weyl-A Weyl-B 12 24 −0.13 −0.13 YTlMg (0.00000, 0.43303, 0.02319) (0.44076, 0.02908, 0.00441) Weyl-A Weyl-B 12 24 −0.05 −0.11 LuAsPd (0.00000, 0.11481, 0.14140) (0.00000, 0.12004, 0.13942) Weyl-A Weyl-A 12 12 +0.18 +0.19 ZrAsOs (0.47365, 0.02591, 0.04792) 0.47406, 0.01215, 0.04789 Weyl-B Weyl-B 24 24 +0.02 +0.02 TiGePd (0.00000, 0.00000, 0.16495) (0.00000, 0.00000, 0.20775) Triple Triple 2 2 +0.14 +0.22 VAsFe (0.00000, 0.000000, 0.32279) (0.00000, 0.000000, 0.47625) (0.00000, 0.38339, 0.17269) Triple Triple Weyl-A 2 2 12 +0.14 +0.19 +0.09 TABLE I : IList of noncentrosymmetric hexagonal compounds in the p62m (# 189) space group with the ZrNiAl structure studied in this work. The compounds containing Lanthanide element are explicitly excluded from theTable.Class X = Class X = CrAsX Ti, Pd, Fe, Co, Ni, Rh XPtIn Sc, Y MnAsX Ti, Ni, Rh, Fe, Pd, Ru TiGeX Co, Pd ScGeX Fe, Rh, Cu, Os, Pd, Ru ZrCoX Ga, Sn XSiRe Hf, Ta, Ti, Zr ZrGeX Os, Zn HfGeX Fe, Os, Rh, Ru XNiGa Hf, Zr FeAsX Ti, Co, V , Ni ScPX Ir, Na XPNi Fe, Mo, W , Co MnGeX Pd, Rh XGeMn Hf, Nb, Sc, Ta CrPX Pd, Ni TiPX Cr, Os, Ru HfXRu P, As ZrPX Os, Mo, Ru XAsOs Hf, Zr MnPX Rh, Pd, Ni XPdPb Ca, Y ScSiX Cu, Ru, Mn XSiMn Nb, Ta CaXCd Ge, Sn, Pb HfSiX Os, Ru XAsPd Hf, Ti, Zr CaXAg P, As XNiAl Hf, Y , Zr ZrXRu Si, As XBFe Nb, Ta NbCrX Ge, Si Other: YRhSn, YAuCd , YPdMg , YNiIn , ScGeAg YPdAl , YInMg , YAuZn , YPbAg , YPdTl YAuMg , YPdZn , YPdTl , YSiAg , YRhIn YTlMg , YAgMg , YAuIn , YCuIn , YGaMg HfIrSn , YPdIn , YCuAl , YGeLi , YPtSn YCuMg , BaBiNa, YAlMg, ScSnAg, YSiLi Data for Topological Points Colloquium: Topological insulators. M Z Hasan, C L Kane, Rev. Mod. Phys. 823045For a review, see, e.g., M. Z. 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[ "Capacity Variation in the Many-to-one Stable Matching", "Capacity Variation in the Many-to-one Stable Matching", "Capacity Variation in the Many-to-one Stable Matching", "Capacity Variation in the Many-to-one Stable Matching" ]	[ "Gerhard To ", "Woeginger ", "Gerhard To ", "Woeginger " ]	[]	[]	The many-to-one stable matching problem provides the fundamental abstraction of several real-world matching markets such as school choice and hospital-resident allocation. The agents on both sides are often referred to as residents and hospitals. The classical setup assumes that the agents rank the opposite side and that the capacities of the hospitals are fixed.It is known that increasing the capacity of a single hospital improves the residents' final allocation. On the other hand, reducing the capacity of a single hospital deteriorates the residents' allocation. In this work, we study the computational complexity of finding the optimal variation of hospitals' capacities that leads to the best outcome for the residents, subject to stability and a capacity variation constraint.First, we show that the decision problem of finding the optimal capacity expansion is NPcomplete and the corresponding optimization problem is inapproximable within a certain factor. This result holds under strict and complete preferences, and even if we allocate extra capacities to disjoint sets of hospitals. Second, we obtain analogous computational complexity results for the problem of capacity reduction. Finally, we study the variants of these problems when the goal is to maximize the size of the final matching under incomplete preference lists.1 Not all the agents are ranked. In the case of incomplete preference lists, the Rural Hospital Theorem holds[40,22,42,32], which states that all the stable matchings have the same cardinality.2 Some agents in the preference list are ranked equally. In the case of preference lists with ties, all the weakly stable matchings are complete (under the assumption that the cardinalities on the two sides of the bipartition are equal). Weakly stability means there is no pair of agents that strictly prefer to be matched to each other rather than being in their current assignment.	10.48550/arxiv.2205.01302	[ "https://arxiv.org/pdf/2205.01302v1.pdf" ]	248,505,758	2205.01302	c6126a9a1503fc4866297f9edbace3a162749909	Capacity Variation in the Many-to-one Stable Matching 1964-2022 Gerhard To Woeginger Capacity Variation in the Many-to-one Stable Matching 1964-2022 The many-to-one stable matching problem provides the fundamental abstraction of several real-world matching markets such as school choice and hospital-resident allocation. The agents on both sides are often referred to as residents and hospitals. The classical setup assumes that the agents rank the opposite side and that the capacities of the hospitals are fixed.It is known that increasing the capacity of a single hospital improves the residents' final allocation. On the other hand, reducing the capacity of a single hospital deteriorates the residents' allocation. In this work, we study the computational complexity of finding the optimal variation of hospitals' capacities that leads to the best outcome for the residents, subject to stability and a capacity variation constraint.First, we show that the decision problem of finding the optimal capacity expansion is NPcomplete and the corresponding optimization problem is inapproximable within a certain factor. This result holds under strict and complete preferences, and even if we allocate extra capacities to disjoint sets of hospitals. Second, we obtain analogous computational complexity results for the problem of capacity reduction. Finally, we study the variants of these problems when the goal is to maximize the size of the final matching under incomplete preference lists.1 Not all the agents are ranked. In the case of incomplete preference lists, the Rural Hospital Theorem holds[40,22,42,32], which states that all the stable matchings have the same cardinality.2 Some agents in the preference list are ranked equally. In the case of preference lists with ties, all the weakly stable matchings are complete (under the assumption that the cardinalities on the two sides of the bipartition are equal). Weakly stability means there is no pair of agents that strictly prefer to be matched to each other rather than being in their current assignment. Introduction The stable matching problem has found multiple applications such as daycare admission in Denmark [27], school and hospital-resident allocation in the USA [2,3,1,40,43], school and university admission in Hungary [13,14], school admission in Singapore [49], university admission in China [53], Germany [16] and Spain [37], faculty recruitment in France [8]. The many-to-one stable matching problem (HR) consists of two sides-henceforth referred to as hospitals and residentswhere hospitals have fixed and known capacities. Both sides have preferences over each other, and the goal of the decision-maker is to find an assignment such that, in each pair, both agents simultaneously prefer each other over any other agent. The HR problem, and its multiple variants, have been widely studied in the literature by different disciplines: From a polyhedral [7,9] and algorithmic [21] perspective, to geometry [46], mathematical programming [50], combinatorics [29], fixed-point methods [48] and graph theory [10]. As mentioned, in the standard version of HR, the capacity of the hospitals are fixed and known in advance. The decision-maker in charge of the final assignment does not have control over these quota. However, there are multiple real-life situations in which the variation of the size of the market, expansion or reduction, could play a significant role. For example, when allocating couples in hospitals [40], siblings in school [19], scholarships or expenses reduction. The idea of introducing new participants in the matching market has been previously studied through the lens of game theory and econonomics. This problem is known as entry comparative static, and is usually assumed that the introduced agent is an independent entity with a certain preference list; the participants of the opposite side also rank this new agent. It has been shown that when a new agent is introduced, then the resulting matching is weakly better (i.e., equal or better) for the agents of the opposite side [26,22,44]. On the mathematical programming domain, for the HR, the problem of deciding simultaneously capacity expansions on the hospitals' side and a stable matching was first proposed in [15]. Using integer programming, the authors demonstrated empirically that significantly better matchings for the residents can be obtained through the allocation of a few extra spots. In this work, we study the computational complexity of the problem proposed in [15] as well as its counterpart, i.e., when reduction of the hospitals' capacity is required. Roughly speaking, for the expansion of the market we study the following question: Given a non-negative integer number B ∈ Z + of extra spots, which hospitals should the decision-maker expand the capacity of to obtain the best stable matching for the residents? In the second part of this work, we focus on the reduction of the market. Simply, we study the following question: Given a non-negative integer number B ∈ Z + of spots to be removed, which hospitals should the decision-maker reduce the capacity of in order to obtain the best stable matching for the residents? We primarily focus on a rank-based metric to choose the best matching for residents. We also study the variants of the problems above under a cardinality-based metric, which has been widely studied in the literature [40,22,42,32]. Related Work In their seminal paper, Gale and Shapley [21] introduced the stable matching problem and provided a polynomial time algorithm known as the deferred acceptance (DA) algorithm. The DA algorithm computes an assignment such that there is no pair of agents that would simultaneously prefer to be paired to each other rather than being in their current assignment; this is known as a stable matching. In practice, the DA mechanism has been extensively used to improve admission processes, e.g., see [1,13]. For further details on stable matching mechanisms, see [44,32]. In general, the main focus of the literature has been on finding the maximum cardinality stable matching, which can be efficiently obtained when there are incomplete preference lists 1 without ties or complete preference lists that include ties. 2 Once we assume both, incomplete lists and ties, the problem of finding the maximum cardinality stable matching becomes NP-hard even under very restrictive conditions [33]. In terms of approximation ratios, the best known factor is 3 2 [28] and the best lower bound is 33 29 [52]. The design of a stable matching mechanism, when the number of participants of one side is increased, has already been investigated in the past. If hospitals have capacity one, this is known as the entry comparative static in the stable marriage problem. In this setting, the two sides are traditionally called women and men. In [26,22,44], the authors proved that when a new woman is added to the instance, all men are matched weakly better. Recently, Kominers [30] extended this result to the many-to-one stable matching problem. On a similar path, Balinski and Sonmez [11] proved that the DA method is invariant with respect to residents who improve their score in the ranking lists, i.e., instead of introducing a new agent, the ranking of an existing agent is improved. A substantial part of the literature has focused on strategy-proof matching mechanisms, i.e., on matching mechanisms that incentivize participants to reveal their true preferences. Sonmez [47] proved that hospitals can manipulate the stable matching in their favor by falsely reporting a reduced capacity. Moreover, Romm [38] proved that the stable matching mechanism can still be manipulated even if the reported capacities are enforced during the admission process. Digressing from the entry comparative static approach and strategy-proof mechanisms, Bobbio et al. [15] considered the allocation of extra capacities to hospitals as a decision variable rather than a parameter. Alternatively to the integer programming approach proposed by the authors, a solution methodology for optimizing the outcome for the residents is devised in [4]. Our work focuses on providing the computational complexity landscape of the problem tackled in [15,4], its counterpart where existent hospital spots are removed, and other variants. In [17,6], the problem of capacity expansion was addressed in the framework of residents' interviews for the hospital admission and was solved through an optimal portfolio choice. In [31], the authors studied how the expansion of interviews (only on the residents' side) impacts the final matching. Kamada and Kojima [25] studied matching mechanisms that impose regional quotas for the Japan Residency Matching Program. Our work differs from this, since we look to optimize the quotas rather than imposing them. A problem related to ours was addressed in [51]. As part of their problem's input, they considered a profile of "resources" that can be allocated to "projects" (which would be the hospitals). The authors concentrated on designing strategy-proof and efficient mechanisms. In the capacity variation problems considered in our work, the resources are decision variables rather than part of the input. Even if we translate the capacity expansion problem into the setting in [51], the input size would be exponential. In [35], the authors studied the presence of couples in matching markets. The authors proved that by adding at most 9 extra capacities in a market with couples, the existence of a stable matching is guaranteed. The problems studied in our work also relate to the literature on resource augmentation [36,24,45], where the goal is to design algorithms whose performance is compared to the benchmark that takes decisions with complete information but with a deficit in resources. For more details, we refer the interested reader to [45] and the references therein. Finally, in the context of ride sharing, it has been shown that the expansion of the capacities of the drivers by ǫ leads to a substantial reduction in the cost of the matching [5], which, however, is not required to be stable. Contributions and Organization This paper is organized as follows. In Section 2, we introduce the formal notation, the problem of expanding capacities (Problem 1), and the problem of reducing capacities (Problem 2). In Section 3, our main focus is to establish the complexity of Problem 1. To achieve this result, we first prove, in Corollary 3.2, that determining the resident-optimal stable matching in the presence of ties is NP-hard and is not approximable withinn 1−ε , for any ε > 0, wheren is the number of residents. This result puts a boundary on the computability of the resident-optimal stable matching, which is well known to be polynomially solvable when there are no ties. The remainder of Section 3 is devoted to study the complexity of the capacity expansion problem. All our results are proven in the special case in which the capacity of every hospital is at most 1. Indeed, even under very restrictive assumptions, finding the allocation of extra capacities to the hospitals that minimizes the average hospital rank of the residents is NP-hard, and for any ε > 0, it cannot be approximated within a factor of (n/2) (1−ε)/2 unless P=NP (Theorem 3.1). This result may be counter-intuitive because, in the vanilla version of HR, we can compute in polynomial-time the resident-optimal stable matching, which is equivalent to the one that minimizes the average hospital rank. Our complexity proof is based on a new structure that we call village. Each village is assigned some extra capacities, and the preferences of the hospitals and residents in a village ensure that the extra capacities can be optimally allocated only in a specific way. In Section 4, we study the capacity reduction problem. We prove that this problem is NPhard, and for any ε > 0, it cannot be approximated within a factor of (n/2) (1−ε)/2 unless P=NP (Theorem 4.1). The proof follows a similar reasoning as in Theorem 3.1. We exploit again the structure of the village, in which now every relevant hospital has capacity 1. In Section 5, we study several variants of Problems 1 and 2. Specifically, we partition the set of hospitals and allocate (remove) a certain amount of capacities to (from) each set of the partition. Theorem 5.2 shows that, even when we partition the set of hospitals and we allocate to or remove from each set at most one spot, finding the optimal allocation is an NP-hard problem. Moreover, we prove that the optimization version of the problem is not approximable within a factor ofn 1−ε , for any ε > 0 (Theorem 5.2). The equivalent results for the reduction problem are shown in Theorem 5.4. Finally, we provide similar results to the variant of the problems that consider as an objective function the cardinality of the matching, Theorems 5.3 and 5.5, respectively. Finally, some conclusions are drawn in Section 6. A summary of our results and relevant results from the literature can be found in Table 1. Preliminaries and Problem Definition The many-to-one stable matching problem consists of a set of residents R = {i 1 , . . . , i \|R\| }, a set of hospitals H = {j 1 , . . . , j \|H\| } and a set of edges E between R and H. A resident and a hospital are linked by an edge in E if they deem each other acceptable. In this work, we assume (if not otherwise stated) that every resident-hospital pair is acceptable, i.e., E = R × H. Each hospital j ∈ H has an non-negative integer capacity c j ∈ Z + that represents the maximum number of residents that hospital j can admit. In this setting, a matching M is a subset of E in which each hospital j appears in at most c j pairs and each resident appears in at most 1 pair. We denote by M (i) and M (j) the hospital assigned to resident i and the subset of residents assigned to hospital j, respectively. An instance Γ of the HR problem corresponds to a tuple Γ = R, H, ≻, c , where c ∈ Z H + is the vector of capacities and ≻ corresponds to the profile of preferences that residents have over hospitals and vice-versa. Specifically, we assume that the preference list of each resident is a linear order. We use the notation j ≻ i j ′ to describe when resident i prefers hospital j over hospital j ′ . We assume that every agent is individually rational, i.e., every agent prefers the proposed assignment than to be unmatched. Concerning the preference list of every hospital, we assume it is a responsive linear order over the power-set of the residents [41]. 3 A responsive linear order facilitates the description of the preference list, since we only have to focus on the linear order over single residents. We write i ≻ j i ′ to denote when hospital j prefers resident i over i ′ . Whenever the context is clear, we drop the subscript in ≻. We emphasize that in the HR problem, unless otherwise stated, the preference lists are complete and strict (there are no ties). Under these assumptions, the length of the preference list of each agent, hospital or resident, is exactly the size of the other side of the bipartition. Therefore, preference lists can be interpreted in terms of rankings. Formally, for each resident i ∈ R and hospital j ∈ H, we denote by rank i (j) ∈ {1, . . . , \|H\|} the rank of hospital j in the list of resident i. This means, for example, that the most preferred hospital has the lowest ranking. Analogously, we define rank j (i) ∈ {1, . . . , \|R\|} for all j ∈ H, i ∈ R. Given a matching M , we say that a pair (i, j) ∈ E is a blocking pair if the following two conditions are satisfied: (1) resident i is unassigned or prefers hospital j over M (i), and (2) \|M (j)\| < c j or hospital j prefers resident i over at least one resident in M (j). The matching M is said to be stable if it does not admit a blocking pair. Gale and Shapley [21] showed that every instance of the HR problem admits a stable matching that can be found in polynomial time by the deferred acceptance method, also known simply as the Gale-Shapley algorithm. In particular, this algorithm can be designed to prioritize the residents in the following sense: Let M and M ′ be two different stable matchings, we say that a resident i weakly prefers M over M ′ if M (i) ≻ i M ′ (i) or M (i) = M ′ (i). Then, the DA algorithm can be adapted to compute the unique stable matching that is weakly preferred by all residents over all the other possible stable matchings. Such unique stable matching is called resident-optimal. Notation To ease the exposition, we avoid using the symbol ≻ when presenting a preference list, instead we simply separate agents by "," and use the convention that the leftmost agents are the most preferred. For instance, we will represent the preference list w ≻ w ′ ≻ w ′′ as w, w ′ , w ′′ . Throughout this work, for a given integer k ≥ 1, we use the shorthand [k] := {1, . . . , k}. Finally, otherwise stated, we use indices i for residents and j for hospitals. Problem Definition In this work, we focus on the stable matchings that minimize the average hospital rank. Recall that we denote by rank i (j) the position of hospital j in the list of resident i. The average hospital rank of a matching M is defined as AvgRank(M ) := (i,j)∈M rank i (j),(1) where, to ease the exposition, we do not divide by the total number of hospitals. We consider Expression (1) as the objective function, since a basic result states that a stable matching M is resident-optimal if, and only if, it is a stable matching of minimum average hospital rank [15]. In our first problem, proposed in [15], we aim to improve the allocation of residents by increasing the capacity of the hospitals. For a non-negative vector t ∈ Z H + , we denote by Γ t = R, H, ≻, c + t an instance of the CA problem in which the capacity of each hospital j ∈ H is c j + t j . Observe that Γ 0 corresponds to the original instance Γ with no capacity expansion. Formally, we define the capacity expansion problem as follows. Problem 1 (Min-Avg exp HR). instance: A CA instance Γ = R, H, ≻, c , a non-negative integer expansion budget B ∈ Z + , and a target value K ∈ Z + . question: Is there a non-negative vector t ∈ Z H + and a matching M t such that AvgRank(M t ) ≤ K, where t satisfies j∈H t j ≤ B and M t is a stable matching in instance Γ t ? Given parameters B and K, Problem 1 aims to determine the existence of an allocation of B extra spots through vector t such that there is a stable matching with an average hospital rank of at most K. Throughout the paper, we assume that the total capacity of the hospitals is at least the total number of residents, i.e., j∈H c j ≥ \|R\|. If this assumption does not hold, we must define the cost of an un-assigned resident. A natural option is to add an artificial hospital with large capacity such that is ranked last by every resident. Therefore, un-assigned residents will be allocated in the artificial hospital whose rank is \|H\| + 1. Note that as a consequence of our assumption, j∈H c j ≥ \|R\|, there may be hospitals that do not fill their quota. In our second problem, we aim to find the reduction of the hospitals' capacities such that the final average hospital rank is the lowest possible, i.e., that has the least impact on the allocation of residents. As before, for a non-negative vector t ∈ Z H + , we denote by Γ −t = R, H, ≻, c − t an instance of the CA problem in which the capacity of each hospital j ∈ H is c j − t j . Formally, we define our second problem as follows. Problem 2 (Min-Avg red HR). instance: A CA instance Γ = R, H, ≻, c , a non-negative integer reduction budget B ∈ Z + such that −B + j∈H c j ≥ \|R\| and a target value K ∈ Z + . question: Is there a non-negative vector t ∈ Z H + and a matching M t such that AvgRank(M t ) ≤ K, where t satisfies j∈H t j ≥ B and (c j − t j ) ≥ 0 for every j ∈ H, and M t is a stable matching in instance Γ −t ? Note that in Problem 2, we have the additional constraint that the capacity of every hospital should remain non-negative after removing spots, i.e., c j − t j ≥ 0 for all j ∈ H. We further assume that the sum of the reduced hospitals' capacities is greater or equal than the number of residents, i.e., −B + j∈H c j ≥ \|R\|. As in Problem 1, if this assumption does not hold, we can transform the instance by adding an artificial hospital with a large capacity (which is ranked last in every resident's list) and by allowing the reduction of capacities to the original hospitals only. The Capacity Expansion Problem Our main result in this section establishes the computational complexity and inapproximability of Problem 1. Denote by Min-Avg exp HR opt the optimization version of Problem 1, i.e., the problem of finding the allocation of extra spots and the stable matching in the expanded instance that minimizes AvgRank. Formally, our main result is the following. Theorem 3.1. Min-Avg exp HR is NP-complete. Moreover, for any ε > 0, Min-Avg exp HR opt cannot be approximated within a factor of (n/2) (1−ε)/2 , wheren is the number of residents, unless P=NP. To give some insights on the difficulty of Problem 1, we first present an intuitive approach when B = 1 and we show that it does not always provide an optimal solution. In real life instances, certain hospitals may be "more popular" than others, namely, they are preferred by well-known voting methods such as Majority or Borda count [54]. Thus, when B = 1, a natural approach is to assign the additional spot to the hospital that is preferred by the majority or Borda count. However, as the following example shows, this is not necessarily optimal. Counterexample for the Majority and Borda count Let R = {i 1 , i 2 , i 3 , i 4 , i 5 , i 6 } and H = {j 1 , j 2 , j 3 , j 4 }. We assume that all hospitals have the same preference list: i 1 ≻ i 2 ≻ · · · ≻ i 6 . Hospitals j 1 , j 2 and j 3 have each capacity 1, and hospital j 4 has capacity 3. Resident i 1 ranks hospitals as j 2 ≻ j 1 ≻ j 3 ≻ j 4 . Resident i 2 ranks hospitals as j 2 ≻ j 3 ≻ j 1 ≻ j 4 . Resident i 3 ranks hospitals as j 3 ≻ j 2 ≻ j 4 ≻ j 1 . Residents i 4 , i 5 and i 6 rank hospitals as j 1 ≻ j 4 ≻ j 3 ≻ j 2 . The resident-optimal stable matching is M = {(i 1 , j 2 ), (i 2 , j 3 ), (i 3 , j 4 ), (i 4 , j 1 ), (i 5 , j 4 ), (i 6 , j 4 )} with AvgRank(M ) = 11. Now, consider Problem 1 with B = 1 and K = 9. For this instance, an intuitive solution is allocating the extra spot to j 1 , which is the most preferred hospital according to both Majority vote and Borda vote; the allocation of one extra capacity to j 1 is sub-optimal. Indeed, if we expand the capacity c j 1 = 1 to c j 1 = 2, then resident i 5 would be assigned to hospital j 1 , which leaves an extra spot in hospital j 4 . This solution reduces the average hospital rank by 1 unit and the resulting matching does not meet the target K = 9. Instead, if we expand the capacity of j 2 to 2, then resident i 2 is admitted by hospital j 2 , leaving an empty spot in hospital j 3 that is filled by resident i 3 ; the resulting matching has an average hospital rank of 9. As the previous example shows, the allocation of one extra spot is not trivial when we try to solve it by just looking at the residents' preferences. However, we can still solve this problem in polynomial time by doing an exhaustive search in combination with the DA algorithm. To achieve this, we compute the resident-optimal stable matching using the DA mechanism in the instance Γ t with t = 1 j for each j ∈ H, where 1 j ∈ {0, 1} H is the indicator vector whose j-th component is 1 and the rest is 0. Once we obtain the cost for each j ∈ H, we output the resident-optimal stable matching of minimum average hospital rank. Finally, we compare with our target K to decide if such an allocation exists or not. Since the DA algorithm's runtime complexity is O(\|R\| · \|H\|) [21], then this exhaustive search runs in O(\|R\| · \|H\| 2 ). Whether this can be improved remains an open question. To prove Theorem 3.1, we first study a variant of the egalitarian stable marriage problem [32]. Formally, the stable marriage (SM) problem corresponds to the HR problem where c j = 1 for all j ∈ H. We use SMT to indicate the version of SM when ties are present in the preference lists. A tie appears when an agent allocates in the same position of the list two different participants of the opposite side. For example, if the preference list for resident i is j 4 , (j 1 , j 3 ), j 2 , 4 then the rankings are rank i (j 4 ) = 1, rank i (j) = 2 for j ∈ {j 1 , j 3 } and rank i (j 2 ) = 4. For the SMT problem, stable matchings can be defined in several ways, but in this paper we consider weak stability [32]. Formally, a matching M is weakly stable if there is no pair such that both agents strictly prefer each other over their allocation in M . An egalitarian stable matching is a stable matching that minimizes the total sum of the rankings, i.e., (i,j)∈M [rank i (j) + rank j (i)]. Manlove et al. [33] proved that the problem of finding the egalitarian stable matching for SMT is not approximable withinn 1−ǫ , for any ǫ > 0, unless P = NP, wheren is the size of one side of the bipartition. For more details, we refer to Theorem 7 in [33]. Let us define the following variant of the egalitarian SMT problem. Problem 3 (Min-w SMT). Instance: An SMT instance Γ = R, H, ≻, c with c j = 1 for all j ∈ H and a target value K ∈ Z + . Question: Is there a weakly stable matching M such that AvgRank(M ) ≤ K? We use Min-w SMT opt to denote the optimization version of Min-w SMT, i.e., the problem of finding a weakly stable matching that minimizes AvgRank. Using the ideas in [33], we can obtain the following result for Min-w SMT. Corollary 3.2. Min-w SMT is NP-complete. Moreover, for any ε > 0, Min-w SMT opt is not approximable within a factor ofn 1−ε , unless P = NP, wheren = \|H\|. This result holds even if ties are only in one side, there is at most one tie per list, and each tie is of length two. For completeness, we provide the proof of this corollary in the Appendix. Let us now provide a sketch of the steps to prove Theorem 3.1. Given an instance Γ of Min-w SMT, we construct the following instanceΓ of Min-Avg exp HR: For every hospital in Γ that has ties in its preference list, we create a village of residents and hospitals with different capacities and strict preferences. In Lemma 3.4, we prove that the construction can be done in polynomial time and it selects a special stable matching in the new instance. Let M be the stable matching of minimum average hospital rank in Min-w SMT; in Lemma 3.5, we prove that the stable matchingM t inΓ is in fact the stable matching of minimum average hospital rank in Min-Avg exp HR. Design of the Instance First, we observe that Min-w SMT is NP-complete even if ties occur only among the preference lists of residents and in each preference list there is at most one tie of length 2, and it is positioned at the head of the list. For more details, we refer to Remark A.1 in the Appendix. Throughout this section, we assume that an instance of SMT satisfies these properties. Now, we introduce a polynomial transformation from such an instance of Min-w SMT to an instance of Min-Avg exp HR. Let Γ = R, H, ≻, c be an instance of Min-w SMT such that c j = 1 for all j ∈ H and \|R\| = \|H\| = n. Let L ≤ n be the number of residents with ties in their preference list. The set of residents is partitioned in two sets R = R ′ ∪ R ′′ where R ′ is the set of residents with a tie of length two at the head of the preference list and R ′′ is the set of residents with a strict preference list. Henceforth, we fix an ordering of the residents in R and denote R ′ = {i 1 , . . . , i L } and R ′′ = {i L+1 , . . . , i n }. Since preference lists are complete, observe that in any weakly stable matching every resident is matched. 5 In the following, we create an instanceΓ = R ,Ĥ,≻,ĉ of Min-Avg exp HR with a specific target value and budget. • a resident y ℓ ; • two sets of hospitals V ℓ,0 = {v 0 ℓ,h } h∈[n] and V ℓ, 1 = {v 1 ℓ,h } h∈[n] . Let V ℓ = V ℓ,0 ∪ V ℓ,1 . We denote as B i ℓ the village associated with resident i ℓ ∈ R ′ and V := L ℓ=1 V ℓ . In summary, we haveR = R ′′ ∪ {W ℓ , y ℓ } ℓ∈[L] andĤ = H ∪ H 0 ∪ X ∪ V ∪ Z. Capacity vector. Now, let us construct the capacity vector: For each hospital v ∈ V ∪ H 0 ∪ Z, we considerĉ v = 0; for every other hospital j ∈ H ∪ X , we takeĉ j = 1. Preference lists. We now proceed to construct the preference lists inΓ. Given a resident i ℓ ∈ R ′ with ℓ ∈ [L], let (j σ 1 , j σ 2 ), j σ 3 . . . be her ranking of the hospitals in the original instance Γ (the parenthesis symbolizes the tie at the head of the list). We provide the preference lists of the residents and hospitals in village B i ℓ with ℓ ∈ [L], namely w ℓ,1 : v 1 ℓ,1 , V ℓ,0 \ {v 0 ℓ,1 }, H 0 n−1 , j σ 1 , Z, . . . , X w ℓ,2 : v 1 ℓ,2 , V ℓ,0 \ {v 0 ℓ,2 }, H 0 n , j σ 2 , Z, . . . , X w ℓ,h : v 1 ℓ,h , V ℓ,0 \ {v 0 ℓ,h }, H 0 (n−1)·h , j σ h , Z, . . . , X h ∈ {3, . . . , n} y ℓ : v 0 ℓ,2 , v 0 ℓ,1 , v 0 ℓ,3 , . . . , v 0 ℓ,n , Z, . . . , X v 1 ℓ,h : w ℓ,h , . . . h ∈ [n] v 0 ℓ,h : W ℓ \ {w ℓ,h }, y ℓ , . . . h ∈ [n] where H 0 h = {j 0 1 , . . . , j 0 h }. The purpose of positioning of set H 0 h in the preference lists of the residents w ℓ,h for h ∈ [n], is to ensure that we can mimic rank i ℓ (j), for every j ∈ H, of the original instance. The symbol ". . . " means that the remaining agents on the other side of the bipartition are ranked strictly and arbitrarily. Now, we present the preference list of the copy of every hospital j ∈ H and every resident i ∈ R ′′ in the new instance. We modify the original preference list of j ∈ H by substituting every resident i ℓ ∈ R ′ (for ℓ ∈ [L]) with resident w ℓ,r , where r = rank i ℓ (j) is the rank of j in the list of i ℓ . If j is ranked first by i ℓ , then we substitute i ℓ with resident w ℓ,1 when j is the first hospital listed in the tie, otherwise with w ℓ,2 . Then, hospital j ranks arbitrarily a strict ordering of the remaining residents. Let i ∈ R ′′ and let j σ 1 , . . . , j σn be her strict and complete preference list in Γ. The preference list of i in our new instanceΓ is H 0,1 n−1 , j σ 1 , H 0,2 n−1 , j σ 2 , . . . , H 0,n n−1 , j σn , Z . . . The preference lists of hospitals in X , H 0 and Z are arbitrary. The sole purpose of the hospitals in X is to ensure that there are sufficient capacities for all the residents. The scope of the set H 0 is to help mimic the original ranking of the copies of the hospitals. The set of hospitals Z is introduced to make costly certain re-allocation of extra spots. The set V is used to leverage the stability and ensure that different allocations of extra spots yield sub-optimal results. Target value and budget. Given target K and L residents with a tie in their list in an instance of Min-w SMT, we define targetK = n · K + 2n · L and budget B = L · n for Min-Avg exp HR, where L is the number of residents with a tie in their list. Finally, note thatR = R ′′ ∪ {W ℓ , y ℓ } ℓ∈[L] andĤ = H ∪ H 0 ∪ X ∪ V ∪ Z. Therefore, the instancê Γ consists of ((n − L) + L · (n + 1)) + (n + n 2 + n · L + L · 2n + n 3 ) = n 3 + n 2 + 4n · L + 2n residents and hospitals, which is O(n 3 ); therefore the construction can be done in polynomial time. Remark 3.3. Note that if no extra spots are assigned in our new instance, the set of hospitals X ensures that the residents are always matched. 6 Matching the residents to the hospitals in X leads to a higher average hospital rank. In the following section, we will prove that it is optimal to assign L · n extra capacities to the hospitals in V, whose initial capacity is zero. Useful Lemmata For this section, recall that we are considering budget B = L · n, where L = \|R ′ \| is the number of residents with a tie in their preference list. Proof. Let M be a (complete) weakly stable matching in Γ. Recall that R ′ = {i 1 , . . . , i L } is the set of residents in Γ with a single tie at the head of the list. Define the following set of indices in M : Idx(M ) = {(ℓ, r) : r = rank i ℓ (j), (i ℓ , j) ∈ M ∩ (R ′ × H)}. The set Idx(M ) contains the information of the pairs R ′ × H that are matched in M . Given M and Idx, we now define the following sets that will be helpful in this proof. First, define the set of residents W M = {w ℓ,r ∈ W : (ℓ, r) ∈ Idx(M )}, and define the sets of hospitals V M,0 = {v 0 ℓ,r ∈ V : (ℓ, r) ∈ Idx(M )}, V M,1 = {v 1 ℓ,h : (ℓ, r) ∈ Idx(M ), h ∈ [n] \ {r}}. We now provide an allocation of extra spots t with a total budget B = L · n and a stable matchinĝ M t inΓ t . • Allocation of extra spots. We assign one extra position to each hospital in V M,0 ∪ V M,1 . For the rest of the hospitals, we assign 0 extra capacity. We denote this allocation t. Formally, we have t u = 1 u ∈ V M,0 ∪ V M,1 0 otherwise Since L = \|R ′ \|, all of the extra positions B = L · n were used. • Matching. For each (ℓ, r) ∈ Idx(M ) with j such that r = rank i ℓ (j) in Γ, we match the following pairs inM t : (w ℓ,r , j), (y ℓ , v 0 ℓ,r ), and (w ℓ,h , v 1 ℓ,h ) for h ∈ [n] \ {r}. Note that if j is ranked first by i ℓ , the hospital is listed first or second in the tie. If j is listed first, then r = 1 and we match the pair (w ℓ,1 , j), otherwise, r = 2 and we match the pair (w ℓ,2 , j). For each (i, j) ∈ M with i ∈ R ′′ , we match the pair (i, j) inM t , where j is the corresponding copy in H; recall that R ′′ is the set of residents with a strict preference list. Formally, matchingM t is as follows:M t ={(i, j) : (i, j) ∈ M ∩ (R ′′ × H)} ∪ {(w ℓ,r , j) : r = rank i ℓ (j), (i ℓ , j) ∈ M ∩ (R ′ × H)} ∪ {(y ℓ , v 0 ℓ,r ) : (ℓ, r) ∈ Idx(M )} ∪ {(w ℓ,h , v 1 ℓ,h ) : (ℓ, r) ∈ Idx(M ), h ∈ [n] \ {r}}. Let us verify thatM t is a stable matchingΓ t . First, note that residents i ∈ R ′′ and hospitals j ∈ H cannot create blocking pairs because of their stability in M . Now, let us check the stability of the pairs in each village B i ℓ , where i ℓ ∈ R ′ with ℓ ∈ [L]. Consider j ∈Ĥ and assume for now that j is ranked first by i ℓ , i.e., j is listed first or second in the tie: j = j r with r = 1, 2. The pairs matched in village B i ℓ are (w ℓ,r , j), (y ℓ , v 0 ℓ,r ) and (w ℓ,h , v 1 ℓ,h ) for h ∈ [n] \ {r}. • The pair (w ℓ,r , j) is clearly stable; in fact, w ℓ,r cannot be matched to any of the hospitals in v 1 ℓ,r and V ℓ,0 \ {v 0 ℓ,r } because they have capacity 0. If r = 2, w ℓ,r cannot be matched to any hospital in H 0 n because they all have capacity 0. Also, j cannot create a blocking pair. Indeed, all the residents w ℓ ′ ,r ′ ranked in its preference list before w ℓ,r are matched to hospitals of the form v 1 ℓ ′ ,r ′ that they rank first. The case in which r = 1 is analogous. • For h ∈ [n] \ {r}, w ℓ,h ranks v 1 ℓ,h first and vice-versa, hence the n − 1 pairs (w ℓ,h , v 1 ℓ,h ) are stable. • If r = 2, then y ℓ ranks v 0 ℓ,r first, and v 0 ℓ,r cannot be matched to any of the residents in W ℓ \ {w ℓ,r } because of the previous point; therefore, pairs (y ℓ , v 0 ℓ,r ) are stable when r = 2. If r = 1, then y ℓ ranks v 0 ℓ,r second, and y ℓ cannot be matched to v 0 ℓ,2 because it has capacity 0. As before, v 0 ℓ,r cannot create a blocking pair with any of the residents in W ℓ \ {w ℓ,1 } because they are matched to their most preferred hospital. Therefore, the pair (y ℓ , v 0 ℓ,r ) is also stable when r = 1. The case in which j is ranked third or more by i ℓ is analogous to the case in which j is of the form j r with r = 2 in i ℓ 's preference list. Therefore,M t is a stable matching inΓ t . Next, we compute the average hospital rank in M andM t . In M , we can distinguish whether a resident is matched to a hospital ranked first or not, and we can distinguish if a resident is in R ′ or R ′′ . Let K ′′ be the average hospital rank of residents in R ′′ , K t be the average hospital rank of the residents in R ′ that are matched to a hospital in their ties, and K s be the average hospital rank of the residents in R ′ that are matched to a hospital they rank third or more. Note that K t is also the number of residents matched to a hospital they rank first. The average hospital rank of M is K M = K ′′ + K t + K s . We now show that AvgRank(M t ) is n · K ′′ + n · K t + n · K s + (n + 1) · (K t + L − K t ) + L · (n − 1): • The first term, n · K ′′ , is given by the contribution from the residents in R ′′ . • The second term is given by the pairs (w ℓ,r , j), (y ℓ , v 0 ℓ,r ) (for r = 1 or r = 2) in the villages B i ℓ of the residents i ℓ in M that are matched to a hospital they rank first. • The third contribution is given by the pairs (w ℓ,r , j), (y ℓ , v 0 ℓ,r ) (for r ≥ 3) in the villages B i ℓ of the residents i ℓ in M that are matched to a hospital they rank third or more. • The forth term, L · (n − 1), is given by the pairs of the form (w ℓ,h , v 1 ℓ,h ) for h = r, of which there are n − 1 in each of the L villages. If we rearrange the terms, we obtain AvgRank(M t ) = n · K ′′ + n · K t + n · K s + (n + 1) · (K t + L − K t ) + L · (n − 1) = n · K M + 2n · L. In the next result, we show that the allocation vector and the stable matching constructed in Lemma 3.4 correspond to the solution with the minimum average hospital rank, as long as the original matching is of minimum average hospital rank. Proof. Let M be a stable matching in Γ of minimum average hospital rank. Recall the instanceΓ constructed in Section 3.1, the allocation t u = 1 u ∈ V M,0 ∪ V M,1 0 otherwise, and the matchingM t = {(i, j) : (i, j) ∈ M ∩ (R ′′ × H)} ∪ {(w ℓ,r , j) : r = rank i ℓ (j), (i ℓ , j) ∈ M ∩ (R ′ × H)} ∪ {(y ℓ , v 0 ℓ,r ) : (ℓ, r) ∈ Idx(M )} ∪ {(w ℓ,h , v 1 ℓ,h ) : (ℓ, r) ∈ Idx(M ), h ∈ [n] \ {r}}, constructed in Lemma 3.4. Denote byK = nK M + 2nL, which is the average rank ofM t inΓ. Now, we will prove that any other feasible allocationt with total budget B = L · n and any stable matchingMt in the expanded instanceΓt have AvgRank(Mt) ≥K. 7 Given allocation t, note that it is not optimal to move one extra capacity from a hospital v s ℓ,h to a hospital j ∈ H ∪ H 0 ∪ X ∪ Z for s = 0, 1. Indeed, X already has B positions available, but since it is at the end of the preference list of every resident, it would be sub-optimal to match a resident to a hospital in it. Similarly, it would be sub-optimal to allocate an extra-capacity to Z, since all the residents are already matched to a hospital they prefer to any hospital in Z. Regarding the hospitals in H 0 , let us assume we move a capacity from v s ℓ,h to a hospital h 0 ∈ H 0 with s = 0, 1 and h ∈ [n]. If v s ℓ,h was matched to y ℓ , then y ℓ will be matched to some hospital ranked after Z with a rank of at least n 3 , making the transfer sub-optimal. Otherwise, v s ℓ,h was matched to resident w ℓ,h ; therefore, resident w ℓ,h will be matched to a certain v 0 ℓ,h ′ , which was previously matched to y ℓ ; hence, the same reasoning just seen for y ℓ applies, making the transfer of the extra spots sub-optimal. The additional cost in the average rank obtained by moving a position from v s ℓ,h to a hospital in H follows the same reasoning just outlined for H 0 . Therefore, this is also a sub-optimal reallocation, and it is optimal to assign all the extra capacities to hospitals in V. Consequently, B residents are matched to hospitals in V and the remaining residents are matched to hospitals in H ∪ H 0 . Given the fact above, we only have to focus on feasible allocations in hospitals that belong to V. In the following, we analyze why a different allocation of extra capacities in V does not lead to a stable matching with a lower average hospital rank. Since M is a stable matching of minimum average hospital rank in Γ and each resident in the new instanceΓ ranks in the first n 3 positions only one hospital in H, a re-allocation of extra capacities within the same village would result in a matching with a worse objective within the village. Therefore, the re-allocation of extra capacities that could improve the objective is the one obtained by transferring extra positions from one village to another village. Consider i ℓ , i ℓ ′ ∈ R ′ . We now analyze the effects of moving one extra capacity from village B i ℓ ′ to village B i ℓ . The reason why we are analyzing these transfers of extra capacities is because the corresponding residents are not necessarily matched with their top choice so their ranking and the overall average ranking may improve. • From v 0 ℓ ′ ,r ′ to v 0 ℓ,2 . Note that, by assumption, both residents y ℓ ′ and y ℓ are matched inM t to v 0 ℓ ′ ,r ′ and v 0 ℓ,r (for a certain r ≤ n), respectively. If we move one extra position from v 0 ℓ ′ ,r ′ to v 0 ℓ,2 , then w ℓ,r un-matches from j (r = rank i ℓ (j)) and matches to v 0 ℓ,2 , thus providing a reduction in the objective value between n and n 2 . The preference list of v 0 ℓ,2 prevents y ℓ to be matched to v 0 ℓ,2 because w ℓ,r is more preferred and w ℓ,r prefers v 0 ℓ,2 over j. As a consequence, j is un-matched and is re-assigned to y ℓ ′ (who was un-matched as a consequence of removing the extra spot of v 0 ℓ ′ ,r ′ ). By matching y ℓ ′ to j, the objective value increases by at least n 3 . Therefore, it is sub-optimal to move an extra capacity in this way. • All the remaining cases follow a similar reasoning. Therefore, there is no allocationt = t and a stable matchingMt with an objective value strictly lower thanK. Min-Avg exp HR is NP-complete In the following, we prove the main result of this section, Theorem 3.1. Proof of Theorem 3.1. Min-Avg exp HR is clearly in NP since given t and a matchingM t in in-stanceΓ t , we can verify in polynomial time whetherM t is stable, whether the budget B = L · n is satisfied and whether its objective value is less than the target value. We now show that Min-Avg exp HR is NP-complete. From Corollary 3.2, we know that Min-w SMT is NP-complete. Consider the reduction given in Section 3.1. In the constructed instanceΓ of Min-Avg exp HR, we set the budget to B = L · n and the target value toK = n · K + 2n · L, where K is the target value of the instance of Min-w SMT. First, suppose that the answer to the instance of Min-w SMT is NO, i.e., there is no weakly stable matching M with an average hospital rank less or equal than K. Let M be a weakly stable matching with minimum average hospital rank K M ; note that K M > K. Next, we prove that there is no allocation of extra positions and a stable matching in the respective instanceΓ of Min-Avg exp HR with an objective value less or equal than nK + 2nL. Indeed, in Lemma 3.4, we show that there is an allocation t and a stable matchingM t with AvgRank(M t ) = nK M + 2nL. In Lemma 3.5, we prove that these are the best solutions since M is the optimal matching. Therefore, nK M + 2nL > nK + 2nL =K, which means that the answer for the instance of Min-Avg exp HR is also NO. On the other hand, consider a YES instance of Min-w SMT. Then, there is a weakly stable matching M with an average hospital rank of K M ≤ K. Therefore, the allocation t and the stable matchingM t inΓ constructed in Lemma 3.4 have an objective value of nK M +2nL ≤ nK+2nL =K. Hence, the instance of Min-Avg exp HR has a YES answer. Let us prove now that, for any ε, Min-Avg exp HR opt is not approximable within a factor of (n/2) 1−ǫ 2 , wheren is the number of residents, unless P=NP. Consider an instance Γ of Min-w SMT with n residents and L ≥ 1 of them with a tie in their preference list. Let M yes and M no be the stable matchings of minimum average hospital rank for the cases in which the answer of the decision problem Min-w SMT is YES and NO, respectively. Corollary 3.2 implies that, for any ε > 0, AvgRank(M no ) ≥ n 1−ǫ · AvgRank(M yes ). Now, consider the reduction presented in Section 3.1 from instance Γ to an instanceΓ of Min-Avg exp HR. Lemma 3.5 implies that there are allocations t and t ′ , and matchingsM yes t andM no t ′ for the respective YES and NO answers of Min-Avg exp HR such that AvgRank(M yes t ) = n · AvgRank(M yes ) + 2nL AvgRank(M no t ′ ) = n · AvgRank(M no ) + 2nL. Recall that the reduction in Section 3.1 constructsΓ withn := \|R\| ≤ 2n 2 residents. Then, for any ε > 0, we have AvgRank(M no t ′ ) AvgRank(M yes t ) = n · AvgRank(M no ) + 2nL n · AvgRank(M yes ) + 2nL ≥ AvgRank(M no ) AvgRank(M yes ) ≥ n 1−ǫ ≥ n 2 1−ε 2 , where the first inequality is because f (x) = (a+ x)/(b+ x) is increasing when b > a. This completes the proof. Note that the proof can be slightly modified to obtain a similar result for HR with incomplete preference lists, as long as the condition j∈H c j ≥ \|R\| is met (e.g., we can remove one hospital in X from the preference list of a resident). The Capacity Reduction Problem In this section, we focus on Problem 2 that looks for the reduction of capacities such that the residents' allocations are impacted the least. Our main result establishes the computational complexity of this problem. Formally, our result is the following. Theorem 4.1. Min-Avg red HR is NP-complete. Moreover, for any ε > 0, Min-Avg red HR opt cannot be approximated within a factor of (n/2) (1−ε)/2 , wheren is the number of residents, unless P=NP. Proof. We restrict our analysis to the case in which hospitals' capacities are all 1. Recall that Problem 2 assumes that reducing the capacities of hospitals does not leave any resident un-assigned. First, clearly Min-Avg red HR is in NP, since for a given vector t and a matching M t , we can verify in polynomial time whether t satisfies the lower bound on the number of spots to be removed, whether M t is stable in Γ −t and if the target value is attained. Now, we concentrate on showing that the problem is NP-complete. The rest of the proof follows the same idea than the proof of Theorem 3.1. We build a reduction from an instance Γ of Min-w SMT into an instanceΓ of Min-Avg red HR. We assume that Γ satisfies \|R\| = \|H\| = n, ties occur only in residents' lists, and each of their preference list has at most one tie of length 2 positioned at the head of it. Recall also that we denoted by R ′ the set of residents with a tie in their preference list and by R ′′ the set of residents with strict preference lists. The correspondingΓ is defined as in the reduction presented in the proof of Theorem 3.1, with the following difference: • For every village B ℓ defined for i ℓ ∈ R ′ : Each hospital v 1 ℓ,h for h ∈ [n] has capacity 1 and each hospital in V ℓ,0 has capacity 1. All the remaining preferences and capacities remain as in Section 3.1. Given a weakly stable matching M in the instance Γ with an average hospital rank K M , we provide a reduction of the capacities t that respects the budget B = n · L and we build a stable matchingM t inΓ t with an average hospital rankK = nK M + 2nL. • Reduction of capacities. We remove n spots from each village B ℓ in the following way: Assume in M we have the pair (i ℓ , j), where j is such that r = rank i ℓ (j). Then, we reduce by 1 the capacities of v 1 ℓ,r and of each hospital in V ℓ,0 \ {v 0 ℓ,r }. • Matching. We build the matchingM t as follows: We match (w ℓ,r , j), (y ℓ , v 0 ℓ,r ), {(w ℓ,h , v 1 ℓ,h )} h =r . The remaining pairs are the same as in the proof of Lemma 3.4. The rest of the proof is analogous to the proofs of Lemma 3.4, Lemma 3.5, and Theorem 3.1. Note that the proof can be slightly modified to obtain a similar result for HR with incomplete preference lists, as long as the condition −B + j∈H c j ≥ \|R\| is met. Extensions In this section, we investigate the variants of Problems 1 and 2 where the decision-maker has budgets for different subsets of hospitals. In the remainder of this section, we say that P = {H 1 , . . . H q } is a partition of the set of hospitals H if ∪ k∈[q] H k = H and H k ∩ H k ′ = ∅ for all k, k ′ ∈ [q] with k = k ′ . Allocating Extra Spots to a Partition of Hospitals We generalize Problem 1 to the setting where the set of hospitals is partitioned and we seek to find an allocation of extra spots such that each part has a specific budget. Formally, we study the following problem. Problem 4 (Min-Avg sub exp HR). instance: A CA instance Γ = R, H, ≻, c , a partition P = {H 1 , . . . H q } of H, budget for each part {B k ∈ Z + : k ∈ [q]}, and a non-negative integer target value K ∈ Z + . question: Is there a non-negative vector t ∈ Z H + and a matching M t such that AvgRank(M t ) ≤ K, where t is such that j∈H k t j ≤ B k for each k ∈ [q] and M t is a stable matching in instance Γ t ? The next result can be directly obtained by considering a single set of hospitals in the partition, i.e., q = 1 and P = H, and by using Theorem 3.1. Before proving Theorem 5.2, we need to introduce a variant of Problem 4, where the goal is to find a stable matching whose size is at least a certain target. The problem of finding the maximum cardinality stable matching is one of the main focus of the literature [32]. We investigate it in relation with capacity expansion when there are incomplete preference lists. Recall that a CA instance with incomplete preference lists means that there is at least one resident or one hospital that does not rank completely the opposite side. Formally, we consider the following problem. \|M t \| ≥ K, where t is such that j∈H k t j ≤ B k for each k ∈ [q] and M t is a stable matching in instance Γ t ? Recall that if we consider complete preference lists, the problem above becomes trivial since all stable matchings have the same size. We prove the following result. Proof of Theorem 5.2. Let ǫ > 0 and define a = ⌈(3/ε)⌉. We consider an instance Γ of Max-Card sub exp HRI in which the set of hospitals is H, the set of residents is R (we assume \|R\| = n), every H k is of size at most two and B k ∈ {0, 1} for every k ∈ [q]. We denote by O j (resp. O i ) the preference list of hospital j (resp. resident i). We assume that the target value K is equal to n. We now build an instanceΓ of Min-Avg sub exp HR opt. Let us define A = n a−1 . In this instance, the set of hospitals is where the dots ". . . " in the preference lists mean that the remaining agents on the other side of the bipartition are ranked strictly and arbitrarily. Our Min-Avg sub exp SM instance comprises 2n a residents, so thatn := 2n a ; the target value is K ′ = n a+2 /2. The remainder of the proof follows the same reasoning as of Corollary 3.2; the proof can be found in the Appendix. A h=1 H h ∪ H 0 with H h = {j h 1 , . . . , j h n } and H 0 = {j 0 1 , . . . , j 0 n a }. The set of residents is A h=1 R h ∪ R 0 with R h = {i h 1 , . . . , i h n } and R 0 = {i 0 1 , . . . , i 0 n a }. Removing Spots from a Partition of Hospitals Similar to the problems presented in the previous section, we now study the generalization of Problem 2 where the set of hospitals is partitioned in q parts and each part has a budget for the removal of spots. Specifically, we consider the following problem. AvgRank(M t ) ≤ K, where t is such that j∈H k t j ≥ B k and c j − t j ≥ 0 for k ∈ [q], and M t is a stable matching in instance Γ t ? For Problem 6, we prove the following inapproximability result. Theorem 5.4. For any ε > 0, Min-Avg sub red HR opt is not approximable within a factor of n 1−ε , unless P=NP, where n is the number of residents. This result holds even with a partition in which each part H k contains at most two hospitals and B k ∈ {0, 1} for every k ∈ [q]. To prove Theorem 5.4, we need to study the analogous version of Problem 5 for the capacity reduction setting. Formally, we define the following problem. Problem 7 (Max-Card sub red SMI). instance: A CA instance Γ = R, H, ≻, c with incomplete preference lists, a partition P = {H 1 , . . . H q } of H, budget for each part {B k ∈ Z + : k ∈ [q]}, and a non-negative integer target value K ∈ Z + . question: Is there a non-negative vector t ∈ Z H + and a matching M t such that \|M t \| ≥ K, where t is such that j∈H k t j ≥ B k and c j − t j ≥ 0 for k ∈ [q], and M t is a stable matching in instance Γ t ? In particular, we show the following result. Theorem 5.5. Max-Card sub red SMI is NP-complete. This result holds even with a partition in which each part H k is of size at most two and B k ∈ {0, 1} for every k ∈ [q]. Proof. The proof is analogous to the proof of Theorem 5.3 with the difference that every hospital in each part H k has capacity 1. Proof of Theorem 5.4. The proof follows a similar reasoning as the proof of Theorem 5.2 with the difference that every hospital in each part H k has capacity 1. Conclusions In this work, we have investigated the following question: How should a centralized institution optimally manage a variation in the capacities of the hospitals? We addressed this question from two points of view: Capacity expansion and capacity reduction. Our analysis is focused on the computational complexity of these problems and some of its variations. Our first result established the approximation hardness of the problem of finding the residentoptimal stable matching in the presence of ties. Our theorem defined a boundary on the complexity of the resident-optimal stable matching, which is well known to be polynomial-time solvable when there are no ties. We used this result as the first building block in the construction of the main proof of the paper: The approximation hardness of the problem of allocating optimally extra capacities to the hospitals to reduce the average hospital rank. Our proof introduced a crucial structure, the village, that enabled us to manage the subtleties of the allocation of extra capacities. The problem of allocating extra resources is not easier when we restrict the distribution of capacities to a partition of the hospitals. If the objective of the problem is the cardinality of the stable matching, we proved that it is NP-complete when the problem has incomplete lists. If the objective is the average hospital rank, the corresponding optimization problem cannot be approximated within a certain factor. The problem of reducing the capacities is equally interesting. Indeed, we showed that the capacity reduction problem is NP-complete. We generalized this result to the case in which the set of hospitals is partitioned and there is a budget for each part. For this problem, we proved that its optimization version is also inapproximable within a certain factor. Finally, we studied the variant of the problem that seeks to maximize the cardinality of the matching when the preference lists are incomplete. We believe these results are significant because they emphasize the existence of an underlying structure in the stable matching problem which governs both the capacity expansion and reduction. Unveiling the properties of this structure is certainly an open question worth being explored. Another interesting future direction of research is understanding what is the role of meta-rotations [23,12,18] in the capacity variation problem. • the set of men is H ′ = H 0 ∪ C A.2 Proof of Theorem 5.3 In this section, we prove that Max-Card sub exp HRI is NP-complete. Proof of Theorem 5.3. We build a polynomial reduction from an instance of Max-Card HRTI where ties are only on the hospital side, they are at the head of the preference list and are of length two. Let H and R be the set of hospitals and residents in Γ, respectively; H = H ′ ∪ H ′′ , where H ′ is the set of hospitals with a tie at the head of the preference list and H ′′ is the set of hospitals with a strict preference list. We build an instanceΓ = R ,Ĥ,≻,ĉ of Max-Card sub exp HRI as follows: • The set of residents is a copy ofR = R; • The set of hospitalsĤ consists of a copy of H ′′ and the setH = {j ′ : j ∈ H ′ } ∪ {j ′′ : j ∈ H ′ }, i.e., we make two copies per hospital in H ′ . Each hospital inH has capacity 0 and each hospital in H ′′ has capacity 1; • For each resident inR, we keep the preference list that she has in the original instance Γ, with the exception that each j ∈ H ′ in her preference lists is replaced by j ′ if she does not appear in the tie. If she is the first resident listed in the tie of j ∈ H ′ , then we replace the hospital j in the preference list by j ′ ; otherwise, if the resident is listed second in the tie of j ∈ H ′ , then we replace the hospital j in the preference list by j ′′ ; • For the hospitals in H ′′ , we maintain their preference lists of Γ over the residents in R. For a hospital j ∈ H ′ with a preference list (i σ 1 , i σ 2 ), i σ 3 , . . . , i σs , the preference list of j ′ becomes i σ 1 , i σ 3 , . . . , i σs and that of j ′′ becomes i σ 2 , i σ 3 , . . . , i σs ; • For each hospital j ∈ H ′′ , we create a set H j = {j} with B j = 0. For every hospital j ∈ H ′ , we create a set H j = {j ′ , j ′′ } with B j = 1. Clearly, the sets H j induce a partition of the set of hospitalsĤ. • The target value is K, i.e., the same as in the Max-Card HRTI instance. Let M be a weakly stable matching of the Max-Card HRTI instance. We will show that there is a feasible allocation of the capacities t and a stable matching M t inΓ t with the same cardinality, and thus, establishing the problems equivalence. For every pair (i, j) in M , we have to distinguish whether j ∈ H ′ or j ∈ H ′′ . If j ∈ H ′′ , then we add the corresponding pair (i, j) to M t ; recall that for a hospital j ∈ H ′′ , B j = 0. Otherwise, j ∈ H ′ . If i = i σ 2 , then we allocate the extra capacity of part H j to j ′ and we match the pair (i, j ′ ). If, instead, i = i σ 2 , then we match the pair (i, j ′′ ) by assigning the extra capacity of part H j to j ′′ . If there is a hospital j ∈ H ′ that has not been assigned to any resident, then we may allocate the extra capacity of part H j to j ′ . 8 Note that M t is stable indeed. If not, there must be a blocking pair (i, j) where both the resident and the hospital have a capacity of 1 (otherwise, a hospital with capacity 0 could not create a blocking pair). Note that j must be in some H k given that those subsets form a partition ofĤ. Indeed, in each set H k exactly one hospital has capacity 1, and for k = j, j is exactly such hospital. If \|H j \| = 1, then j ∈ H ′′ and, thus, it has exactly the same preference list that it has in the instance Γ; therefore the corresponding pair (i, j) in M is a blocking pair, which yields a contradiction. If \|H j \| = 2, then we have to distinguish whether j = j ′ or j = j ′′ . If j = j ′ , then we find that (i, j) is a blocking pair in M . Otherwise, if j = j ′′ , then (i, j) is a blocking pair if and only if i = i σ 2 since it is the only resident ranking j ′′ inΓ. The pair (i, j ′′ ) could be a blocking pair only if j ′′ has capacity 1; the extra capacity B j = 1 was assigned to j ′′ in accordance with the reduction. Therefore (i, j ′′ ) is already matched in M t and (i, j ′′ ) cannot be a blocking pair. Note that we have created a bijection between the set of stable matchings in the Max-Card HRTI instance and the allocation of extra spots as well as the set of stable matchings in the Max-Card sub exp HRI instance modulo the stable matchings in the Max-Card sub exp HRI instance that have some unassigned hospitals of the form j ′ or j ′′ . Moreover, this correspondence preserves the cardinality of the stable matching. To conclude, note that the created instance introduces a polynomial number of hospitals, residents, preferences and pairs {(H j , B j )} j∈H in the input. Moreover, it can be verified in polynomial time that: (1) the vector of allocation t satisfies the corresponding constraints and (2) the constructed stable matching has a cardinality greater or equal than the target value. Hospitals and residents. First, we add a copy of the hospitals in H = {j 1 , . . . , j n } and residents in R ′′ = {i L+1 , . . . , i n } We also introduce a set of hospitals H 0 = {j 0 1 , . . . , j 0 n 2 } = H 0,1 ∪ . . . ∪ H 0,n of size n 2 (where each H 0,h has size n), a set Z = {z 1 , . . . , z n 3 } of hospitals of size n 3 , and a set X = {x 1 , . . . , x n·L } of hospitals of size n · L. Recall that we index the residents in R ′ as i 1 , . . . , i L . For every resident i ℓ ∈ R ′ , where ℓ ∈ [L], we introduce additional hospitals and residents, to form a structure that we call village, namely• a set of residents W ℓ = {w ℓ,h } h∈[n] ; Lemma 3 . 4 . 34For every weakly stable matching M in Γ with AvgRank(M ) = K M , there is an allocation t respecting the budget B = L · n and a stable matchingM t inΓ t = R ,Ĥ,≻,ĉ + t with AvgRank(M t ) = n · K M + 2n · L. Lemma 3 . 5 . 35Consider a weakly stable matching M in Γ of minimum average hospital rank. Then, the allocation t and the stable matchingM t constructed in Lemma 3.4 are the solutions of minimum average hospital rank forΓ when B = L · n. Corollary 5 . 1 . 51Min-Avg sub exp HR is NP-complete.Denote by Min-Avg sub exp HR opt the optimization version of Min-Avg sub exp HR, i.e., the problem of finding an allocation of extra capacities and a stable matching in the expanded instance of minimum average hospital rank. In the following result, we show the approximation complexity of Min-Avg sub exp HR opt. Problem 5 ( 5Max-Card sub exp HRI). instance: A HR instance Γ = R, H, ≻, c with incomplete preference lists, a partition P = {H 1 , . . . , H q } of H, budget for each part {B k ∈ Z + : k ∈ [q]} and a non-negative integer target value K ∈ Z + . question: Is there a non-negative vector t ∈ Z H + and a matching M t such that Theorem 5 . 3 . 53Max-Card sub exp HRI is NP-complete, even if the partition P = {H 1 , . . . , H q } is such that each H k is of size at most two and B k ∈ {0, 1} for every k ∈ [q]. The proof of this result can be found in the Appendix. Let us now focus on the proof of Theorem 5.2. Problem 6 ( 6Min-Avg sub red HR). instance: A CA instance Γ = R, H, ≻, c , a partition P = {H 1 , . . . H q } of H, budget for each part {B k ∈ Z + : k ∈ [q]}, and a non-negative integer target value K ∈ Z + . question: Is there a non-negative vector t ∈ Z H + and a matching M t such that , unless P=NP, where n is the number of residents. This result holds even if the partition P = {H 1 , . . . , H q } is such that each H k contains at most two hospitals and B k ∈ {0, 1} for every k ∈ [q].Theorem 5.2. For any ε > 0, Min-Avg sub exp HR opt is not approximable within a factor of n 1−ε For every pair {H k , B k } for k ∈ [q] in Γ, we introduce copies inΓ of the form {H h k , B h k } in H h for k ∈ [q] and h ∈ [A]. The hospitals in H 0 have all capacity 1; the other hospitals have the same capacities of the original hospitals in Γ. For j ∈ H and h ∈ [A], we denote by O h j the preference list obtained by substituting in the preference list O j the residents in R with the residents in R h . We define similarly O h i . The preference lists of the hospitals and residents inΓ are as follows:j 0 h : i 0 h , . . . h ∈ [n a ] j h s : O h js , R 0 , . . . s ∈ [n], h ∈ [A] i 0 h : j 0 h , . . . h ∈ [n a ] i h s : O h is , H 0 , . . . s ∈ [n], h ∈ [A], For any two subsets of residents R ′ , R ′′ , we denote that hospital h prefers R ′ over R ′′ as R ′ ≻ h R ′′ . A preference relation of a hospital is responsive if for everyR ′ ⊆ R with \|R ′ \| ≤ c h , s ′ ∈ R ′ and s ′′ / ∈ R ′ , we have that (i) R ′ ≻ h R ′ ∪ {s ′′ } \ {s ′ } if and only if {s ′ } ≻ h {s ′′ }, and (ii) R ′ ≻ h R ′ \ {s ′ } if and only if {s ′ } ≻ h ∅.Therefore, a responsive preference list can be obtained from the linear order over singletons. Since responsive preferences are substitutable and satisfy the law of aggregated demand, our results hold also under these more relaxed assumptions. Throughout this paper, round brackets denote a tie. This follows from the hypothesis that preference lists are complete and that the total capacity of the hospitals can accommodate all the residents. This could be done also by adding a copy of themselves at the end of their list, which is usually referred to as individual rationality. Matching with oneself means being unassigned. Note that the optimal solution may not use the entire budget, for instance, when every resident is matched with her top choice. However, we can always arbitrarily assign the remaining extra spots without affecting the final average hospital rank. Alternatively, we may leave un-assigned the extra capacity Bj for every unassigned hospital j ∈ H ′ in M . AcknowledgmentsThis work was funded by the Institut de valorisation des données and Fonds de Recherche du Québec through the FRQ-IVADO Research Chair in Data Science for Combinatorial Game Theory, and the Natural Sciences and Engineering Research Council of Canada through the discovery grant 2019-04557. Part of the work was conducted when the third author was Canada Excellence Research Chair at Polytechnique Montréal, the generous support of the CERC grant being warmly acknowledged.A Missing proofsThe following problem will be useful for the proofs that we provide in this Appendix.Problem 8 (Max-Card HRTI). instance: An HRTI instance Γ = R, H, ≻, c with c j ∈ {0, 1}, for all j ∈ H, \|H\| = \|R\| and a non-negative integer target value K ∈ Z + . question: Is there a weakly stable matching M such that \|M \| ≥ K?Recall that HRTI corresponds to the problem with ties and incomplete preference lists. In[33], the authors proved that Max-Card HRTI is NP-complete. As the next remark states, this result holds even if ties are at the head of the preference list, only on one side of it, at most one tie per list, and each tie is of length 2.Remark A.1. After the proof of Lemma 1 in[33], the authors showed that the problem Max-Card HRTI can be simplified to the case in which ties are only on one side of the bipartition and are at the end of the preference list. Since the ties of the new instance created in Lemma 1 from[33]are at most two, we can use the same reasoning to assume instead, without loss of generality, that in an instance of Max-Card HRTI and the corresponding Min-w SMT instance of Corollary 3.2 ties occur only at the head of a preference list.A.1 Proof of Corollary 3.2In this section, we prove that Min-w SMT is NP-complete and its optimization version cannot be approximated within a certain factor. The proof is inspired by the proof of Theorem 7 in[33]. The result in[33]is stated in the traditional notation of the stable marriage problem where both sides are defined as women and men, instead of residents and hospitals. To keep coherence with the previous work, for this proof we also denote both sides as women and men.Proof of Corollary 3.2. Clearly, Min-w SMT is in NP. Given ε > 0, let a = ⌈(3/ε)⌉. From Theorem 2 in[33], we know that, when ties occur on the women's side only, and each tie has length two, Max-Card HRTI is NP-complete. Consider an instance of Problem 8 with H = {m 1 , m 2 , . . . , m n } and R = {w 1 , w 2 , . . . , w n }. We assume that the target value K is equal to n, since it was shown that even for this target value the problem is NP-complete. Let O h (resp. R h ) denote the preference list of man m h (resp. woman w h ) for h ∈ [n]. Next, we build an instance of Min-w SMT. Let • the set of women iswhere the dots ". . . " in the preference lists mean that the remaining agents on the other side of the bipartition are ranked strictly and arbitrarily, and the sets mean that the agents within are ranked according to their indices;• the target value is K ′ = (n a+2 )/2.Our Min-w SMT instance comprises 2n a men and 2n a women, so thatn := 2n a . Note also that the only ties in Min-w SMT occur in the preference lists of women w s h for h ∈ [n], s ∈ [C]. Moreover, there is at most one tie per list, and each tie has length 2.Suppose that we have a YES instance for Max-Card HRTI, i.e., there is a stable matching M with \|M \| = n. We create a matching M ′ in Min-w SMT as follows: For every h ∈ [n a ], we add the pair m 0 h , w 0 h to M ′ , and for each s ∈ [n], we add the pair m ℓ s , w ℓ k to M ′ for all ℓ ∈ [C], where (m s , w k ) ∈ M . Note that M ′ is stable for our Min-w SMT instance. We also have that AvgRank M ′ ≤ n a + n a−1 n 2 ≤ n a+2 2 = K ′ , since, without loss of generality, we may choose n ≥ 3. Therefore, the objective value in Min-w SMT satisfies the target of K ′ . On the other side, let us suppose that we have a NO instance for Max-Card HRTI, i.e., it does not have a stable matching of cardinality n. Then, in any stable matching M ′ of Min-w SMT, it holds that, for every s ∈ [C], there is some h ∈ [n] for which w s h is not matched to one of her proper men. Nonetheless, in M ′ , m 0 h and w 0 h must be partners, for every h ∈ [n a ]. Therefore, there is some h ∈ [n] such that rank w s h (M ′ (w s h )) > n a . Hence, AvgRank (M ′ ) > n 2a−1 > K ′ for any stable matching of our Min-w SMT instance.Therefore, the existence of a polynomial-time approximation algorithm for Min-w SMT opt whose approximation ratio is as good as 2n 2a−1 /n a+2 = 2n a−3 would give a polynomial-time algorithm for determining whether Max-Card HRTI has a stable matching in which everybody is matched (i.e., K = n). 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[ "Characterization of the hot Neptune GJ 436 b with Spitzer and ground-based observations ⋆", "Characterization of the hot Neptune GJ 436 b with Spitzer and ground-based observations ⋆" ]	[ "B.-O Demory \nObservatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland\n\nObservatoire François-Xavier Bagnoud -OFXB\n3961Saint-LucSwitzerland\n", "M Gillon \nObservatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland\n\nInstitut d'Astrophysique et de Géophysique\nUniversité de Liège\n4000LiègeBelgium\n", "T Barman \nLowell Observatory\n1400 West Mars Hill Road86001FlagstaffAZUSA\n", "X Bonfils \nObservatório Astronómico de Lisboa\nTapada da Ajuda\nP-1349-018LisboaPortugal\n", "M Mayor \nObservatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland\n", "T Mazeh \nSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences\nTel Aviv University\nTel AvivIsrael\n", "D Queloz \nObservatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland\n", "S Udry \nObservatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland\n", "F Bouchy \nInstitut d'Astrophysique de Paris\nCNRS\nUniversité Pierre & Marie Curie\n98bis Bd. AragoUMR7095, 75014ParisFrance\n", "X Delfosse \nLaboratoire d'Astrophysique de Grenoble\nObservatoire de Grenoble\nUMR5571 de l'Université J.Fourier et du CNRS\nBP53, 38041GrenobleFrance\n", "T Forveille \nLaboratoire d'Astrophysique de Grenoble\nObservatoire de Grenoble\nUMR5571 de l'Université J.Fourier et du CNRS\nBP53, 38041GrenobleFrance\n", "F Mallmann \nObservatoire François-Xavier Bagnoud -OFXB\n3961Saint-LucSwitzerland\n", "F Pepe \nObservatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland\n", "C Perrier \nLaboratoire d'Astrophysique de Grenoble\nObservatoire de Grenoble\nUMR5571 de l'Université J.Fourier et du CNRS\nBP53, 38041GrenobleFrance\n" ]	[ "Observatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland", "Observatoire François-Xavier Bagnoud -OFXB\n3961Saint-LucSwitzerland", "Observatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland", "Institut d'Astrophysique et de Géophysique\nUniversité de Liège\n4000LiègeBelgium", "Lowell Observatory\n1400 West Mars Hill Road86001FlagstaffAZUSA", "Observatório Astronómico de Lisboa\nTapada da Ajuda\nP-1349-018LisboaPortugal", "Observatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland", "School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences\nTel Aviv University\nTel AvivIsrael", "Observatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland", "Observatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland", "Institut d'Astrophysique de Paris\nCNRS\nUniversité Pierre & Marie Curie\n98bis Bd. AragoUMR7095, 75014ParisFrance", "Laboratoire d'Astrophysique de Grenoble\nObservatoire de Grenoble\nUMR5571 de l'Université J.Fourier et du CNRS\nBP53, 38041GrenobleFrance", "Laboratoire d'Astrophysique de Grenoble\nObservatoire de Grenoble\nUMR5571 de l'Université J.Fourier et du CNRS\nBP53, 38041GrenobleFrance", "Observatoire François-Xavier Bagnoud -OFXB\n3961Saint-LucSwitzerland", "Observatoire de Genève\nUniversité de Genève\n1290SauvernySwitzerland", "Laboratoire d'Astrophysique de Grenoble\nObservatoire de Grenoble\nUMR5571 de l'Université J.Fourier et du CNRS\nBP53, 38041GrenobleFrance" ]	[]	We present Spitzer Space Telescope infrared photometry of a secondary eclipse of the hot Neptune GJ 436 b. The observations were obtained using the 8-µm band of the InfraRed Array Camera (IRAC). The data spanning the predicted time of secondary eclipse show a clear flux decrement with the expected shape and duration. The observed eclipse depth of 0.58 mmag allows us to estimate a blackbody brightness temperature of Tp = 717±35 K at 8 µm . We compare this infrared flux measurement to a model of the planetary thermal emission, and show that this model reproduces properly the observed flux decrement. The timing of the secondary eclipse confirms the non-zero orbital eccentricity of the planet, while also increasing its precision (e = 0.14 ± 0.01). Additional new spectroscopic and photometric observations allow us to estimate the rotational period of the star and to assess the potential presence of another planet.	10.1051/0004-6361:20078354	[ "https://arxiv.org/pdf/0707.3809v3.pdf" ]	15,889,308	0707.3809	6c27f0633f5229f7bcac56b23ec2ec8d25b07f0c	Characterization of the hot Neptune GJ 436 b with Spitzer and ground-based observations ⋆ 14 Sep 2007 February 1, 2008 B.-O Demory Observatoire de Genève Université de Genève 1290SauvernySwitzerland Observatoire François-Xavier Bagnoud -OFXB 3961Saint-LucSwitzerland M Gillon Observatoire de Genève Université de Genève 1290SauvernySwitzerland Institut d'Astrophysique et de Géophysique Université de Liège 4000LiègeBelgium T Barman Lowell Observatory 1400 West Mars Hill Road86001FlagstaffAZUSA X Bonfils Observatório Astronómico de Lisboa Tapada da Ajuda P-1349-018LisboaPortugal M Mayor Observatoire de Genève Université de Genève 1290SauvernySwitzerland T Mazeh School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Tel AvivIsrael D Queloz Observatoire de Genève Université de Genève 1290SauvernySwitzerland S Udry Observatoire de Genève Université de Genève 1290SauvernySwitzerland F Bouchy Institut d'Astrophysique de Paris CNRS Université Pierre & Marie Curie 98bis Bd. AragoUMR7095, 75014ParisFrance X Delfosse Laboratoire d'Astrophysique de Grenoble Observatoire de Grenoble UMR5571 de l'Université J.Fourier et du CNRS BP53, 38041GrenobleFrance T Forveille Laboratoire d'Astrophysique de Grenoble Observatoire de Grenoble UMR5571 de l'Université J.Fourier et du CNRS BP53, 38041GrenobleFrance F Mallmann Observatoire François-Xavier Bagnoud -OFXB 3961Saint-LucSwitzerland F Pepe Observatoire de Genève Université de Genève 1290SauvernySwitzerland C Perrier Laboratoire d'Astrophysique de Grenoble Observatoire de Grenoble UMR5571 de l'Université J.Fourier et du CNRS BP53, 38041GrenobleFrance Characterization of the hot Neptune GJ 436 b with Spitzer and ground-based observations ⋆ 14 Sep 2007 February 1, 2008Received date / accepted datearXiv:0707.3809v3 [astro-ph] Astronomy & Astrophysics manuscript no. gj436˙spitzer2˙rev˙v1 (DOI: will be inserted by hand later)techniques: photometric -techniques: spectroscopic -eclipses -stars: individual: GJ 436 -planetary systems -infrared: general We present Spitzer Space Telescope infrared photometry of a secondary eclipse of the hot Neptune GJ 436 b. The observations were obtained using the 8-µm band of the InfraRed Array Camera (IRAC). The data spanning the predicted time of secondary eclipse show a clear flux decrement with the expected shape and duration. The observed eclipse depth of 0.58 mmag allows us to estimate a blackbody brightness temperature of Tp = 717±35 K at 8 µm . We compare this infrared flux measurement to a model of the planetary thermal emission, and show that this model reproduces properly the observed flux decrement. The timing of the secondary eclipse confirms the non-zero orbital eccentricity of the planet, while also increasing its precision (e = 0.14 ± 0.01). Additional new spectroscopic and photometric observations allow us to estimate the rotational period of the star and to assess the potential presence of another planet. Introduction GJ 436 b is one of the few known Neptune-mass extrasolar planets. It was discovered by radial-velocity measurements (Butler et al. 2004) as a planet with a period of 2.6 days and a minimum mass of 21 M ⊕ . Follow-up Doppler observations of GJ 436 refined the planetary mass and the orbital parameters, including an eccentricity of 0.16 ± 0.02 (Maness et al. 2007, hereafter M07). Our team (Gillon et al. 2007a, hereafter G07a) discovered the transiting nature of GJ 436 b, enabling us to measure a planetary radius ∼ 4 R ⊕ . This discovery and the corresponding Send offprint requests to: [email protected] ⋆ Our final secondary eclipse, photometric and Ca II H+K index time series are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strabg.fr/cgi-bin/qcat?J/A+A/ measurements of the planetary radius and mass indicated a planet composed mostly of ice, probably surrounded by a small H/He envelope. Because of the small size of the parent star (R ∼ 0.4 R ⊙ ) and the short orbital period of GJ 436 b, the planet-to-star luminosity ratio in the infrared is comparable to that of many known hot Jupiters, despite the planet's much smaller radius. Furthermore, the M dwarf GJ 436 is rather bright in the infrared (K ∼ 6). Detection of the thermal emission from this small planet had thus been expected to be within the reach of the Spitzer Space Telescope. Following our transit discovery, we submitted a Discretionary Director Time (DDT) Spitzer proposal to better characterize this interesting planet. We applied for photometric observations of the primary transit using the 8-µm band of the InfraRed Array Camera IRAC (Fazio et al. 2004) in order to get a very accurate radius measurement and constrain the bulk composition of the planet. We also applied for photometric observations of the secondary eclipse in the four bands of IRAC (3.6,4.5,5.8 and 8 µm), in the 16-µm band of the InfraRed Spectrograph IRS (Houck et al. 2004) and the 24-µm band of the Multiband Imaging Photometer MIPS (Rieke et al. 2004) to assess the atmospheric temperature, albedo, heat distribution efficiency and composition. However, the observations were actually triggered and performed as part of an existing Target of Opportunity (ToO) program (ID 30129, PI J. Harrington) which has a total priority for the observations of transiting planets. The main goal of this ToO is to deliver to the community without any proprietary period optimal Spitzer observations of transiting planets. Spitzer observed the transit and the secondary eclipse of GJ 436 in the 8-µm IRAC band on June 29 and 30 respectively. The data of the primary transit were made publicly available on July 13th 2007. Our direct analysis of these data allowed us to determine a very accurate radius for GJ 436 b (Rp = 4.2 R ⊕ , Gillon et al. 2007b, hereafter G07b) and to confirm the presence of an H/He envelope. Spitzer data of the secondary eclipse were not released to the community until July 17th 2007, due to an oversight that occurred at the Spitzer Science Center. This explains why we separated our analysis and present here our results regarding the secondary eclipse data. During the writing of this study, a paper by Deming et al. (2007) reporting primary and secondary eclipses analyses has been submitted to ApJ and put on astro-ph. The present analysis has been conducted independently from their work. Their results are consistent with the ones presented here. Analyzing the secondary eclipse data, we report here the detection of a secondary eclipse and draw conclusions about the thermal emission of GJ 436 b and refine its orbital parameters, allowing a better understanding of GJ 436 dynamics by exploring the contingency of a supplementary planet. In addition, we report here on additional ground based observations to determine the stellar rotational period. We followed the photometric intensity and the Ca II H+K activity index of GJ 436. Although the photometric data are sparse and cover only 50 days, we find some evidence that the stellar rotational period is of the order of 50 days, which is also consistent with long-term CaII measurements. Section 2 describes the observations and the reduction procedure. Our analysis of the obtained secondary eclipse time series is described in Section 3. In Section 4, we analyze the infrared emission from the planet and draw some conclusions about its atmosphere composition. We detail an orbital analysis, encompassing the possibility of a perturbing planet, stellar activity and GJ 436 b orbital parameters refinements in Section 5. Our conclusions are presented in Section 6. Observations and data reduction Spitzer IRAC observations GJ 436 has been observed on June 30th UT for 6 hours, to cover the secondary eclipse, resulting in 49920 frames. Observations were made so as to encompass the expected secondary eclipse window, whose timing calculations were made by taking into account transit timing and orbital eccentricity. Due to the uncertainties on eccentricity and argument of periastron, a larger time-window was chosen to ensure the detection of the secondary eclipse. Data acquisition was made using IRAC in its 8-µm band with the same mode and strategy employed for the primary transit (G07b). We combine each set of 64 images using a 3-σ clipping to get rid off transient events in the pixel grid, yielding 780 stacked images for the secondary eclipse, with a temporal sampling of ∼ 28s. Heliocentric Julian Day (HJD) conversion was made according to the mean Spitzer orbital position at the time of each exposure and GJ 436 apparent position. Spitzer position ephemerides were obtained through JPL-Horizons web interface (Giorgini et al. 1996) and converted from TT (Terrestrial Dynamic Time) to UTC. We faced the same instrumental rise issue noticed in our work on primary transit. To mitigate its effect, we zero weight the eclipse and the first 100 points of the timeseries. We then divide the lightcurve by the best fitting asymptotic function with three free parameters and evaluate the average flux outside the eclipse to normalize the time series, exactly as for the primary transit. The rms of the resulting time series evaluated outside the eclipse is the same as for the primary (G07b): 0.7 mmag, which is 1.2 times GJ 436's photon noise. Ground-based photometry To assess the variability of the star, we observed GJ 436 with the Euler Swiss telescope located at La Silla Observatory (Chile) and the François-Xavier Bagnoud Observatory's (OFXB) 0.6m telescope located at Saint-Luc (Switzerland). Observations occurred in 14 nights from May 4th to May 21th. A sequence of 10 exposures was done every night. The same strategy used for our observation of the May 2nd transit (G07a) was applied (V-band filter, 80s exposure time, defocus to ∼ 9"). The data reduction was also similar. We also use for our analysis of the GJ 436 variability the May 2nd out-of-transit data and the photometric lightcurves obtained with the OFXB 0.6m telescope during our search for the transits of GJ 436 b (G07a). We scale OFXB points with Euler ones because of the filters slightly different bandpasses. At the end, our data amounts to 24 points spanning 48 days. The lightcurve is represented in Fig. 6, and discussed in Sect. 5.3. Ground-based spectroscopy Since the discovery of GJ 436 b (Butler et al. 2004), we obtained additional spectra of the star with the ESO Harps spectrograph (Mayor et al. 2003). Harps is mounted on ESO 3.6m telescope and is dedicated to high precision radial-velocity measurements thanks to its resolution of 110'000 and a wavelength range coverage between 3800 and 6800Å. To assess the stellar activity and rotation we used 23 high SNR spectra from which we measured the Ca II H+K index. Results are discussed in Sect. 5.3 Analysis of secondary eclipse time series We fit a non-limb-darkened eclipse profile to the secondary eclipse data using the Mandel & Agol (2002) algorithm. The eccentricity of the orbit is considered as described in G07b, taking the values for the eccentricity e and the argument of periastron ω from M07. The formula connecting ω to the true anomaly f at the orbital location of the secondary eclipse is: f = π 2 + ω.(1) We fix the stellar and orbital parameters to the values mentioned in G07a. The free parameters are the central epoch of the secondary eclipse T s and the flux decrement ∆F s . The fit procedure and the error bars estimation is similar to the one described in G07b. The obtained value for T s and ∆F s , including their respective error bars are given in Table 1. Figure 1 shows the best-fit theoretical curve superimposed on the lightcurve (zoomed on secondary eclipse center, binned for clarity) and the residuals of the fit. After having derived an accurate value for the eccentricity (see Section 5), we perform a new fit to the secondary eclipse, taking into account the new values for the orbital eccentricity and the true anomaly at the orbital location of the eclipse, and their new error bars. The obtained values are in excellent agreement with the one given in Table 1. Infrared radiation While GJ 436 b is properly classified as a hot Neptune, the irradiation by the host star is weaker than for most hot Jupiters. Consequently, the contrast measurement reported here is that of the coolest exoplanet atmosphere detected so far. The atmospheric temperatures are predicted to be low enough for carbon to be bound in CH 4 (instead of CO as is the case for most hot Jupiters), placing GJ 436 b in a yet unexplored exoplanet atmospheric regime. The temperatures should also be cool enough for NH 3 absorption to appear between 10 and 11 µm. This situation is comparable to T dwarfs, which have prominent absorption bands of NH 3 at 10.5 µm as seen in recent Spitzer IRS observations (Cushing et al. 2006). In Fig. 2 we compare our 8 µm contrast measurement to Model planet-star flux ratios for GJ 436 b assuming that the absorbed stellar flux is redistributed across the dayside only (top curve) and uniformly redistributed across the entire planetary atmosphere (lower curve). In both models the composition is equal to that of the host star. For the wavelength range shown, the majority of the planet spectral features are produced by water, methane, and ammonia absorption. The filled black diamond is our Spitzer contrast measurement at 8 µm with its associated error bars while white diamonds are the model contrast values in the Spitzer IRAC and IRS bandpasses. The dashed line is the contrast curve for 700K blackbody planet spectrum. synthetic planet-star flux ratios calculated following the methods described in Barman et al. (2001Barman et al. ( , 2005 for two different assumptions for the day-to-night energy redistribution. The hotter dayside model corresponds to no redistribution of energy to the night side, while the second (lower flux) model assumes very efficient redistribu- tion of energy capable of completely homogenizing the day and night sides. As can be seen, our 8 µm measurement agrees very well with the hotter of the two models suggesting that redistribution is fairly inefficient. However, it is impossible to constrain the bolometric flux emerging from the planet (and thus the true energy budget of the day and night sides) with a single flux measurement in one bandpass. If energy redistribution is highly depthdependent, as indicated by recent dynamical simulations (Cooper & Showman 2005), then it remains possible that significant amounts of energy is being transported to the nightside, resulting in a warm nightside and cooler dayside at depths above or below the 8 µm photosphere. The agreement with the model spectrum suggests that observations at other Spitzer bandpasses should be possible and will allow further valuable constraints on both the atmospheric composition and the energy redistribution. In particular, the 700K blackbody planet spectrum (dashed line, Fig. 2) illustrates the value of observations at 4.5 and 16 µm as helpful probes of different atmospherics depths having different brightness temperatures. Here, we estimate a temperature of T b = 717 ±35 K at 8 µm, by comparing the observed contrast to blackbody SEDs divided by a synthetic stellar spectrum (Teff = 3350 K, M07), weighted by the radii ratio squared. We then varied the blackbody temperature until the 8 µm integrated contrast matched the observed contrast value. Orbital analysis The non-zero eccentricity One noticeable characteristic of GJ 436 b is its non-zero eccentricity (e=0.16±0.02 -M07). It contrasts with most known short-period exoplanets (P < 5 days) which have very small eccentricities, often indistinguishable from zero. Unfortunately, moderate eccentricities are difficult to constrain with radial-velocity measurements, and M07 warn that the quoted errors of the orbital parameters, based on the bootstrap technique, may lead to wrong estimates in some cases. To assess the statistical significance, they choose to use a rigorous Bayesian analysis and found the eccentricity to be greater than 0 with a high confidence level. Still, GJ 436 b's eccentricity is only known with a large uncertainty. To improve the determination of GJ 436 b's eccentricity we combine Spitzer eclipse timings with M07 radial velocities and perform a combined fit. As M07 have shown with a high confidence level, a positive radial-velocity trend is present in their data, we choose a model made of a planet plus a linear drift. Our minimization is based on the Levenberg-Marquardt algorithm (Press et al. 1992) and, as a maximum likelihood approximation, minimizes the following χ 2 : χ 2 = i ( v i − v i ǫ v,i ) 2 + ( T p − T p ǫ T p ) 2 + ( T s − T s ǫ T s ) 2 ,(2) where v i is the i th radial velocity given in M07 and T p and T s are respectively the timings of the Spitzer primary transit and secondary eclipse reported in this paper. The corresponding error estimates are ǫ v,i , ǫ T p and ǫ T s , and v i , T p and T s are their corresponding computed value, according to the chosen model. We find the χ 2 to be minimum with an orbital period P = 2.643859 days, a semi-amplitude K = 18.2 m s −1 , a date of the passage at periastron T 0 = 2454198.2056714 HJD, an argument of periastron ω = 350 • , an orbital eccentricity e = 0.14±0.01, a radial-velocity offset γ = 4.2 m s −1 and slope dv/dt = 1.4 m s −1 yr −1 . For this fit, the squared root of the reduced χ 2 is 1.84, marginally higher than a fit with radial velocities alone ( χ 2 \| rv only = 1.81). To derive the error of fitted orbital parameters we simulate 1000 virtual sets of new radial velocities and new eclipses timings. In each set, the radial-velocity data are randomized with a bootstrap algorithm (Press et al. 1992) and the eclipses timings are randomly generated according to a normal distribution, with mean and standard deviation given by the actual timing values and their error, respectively. Figure 3 shows these probability distributions for the eccentricity in both cases, when only the radialvelocity data are used and when a combined fit of radialvelocity and eclipses timings data is performed. The determination of eccentricity is clearly improved by the addition of eclipses timings, which bring the 1-σ error on e down to 0.01. Spitzer observations therefore strongly confirmed GJ 436 b unusual eccentricity. M07 pointed out that it may be due either to its own structure (i.e. a high tidal-quality factor Q) or to an additional long-period companion periodically interacting with the planet and pumping up its eccentricity. GJ 436 b has since been caught in transit and we now have a precise measurement of its radius. Considering GJ 436 is probably more than few billion years old, we can estimate what Q would dissipate the tidal circularization up to this age. To match an age >2 Gyr, a Q > 10 6 is necessary (Adams & Laughlin 2006), which is much more than Neptune in the solar system for which Banfield & Murray (1992) give 1.210 4 < Q < 3.310 5 . Thus, interaction with another companion is the most likely explanation for GJ 436 b's large eccentricity, probably due to the long period companion suspected from the radial-velocity trend in M07 data. Looking for additional planets The improvement in the determination of orbital parameters provides an opportunity to look for additional planets in the radial-velocity data. Such analysis is also motivated by the √ χ 2 of our solution, which is larger than one. A period analysis of the residuals around the best solution (Fig. 4) shows no significant power excess at any period. The highest peak is found at P ∼ 5.602 days and is attributed a 92% false alarm probability by bootstrap randomizations. In conclusion, except the companion suspected in Sect. 5.1, the present data set shows no evidence for additional low mass exoplanets in the GJ 436 system. Investigating residuals: the stellar activity An alternative way to explain that the dispersion of the radial-velocity residuals is in excess compared to the internal errors is to invoke the stellar activity. If present on the stellar surface, spots are known to modulate the Doppler measurement and to introduce 'jitter' or additional coherent signal in radial-velocity measurement (Saar & Donahue 1997). Earlier this year we published the discovery of a m sin i = 11 M ⊕ planet orbiting the nearby M dwarf GJ 674 (Bonfils et al. 2007). In addition to the Doppler signal induced by the planet, we clearly identified a sec- ond signal of period ∼ 35 days in the residuals of the one-planet fit. We have shown that Ca II H&K emission lines were varying in phase with this second signal, demonstrating it was due to a spot rather than a planet. This analysis was given further credit by a clear photometric counterpart to the spectral-index variation. To investigate the activity of GJ 436, we can thus apply the same spectroscopic diagnostic as we did for GJ 674, thanks to Harps spectra we obtained since 2004. Figure 5 hence represents the periodogram of Ca II H+K index measured on 23 high SNR spectra of GJ 436. It displays a power excess around P ∼ 48 days that identifies the rotation period of GJ 436. Bootstrap randomizations give a false alarm probability < 1% for this peak. Moreover, complementary photometric observations we did to monitor the long-term activity of GJ 436 (Fig. 6) confirm that a spot is present on GJ 436 surface and that the rotational period is likely more than 40 days. On a 50 day-time span the variation of the flux has an amplitude of ∼ 1%. We know from the spectral index variation that 50 days is close to the rotational period and it is thus reasonable to assume this amplitude for the photometric signal. With an estimate of the amplitude of the photo- metric variation, plus an approximate rotational period, it becomes possible to estimate the amplitude of the activity induced radial-velocity variation. Saar & Donahue (1997) have done some simulations and found the radial-velocity amplitude K s induced by a spot follow approximately the relation : K s [m s −1 ] ∼ 6.5 × f 0.9 s × v sin i,(3) where f s is the size of the spot (expressed in percent of the stellar disk) and v sin i is the projected rotational velocity of the star. In the case of GJ 674, considering its radius (0.34 R ⊙ ) and its rotational period (34.8 days), we calculate a v sin i of 0.5 km s −1 . Equation 3 then converts the observed flux variation (∼ 2.6%) into a radial-velocity amplitude K s ∼ 8 m s −1 , close to the measured amplitude (6 m s −1 ). The same numerical application for GJ 436, with a radius R ⋆ = 0.463 R ⊙ (G07b), a rotational period P rot ∼ 45 days, and a filling factor f s ∼ 1% lead to K s ∼ 3 m s −1 . The spot is thus responsible for a typical dispersion of ∼ 2 m s −1 , which, co-added to the typical radial-velocity errors (∼ 2.4 m s −1 ), explains most (if not all) of the dispersion observed for the residuals around our best solution (∼ 4 m s −1 ). Ultimately, to better weight the errors between radial-velocity data and eclipses timings data, we introduce this 'jitter' in our fitting procedure. Its impact is negligible as the estimated parameters remain unchanged. Conclusions Since the discovery, GJ 436 b has showed itself as a peculiar planet and has risen a strong interest from the community regarding its composition or supplementary planets in the system. Spitzer data gathered from the primary and secondary eclipse are of great help to answer some of those questions as discussed in G07b and in this present study. We especially learn from the infrared emission measurements at 8 µm and planetary atmospheres models that GJ 436 b is characterized by an envelope composed of H, He, H 2 O and CH 4 . Also, our contrast measurement is consistent with a model planet that has very inefficient day-to-night redistribution at 8 µm photospheric depths on the dayside. Moreover, transit and secondary eclipse respective timings combined with radial velocities prove that GJ 436 b has an eccentricity significantly greater than zero. Considering a reasonable tidal dissipative factor, we estimated the orbital circularization timescale to be likely shorter than GJ 436 age. We therefore conclude that the non-zero eccentricity is probably the result of a dynamical interaction with an additional companion in the system, maybe the long period companion suspected from the radial-velocity trend in M07 data. In the course of our orbital analysis we try to find an additional planet around GJ 436, but no significant periodicity is found in the residuals of our best fit. Conversely, we identify that GJ 436 has a spotted surface and probably rotates with a period P rot ∼48 days. We estimate that this magnetic activity noises the radial-velocity signal at a level of ∼ 2 m s −1 , therefore explaining most (if not all) the residual dispersion around our best solution. Nevertheless, the full potential of Spitzer concerning GJ 436 b has not been explored yet, especially regarding thermal emission spectral coverage. Complementary observations in the 3.6, 4.5, 5.8-µm IRAC, 16-µm IRS and 24-µm MIPS channels are due between Nov. 2007 andFeb. 2008. They will certainly bring new constraints on the atmosphere composition of this planet. Fig. 1 . 1T op: Zoomed binned time series for the secondary eclipse. The best-fit theoretical curve is superimposed. Although unbinned data were used for the fit, points are binned by 5 for plotting purposes. Bottom: The unbinned residuals of the fit. Their rms is 0.7 mmag. Fig. 2 . 2Fig. 2. Model planet-star flux ratios for GJ 436 b assuming that the absorbed stellar flux is redistributed across the dayside only (top curve) and uniformly redistributed across the entire planetary atmosphere (lower curve). In both models the composition is equal to that of the host star. For the wavelength range shown, the majority of the planet spectral features are produced by water, methane, and ammonia absorption. The filled black diamond is our Spitzer contrast measurement at 8 µm with its associated error bars while white diamonds are the model contrast values in the Spitzer IRAC and IRS bandpasses. The dashed line is the contrast curve for 700K blackbody planet spectrum. Fig. 3 . 3Probability distributions for the eccentricity resulting from randomly generated datasets including: Top: Radial velocity data only. Bottom: Radial velocities + transit and secondary eclipse timings. Table 1. Parameters derived from the secondary eclipse for GJ 436 b. SE stands for Secondary Eclipse.Mid-SE timing [HJD] 2454282.333 ±0.001 Flux decrement [∆Fs] 0.00054 ±0.00007 T b at 8 µm [K] 717 ±35 Orbital eccentricity 0.14±0.01 Fig. 6. Long-term lightcurve obtained with the Euler 1.2m telescope at ESO La Silla Observatory and the 0.6m telescope at FXB Observatory.0.99 1 1.01 Relative flux Relative flux 54180 54200 54220 54240 54260 Julian date −2,400,000 [day] Julian date −2,400,000 [day] Acknowledgements. This work is based on observations made with the Spitzer Space T elescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under NASA contract 1407. XB acknowledges support from the Fundação para a Ciência e a Tecnologia (Portugal) in the form of a fellowship (references SFRH/BPD/21710/2005). TB acknowledges support by NASA's Origins of Solar Systems grant NNX07AG68G, a Spitzer Theory Grant, and the NAS computing facility. TM acknowledges a grant from the Smithsonian Institution that supports his stay at the CfA, when this work was done. This study has also the support of the Fonds National Suisse de la Recherche Scientifique. . F C Adams, G Laughlin, ApJ. 6491004Adams, F. C. & Laughlin, G. 2006, ApJ, 649, 1004 . D Banfield, N Murray, Icarus. 99390Banfield, D. & Murray, N. 1992, Icarus, 99, 390 . T S Barman, P H Hauschildt, F Allard, ApJ. 556885Barman, T. S., Hauschildt, P. H., & Allard, F. 2001, ApJ, 556, 885 . T S Barman, P H Hauschildt, F Allard, ApJ. 6321132Barman, T. S., Hauschildt, P. H., & Allard, F. 2005, ApJ, 632, 1132 . 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K., Chamberlin, A. B., et al. 1996, in BAAS, Vol. 28, 1158-+ . J R Houck, T L Roellig, J Van Cleve, ApJS. 15418Houck, J. R., Roellig, T. L., van Cleve, J., et al. 2004, ApJS, 154, 18 . K Mandel, E Agol, ApJ. 580171Mandel, K. & Agol, E. 2002, ApJ, 580, L171 . H L Maness, G W Marcy, E B Ford, PASP. 11990Maness, H. L., Marcy, G. W., Ford, E. B., et al. 2007, PASP, 119, 90 . M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20 W H Press, S A Teukolsky, W T Vetterling, B P Flannery, c1992Numerical recipes in FORTRAN. The art of scientific computing. Cambridge: University Press2nd ed.Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in FORTRAN. The art of scientific computing (Cambridge: University Press, -c1992, 2nd ed.) . G H Rieke, E T Young, C W Engelbracht, ApJS. 15425Rieke, G. H., Young, E. T., Engelbracht, C. W., et al. 2004, ApJS, 154, 25 . 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[ "Hairy black hole entropy and the role of solitons in three dimensions", "Hairy black hole entropy and the role of solitons in three dimensions" ]	[ "Francisco Correa \nCentro de Estudios Científicos (CECs)\nAv. Arturo Prat 514ValdiviaChile\n", "Cristián Martínez \nCentro de Estudios Científicos (CECs)\nAv. Arturo Prat 514ValdiviaChile\n\nUniversidad Andrés Bello\nAv. República 440SantiagoChile\n", "Ricardo Troncoso \nCentro de Estudios Científicos (CECs)\nAv. Arturo Prat 514ValdiviaChile\n\nUniversidad Andrés Bello\nAv. República 440SantiagoChile\n" ]	[ "Centro de Estudios Científicos (CECs)\nAv. Arturo Prat 514ValdiviaChile", "Centro de Estudios Científicos (CECs)\nAv. Arturo Prat 514ValdiviaChile", "Universidad Andrés Bello\nAv. República 440SantiagoChile", "Centro de Estudios Científicos (CECs)\nAv. Arturo Prat 514ValdiviaChile", "Universidad Andrés Bello\nAv. República 440SantiagoChile" ]	[]	Scalar fields minimally coupled to General Relativity in three dimensions are considered. For certain families of self-interaction potentials, new exact solutions describing solitons and hairy black holes are found. It is shown that they fit within a relaxed set of asymptotically AdS boundary conditions, whose asymptotic symmetry group coincides with the one for pure gravity and its canonical realization possesses the standard central extension. Solitons are devoid of integration constants and their (negative) mass, fixed and determined by nontrivial functions of the self-interaction couplings, is shown to be bounded from below by the mass of AdS spacetime. Remarkably, assuming that a soliton corresponds to the ground state of the sector of the theory for which the scalar field is switched on, the semiclassical entropy of the corresponding hairy black hole is exactly reproduced from Cardy formula once nonvanishing lowest eigenvalues of the Virasoro operators are taking into account, being precisely given by the ones associated to the soliton.This provides further evidence about the robustness of previous results, for which the ground state energy instead of the central charge appears to play the leading role in order to reproduce the hairy black hole entropy from a microscopic counting. * Electronic address: correa, martinez, [email protected]	10.1007/jhep02(2012)136	[ "https://arxiv.org/pdf/1112.6198v2.pdf" ]	119,265,357	1112.6198	f536ef93d7095958f8f2592b51e136248d8a1685	Hairy black hole entropy and the role of solitons in three dimensions 13 Feb 2012 Francisco Correa Centro de Estudios Científicos (CECs) Av. Arturo Prat 514ValdiviaChile Cristián Martínez Centro de Estudios Científicos (CECs) Av. Arturo Prat 514ValdiviaChile Universidad Andrés Bello Av. República 440SantiagoChile Ricardo Troncoso Centro de Estudios Científicos (CECs) Av. Arturo Prat 514ValdiviaChile Universidad Andrés Bello Av. República 440SantiagoChile Hairy black hole entropy and the role of solitons in three dimensions 13 Feb 2012 Scalar fields minimally coupled to General Relativity in three dimensions are considered. For certain families of self-interaction potentials, new exact solutions describing solitons and hairy black holes are found. It is shown that they fit within a relaxed set of asymptotically AdS boundary conditions, whose asymptotic symmetry group coincides with the one for pure gravity and its canonical realization possesses the standard central extension. Solitons are devoid of integration constants and their (negative) mass, fixed and determined by nontrivial functions of the self-interaction couplings, is shown to be bounded from below by the mass of AdS spacetime. Remarkably, assuming that a soliton corresponds to the ground state of the sector of the theory for which the scalar field is switched on, the semiclassical entropy of the corresponding hairy black hole is exactly reproduced from Cardy formula once nonvanishing lowest eigenvalues of the Virasoro operators are taking into account, being precisely given by the ones associated to the soliton.This provides further evidence about the robustness of previous results, for which the ground state energy instead of the central charge appears to play the leading role in order to reproduce the hairy black hole entropy from a microscopic counting. * Electronic address: correa, martinez, [email protected] I. INTRODUCTION The microscopic origin of black hole entropy, a question raised right after the pioneering work of Bekenstein and Hawking during the 1970's, still remains as a challenging unsolved puzzle. Nevertheless, the most compelling current proposals appear to converge in the sense that, regardless the precise mechanisms and assumptions, an emergent conformal symmetry in two dimensions, endowed with a suitable central extension, allows to reproduce the semiclassical entropy of different classes of black holes from a microscopic counting (see e.g. [1][2][3][4][5], as well as [6][7][8] and references therein). One of the simplest and clear examples is the one provided by Strominger [2]. This proposal relies on an observation pushed forward by Brown and Henneaux during the 1980's [9] and currently interpreted in terms of the AdS/CFT correspondence [10]. As follows from [9], since the asymptotic symmetries of General Relativity with negative cosmological constant in three dimensions correspond to two copies of the Virasoro algebra, a consistent quantum theory of gravity should then be described in terms of a conformal field theory in two dimensions, with a central charge given by c = 3l 2G ,(1) where G and l stand for the Newton constant and the AdS radius, respectively. Thus, in [2] it was assumed that if the CFT fulfills some physically sensible properties, the physical states form a consistent unitary representation of the conformal algebra, so that the asymptotic growth of the number of states must be given by Cardy formula [11]. Remarkably, the result precisely agrees with semiclassical entropy of the BTZ black hole [12,13] provided the central charge is exactly given by (1). Nonetheless, there are known examples for which this proposal has to be refined, since for them the central charge does not play the leading role in order to reproduce the semiclassical black hole entropy from a microscopic counting. Indeed, as explained in [14] the asymptotic growth of the number of states can be expressed only in terms of the spectrum of the Virasoro operators without making any explicit reference to the central charges, so that the relevant quantities that allow to reproduce the black hole entropy turn out to be the lowest eigenvalues of the Virasoro operators. This can be seen as follows. If the spectrum of the Virasoro operators L ± 0 , whose eigenvalues are given by ∆ ± , is such that their lowest eigenvalues, denoted by ∆ ± 0 , are nonvanishing (i.e., for ∆ ± 0 = 0), Cardy formula reads (see e.g. [11,[15][16][17]) S = 2π c + − 24∆ + 0 6 ∆ + − c + 24 + 2π c − − 24∆ − 0 6 ∆ − − c − 24 ,(2) where it is assumed that the ground state is non degenerate. As pointed out in [14], on a cylinder, the zero mode of the Virasoro operators gets shifted according tõ L ± 0 := L ± 0 − c ± 24 ,(3) so that formula (2) can be naturally written as S = 4π −∆ + 0∆ + + 4π −∆ − 0∆ − ,(4) where∆ ± correspond to the eigenvalues ofL ± 0 , while∆ ± 0 stand for the lowest ones. Therefore, from (4) it is apparent that the asymptotic growth of the number of states can be precisely obtained if one only knew the spectrum ofL ± 0 without knowledge about the central charges. Note that when the lowest eigenvalues of the Virasoro operator vanish, i.e. for ∆ ± 0 = 0, or equivalently∆ ± 0 = − c ± 24 , formula (4) reduces to its more familiar form, given by, S = 2π c + 6∆ + + 2π c − 6∆ − .(5) In this case, as shown in [2], assuming that the eigenvalues ofL ± 0 are given by the corresponding canonical generators according tõ ∆ ± = 1 2 (Ml ± J) ,(6) where M and J stand for the mass and the angular momentum, respectively, then formula (5) precisely reproduces the semiclassical entropy of the BTZ black hole. Therefore, for instances such that the lowest eigenvalues of the Virasoro operators do not vanish, i.e., for ∆ ± 0 = 0, formula (5) does not apply. Remarkably, as explained in [14], in these cases the semiclassical black hole entropy can still be successfully reproduced by virtue of the generic formula given by (4) once the ground state configuration is suitably identified. A concrete example where this effect occurs is provided by the existence of hairy black holes found in [18] for General Relativity minimally coupled to a self-interacting scalar field in three dimensions. The action is given by I[g µν , φ] = 1 πG d 3 x √ −g R 16 − 1 2 (∇φ) 2 − V (φ) ,(7) and the self-interaction potential, expanded around φ = 0, is assumed to be of the form V (φ) = − 1 8l 2 − 3 8l 2 φ 2 − 1 2l 2 φ 4 + O(φ 6 ) ,(8) so that the first term corresponds to the cosmological constant Λ = −1/l 2 , while the second one is the mass term, with m 2 = −3/(4l 2 ), fulfilling the Breitenlohner-Freedman bound [19,20]. As shown in [18], the scalar field is able to acquire slow fall-off at infinity, so that the action (7) admits a suitable set of asymptotically AdS boundary conditions in a relaxed sense as compared with the one of Brown and Henneaux [9], which nevertheless possesses the same asymptotic symmetries, i.e., they are also left invariant under the conformal group in two dimensions, spanned by two copies of the Virasoro algebra. The asymptotic conditions are given by: φ = χ r 1/2 + α χ 3 r 3/2 + O(r −5/2 ) (9) g rr = l 2 r 2 − 4l 2 χ 2 r 3 + O(r −4 ) g tt = − r 2 l 2 + O(1) g tr = O(r −2 ) g ϕϕ = r 2 + O(1) g ϕr = O(r −2 ) g tϕ = O(1)(10) where χ = χ(t, ϕ) is an arbitrary function, and α is an arbitrary constant. One of the effects of relaxing the asymptotic conditions is reflected through the fact that the generators of the asymptotic symmetries acquire a nontrivial contribution from the scalar field. Following the Regge-Teitelboim approach [21], the canonical generators were found to be given by Q(ξ) = 1 16πG dϕ ξ ⊥ lr (g ϕϕ − r 2 )−2r 2 (lg −1/2 − 1) +2ξ ϕ π r ϕ +ξ ⊥ 2r l φ 2 − 2l φ∂ r φ √ g rr ,(11) where the reference background is chosen to be the massless BTZ black hole. The corresponding Poisson brackets were shown to span two copies of the Virasoro algebra with the standard central charges c + = c − = c, where c is given by eq. (1). An exact analytic hairy black hole solution whose asymptotic behavior fits within the boundary conditions given by eqs. (9), (10) was found in [18] for the following self-interaction potential V 1,ν (φ) = − 1 8l 2 cosh 6 φ + ν sinh 6 φ ,(12) which belongs to the class defined in (8). As shown in [14], for this specific potential, the field equations corresponding to (7) also admit an exact analytic soliton solution, being such that the metric and the scalar field are regular everywhere and fulfill the boundary conditions (9) and (10). The soliton turns out to be devoid of integration constants, and it has a precise fixed (negative) mass M 0 determined by the Newton constant and the selfinteraction parameter ν. This fact naturally suggests to regard the soliton as the ground state of the "hairy sector", for which the scalar field is switched on. Remarkably, assuming that the lowest eigenvalues of the Virasoro operators are determined by the global charges of the soliton, according to eq. (6), i.e.,∆ ± 0 = l 2 M 0 , the asymptotic growth of the number of states, given by eq. (4), reduces to S = 4πl −M 0 M ,(13) where M = 2 l∆ ± corresponds to the mass of the hairy black hole, which exactly reproduces its semiclassical entropy S = A 4G . One may wonder whether this result is just a curiosity of the particular model specified by the potential in (12), or actually corresponds to a generic feature of hairy black holes. In this article we construct new examples that strongly support the latter possibility. This is carried out for different self-interaction potentials within the class in eq. (8), being simultaneously involved enough so as to provide non trivial lowest eigenvalues for the Virasoro operators in the hairy sector, as well as sufficiently simple in order to find exact analytic hairy black holes and their corresponding solitons. In what follows we show the existence of new analytic hairy black hole and soliton solutions for different classes of self-interaction potentials. In the next section a one-parameter family of potentials which differs from (12) is considered, while in section III a class of potentials that depend on two parameters is discussed. Section IV is devoted to the explicit microscopic computation of the entropy of the hairy black holes mentioned above in terms of their corresponding solitons by means of formula (4) which reduces to eq. (13) in the static case. Final remarks are given in section V. Appendix A includes a description of certain analytic functions that become relevant in order to describe the properties of hairy black holes and solitons, and finally, appendix B is devoted to discuss the new exact solutions in the conformal (Jordan) frame. II. CASE 1: HAIRY BLACK HOLES AND SOLITONS FOR A UNIPARAMETRIC FAMILY OF POTENTIALS Let us consider the following class of self-interaction potentials, V 0,ν (φ) = − 1 16l 2 (ν cosh 8 φ − ν cosh 4 φ + 2 cosh 6 φ 1 − ν ln cosh 2 φ ) ,(14) which, apart from the AdS radius l, depends on a single parameter ν. Around φ = 0, the potential behaves as V 0,ν (φ) = − 1 8l 2 − 3 8l 2 φ 2 − 1 2l 2 φ 4 − 94 + 5ν 240l 2 φ 6 + O(φ 8 ) ,(15) and it then falls within the family defined in eq. (8) that is consistent with the boundary conditions given by (9) and (10). Note that V 0,ν (φ) does not overlap with V 1,ν (φ) in eq. (12) for any value of ν. As mentioned above, the self interaction (14) turns out to be involved enough so as to possess a ground state in the hairy sector with nontrivial lowest eigenvalues for the Virasoro operators, but nevertheless it becomes simple in order to produce exact analytic hairy black holes and solitons. This is discussed next. A. Black hole The field equations that correspond to the action (7) with V = V 0,ν (φ) admit an exact solution, whose the line element reads ds 2 = − r 2 l 2 h(r)dt 2 + l 2 r 2 dr 2 (r + a) 4 h(r) + r 2 dϕ 2 ,(16) with h(r) = 1 + ν a r + a + ln r r + a ,(17) and the scalar field is given by φ = arctanh a r + a .(18) The coordinates range as −∞ < t < ∞, r > 0, 0 ≤ ϕ < 2π, and the solution depends on a single non-negative integration constant a. Remarkably, the scalar field is regular everywhere and the solution describes a hairy black hole provided ν > 0, otherwise the singularity at the origin r = 0 becomes naked. There is a single horizon located at r + = aΦ ν ,(19) where Φ ν depends only on the parameter ν and it is given by Φ ν := −W (−e −1− 1 ν ) 1 + W (−e −1− 1 ν ) .(20) Here W stands for the Lambert W function, defined as W (z)e W (z) = z ,(21) which for −1/e < z < 0 has an upper branch that ranges according to −1 < W (z) < 0 (see e.g. [22]) [40]. For latter purposes it is worth pointing out that for 0 < ν < ∞, the function Φ ν is bounded as 0 < Φ 2 ν < ν 2 .(22) Further details about Φ ν are included in appendix A. The Hawking temperature of the hairy black hole (16), (18) can be obtained demanding regularity of the Euclidean solution at the horizon and it is found to be proportional to the integration constant a, which reads T = aν 4πl 2 Φ ν ,(23) while its entropy is given by S = A 4G = πr + 2G = πΦ ν 2G a .(24) Asymptotically, this hairy black hole behaves as φ = a 1/2 r 1/2 − 1 6 a 3/2 r 3/2 + O 1 r 5/2 ,(25) with g tt = − r 2 l 2 + a 2 ν 2l 2 + O 1 r ,(26)g rr = l 2 r 2 − 4l 2 a r 3 + O 1 r 4 ,(27)g ϕϕ = r 2 ,(28) and then falls within the set defined by eqs. (9), (10) with χ = √ a and α = −1/6. Therefore, the solution is asymptotically AdS in a relaxed sense as compared with the standard one [9]. The mass of the hairy black hole can be readily computed by virtue of eq. (11), yielding [41] M = Q(∂ t ) = νa 2 16Gl 2 .(29) Note that the scalar field cannot be switched off keeping the mass fixed. Indeed, the solution depends on a single integration constant, so that for φ → 0, the geometry approaches the one of the massless BTZ black hole. B. Soliton For the self-interaction potential V 0,ν (φ) in eq. (14), the field equations also admit the following solution: ds 2 = − r 2 l 2 dt 2 + l 2 r 2 dr 2 (r + 2lΦν ν ) 4 H(r) + r 2 H(r)dϕ 2 ,(30) with H(r) = 1 + 2lΦ ν r + 2lΦν ν + ν ln r r + 2lΦν ν ,(31) and φ(r) = arctanh 1 1 + νr 2lΦ ν ,(32) where the coordinates range according to −∞ < t < ∞, 0 ≤ ϕ < 2π, and 2lΦ 2 ν ν ≤ r < ∞ ,(33) with ν > 0. Note that the solution is devoid of integration constants and depends only on the self-interaction parameter ν and the AdS radius l. It is simple to verify that the solution is smooth and regular everywhere. Indeed, the behaviour of the solution around the origin, located at r = 2lΦ 2 ν ν can be seen from the expansion of (31), given by H (r) = ν 2 2lΦ 2 ν (1 + Φ ν ) 2 r − 2lΦ 2 ν ν + O r − 2lΦ 2 ν ν 2 , so that definingt = 2Φ 2 ν ν t, and ρ 2 = r 2 H(r), the metric approaches to the one of Minkowski spacetime, ds 2 → −dt 2 + dρ 2 + ρ 2 dϕ 2 . Analogously, the form of the scalar field around ρ = 0 is given by φ(ρ) = arctanh 1 1 + Φ ν − ν (1 + Φ ν ) 3/2 8l 2 Φ 4 ν ρ 2 + O ρ 4 . The asymptotic behavior of (32) and (30) is given by φ = 2lΦ ν ν 1/2 1 r 1/2 − 1 6 2lΦ ν ν 3/2 1 r 3/2 + O 1 r 5/2 ,(34) with g tt = − r 2 l 2 , g rr = l 2 r 2 − 2lΦ ν ν 4l 2 r 3 + O 1 r 4 ,(35)g ϕϕ = r 2 + O(1) , and then belongs to the class of relaxed asymptotically AdS conditions defined by eqs. (9) and (10) with χ = 2lΦν ν 1/2 and, as for the hairy black hole discussed in II A, α = −1/6. The mass of this solution can then be obtained by virtue of eq. (11) which yields M sol = − Φ 2 ν 4Gν .(36) In sum, this solution is regular everywhere, shares the same causal structure with AdS spacetime, and since it has a fixed finite mass, it describes a soliton. Note that by virtue of (22), which holds for the allowed range of the self-interaction coupling, ν > 0, the soliton mass M 0 = M sol becomes bounded according to Here we consider a wider class of self-interaction potentials being such that not only interpolates, but generalizes the ones considered above. This is given by − 1 8G < M 0 < 0 .(37)V λ,ν (φ) = − ν 16l 2 sinh 2 φ λ(λ + 1) − 2λ(λ + 2) cosh 2 φ + (1 + 4λ + λ 2 ) cosh 4 φ −(λ − 1) cosh 6 φ − cosh 6 φ − λ 2 sinh 6 φ 8l 2 (λ − 1) (λ − 1) + ν ln(λ − (λ − 1) cosh 2 φ) ,(38) which depends on two parameters ν, λ, and l stands for the AdS radius. The behavior of V λ,ν around φ = 0 reads V λ,ν (φ) −−→ φ→0 − 1 8l 2 − 3 8l 2 φ 2 − 1 2l 2 φ 4 − 94 − 30λ 2 + 5λ 2 ν − 10νλ + 5ν 240l 2 φ 6 + O(φ 8 ) ,(39) so that it belongs to the class defined in eq. (8). Note that for λ = 0 the self interaction (38) reduces to V 0,ν (φ) in Eq. (14), while after redefining ν = 6ν +1 (λ−1) 2 , in the limit λ → 1, the potential (38) acquires the form of V 1,ν (φ) in eq. (12). In what follows, exact hairy black holes and soliton solutions for this self interaction are explicitly found. A. Black hole In the case of V = V λ,ν (φ), the field equations possess an analytic solution. The metric is given by ds 2 = Ω 2 (r) −f (r)dt 2 + dr 2 f (r) + r 2 dϕ 2 ,(40) where Ω 2 (r) = λ(r + b) − b λ(r + b) 2 , f (r) = r 2 l 2 − ν λ(λ − 1)l 2 (λ − 1) 2 2 b 2 − b(λ − 1)r + λr 2 ln 1 + b λ − 1 λr ,(41) and the scalar field reads, φ = arctanh b λ(r + b) . The solution depends on a single integration constant b, and the coordinates range as −∞ < t < ∞, 0 ≤ ϕ < 2π, and r > r s , where r = r s stands for the location of the curvature singularity specified below. The asymptotic conditions (9), (10) are also fulfilled with α = − 1 6 (1 + 3λ) and χ = (b/λ) 1/2 , which means that the hairy solution is well defined provided b/λ > 0. Indeed, making the shift r → r + b λ , the asymptotic behavior of the solution reads φ = (b/λ) 1/2 r 1/2 − 1 6 (1 + 3λ) (b/λ) 3/2 r 3/2 + O 1 r 5/2(43)g rr = l 2 r 2 − b λ 4l 2 r 3 + O(r −4 ) g tt = − r 2 l 2 + O(1) g tr = O(r −2 ) g ϕϕ = r 2 + O(1) g ϕr = O(r −2 ) g tϕ = O(1) The mass can then be computed by virtue of eq. (11), which gives M = b 2 16Gl 2 (λ − 1) 2 λ 2 ν .(44) As in the case of the solution found in the previous section, this one depends on a single integration constant b, and it is such that the massless BTZ black hole in vacuum (φ = 0) is recovered for b → 0. It is simple to verify that the solution describes a hairy black hole with a regular scalar field on and outside the event horizon provided ν > 0, which ensures the mass (44) is positive. Thus, remarkably, although the self interaction looks somehow involved, positivity of the hairy black hole energy still goes by hand with cosmic censorship. The event horizon is located at r = r + , with r + = b λ (1 − λ)R λ,ν ,(45) where the function R λ,ν is defined as the real root of R 2 λ,ν − ν (λ − 1) λ 2 + R λ,ν + R 2 λ,ν log 1 − 1 R λ,ν = 0 ,(46) which only holds for ν > 0. In the case of λ > 1 there is a curvature singularity at r s = 0, and since the function R λ,ν ranges as −∞ < R λ,ν < 0, it is always surrounded by the event horizon. For λ < 1 the curvature singularity is located at r s = b 1−λ λ , and it is also always cloaked by the event horizon since in this case the function R λ,ν ranges according to 1 < R λ,ν < ∞. The Hawking temperature and hairy black hole entropy are given by T = ν 4πl 2 b λ 1 − λ Υ λ,ν ,(47)S = A 4G = π 2G (1 − λ) b λ Υ λ,ν ,(48) respectively, where the function Υ λ,ν := (1 − λ)R λ,ν (R λ,ν − 1) λ + (1 − λ)R λ,ν ,(49) fulfills Υ λ,ν (1 − λ) > 0, so that the temperature and the entropy are manifestly positive, and it is bounded as Υ 2 λ,ν < ν 2 .(50) Further details about the functions R λ,ν and Υ λ,ν are revisited in Appendix A. B. Soliton The field equations for the self-interaction potential V λ,ν in (38), with ν > 0, also admit the following soliton solution: φ(r) = arctanh γ λ,ν r + (1 + λ)γ λ,ν ,(51) with ds 2 = − (r + γ λ,ν ) 2 (r + λγ λ,ν ) 2 l 2 (r + (1 + λ)γ λ,ν ) 2 dt 2 + r + λγ λ,ν r + (1 + λ)γ λ,ν 2 dr 2 g(r) + l 2 g(r)dϕ 2 ,(52) where g(r) = ν γ λ,ν l 2 r + (2 + λ − λ 2 ) γ λ,ν 2 + 1 l 2 (r + γ λ,ν ) 2 1 + ν 1 − λ ln r + λγ λ,ν r + γ λ,ν ,(53) and γ λ,ν := 2lΥ λ,ν (1−λ)ν is a two-parametric constant. The coordinates range according to −∞ < t < ∞, 0 ≤ ϕ < 2π, and 2lΥ 2 ν,λ ν ≤ r < ∞ .(54) This solution depends only on the parameters of the potential, ν, λ and the AdS radius l. Thus, as in the case discussed in Sec. II B, the soliton has no integration constants and it is simple to verify that the solution is smooth and regular everywhere. The soliton also fulfills the asymptotic conditions in eqs. (9) and (10), with χ = (γ λ,ν ) 1/2 and α = − 1 6 (1 + 3λ). Indeed, for r → ∞ the solution behaves as φ = (γ λ,ν ) 1/2 r 1/2 − 1 6 (1 + 3λ) (γ λ,ν ) 3/2 r 3/2 + O 1 r 5/2 g rr = l 2 r 2 − γ λ,ν 4l 2 r 3 + O(r −4 ) g tt = − r 2 l 2 + O(1) g tr = O(r −2 ) g ϕϕ = r 2 + O(1) g ϕr = O(r −2 ) g tϕ = O(1) Note that the constant α coincides with the one of the hairy black hole for the same potential V λ,ν . The soliton mass can then be readily obtained from eq. (11), which gives M sol = − Υ 2 ν,λ 4Gν ,(55) and by virtue of (50) turns out to be bounded exactly as in eq. (37), i.e., − 1 8G < M 0 < 0 . As a concluding remark of this section, one can verify that in the limits λ → 0 and λ → 1, not only the potentials V 0,ν , and V 1,ν in eqs. (14) and (12) are recovered from V λ,ν , respectively, but also their corresponding hairy black holes and solitons described above, as well as those in Refs. [14,18]. In the case λ → 0, this can be explicitly see as follows [42]: Taking the limit λ → 0 with b λ → a, the solution defined by the metric (40) and the scalar field (42) becomes the black hole solution of section II A given by (16) and (18), provided the radial coordinates is shifted as r → r + a. In the case of the soliton solution, the function Φ ν is exactly recovered from the limit λ → 0 in (46) and (49), i.e. Υ 0,ν = Φ ν . In this way, the soliton that corresponds to the two parametric potential case, described by eqs. (52) and (51), reduces to the uniparametric one in (30) and (32). In sum, the self-interaction potentials considered here were shown to be simple enough so as to obtain exact and physically sensible hairy black hole solutions, and at the same time, sufficiently involved in order to provide analytic solitons whose masses are fixed and determined by nontrivial functions of the self-interaction parameters, as it can be explicitly seen from eqs. (36), (55). The link between soliton masses and the entropy of their corresponding hairy black hole entropies is discussed next. IV. MICROSCOPIC ENTROPY OF HAIRY BLACK HOLES: SOLITON MASS AND ITS ROLE IN THE ASYMPTOTIC GROWTH OF THE NUMBER OF STATES. The class of hairy black holes found here was shown to have positive mass and it can be seen that they share many of the features with the one previously found in [18]. In fact, for any of the self interactions discussed above, V 0,ν and V λ,ν , it occurs that for some fixed value of the energy, the same theory admits the existence of at least two different static and circularly symmetric black holes. Namely, for a precise value of the mass, apart from the hairy black hole that is dressed with a nontrivial scalar field, in vacuum one may also have the static BTZ black hole. Furthermore, it is worth highlighting that, since both black holes depend on a single integration constant, the hairy and BTZ black holes cannot be smoothly deformed into each other, due to that fact that for a fixed mass, the scalar field cannot be switched off. Following [14], this observation naturally suggests that the hairy and the vacuum black holes belong to different disconnected sectors. In the vacuum sector, the energy spectrum of the static BTZ black hole possesses a continuous part bounded from below by zero, a gap describing naked conical singularities, and a ground state that corresponds to AdS spacetime, having a negative mass given by M 0 = 2 l∆ ± 0 = − c ± 12l = − 1 8G .(56) In the hairy sector the situation is similar, since the energy spectrum also consists of a continuous part being bounded from below by zero that describes the hairy black holes, a gap that corresponds to naked singularities, and remarkably, a ground state that turns out to be consistently identified with the solitons described above. Indeed, the solitons possess negative fixed masses, given by eqs. (36) and (55), being completely determined by the fundamental constants of the theory. The soliton solutions were also found to be smooth, regular everywhere, and devoid of integration constants. Furthermore, they naturally provide the completion of the hairy sector spectrum, since they not only fulfill the same asymptotic conditions as the hairy black holes, but they also have precisely the same boundary conditions, because the value of the constant α, in eq. (9) coincides for both kinds of configurations. Besides, unitarity of the dual theory for c ± > 1 (see e.g. [31]), together with the fact that the asymptotic growth of the number of states, given by (4), is well defined only for negative lowest eigenvalues of the shifted Virasoro operators, impose the following bounds on∆ ± 0 : − c ± 24 ≤∆ ± 0 < 0 .(57) Remarkably, full agreement is found from the bulk theory, since as expressed by eq. (37) this bound is precisely fulfilled by the soliton mass that corresponds to the ground state of the hairy sector, and according to (56) it is saturated in vacuum. According to [14], the semiclassical entropy of the black holes under consideration can then be suitably reproduced in terms of the microscopic counting provided the ground state configuration is identified as the soliton or the AdS spacetime, for the hairy and vacuum sectors, respectively. Therefore, in the vacuum sector, since the lowest eigenvalues of the Virasoro operators ∆ ± 0 , are given by eq. (56), as explained in introduction, the asymptotic growth of the number of states given by (4) reduces its standard form in eq. (5). Thus, by virtue of (6) the semiclassical entropy of the BTZ black hole is exactly reproduced as in Ref. [2]. The microscopic entropy of the hairy black holes discussed here can then be obtained assuming that∆ ± 0 are determined by the global charges of their corresponding solitons, which according to eq. (6) are given by∆ ± 0 = l 2 M 0 , where M 0 stands for the soliton mass. In the case of static hairy black holes, as the ones discussed here, the asymptotic growth of the number of states, given by eq. (4), reduces to S = 4πl −M 0 M ,(58) where M is the hairy black hole mass. For the uniparametric potential V 0,ν , the semiclassical entropy of the hairy black hole, given by (24) is then successfully reproduced from a microscopic counting once the black hole and soliton masses, given by (29) and (36) are replaced into eq. (58). Explicitly, this reads S = 4πl Φ 2 ν 4Gν × νa 2 16Gl 2 = πΦ ν 2G a = A 4G . Analogously, for the more generic potential V λ,ν , taking into account that the black hole and soliton masses are given by (44) and (55) respectively, formula (58) reduces to S = 4πl Υ 2 λ,ν 4Gν × b 2 16Gl 2 (λ − 1) 2 λ 2 ν = π 2G (1 − λ) b λ Υ λ,ν = A 4G , in full agreement with (48). V. FINAL REMARKS New asymptotically AdS hairy black holes and solitons were shown to exist for General Relativity minimally coupled to a self-interacting scalar field in three dimensions. Different self-interaction potentials were engineered in order to obtain nontrivial analytic results, with the purpose of testing the robustness of regarding the soliton as the ground state of the hairy sector, and its key role in a microscopic counting of hairy black hole entropy. Our results then confirm that this proposal successfully goes beyond the example previously discussed in [14] and naturally point towards the fact that this mechanism should correspond to a generic feature of hairy black holes [43]. In the microcanonical ensemble, i.e., for a fixed value of the mass, since the theory admits hairy and vacuum black holes, it is natural to wonder which is the preferred configuration. By virtue of eqs. (58) and (37) (or equivalently (57)), the quotient of the entropies of the vacuum and hairy black holes fulfills S BT Z S hbh = M AdS M sol > 1 , where M AdS , and M sol stand for the mass of AdS and the soliton, respectively. Therefore, the vacuum black hole turns out to be the thermodynamically preferred configuration. This result could be readily extended for the (grand) canonical ensemble, as well as for the rotating case through applying a boost in the "t − ϕ" cylinder. It would then also be interesting to compare it with the one that could be obtained from the mechanical stability of the hairy solutions. As an ending remark, an interesting feature of the hairy black holes reported here and their corresponding solitons that is worth to be highlighted, is that their Euclidean continuations turn out to be diffeomorphic provided their temperatures are related by an S-modular transformation. solitons that correspond to the self interactions V 0,ν and V λ,ν , respectively. Some of their useful properties in this context are detailed in this appendix. · The function Φ ν is defined in terms of the Lambert W function according to eq. (20). It is a monotonically increasing function of the parameter ν, as it is depicted in Fig. 1. Its behaviour around ν → 0 is given by Φ ν −−→ ν→0 e −1− 1 ν + 2e −2− 2 ν + 9 2 e −3− 3 ν + · · · ,(A1) while for ν → ∞, reads Φ ν − −− → ν→∞ ν 2 − 2 3 + O ν −1/2 .(A2) Therefore, by virtue of eqs. (A1) and (A2), it can be readily checked that the bound (22) is fulfilled. · The function Υ λ,ν is defined in terms of the function R λ,ν through eq. (49), where R λ,ν stands for the real root of (46). Further details about R λ,ν are revisited in Fig. 2. It is simple to verify that it fulfills Υ λ,ν (1 − λ) > 0. In order to prove that the bound (50) holds, which reads Υ 2 λ,ν < ν 2 ,(A3) it is useful to recall eq. (46), so that this inequality is equivalently expressed as: F (R λ,ν ) > 0 if λ > 1 ,(A4)F (R λ,ν ) < 0 if λ < 1 ,(A5)R λ,ν 0 1 νλ 2(λ−1) FIG. 2: The plot represents the real zeros of eq. (46), which reads R 2 λ,ν = ν (λ−1) λ 2 + R λ,ν + R 2 λ,ν log 1 − 1 R λ,ν . The solid line corresponds to R 2 λ,ν , while the function at the right hand side, for the case λ > 1, is depicted by the dotted line, and for λ < 1 is given by the dashed line. The intersection of these curves then shows the existence of a real root of (46), which for λ > 1 lies within the range −∞ < R λ,ν < 0. This root approaches to νλ 2(λ−1) for ν → 0. For λ < 1, the plot shows that the real root is in the range 1 < R λ,ν < ∞. where F (R λ,ν ) := (λ + (1 − λ)R λ,ν ) 2 2(λ − 1)(R λ,ν − 1) 2 − λ 2 − R λ,ν − R 2 λ,ν log 1 − 1 R λ,ν .(A6) In the case of λ > 1, when R λ,ν → −∞ the function F (R λ,ν ) approaches to − 2 3R λ,ν > 0, while for R λ,ν → 0 tends to F (R λ,ν ) → λ 2(λ−1) > 0. Hence, since the function F (R λ,ν ) is monotonically increasing in the range −∞ < R λ,ν < 0, the inequality (A4) holds. Finally, for λ < 1, when R λ,ν → ∞ the function F (R λ,ν ) tends to − 2 3R λ,ν < 0, while if R λ,ν → 1 it approaches to F (R λ,ν ) → 1 2(λ−1)(R λ,ν −1) 2 < 0. Therefore, as F (R λ,ν ) is a monotonically increasing function in the range 1 < R λ,ν < ∞, the inequality (A5) is satisfied. It is simple to verify that in the case of λ = 0, the function Υ λ,ν reduces to Φ ν defined above, i.e., Υ 0,ν = Φ ν .(A7) Following the same procedure that allows to recover the potential V 1,ν in (12) from V λ,ν in (38), that consists on redefining ν = 6ν +1 (λ−1) 2 , and then taking the limit λ → 1, the function Υ λ,ν can be shown to fulfill lim λ→1 (λ − 1) 2 Υ 2 λ,6(1+ν)/(λ−1) 2 = Θ 2 ν ,(A8) where Θν := 2(zz) 2 3 z 2 3 −z 2 3 z −z , with z = 1 + i √ν .(A9) As shown in [18] and [14], the function Θν is the relevant one in order to obtain an analytic description of the hairy black holes and solitons that correspond to the self interaction V 1,ν . Appendix B: Solutions in the conformal (Jordan) frame The three-dimensional hairy black hole and soliton solutions for a self-interacting scalar field minimally coupled to General Relativity discussed here, acquire an appealing form in the conformal frame. The action in eq. (7), after applying a conformal transformation, followed by a scalar field redefinition of the form g µν = 1 −φ 2 −2 g µν andφ = tanh (φ) ,(B1) reduces to the one for General Relativity with cosmological constant and a conformally coupled self-interacting scalar field, given by I[ĝ,φ] = 1 πG d 3 x −ĝ R + 2l −2 16 − 1 2 (∇φ) 2 − 1 16Rφ 2 −V (φ) .(B2) It can be shown that the potentialsV 0,ν ,V 1,ν , andV λ,ν coming from their corresponding counterparts in the Einstein frame discussed above, given by eqs. (14), (12) and (38), respectively, are the only three branches of self interactions that are compatible with static and spherically symmetric solutions of the form dŝ 2 = −f (ρ)dt 2 + dρ 2 f (ρ) + ρ 2 dθ 2 ,φ =φ(ρ). As shown below, by means of (B1), this class of solutions corresponds to the hairy black holes described in sections II A and III A as well as the ones in Refs. [39] and [18] in the conformal frame. The explicit form of the self interactionsV 0,ν ,V 1,ν , andV λ,ν , including also their corresponding soliton solutions are discussed in what follows. Solutions forV 1,ν In the conformal frame, the self interaction (12) is mapped intô V 1,ν = − ν 8l 2φ 6 .(B3) This potential is the only one that turns out to be singled out by requiring the matter piece of the action (B2) to be conformally invariant; i.e., unchanged under local rescalings of the formĝ µν → λ 2 (x)ĝ µν , andφ → λ −1/2 (x)φ. In the case of ν ≥ −1, hairy black holes solutions were found in [18], which reduces to the one previously found in [39] for ν = 0. Solutions describing solitons were also found for the same range of the self-interaction coupling in [14]. Solutions forV 0,ν The self interaction (14) in the conformal frame, is mapped intô V 0,ν = ν 16l 2φ 2 − 2 1 −φ 2φ 2 − ν 8l 2 ln 1 −φ 2 ,(B4) whose behavior aroundφ = 0 readŝ V (φ) ∼ − ν 48l 2φ 6 − ν 32l 2φ 8 + O(φ 10 ) .(B5) Hairy black hole: In this case the metric is given by dŝ 2 = −f (ρ)dt 2 + dρ 2 f (ρ) + ρ 2 dθ 2 ,(B6) with f (ρ) = ρ 2 l 2 + ν l 2 aρ + ρ 2 ln 1 − a ρ , where the coordinates range as −∞ < t < ∞, a < ρ < ∞, 0 ≤ ϕ < 2π. The scalar field also acquires a very simple form, that readsφ (ρ) = a ρ .(B8) The hairy black hole solution, then depends on a single non-negative integration constant a, that parametrizes the location of the event horizon, for ν > 0, at ρ = ρ + , with ρ + = a 1 + W (−e −1− 1 ν ) ,(B9) where W stands for the Lambert W function. Soliton: The potential (B4) also admits an additional exact solution. The scalar field is given byφ (ρ) = 1 ρ ,(B10) and the metric reads dŝ 2 = −ρ 2 dt 2 + dρ 2 g(ρ) + 4l 4 Φ 2 ν ν 2 g(ρ)dϕ 2 ,(B11) with g(ρ) = ρ 2 l 2 + ν l 2 ρ + ρ 2 ln 1 − 1 ρ (B12) Note that the solution is devoid of integration constants. The coordinates range according to 1 + Φ ν ≤ ρ < ∞, −∞ < t < ∞, and 0 ≤ ϕ < 2π. The metric and the scalar field are regular everywhere. Since the mass does not depend on the choice of frame, it is given by (36) and the solution then describes a soliton. Solutions forV λ,ν In the conformal frame, the two-parametric potential (38) is mapped intô V λ,ν (φ) = λ 2 8l 2φ 6 − ν 8l 2 (λ − 1) 1 − λ 2 φ 6 ln 1 − λφ 2 1 − φ 2 − ν 8l 2 1 1 − φ 2 φ 2 + λ − 1 2 φ 4 + λ(λ − 1) 2 φ 6 − λ(λ + 1) 2 φ 8 ,(B13) whose behavior aroundφ = 0 is of the form V λ,ν (φ) = − ν(λ − 1) 2 − 6λ 2 48l 2φ 6 + ν(λ − 1) 3 32l 2φ 8 + O(φ 10 ) . Hairy black hole: The field equations derived from (B2) admit a static circularly symmetric solution whose metric is given by dŝ 2 = −f (r)dt 2 + dr 2 f (r) + r 2 dϕ 2 ,(B14) where f (r) is expressed in eq. (41), and the scalar field readŝ φ(r) = b λ(r + b) . The solution then depends on a single integration constant b, and it is well-defined provided b/λ > 0. The coordinates range as −∞ < t < ∞, r s < r < ∞, and 0 ≤ ϕ < 2π. This solution describes a hairy black hole for ν > 0, with an event horizon at r = r + , with r + given by (45), which surrounds the singularity at r = r s , located precisely as explained in Section III A. Soliton: The self interaction (B13) also admits soliton solution described by the metric dŝ 2 = −ρ 2 dt 2 + dρ 2 g(ρ) + l 2 λ 2 γ 2 λ,ν g(ρ)dϕ 2 , with g(ρ) = ρ 2 l 2 − ν λ(λ − 1)l 2 (λ − 1) 2 2 − (λ − 1)ρ + λρ 2 ln 1 + (λ − 1) λρ ,(B17) where the scalar field is given byφ (ρ) = 1 λ(ρ + 1) . 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Hereafter the lower branch is not considered since the solution would describe a naked singularity and it would then be ruled out by cosmic censorship. with the ones that could be obtained from different approaches that are adapted to deal with scalar fields and relaxed AdS asymptotics, as the ones in Refs. It would be interesting to compare this result. 23-25. Further results along the lines of [26-30], previously found for the hairy black hole of Ref. [18], would also worth to be explored for the new solutions found hereIt would be interesting to compare this result with the ones that could be obtained from different approaches that are adapted to deal with scalar fields and relaxed AdS asymptotics, as the ones in Refs. [23-25]. Further results along the lines of [26-30], previously found for the hairy black hole of Ref. [18], would also worth to be explored for the new solutions found here. In the limit λ → 1, following a similar procedure, the hairy black hole and soliton solutions of [14, 18] can be recovered from those in Sec. IIIn the limit λ → 1, following a similar procedure, the hairy black hole and soliton solutions of [14, 18] can be recovered from those in Sec. II. It is worth pointing out that similar results have also been recently found in [32, 33] in the context of BHT massive gravity [34, 35]; namely for asymptotically AdS hairy black holes and solitons in vacuum. as well as for black holes and solitons with Lifshitz asymptotics. 3633, 38It is worth pointing out that similar results have also been recently found in [32, 33] in the context of BHT massive gravity [34, 35]; namely for asymptotically AdS hairy black holes and solitons in vacuum [36, 37], as well as for black holes and solitons with Lifshitz asymptotics [33, 38].	[]
[ "New Hard-thresholding Rules based on Data Splitting in High-dimensional Imbalanced Classification", "New Hard-thresholding Rules based on Data Splitting in High-dimensional Imbalanced Classification" ]	[ "Arezou Mojiri \nDepartment of Mathematical Sciences\nDepartment of Statistics and Mathematics\nDepartment of Industrial and Systems Engineering\nIsfahan University of Technology\nMcGill University\nIsfahan University of Technology\n\n", "Abbas Khalili \nDepartment of Mathematical Sciences\nDepartment of Statistics and Mathematics\nDepartment of Industrial and Systems Engineering\nIsfahan University of Technology\nMcGill University\nIsfahan University of Technology\n\n", "Ali Zeinal Hamadani \nDepartment of Mathematical Sciences\nDepartment of Statistics and Mathematics\nDepartment of Industrial and Systems Engineering\nIsfahan University of Technology\nMcGill University\nIsfahan University of Technology\n\n" ]	[ "Department of Mathematical Sciences\nDepartment of Statistics and Mathematics\nDepartment of Industrial and Systems Engineering\nIsfahan University of Technology\nMcGill University\nIsfahan University of Technology\n", "Department of Mathematical Sciences\nDepartment of Statistics and Mathematics\nDepartment of Industrial and Systems Engineering\nIsfahan University of Technology\nMcGill University\nIsfahan University of Technology\n", "Department of Mathematical Sciences\nDepartment of Statistics and Mathematics\nDepartment of Industrial and Systems Engineering\nIsfahan University of Technology\nMcGill University\nIsfahan University of Technology\n" ]	[]	In binary classification, imbalance refers to situations in which one class is heavily underrepresented. This issue is due to either a data collection process or because one class is indeed rare in a population. Imbalanced classification frequently arises in applications such as biology, medicine, engineering, and social sciences. In this paper, for the first time, we theoretically study the impact of imbalance class sizes on the linear discriminant analysis (LDA) in high dimensions. We show that due to data scarcity in one class, referred to as the minority class, and high-dimensionality of the feature space, the LDA ignores the minority class yielding a maximum misclassification rate. We then propose a new construction of * [email protected] † [email protected] ‡ [email protected] hard-thresholding rules based on a data splitting technique that reduces the large difference between the misclassification rates. We show that the proposed method is asymptotically optimal. We further study two well-known sparse versions of the LDA in imbalanced cases.We evaluate the finite-sample performance of different methods using simulations and by analyzing two real data sets. The results show that our method either outperforms its competitors or has comparable performance based on a much smaller subset of selected features, while being computationally more efficient.	null	[ "https://arxiv.org/pdf/2111.03306v3.pdf" ]	245,704,490	2111.03306	1ede25b42952f27dd2efe6d455bb346c3afda986	New Hard-thresholding Rules based on Data Splitting in High-dimensional Imbalanced Classification January 7, 2022 Arezou Mojiri Department of Mathematical Sciences Department of Statistics and Mathematics Department of Industrial and Systems Engineering Isfahan University of Technology McGill University Isfahan University of Technology Abbas Khalili Department of Mathematical Sciences Department of Statistics and Mathematics Department of Industrial and Systems Engineering Isfahan University of Technology McGill University Isfahan University of Technology Ali Zeinal Hamadani Department of Mathematical Sciences Department of Statistics and Mathematics Department of Industrial and Systems Engineering Isfahan University of Technology McGill University Isfahan University of Technology New Hard-thresholding Rules based on Data Splitting in High-dimensional Imbalanced Classification January 7, 2022arXiv:2111.03306v3 [stat.ME] 6 Jan 2022ClassificationHigh-dimensionalityImbalancedLinear Discriminant Anal- ysisThresholding In binary classification, imbalance refers to situations in which one class is heavily underrepresented. This issue is due to either a data collection process or because one class is indeed rare in a population. Imbalanced classification frequently arises in applications such as biology, medicine, engineering, and social sciences. In this paper, for the first time, we theoretically study the impact of imbalance class sizes on the linear discriminant analysis (LDA) in high dimensions. We show that due to data scarcity in one class, referred to as the minority class, and high-dimensionality of the feature space, the LDA ignores the minority class yielding a maximum misclassification rate. We then propose a new construction of * [email protected] † [email protected] ‡ [email protected] hard-thresholding rules based on a data splitting technique that reduces the large difference between the misclassification rates. We show that the proposed method is asymptotically optimal. We further study two well-known sparse versions of the LDA in imbalanced cases.We evaluate the finite-sample performance of different methods using simulations and by analyzing two real data sets. The results show that our method either outperforms its competitors or has comparable performance based on a much smaller subset of selected features, while being computationally more efficient. INTRODUCTION The rise of high-dimensional data has affected many areas of research in statistics and machine learning, including classification. Linear Discriminant Analysis (LDA) has been extensively studied in high-dimensional classification. Bickel and Levina (2004), , and Shao et al. (2011) showed that when the number of features is larger than the sample size, the LDA can perform as badly as a random guess. To deal with the curse of dimensionality, several developments have been made over the last decade or so. For example, among others, new developments include the nearest shrunken centroids Tibshirani et al. (2002), shrunken centroids regularized discriminant analysis Guo et al. (2006), features annealed independence rule (fair) , sparse and penalized LDA Shao et al. (2011); Witten and Tibshirani (2011), regularized optimal affine discriminant (road) Fan et al. (2012), multi-group sparse discriminant analysis Gaynanova et al. (2015), pairwise sure independent screening Pan et al. (2016), and the ultra high-dimensional multiclass LDA Li et al. (2019). The general idea of these methods is to incorporate a feature selection strategy in a classifier in order to obtain certain optimality properties in the sense of misclassification rates. To the best of our knowledge, most of the existing developments in high dimensions focus on problems with comparable class sizes in the training data. However, in applications such as clinical diagnosis (Bach et al., 2017), fraud detection (Bolton and Hand, 2002), drug discovery (Zhu et al., 2006), or equipment malfunction detection (Park et al., 2013), classification often suffers from imbalanced class sizes where, for example, in a binary problem one class (referred to as the minority class) is heavily under-represented. This is due to either a data collection process or because one class is indeed rare in a population. In such situations, the minority class is of primary interest as it carries substantial information, and often has higher misclassification costs compared to the larger class, referred to as the majority class. For example, in a study of a certain rare disease, the cost of misclassifying a positive case is often higher than the cost of misclassifying a negative one (Ramaswamy et al., 2002). In banking or telecommunication studies, few customers are voluntarily willing to terminate their contracts and leave their provider. In these applications, misclassification of a potential churner is more expensive than that of a non-churner for a provider (Verbeke et al., 2012). Due to data scarcity in the minority class, conventional discriminant methods are often biased toward the majority class resulting in much higher misclassification rate for the minority class. This error dramatically increases in high-dimensional cases, as empirically shown by Blagus and Lusa (2010). In this paper, we study imbalanced binary classification with the class sizes n 2 << n 1 , when the number of features, p n , grows to infinity as the total sample size n = (n 1 + n 2 ) grows to infinity. We refer to Class 1 with size n 1 as the majority class, and Class 2 with size n 2 as the minority class. A specific limiting relationship between n 1 and n 2 is given in Section 2.2. Imbalanced classification under various settings have attracted attention in recent years. A common approach to deal with the imbalanced issue is to make virtual class sizes comparable by using resampling methods, for example, the synthetic minority over-sampling technique (smote) of (Chawla et al., 2002). The recent work of Feng et al. (2020) provides a review of the common re-sampling techniques for fixed dimensional imbalanced problems. In other methods, such as the weighted extreme learning machine Zong et al. (2013) and the cost-sensitive support vector machine (svm) Iranmehr et al. (2019), the idea is to strengthen the relative impact of the minority class by either assigning different weights to sample units or different costs to misclassification instances in each class. Bak and Jensen (2016) studied distributional properties of the correct classification probabilities of the minority and majority classes of a hard-thresholding independence rule. Using a non-asymptotic approach, they adjusted the bias of correct classification probabilities which is rooted on the imbalanced class sizes. Huang et al. (2010) and Pang et al. (2013) proposed bias-corrected discriminant functions. Owen (2007) studied limiting form of the logistic regression under a so-called infinitely imbalanced case in which the size of one class is fixed and the other grows to infinity. Qiao and Liu (2009) proposed new evaluation criteria and weighted learning procedures that increase the impact of a minority class. Qiao et al. (2010) developed a distance weighted discrimination method (dwd), originally proposed to overcome the well-known data-piling issue Ahn and Marron (2010) in high-dimensions, by an adaptive weighting scheme to reduce sensitivity to unequal class sizes. Qiao and Zhang (2013) proposed a linear classifier that is a hybrid of dwd and svm, thus haivng advantages of both techniques. Qiao and Zhang (2015) introduced a new family of classifiers including svm and dwd that provides a tradeoff between imbalanced and high-dimensionality. Hall et al. (2005); Nakayama et al. (2017) theoretically showed that under certain conditions, svm suffers from data-piling in highdimensions, meaning that all data points become the support vectors which may result in ignorance of one of the classes. Nakayama et al. (2017) proposed a biased-corrected svm that improves its performance even when the class sizes are imbalanced. Nakayama (2020) proposed a robust svm which is less sensitive to class sizes and choice of a regularization parameter. Xie et al. (2020) used a repeated case-control sampling technique coupled with a fused feature screening procedure to deal with imbalanced and high-dimensionality. The behaviour of LDA in high-dimensional imbalanced classification has often been studied empirically. In this paper, we first theoretically show that in such cases this classifier ignores the minority class, yielding a maximum misclassification rate for this class. On the other hand, a common approach to deal with high-dimensionality is to use a hardthresholding operator for feature selection. However, our simulations show large differences between the misclassification rates of the hard-thresholding rule (hr) in imbalanced settings. Thus, we face both high-dimensionality and an inflated bias in the difference between the two misclassification rates. To address the issues, we propose a new construction of the hr based a multiple data splitting (Msplit) technique as described below, and thus called Msplit-hr. We randomly split the training data in each class into two parts of sizes ⌊n k /2⌋, k = 1, 2, and use one part only for feature selection and the other part is then used to construct a bias-corrected classifier based on the selected features. As shown in Section 3, the splitting facilitates the correction of the inflated bias in the difference between the two misclassification rates. To reduce the effect of randomness in single-split, we repeat the process several (L) times which maximizes the usage of training data in finite-sample situations. In general, as pointed out by Meinshausen et al. (2009), multiple splitting also helps reproducibility of finite sample results. As shown numerically in Figures 1 and 2, respectively discussed in Sections 3.1 and 3.2, the classification results of Msplit-hr corresponding to L ≈ 30 are unsurprisingly more powerful than a single-split (L = 1). We show that our method is asymptotically optimal. We also study asymptotic properties of two well-known linear classifiers, namely the sparse LDA Shao et al. (2011), and the regularized optimal affine discriminant analysis Fan et al. (2012), under the imbalanced setting. Our simulations show that Msplit-hr either outperforms its competitors or has comparable performance based on a much smaller subset of selected features, while being computationally more efficient as discussed in Section 5. The rest of the paper is organized as follows. Section 2 gives the problem setup and investigates the behaviour of the LDA in high-dimensional imbalanced binary classification. Section 3 introduces our proposed method, Msplit-hr. Large-sample properties of the method are also discussed in this section. Two well-known high-dimensional variants of the LDA, under the imbalanced setting, are studied in Section 4. The finite-sample performance of several binary classifiers is examined using simulations in Section 5. Analysis of two real data sets are given in Section 6. A summary and discussion are given in Section 7. Technical Lemmas and proofs of our main results are given in Appendices A-C. Notation: All vectors and matrices are shown in bold letters. For any vector a ∈ R p , a 0 = #{j : a j = 0}, a 1 = p j=1 \|a j \|, a 2 = ( p j=1 a 2 j ) 1/2 , a ∞ = max j=1,...,p \|a j \|. For any symmetric matrix A ∈ R p×p , A 1 = max i=1,...,p p j=1 \|a ij \|, A = max j=1,...,p \|λ j (A)\|, where λ j (A) are the eigenvalues of the matrix A, and A ∞ = max i,j=1,...,p \|a ij \|. A diagonal matrix is denoted by D. For any two sequences a n and b n , we write a n b n or a n = O(b n ), if for sufficiently large n there exists a constant C such that a n ≤ C b n . We write a n ∼ b n , if a n /b n → 1, as n → ∞. And a n ≍ b n , if a n = O(b n ) and b n = O(a n ). Also, a n = o(b n ), when a n /b n → 0 as n → ∞. The notations o p and O p are respectively used to indicate convergence and boundedness in probability. An indicator function is denoted by 1{·}. THE LDA In this section, we first describe the setting of the binary classification problem under our consideration. We then study the effect of dimension and imbalanced class sizes on the LDA, which motives the topics of the remaining sections. Overview We consider the class labels Y ∈ {1, 2}, class prior probabilities π k = Pr(Y = k), and a pdimensional feature vector X = (X 1 , X 2 , . . . , X p ) ⊤ such that X\|Y = k ∼ N p (µ k , Σ), k = 1, 2. The LDA is a well-known classification technique for this setting. More specifically, given the parameter vector θ θ θ = (µ 1 , µ 2 , Σ) and assuming π 1 = π 2 , the optimal rule classifies a subject with an observed feature vector x * = (x * 1 , . . . , x * p ) ⊤ to Class 1 if and only if where Φ is the cumulative distribution function of the standard normal, and ∆ 2 p = µ ⊤ d Σ −1 µ d is referred to as the discriminative power or signal value. It is seen that as ∆ p → ∞, high discriminative power, then Π opt → 0; and as ∆ p → 0, low discriminative power, then Π opt → 1 2 implying that the classifier performs as a random guess. From now on, we assess the performance of other classifiers under consideration by comparing them with the optimal rule. In practice, the parameter vector θ θ θ is unknown and needs to be estimated using a training data D n = {x ik , i = 1, ..., n k , k = 1, 2}, where x ik is the i-th observed value of X in Class k, and the n k are the class sample sizes with the total sample size n = n 1 + n 2 . For a new feature vector x * , a so-called plug-in discriminant function based on the parameter estimates is given by δ lda (x * ; θ θ θ n ) = µ ⊤ d Σ −1 n (x * − µ a ), (2.3) where θ θ θ n = ( µ n,1 , µ n,2 , Σ n ) and µ n,k ≡ µ k = 1 n k n k i=1 x ik , k = 1, 2, (2.4) Σ n = 1 n − 2 2 k=1 n k i=1 (x ik − µ k )(x ik − µ k ) ⊤ . (2.5) The matrix Σ −1 n in (2.3) is a generalized inverse when Σ n is not invertible. Given D n , the conditional MCR of the plug-in linear discriminant rule based on (2.3) corresponding to Class k ∈ {1, 2}, is given by Π lda k (D n ) = Pr (−1) k δ lda (X * ; θ θ θ n ) < 0 Y = k, D n = Φ Ψ lda k ( θ θ θ n ) Υ lda ( θ θ θ n ) , (2.6) where Ψ lda k ( θ θ θ n ) = (−1) k µ ⊤ d Σ −1 n ( µ a − µ k ), and Υ lda ( θ θ θ n ) = µ ⊤ d Σ −1 n Σ Σ −1 n µ d . As is common in the literature, we study large-sample properties of a classifier through its conditional MCR. Impact of the dimension and imbalanced class sizes The effect of the dimension p on the LDA's performance is well studied in the literature. Shao et al. (2011) showed that when p is fixed or diverges to infinity at a slower rate than √ n, the classifier is asymptotically optimal (Shao et al., 2011, Definition 1). When p → ∞ such that p/n → ∞, Bickel and Levina (2004), , and Shao et al. (2011) showed that this classifier performs no better than a random guess. Hence, feature selection is essential when p is large compared to the sample size n. In the aforementioned works, the impact of dimensionality is studied under particular limiting settings on the class sizes n 1 and n 2 . Bickel and Levina (2004) and Shao et al. (2011) respectively considered equal class sizes (n 1 = n 2 ) and unequal sizes where n 1 , n 2 → ∞ such that n 2 n → π, 0 < π < 1. developed their results by considering compatible class sizes, such that c 1 ≤ n 1 n 2 ≤ c 2 , with 0 < c 1 ≤ c 2 < ∞. Bak and Jensen (2016) investigated the case where n 1 , n 2 → ∞, such that n 1 −n 2 n 1 +n 2 = ρ > 0 is fixed. All in all, it is seen that the sizes of the two classes grow similarly and proportional to the total sample size n, that is n k = O(n), k = 1, 2. We refer to these settings as a balanced classification problem. Owen (2007) analyzed the binary logistic regression models with fixed dimension p in a so-called infinitely imbalanced case in which n 1 → ∞ but the class size n 2 is fixed. In this paper, we study imbalanced classification in which n 1 and n 2 grow to infinity such that n 2 = o(n 1 ), implying a different growth rate of the class sizes. In the balanced classification, typically average (over the two classes) MCR (Shao et al., 2011;Fan et al., 2012) or the MCR of one arbitrary class (Bickel and Levina, 2004;) is used as a performance measure of a classifier. However, in imbalanced situations due to data scarcity in the minority class, classification results have a tendency to favour the majority class. Thus, the average MCR is not an appropriate performance measure for a classifier T . This motivated us to adapt the optimality definition of a classifier from Shao et al. (2011) to our setting as follows. Definition 1 Suppose T is a classifier in a binary classification problem. The misclassification rates of T , given the training data D n , are denoted by Π T k (D n ), k = 1, 2. Then, (i) T is asymptotically-strong optimal if Π T k (D n )/Π opt p −→ 1, k = 1, 2, (ii) T is asymptotically-strong sub-optimal if Π T k (D n ) − Π opt p −→ 0, k = 1, 2, (iii) T is asymptotically-strong worst if Π T k (D n ) p −→ 1 2 , k = 1, 2, (iv) T is asymptotically ignorant if min k=1,2 Π T k (D n ) p −→ 0 and max k=1,2 Π T k (D n ) p −→ 1. Note that any classifier T satisfying either of the properties in parts (i)-(iii) of the above definition also satisfies the properties discussed in the corresponding parts of Definition 1 of Shao et al. (2011) for a balanced case, but not vice versa. Part (iv) of the above definition occurs when a classifier completely ignores one of the classes, and more specifically the minority class. We now state our first result. Theorem 2.1 Suppose that the estimator Σ −1 n in δ lda in (2.3) is replaced by Σ −1 , and Σ is known. When n 2 = o(n 1 ), such that p/n 2 → ∞ and n 2 p ∆ 2 p = o(1), as n 1 , n 2 → ∞, then the LDA is asymptotically ignorant, that is, Π lda 1 (D n ) p −→ 0 , Π lda 2 (D n ) p −→ 1. This result implies that in the high-dimensional imbalanced cases, the MCR of the majority class tends to 0 which will be even better than the optimal value Π opt , but the MCR of the minority class approaches 1 which is worse than a random guess. Note that the above result also holds in the case of p/n 1 → c, for some finite constant c ≥ 0. (Hall et al., 2005;Nakayama et al., 2017) showed that, under certain conditions, the svm ignores the minority class in high-dimensional imbalanced problems. Remark 2.1 When p is fixed and n 2 = o(n 1 ), then the LDA is asymptotically-strong optimal. Remark 2.1 illustrates that in the fixed-dimensional case, the impact of imbalanced class sizes asymptotically vanishes and Π lda k (D n ), k = 1, 2, converge to the optimal value Π opt . Hence, by Theorem 2.1, Shao's results, and Remark 2.1, the effects of both dimension and imbalanced class sizes are responsible for ignorance of the minority class. PROPOSED METHOD: MSPLIT HARD-THRESHOLDING RULE (MSPLIT- HR) A common approach to deal with high-dimensionality in the LDA is to incorporate feature selection using a hard-thresholding rule (hr) based on a two-sample t-statistic as in . More specifically, by ignoring the correlation among features, Σ is estimated by the diagonal matrix D n = diag{ σ 2 1 , ..., σ 2 p }, and the discriminant function is given by δ hr (x * ; θ θ θ n ) = p j=1 r j (x * ; θ θ θ n ) h j ( θ θ θ n ), (3.1) where θ θ θ n = ( µ µ µ 1 , µ µ µ 2 , D n ), r j (x * ; θ θ θ n ) = ( µ dj / σ 2 j )(x * j − µ aj ), and h j ( θ θ θ n ) = 1{\|t j \| > τ n } is the thresholding operator based on the t-statistic t j = µ j2 − µ j1 σ j n/n 1 n 2 . (3.2) Here µ jk 's and σ 2 j 's are the entries of µ µ µ k and Σ n in (2.4) and (2.5), respectively. The discriminant function of FAIR proposed by for balanced problems belongs to the class of functions in (3.1). The authors select an optimal number of statistically most significant features, or equivalently the threshold value τ n of t-statistic, by minimizing a common upper bound on its corresponding MCRs. However, for the case of general Σ, such choice of τ n does not necessarily result in an asymptotically optimal classifier Shao et al. (2011). Thus, for generality, in the rest of the paper, for any given sequence of τ n , we refer to a classifier based on (3.1) as an HR unless otherwise is specified. Bak and Jensen (2016) showed that the hr in (3.1) based on a fixed threshold τ n = τ , is asymptotically ignorant when ρ = (n 1 − n 2 )/(n 1 + n 2 ) > 0 is fixed, as n 1 , n 2 → ∞. As stated after Theorem 3.1 below, it is interesting to note that under the imbalanced setting n 2 = o(n 1 ) and by an appropriate choice of τ n , the hr is indeed asymptotically-strong optimal. However, our simulations in Section 5 show an unsatisfactory finite-sample performance of the hr in the sense of both the MCR in the minority class and large difference between the two MCRs. We propose a new construction of the hr which outperforms (3.1) in finite-samples, while maintaining the same desirable large-sample properties, to be discussed below. To fix ideas, we first consider the imbalanced problem with a diagonal Σ = D. The general case of a non-diagonal Σ is discussed in Subsection 3.2, which is based on a feature screening technique. Note that under this case, the hr based on (3.1) is not optimal, as it ignores the correlation among the features. If indeed Σ = D, Msplit-HR under a diagonal Σ As discussed in Section 2, the class specific MCRs of the optimal rule are equal, and are given in (2.2). Our numerical experiments show that, due to the imbalanced class sizes, hr performs well in majority class but underperforms in minority class, though it has largesample optimal property as discussed after Theorem 3.1 below. Thus, the idea in our work is to reduce the difference between two conditional MCRs of hr toward that of the optimal rule which is zero. More specifically, our main goal is to propose a new discriminant function aiming to reduce the difference between the MCRs of the hr, \|Π hr 1 (D n ) − Π hr 2 (D n )\| = \|Φ(ψ 1,n ) − Φ(ψ 2,n )\|, where ψ k,n = Ψ hr k ( θ θ θ n )/ √ Υ hr ( θ θ θ n ) and Ψ hr k ( θ θ θ n ) = (−1) k+1 p j=1 r j (µ k ; θ θ θ n )h j ( θ θ θ n ) , Υ hr ( θ θ θ n ) = p j=1 ( µ dj / σ 2 j ) 2 σ 2 j h j ( θ θ θ n ) for k = 1, 2. To understand the above difference, Bak and Jensen (2016) studied distributional properties of the quantities ψ k,n , k = 1, 2. They focused on reducing the so-called bias B hr n = E Ψ hr 1 ( θ θ θ n ) − Ψ hr 2 ( θ θ θ n ) (3.3) to zero, which results in decreasing the bias between ψ 1,n and ψ 2,n and consequently of that between MCRs. However, it turns out that due to the dependency between the random variables r j and h j , computing B hr n is not an easy task. Bak and Jensen (2016) studied the origin of the bias and proposed methods for its correction. We instead propose a new construction of the hr that facilities the computation of such bias by adapting a samplesplitting strategy as follows. The training sample of each class is randomly partitioned into two sub-samples of sizes n ′ k = ⌊n k /2⌋. The two sub-samples are used for computing two quantities similar to the r j and h j in (3.1), for each j = 1, . . . , p, and then the results are merged. To reduce the effect of randomness due to the data splitting, this process is repeated, say, L times. Our new discrimination function is then constructed by averaging over the hr-type discriminant functions based on each splitting. Thus, we chose the name Msplit-hr for our method. More specifically, at the ℓ-th data splitting, for each ℓ = 1, ..., L, the entire training data D n is partitioned into two parts D (1) n,ℓ and D (2) n,ℓ . The parameter estimates based on each sub-sample are distinguished by the superscripts (1) and (2) , that is, θ θ θ (1) n,ℓ and θ θ θ (2) n,ℓ . A new observation with a feature vector x * is then classified using the discriminant function δ Msplit-hr 0 (x * ; θ θ θ n ) = 1 L L ℓ=1 p j=1 r j (x * ; θ θ θ (2) n,ℓ ) h j ( θ θ θ (1) n,ℓ ), (3.4) where θ θ θ n = {( θ θ θ (1) n,ℓ , θ θ θ (2) n,ℓ ) : for ℓ = 1, . . . , L}. Due to the statistical independence of the two random functions h j and r j in (3.4), for all j = 1, . . . , p, calculation of the bias B n for δ Msplit-hr 0 is straight forward, which is shown below. Recall n ′ k = ⌊n k /2⌋, and let n ′ = n ′ 1 + n ′ 2 and f n ′ = n ′ /2 − 1. E{h j ( θ θ θ (1) n,ℓ )}, wherer n = f n ′ ( 1 n ′ 1 − 1 n ′ 2 ) Γ(f n ′ −1) Γ(f n ′ ) , and Γ(·) is the gamma function. Finally, using the above result, we propose the bias-corrected discriminant function δ Msplit-hr (x * ; θ θ θ n ) = 1 L L ℓ=1 p j=1 r j (x * ; θ θ θ (2) n,ℓ ) −r n 2 h j ( θ θ θ (1) n,ℓ ) (3.5) which has its bias B Msplit-hr n = 0. The termr n is a function of (n ′ 2 − n ′ 1 ) which is negative since n 2 < n 1 . Hence, for any new feature vector x * , the resulting discriminant function (3.5) tends to be more positive compared to the rule in (3.4). This increases the chance (or probability) of classifying a new observation to the minority class, and hence improving the classification results for this class. In our simulations and the real-data analysis, we evaluate the performance of Msplit-hr based on the bias corrected function δ Msplit-hr . We now describe Algorithm 1 that summarizes the steps for computing (3.5). Algorithm 1 : Computing the discriminant function δ Msplit-hr . Require: Input n ′ 1 = ⌊n 1 /2⌋, n ′ 2 = ⌊n 2 /2⌋, x * , L,r n and τ n . 1: for ℓ = 1, . . . , L do 2: Split D n into D(1) n,ℓ and D (2) n,ℓ 3: for j = 1, . . . , p do 4: Step1: Using D (1) n,ℓ compute h j ( θ θ θ (1) n,ℓ ) 5: Step2: Using D (2) n,ℓ compute r j (x * ; θ θ θ (2) n,ℓ ) −r n 2 6: end for 7: end for 8: return δ Msplit-HR (x * ; θ θ θ n ) = 1 L L ℓ=1 p j=1 r j (x * ; θ θ θ (2) n,ℓ ) −r n 2 h j ( θ θ θ (1) n,ℓ ). In practice, a value of L is required to compute (3.5). Figure 1 shows the class-specific MCRs of (3.5) as a function of L, corresponding to scenario (i) in our simulations in Section 5.1. It can be seen that a value of L between 20 to 30 provides a satisfactory performance of Msplit-hr. We used L = 30 in our numerical experiments. The following results show the asymptotic behaviour of δ Msplit-hr . First, we state Lemma 3.1 that provides conditions under which the t-statistic (3.2) used in the thresholding operator h j in δ Msplit-hr selects all the important features. Since L is fixed, the result of the lemma holds for all ℓ = 1, . . . , L. Lemma 3.1 Assume that the mean difference vector µ d = µ 2 − µ 1 is sparse. Let S = {j : µ dj = 0} be the the corresponding active set with the cardinality s = \|S\|, and define In the above Lemma, if d 0,n = d 0 , for some constant d 0 > 0, then τ n = O( √ n 2 ) and log p = o(n 2 ). On the other hand, if d 0,n ∼ n −γ 2 α n 2 , for 0 < γ < 1 and some α n 2 → ∞, such that d 0,n declines to zero and √ n 2 d 0,n → ∞, then we have τ n = O(n 1/2−γ 2 α n 2 ) and log p = o(n 1−2γ 2 α 2 n 2 ). Therefore, in both cases the divergence rate of the dimension p is smaller than that of the minority class size n 2 , as opposed to the balanced case where log p = o(n), that is, a larger dimension p allowance. d 0,n = min j∈S \|µ dj \|. Under Conditions (C1) and (C2) in Appendix A, if τ n = O( √ n 2 d 0,n ), log s = o(n 2 d 2 0,n ), log(p − s) = o(τ 2 n ), n 2 = o(n 1 ), and √ n 2 d 0,n → ∞, as n 1 , n 2 → ∞, then (a) Pr j ∈S {\|t j \| ≤ τ n } → 1; (b) Pr j∈S {\|t j \| > τ n } → 1.Π Msplit-hr k (D n ) = Φ − 1 2 ∆ p (1 + O p (κ n )) , k = 1, 2 (b) if s∆ 2 p = o(n 2 ) and ∆ 2 p log p/n 1 = o(1), the Msplit-hr is asymptotically-strong opti- mal. Note that the result of Theorem 3.1 also holds for the hr. Part (b) of the theorem implies that the growth rates of both the sparsity size s and the discriminative power ∆ p are controlled by the minority class size n 2 . Msplit-HR under a general Σ When the dimension p is large compared to the sample size n, the sample covariance matrix in (2.5) is ill-conditioned. To deal with the singularity issue, many existing methods in the literature involve a feature selection strategy. In what follows, we use a variable screening method (Fan and Lv, 2008;Pan et al., 2016) to select a subset of features x j 's that have the highest discriminative power. At the ℓ-th data splitting stage of Msplit-hr, we consider the mean difference estimators µ µ µ (1) d,ℓ = µ (1) 2,ℓ − µ(1) 1,ℓ , which are computed based on the training sub-samples D (1) n,ℓ , for ℓ = 1, ..., L. For a given threshold parameter τ n , we select those features x j whose indices belong to the set S (1) n,ℓ = {1 ≤ j ≤ p : \| µ (1) dj,ℓ \| > τ n }, where µ (1) dj,ℓ is the j-th entry of µ µ µ (1) d,ℓ . For any p-dimensional feature vector x * , we define the discriminant function δ Msplit-hr 0 (x * ; θ θ θ n ) = 1 L L ℓ=1μ µ µ ⊤ d,ℓ Σ −1 n,ℓ (x * ℓ −μ µ µ a,ℓ ), (3.6) where θ θ θ n is the vector of corresponding parameter estimates, and x * ℓ = (x * j : j ∈ S (1) n,ℓ ) ⊤ are sub-vectors of the full feature vector x * . Furthermore, for all ℓ = 1, . . . , L, we havẽ µ d,ℓ =μ 2,ℓ −μ 1,ℓ ,μ a,ℓ = (μ 1,ℓ +μ 2,ℓ )/2, such thatμ k,ℓ = ( µ (2) jk,ℓ : j ∈ S (1) n,ℓ ) ⊤ for k = 1, 2, and Σ n,ℓ = [ σ (2) jj ′ ,ℓ : j, j ′ ∈ S (1) n,ℓ ] are respectively the sub-vectors and sub-matrices of the sample means and covariance matrix given in (2.4) and (2.5). Note that for the existence of Σ −1 n,ℓ , for all ℓ = 1, 2, . . . , L, we include at most (n ′ − 2) features in each S (1) n,ℓ . As discussed in Subsection 3.1, the data splitting technique facilitates computation of the bias B n (3.3) corresponding to (3.6). Proposition 3.2 If \|S (1) n,ℓ \| < n ′ − 3, for all ℓ = 1, .., L, then B Msplit-hr 0,n = E{Ψ Msplit-hr 0,1 ( θ θ θ n ) − Ψ Msplit-hr 0,2 ( θ θ θ n )} = 1 L L ℓ=1 E{r n,ℓ }, wherer n,ℓ = 1 n ′ 1 − 1 n ′ 2 n ′ − 2 n ′ − 3 − \|S (1) n,ℓ \| × \|S (1) n,ℓ \|. (3.7) Finally, our bias-corrected discriminant function is δ Msplit-hr (x * ; θ θ θ n ) = 1 L L ℓ=1 {μ µ µ ⊤ d,ℓΣ Σ Σ −1 n,ℓ (x * ℓ −μ µ µ a,ℓ ) −r n,ℓ 2 } (3.8) which has its bias B Msplit-hr n = 0. The termr n,ℓ as a function of (n ′ 2 − n ′ 1 ) makes the corrected discriminant function (3.8) more positive compared to the rule in (3.6). This increases the probability of classifying a new observation to the minority class, and hence improving the results for this class. Algorithm 2 below summarize the steps for computing in (3.8). Figure 2 shows the class-specific MCRs of (3.8) as a function of L, corresponding to scenario (iv) in our simulations in Section 5.2. Based on these results, we used L = 30 in our numerical experiments. Algorithm 2 Computing the discriminant function in (3.8) Require: Input n ′ 1 = ⌊n 1 /2⌋, n ′ 2 = ⌊n 2 /2⌋, x * , L, and τ n . 1: for ℓ = 1, . . . , L do 2: Split D n into D (1) n,ℓ and D (2) n,ℓ 3: Using D (1) n,ℓ , obtain S (1) n,ℓ = {1 ≤ j ≤ p : \| µ (1) dj,ℓ \| > τ n } and computer n,ℓ in (3.7) 4: if \|S (1) n,ℓ \| < n ′ 1 + n ′ 2 − 3 then 5: Using S (1) n,ℓ and D (2) n,ℓ , computeμ µ µ ⊤ d,ℓΣ Σ Σ −1 n,ℓ (x * ℓ −μ µ µ a,ℓ ) 6: else 7: Step 1: Select the first (n ′ 1 + n ′ 2 − 4) features in S (1) n,ℓ with highest value of \| µ (1) dj,ℓ \| 8: Step 2: Using D (2) n,ℓ and the selected features in Step 1, computeμ µ µ ⊤ d,ℓΣ Σ Σ −1 n,ℓ (x * ℓ −μ µ µ a,ℓ ) 9: end if 10: end for 11: return δ Msplit-HR (x * ; θ θ θ n ) = 1 L L ℓ=1 {μ µ µ ⊤ d,ℓΣ Σ Σ −1 n,ℓ (x * ℓ −μ µ µ a,ℓ ) −r n,ℓ 2 }. The following lemma shows that the variable screening method used to obtain the selection sets S (1) n,t have a so-called strong screening consistency property, as discussed in Pan et al. (2016). We then establish the asymptotic optimality of δ Msplit-hr in Theorem 3.2. tion setting (iv) and p = 500. Lemma 3.2 Let β = Σ −1 µ d , and define the active set S = {1 ≤ j ≤ p : β j = 0} with its cardinality denoted by \|S\|. Furthermore, let d 0,n = min j∈S \|µ dj \| and m max = c 1 (max j∈S β 2 j )\|S\|/d 2 0,n , for some constant c 1 > 0 such that m max ≥ \|S\|. Under Condition (C2) in Appendix A, if τ n ≍ d 0,n , log p = o(n 2 d 2 0,n ), n 2 = o(n 1 ), and √ n 2 d 0,n → ∞, as n 1 , n 2 → ∞, for any ℓ = 1, ..., L, we have that (a) Pr S (1) n,ℓ ⊃ S → 1 ; (b) Pr \|S (1) n,ℓ \| ≤ m max → 1. Part (a) implies that that for large sample sizes n, with probability tending to one, all the active features will be included in the selection sets S (1) n,ℓ , for each ℓ = 1, 2, . . . , L. Part (b) shows that the size of each set S (1) n,ℓ is of order m max . These properties are obtained under the conditions that the divergence rate of the dimension p is lower than that of the minority class size n 2 . (1), then the Msplit-hr is asymptoticallystrong optimal. Π Msplit-hr k (D n ) = Φ − 1 2 ∆ p (1 + O p (κ ′ n )) , k = 1, 2 (b) if ∆ 2 p m max = o(n 2 ) and ∆ 2 p m max log p/n 1 = o Condition ∆ 2 p m max = o(n 2 ) in the above theorem implies that the maximum size of the selection sets S n,ℓ , that is m max , is affected by the minority class size n 2 . Note that the results of the theorem also holds for the pairwise sure independence screening of Pan et al. (2016) in the imbalanced binary cases, as well as in the balanced cases which was not studied before. TWO EXISTING HIGH-DIMENSIONAL VARIANTS OF LDA In this section, we investigate conditions under which two well-known sparse variants of the LDA obtain certain optimality properties under the imbalanced setting. Sparse LDA (slda) This method, proposed by Shao et al. (2011), uses thresholding-type estimators for both the mean-difference vector µ d = µ 2 −µ 1 and Σ. In slda, a new feature vector x * is allocated to Class 1 if and only if δ slda (x * ; θ θ θ n ) =μ ⊤ d Σ −1 n (x * − µ a ) < 0, where µ µ µ a = ( µ µ µ 1 + µ µ µ 2 )/2, and ( Σ n ,μ d ) are thresholded estimates of Σ and µ d , respectively, with the entries,σ ij = (1 − 2/n) σ ij 1{(1 − 2/n)\| σ ij \| > t n }, i, j = 1, . . . , p µ dj = µ dj 1{\| µ dj \| > a n }, j = 1, . . . , p, where σ ij is (i, j)-th element of Σ n in (2.5), and µ dj is the j-th entry of µ µ µ d in (2.4). Further, t n = M 1 log p/n with M 1 > 0, and a n = M 2 (log p/n) α , 0 < α < 1/2, M 2 > 0. Shao et al. (2011) derived conditions under which the slda is optimal according to their Definition 1, when p/n → ∞ and n 1 /n → π with 0 < π < 1, as n → ∞. It turns out that their conditions do not yield an optimal slda in the imbalanced case. In Theorem 4.1 below, we investigate conditions under which the slda is asymptotically-strong optimal under the imbalanced case. We then discuss and compare these conditions with those of Shao et al. (2011) under the balanced case. First, for ease of comparison, we recall some notations introduced in Shao et al. (2011). Let q n be the number of features for which the value \| µ dj \| is greater than a n . Further, let q n0 and q n be the number of features for which the value of \|µ dj \| is greater than ra n and a n /r, respectively, for some fixed constant r > 1. Also let D g,p = p j=1 µ 2g dj , 0 ≤ g < 1, and C h,p = max 1≤i≤p p j=1 \|σ ij \| h , 0 ≤ h < 1, be the sparsity measures corresponding to µ d and Σ, respectively. Here, 0 0 is defined to be 0. Furthermore, let d n 1 = C h,p (n 1 −1 log p) (1−h)/2 , and b n 1 = ∆ −1 p max ∆ p d n 1 , a 2(1−g) n D g,p , q n /n 2 , C h,p q n /n 1 , b n 2 = ∆ −1 p max ∆ p d n 1 , a 2(1−g) n D g,p , C h,p q n /n 2 , where ∆ 2 p = µ ⊤ d Σ −1 µ d . Note that under the imbalanced setting n 2 = o(n 1 ), we have d n 1 ∼ d n , where d n = C h,p (n −1 log p) (1−h)/2 . The following Lemma shows that the set {1 ≤ j ≤ p : \| µ dj \| > a n } has indeed a sure screening property, which is essential in Theorem 4.1 for the assessment of slda. Lemma 4.1 Suppose that, (log p) (n 1 / log p) 2α = o(n 2 ), (4.1) and n 2 = o(n 1 ), then as n 1 , n 2 → ∞, (a) Pr j:\|µ dj \|>ran {\| µ dj \| > a n } → 1, (b) Pr j:\|µ dj \|≤an/r {\| µ dj \| ≤ a n } → 1, (c) Pr q n0 ≤ q n ≤ q n → 1. Condition (4.1) replaces the condition log p/n = o(1) in Shao et al. (2011). One implication of (4.1) is log p/n 2 = o(1), which shows the impact of the minority class size n 2 on the dimension allowance p. Theorem 4.1 Suppose that the conditions of Lemma 4.1, and Conditions (C2) and (C3) in Appendix A are satisfied. Then, as n 1 , n 2 → ∞, (a) the MCRs of slda are given by Π slda k (D n ) = Φ − 1 2 ∆ p {1 + O p (b n k )} , k = 1, 2. (b) the slda is asymptotically-strong optimal if i. ∆ 2 p is bounded, and b n 2 = o(1), or ii. ∆ 2 p → ∞, such that ∆ 2 p b n 2 = o(1) holds. The difference between the above theorem and Theorem 3 b n 2 = o(1) is equivalent to s = o((n 1 / log p) α ) . This implies that under the imbalanced setting, the growth rate of the sparsity factor s is smaller than √ n 2 and consequently is smaller than the growth rate of s in the balanced setting. Therefore, due to the data scarcity in the minority class (n 2 ) in the imbalanced setting, in order for the slda to be asymptotically-strong optimal more restrictive conditions are required on both the dimension p and the sparsity size s compared to the balanced case. Next, we compare the optimality conditions of Msplit-hr and slda. The relation between these conditions for a general Σ is not straightforward, and thus to get some insight we consider a diagonal case. Suppose that Σ is diagonal (C 0,p = 1), and g = 0 such that D 0,p = s = \|S\|, where S = {1 ≤ j ≤ p : µ dj = 0}. By condition (4.1), we have log p = o(n 2 ) which implies the necessary conditions of Lemma 3.1 on (s, p), if d 0,n = min j∈S \|µ dj \| = d 0 > 0 and τ n = M √ n 2 , for some constant M > 0. On the other hand, if d 0,n decays, the same conclusion holds when a n = O(d 0,n ) and τ n = M √ n 2 d 0,n . Furthermore, by (4.1) the conditions of Theorem 4.1-(b) are equivalent to s∆ 2 p (log p/n 1 ) 2α = o(1) implying s∆ 2 p = o(n 2 ) which is required for the optimality of Msplit-hr. Therefore, the conditions of Theorem 4.1 for slda on the dimension p and the sparsity size s are more restrictive than those in Theorem 3.1 for Msplit-hr. In terms of feature selection, Lemma 3.2-(b) provides an upper bound m max = o(n 2 ∆ −2 p ) on the size of the set of selected features by Msplit-hr, whereas the slda allows the number of nonzero estimators of µ dj 's or σ l,j 's to be much larger than the class sizes to ensure optimality of the classifier, see Shao et al. (2011). Therefore, the number of selected features by slda could be potentially larger than the class sizes which we have also observed in our numerical study in Section 5. Regularized optimal Affine discriminant (road) This method, proposed by Fan et al. (2012), is constructed based on a sparse estimate of w = Σ −1 µ d , unlike the slda which uses sparse estimates of µ d and Σ, separately. The road assigns x * to Class 1 if and only if δ road (x * ; θ θ θ n , c) = w ⊤ c (x * − µ µ µ a ) < 0, (4.2) where θ θ θ n = ( µ µ µ 1 , µ µ µ 2 , Σ n ), µ µ µ a = ( µ µ µ 1 + µ µ µ 2 )/2, and w c ∈ arg min w 1 ≤c, w ⊤ µ µ µ d =1 w ⊤ Σ n w (4.3) with µ µ µ d = µ µ µ 2 − µ µ µ 1 , and ( µ µ µ k , Σ n ) are the estimates in (2.4)-(2.5). Note that in (4.3) the smaller the c, the sparser the solution w c , and as c → ∞ the solution is equivalent to the regular weight w c ∝ Σ −1 µ d . Fan et al. (2012) studied the asymptotic difference between the average MCR of the road and its oracle version for which the true values of (µ 1 , µ 2 , Σ) are used in (4.3). However, as discussed in Section 2.2, under the imbalanced setting the average MCR is not an appropriate performance measure for a classifier. Therefore, in the following theorem, we study the class-wise MCRs of the road. c , and w c are respectively the solutions of (4.3) when (µ d , Σ), ( µ µ µ d , Σ) and ( µ µ µ d , Σ n ) are used. Furthermore, let Π road k (D n ; c) be the MCR of Class k = 1, 2, associated with road, and Π orc k (c) denotes its oracle value. Under Condition (C2) in Appendix A, if n 2 = o(n 1 ) and log p = o(n 2 ), then as n 1 , n 2 → ∞, Π road k (D n ; c) − Π orc k (c) = O p (e n ), k = 1, 2, (4.4) where e n = max c 2 (log p)/n 1 , (log p)/n 2 × max{s c , s c , s c } . By Theorem 4.2, a necessary condition for convergency of the MCRs of road to their oracle values is that the sparsity size s c of the vector w c and the dimension p are controlled by the minority class size n 2 (similar to the slda), which in turn shows the effect of imbalanced class sizes on the performance of road. In general, the conditions of Theorem 4.2 do not guarantee the optimality of road according to Definition 1. Fan et al. (2012) showed that when the penalty parameter c is chosen as c ≥ ∆ −2 p Σ −1 µ d 1 , then w c ∝ Σ −1 µ d and the oracle MCRs Π orc k (c) reduce to those of the optimal rule in (2.2). Hence, by Definition 1, for such c's, Theorem 4.2 shows that road is asymptotically-strong sub-optimal as long as e n → 0. Furthermore, road becomes asymptotically-strong optimal if ∆ p is bounded. The condition e n → 0 is the same as log p = o(n 1 /c 2 ) and log p = o(n 2 /s max ), where s max = max{s c , s c , s c }. Note that, the larger the c, the larger the quantities s c , s c and s (1) c , and hence more restrictions on (n 1 , n 2 , p) compared to those in Theorem 4.2, and the conditions of Msplit-hr. In our numerical study, we observe that the performance of road in terms of MCR 2 improves for lower dimensions. SIMULATION STUDY In this section, we assess the finite-sample performance of Msplit-hr and several binary classification methods using simulations. We consider two settings of diagonal and general covariance matrix Σ under the model X\|Y = k ∼ N p (µ k , Σ), k = 1, 2. Diagonal Σ We compare the following methods: the bias adjusted independence (bai) and leaveone-out independence rules (loui) Bak and Jensen (2016), diagonal road method (droad) Fan et al. (2012), the bias corrected LDA (blda) Huang et al. (2010), the hr and its undersampling version (us-hr), and our proposed method Msplit-hr. Note that the aforementioned methods use the knowledge of a diagonal Σ. In our comparison, we also include a bias-corrected support vector machines proposed by (Nakayama et al., 2017) coupled with an under-sampling method (us-bcsvm). In regards to over-sampling techniques such as the somte, Bak and Jensen (2016) and Blagus and Lusa (2013) showed that such techniques deduce larger differences between the MCRs in high-dimensional imbalanced problems. For example, we examined the performance of hr and bcsvm coupled with smote (under both diagonal and general Σ) and since their performances were not satisfactory, we did not report the results here. We implemented the methods using R software. The droad results are based on the authors' MATLAB codes available on their website 1 . Our computations are carried out on a computer with an AMD Opteron(tm) Processor 6174 CPU 2.2GHz. The above methods involve certain tuning (threshold) parameters that need to be chosen using data-driven methods. We chose best threshold parameters in blda, bai and loui by a grid search using the techniques outlined by the authors. As in Huang et al. (2010), an F-statistic is used to select the important features in blda method. In both hr and Msplithr, we choose the tuning parameter τ by minimizing MCR of the minority class based on a leave-one-out cross validation. We consider the binary classification problem X\|(Y = k) ∼ N p (µ k , D), k = 1, 2, and D = diag{σ 2 1 , ..., σ 2 p }. We generated training data with different class sizes n 1 and n 2 , and test data sets of size 50 in both classes. We considered two dimensions p = 1000, 3000, and class-wise sample sizes (n 1 , n 2 ) = (25, 5), (50, 10), (100, 10) for the training data. The simulation results are based on 100 randomly generated data sets, and the two parameter settings: (i) µ 1 = (1, 1, 0 p−2 ) ⊤ , µ 2 = (2, 2.2, 0 p−2 ) ⊤ , σ 2 1 = 1.5 2 , σ 2 2 = 0.75 2 , and σ 2 j = 1, for j = 3, ..., p. (ii) µ 1 = (1 9 , 0 p−9 ) ⊤ , µ 2 = (2 * 1 4 , 2.5 * 1 3 , 3 * 1 2 , 0 p−9 ) ⊤ , σ 2 j = 10, for j = 1, ..., 4, σ 2 j = 2.25 2 , for j = 5, 6, 7, σ 2 j = 1.5 2 , j = 8, 9, and σ 2 j = 1, for j = 10, ..., p. The number (s) of active features x j 's that distinguish the two classes, and also the value of ∆ p in the two settings are respectively s = 2, ∆ 2 p = 3 and s = 9, ∆ 2 p = 8.7. Since the signal strength is measured by ∆ p , setting (i) has a weaker signal than (ii). Under these settings, the value of the optimal MCR, Π opt in (2.2), are respectively 19.32% and 7%. Also Discussion of the results The results for the cases (n 1 , n 2 , p) =(25, 5, 1000), (50, 10, 1000) and (100, 10, 1000) are given in Table 1. The results corresponding to dimension p = 3000 are given in Table 2. From Table 1, under both settings (i) and (ii), we can see that droad, hr, and blda have smaller error rates in the majority class (MCR 1 ) compared to the other methods, but the differences between their MCR 1 and MCR 2 are larger. The class-wise error rates corresponding to us-hr and us-bcsvm have smaller differences than those of droad, hr, and blda. Furthermore, the us-hr outperforms us-bcsvm, droad, hr, and blda in terms of MCR 2 . Under setting (i), Msplit-hr outperforms all the other methods in terms of MCR 2 ; for example, its MCR 2 is better than the next best method loui up to about 8%, depending on class sizes (n 1 , n 2 ) and dimension p, while having balanced results for both classes. In setting (ii), Msplit-hr behaves similarly to loui and bai, with its MCR 2 better than loui and bai respectively up to about 3% and 7%. Note that in (i), we have a weaker signal strength (∆ 2 p ) and fewer number of active features (s) than (ii), which matches the conditions of Theorem 3.1 for Msplit-hr on controlling the size of s∆ 2 p . In other words, we can see that the weaker the signal, the better the performance of Msplit-hr in terms of MCRs in both classes. On the other hand, from the columns A and N of Table 1, Msplit-hr tends to select fewer number of inactive (noise) features compared to the two its competitors bai and loui. In bcsvm, the bias caused by dimension is corrected by using all features in the model and therefore this method does not perform any feature selection. Table 2 consists of the results for dimension p = 3000. As expected, the class-specific MCRs of all the methods increase compared to p = 1000. Msplit-hr outperforms all the other techniques in terms of MCR 2 while having balanced misclassification rates. For example, the MCR 2 of Msplit-hr is smaller than the next best method loui up to about 7%. In addition, we observe that Msplit-hr has better performance than bai and loui even in setting (ii) in which they have comparable performance for p = 1000. We now assess the computational efficiency of the different methods. For a fixed threshold, the computational complexity of bai and loui is O(n 2 p) and that of all the other methods is O(np). In our simulations, the threshold (or tuning) parameter in each method was chosen using a cross validation criterion. Table 3 provides the average computational time (in seconds) taken by each method to complete per-sample results. Note that since usbcsvm does not involve any feature selection step, as expected, this method is among the faster methods discussed here. It can be seen that the hr and blda, followed by us-hr and us-bcsvm, are the fastest among all the methods we considered, but they are outperformed by the other methods in terms of the error rate in the minority class. In addition, while bai and loui's performances in terms of the error rates in the minority class are comparable to our proposed method Msplit-hr; the former are slower in terms of computational time. General Σ We considered the same binary classification problem as in Section 5.1, i.e. X\|(Y = k) ∼ N p (µ k , Σ), k = 1, 2, but with a general non-diagonal Σ. We generated training data with different class sizes n 1 and n 2 , and test data sets of sizes 50 in both classes. The simulation results are based on 100 randomly generated data sets. The parameter settings are: (iii) µ 1 = 0 p , µ ⊤ 2 = (1, 0.5 * 1 ⊤ 5 , 0.1 * 1 ⊤ 5 , 0 ⊤ p−11 ), (Σ) ij = 0.8, for i = j, (Σ) ii = 4, for i = 1, ..., p and ∆ 2 p = 0.71. (iv) µ 1 = 0 p , µ ⊤ 2 = (1, 0 ⊤ 4 , 0.1, 0 ⊤ p−6 ), Σ = Σ 1 0 Σ 2 0 . . . , where (Σ 1 ) ij = 0.3, and (Σ 2 ) ij = 0.8, for i = j, (Σ 1 ) ii = (Σ 2 ) ii = 1, for i = 1, ..., 5 and ∆ 2 p = 1.27. In what follows, using the same performance measures described in Section 5.1, we compare these methods: fair, slda, road, Msplit-hr, a binary version of the pairwise sure independent screening (psis) method by Pan et al. (2016), bias adjusted road (ba-road) and leave-one-out road (lou-road) by Bak and Jensen (2016), and us-bcsvm mentioned in Section 5.1. For the fair, road, ba-road, lou-road, we used the techniques based on cross-validation described in the related papers for selecting tuning parameters. We applied the bi-section method of Li and Shao (2015) for tuning parameter selection in slda by minimizing the MCR of the minority class (called slda mcr 2 , in the tables). All the aforementioned methods provide sparse estimates, say β, of the vector β = (β j : 1 ≤ j ≤ p) ⊤ = Σ −1 µ d by either plugging in particular sparse estimates of µ d and Σ, or by directly finding sparse estimate of β. Thus, in our simulation results for each method, we also report the number of j's for which β j = 0, denoted by S in the tables. For Msplithr, we report the cardinality of the set S n = {1 ≤ j ≤ p : f j L ≥ 0. 5}, where f j is the selection frequency corresponding to index j over the splits ℓ = 1, ..., L. Table 4 contains the simulation results for (n 1 , n 2 , p) = (25, 5, 200), (50, 10, 200) and (100, 10, 200), and the results for the dimension p = 500 are given in the Table 5. Discussion of the results From Tables 4 and 5, under both settings (iii) and (iv), we can see that fair, slda, psis and road tend to classify more observations to the majority class, and resulting in large differences between the two MCRs. Overall, the techniques us-bcsvm, ba-road, louroad and Msplit-hr perform better than fair, slda, psis and road in terms of MCR 2 and the geometric mean. For the setting (iii), in the case (n 1 , n 2 ) = (25, 5), Msplit-hr outperforms others, and in the cases, (n 1 , n 2 ) = (50, 10) and (100, 10), the us-bcsvm and lou-road have better performance than others; for example, when (n 1 , n 2 ) = (100, 10), lou-road outperforms Msplit-hr about 4%. For the setting (iv), Msplit-hr outperforms all the other techniques in terms of MCR 2 ; for example outperforms bc-svm and lou-road respectively up to about 10% and 12% depending on the values of (n 1 , n 2 , p). Moreover, this performance of Msplit-hr is based on a much smaller set of selected features compared to its competitors. In summary, Msplit-hr has better performance in the setting (iv) which includes more features with weak signals than (iii). Next, we assess the computational efficiency of different methods by studying the average computational time (in seconds) taken by each method to complete per-sample results, which are given in Table 6. We can see that psis is the fastest method followed by fair and usbcsvm. However, as seen above, these methods do not perform well in terms of the MCRs. As mentioned before, us-bcsvm is computationally fast, since it does not involve any feature selection step. The slda is slower than the Msplit-hr when the dimension p is increased from p = 200 to 500. On the other hand, Msplit-hr is computationally more efficient than its two competitors ba-road and lou-road. Note that, for a fixed value of tuning parameter, the computational complexity of ba-road and lou-road is O(n 2 p 2 ), and that of Msplit-hr is O(np 2 ). Therefore, even without a tuning selection procedure, our technique has lower computational cost. In summary, given the difficulty of the imbalanced problem, our current simulation study shows that (considering all the three factors: misclassification rates, feature selection, and computational efficiency) Msplit-hr has a good performance compared to the methods discussed here, and is yet another reliable technique for high-dimensional imbalanced problems. REAL-DATA ANALYSIS We now demonstrate the performance of different methods on two real data sets. 2 The first data set, on breast cancer (Gravier et al., 2010), consists of the expression profiles of 2905 genes for 168 patients of whom 111 patients with no event after diagnosis were labelled as "good" and the remaining 57 patients with early metastasis were labelled as "poor". In our analysis, we randomly split the data into training data of sizes 56 and 28 of respectively good cases (the majority Class 1) and poor cases (the minority Class 2). The rest of the data is used for testing. The classification results, under the assumptions of (a) uncorrelated and (b) correlated features, are given in Table 7. Under (a), the results suggest that bai, loui, Msplit-hr, and us-hr have comparable performance, with bai and loui performing slightly better than the other two in terms of the MCR of the minority class (MCR 2 ). Under (b), ba-road, lou-road, and Msplit-hr perform similar in terms of the MCRs. us-bcsvm has smaller MCRs compared to the others but by using the set of all features as it is not able to perform any feature selection. Note that in both cases, Msplit-hr selects a much smaller number of features toward the classification task. The second data set, on multiple-myeloma cancer (Tian et al., 2003), consists of the expression profiles of 12, 2625 genes for 173 patients with newly diagnosed multiple-myeloma, of whom 137 were with bone lytic lesions and the remaining 36 patients were without bone lytic lesions. We randomly choose a training set containing 18 observations from patients labelled by MRI-no-lytic-lesion (the minority Class 2), and 72 observations from patients labelled by MRI-lytic-lesion (the majority Class 1). The rest of the data were used for testing. Table 8 For this data set, the overall performances of the aforementioned three methods are better than us-bcsvm. Note that in both cases, Msplit-hr selects a smaller number of features toward the classification task. To reduce the computational cost of each method, and by using a t-statistic, we screened the initial number of features in each of the above data sets by selecting a subset of p = 1500 genes. CONCLUSION In this paper, we have studied linear discriminant analysis (LDA) in high-dimensional imbalanced binary classification. To the best of our knowledge, this is the first work that rigorously investigates such problems which frequently arise in a wide range of applications. First, we showed that in the aforementioned settings the standard LDA asymptotically ignores the so-called minority class. Second, using a multiple data splitting technique, we proposed a new method, called Msplit-hr, that obtains desirable large-sample properties. Third, we derived conditions under which two well-known sparse versions of the LDA in our setting obtain certain desirable large-sample properties. We then examined the finite-sample performance of different methods via simulations and by analyzing two real data sets. In our simulations, the Msplit-hr either outperforms competing methods or has comparable performance in terms of misclassification rate in the minority class, while it has a lower computational cost. The methodology (Msplit-hr) and theory developed in this paper are based on normal distribution for the feature vector X. The normality is used for bias calculations in Propositions 3.1-3.2, and to establish feature selection consistency in Lemmas 3.1-3.2. On the other hand, Delaigle and Hall (2012) showed that feature selection methods based on mean-differences are sensitive to heavy-tailed distributions for X, and they suggested transformation approaches in feature space which are more resistant to extreme observations from heavy-tailed distributions. Properties of such transformations with respect to our theoretical guidelines, and in general, extension of our results to non-normal models require further investigation and is a topic of future research. If the covariance matrix differs between the two classes, i.e. X\|(Y = k) ∼ N(µ k , Σ k ), k = 1, 2, the optimal (Bayes) rule is the quadratic discriminant analysis (QDA). Our limited numerical experiment shows that the QDA in imbalanced high-dimensional problems behaves similarly to the LDA ignoring the minority class. A potential approach to alleviate the im-pact of imbalanced class sizes is to reduce the difference between MCRs of an empirical QDA toward that of the optimal rule. However, the main challenge is that none of the aforementioned MCRs have workable closed forms. Li and Shao (2015) studied such differences for sparse QDA, and their results might be useful toward imbalanced problems in the context of QDA. This, however, requires a careful investigation and is a subject of future work. Another possible future research direction is to investigate the possibility of extending the methodology and theory developed in this paper to imbalanced multi-class classification problems. ACKNOWLEDGEMENTS We would like to thank the editor, an associate editor, and two referees for their insightful comments and suggestions that improved the quality of this paper. We thank the National A TECHNICAL LEMMAS In this Appendix, we first state the technical conditions (C1)-(C3) required in our theoretical developments. Next, we state several lemmas that are used in the proofs of our main results. Lemmas A.1 and A.2 are from Bickel and Levina (2008b) and Shao et al. (2011). Lemmas A.3-A.5 are the results from other papers adapted to the imbalanced setting under our consideration. Lemma A.6 states an upper bound for the tail of Student's t-distribution. Technical Conditions: (C1) log p = o(n 1 ), where n 1 is the majority class size. (C2) 0 < c −1 0 < λ min (Σ) ≤ λ max (Σ) < c 0 < ∞, for a constant c 0 > 0. (C3) 0 < c −1 0 < max j=1,...,p µ 2 dj < c 0 < ∞, where µ d = µ 2 − µ 1 . Lemma A.1 (Bickel and Levina, 2008b, Lemma A.3) Let Z i be independent and identically random variables from N p (0, Σ) and λ max (Σ) ≤ ε −1 0 < ∞. Then, P \| n i=1 (Z ij Z ik − σ jk ) \|> nν ≤ C 1 exp(−C 2 nν 2 ) for all \|ν\| ≤ δ where σ jk 's are entries of Σ, and C 1 , C 2 , and δ depend on ǫ 0 only. Shao et al., 2011, Lemma 1) Let ξ n and ν n be two sequence of positive numbers such that ξ n → ∞ and ν n → 0 as n → ∞. If lim n→∞ ξ n ν n = γ, where γ may be 0, Lemma A.2 (positive or ∞, then lim n→∞ Φ(− √ ξ n (1 − ν n )) Φ(− √ ξ n ) = e γ . Lemma A.3 Denote the sets U τ (h, c 0 (p), M) = Σ : σ ii ≤ M, p j=1 \|σ ij \| h ≤ c 0 (p), ∀i, 0 ≤ h < 1 , U τ (h, c 0 (p), M, ǫ 0 ) = Σ : Σ ∈ U τ (h, c 0 (p), M), λ min (Σ) ≥ ǫ 0 > 0 . Let Σ n be a thresholded version of the pooled sample covariance matrix Σ n in (2.5), such , and uniformly on U τ (h, c 0 (p), M, ǫ 0 ), thatσ ij = (1 − 2/n) σ ij 1{(1 − 2/n)\| σ ij \| > t n },Σ −1 n − Σ −1 = O p c 0 (p) (log p/n 1 ) 1−h 2 . Proof. The proof is a straight forward extension of Theorem 1 of Bickel and Levina (2008a) to imbalanced case, and thus omitted here. Lemma A.4 Let X ik = (X i1k , ..., X ipk ) ⊤ , for i = 1, ..., n k , and k = 1, 2, be random samples from p-variate normal distribution with mean vector 0 and diagonal covariance matrix D = diag{σ 2 1 , ..., σ 2 p }. If the Conditions (C1) and (C2) are satisfied and n 2 = o(n 1 ), then as n 1 , n 2 → ∞, we have max 1≤j≤p \| σ 2 j − σ 2 j \|= O p ( (log p)/n 1 ), where σ 2 j , j = 1, . . . , p, are the diagonal elements of the pooled sample variance Σ n in (2.5). Proof. Let X jk = 1 n k n k i=1 X ijk , for k = 1, 2, j = 1, ..., p. We have, Pr max 1≤j≤p \| σ 2 j − σ 2 j \|> η ≤ p j=1 Pr \| σ 2 j − σ 2 j \|> η ≤ 2 k=1 p j=1 Pr 1 √ n k \| n k i=1 (X 2 ijk − σ 2 j ) \|> 1 √ n k η 4 (n 1 + n 2 − 2) + 2 k=1 p j=1 Pr \| n k X 2 jk − σ 2 j \|> η 4 (n 1 + n 2 − 2) ≤ 2 k=1 pC 1 exp − C 2 η 2 16 (n − 2) 2 n k + pC 3 exp − C 4 η 2 16 (n − 2) 2 for \|η\| < δ, where C 1 , C 2 , C 3 , C 4 , and δ are constants depending only on c 0 . The last inequality follows from Lemma A.1. By taking η = M log p/n 1 , for sufficiently large M > 0, under the imbalanced setting and the Condition (C1), the result holds. Lemma A.5 Under conditions of Lemma 3.2 and the imbalanced setting n 2 = o(n 1 ), assume that m max log p/n 1 = o(1). Then for ℓ = 1, . . . , L, as long as n 1 , n 2 → ∞, Σ n,ℓ − Σ ℓ = O p m max log p/n 1 . where Σ n,ℓ = [ σ (2) jj ′ ,ℓ : j, j ′ ∈ S (1) n,ℓ ] and Σ ℓ = [σ jj ′ : j, j ′ ∈ S (1) n,ℓ ]. Proof. Note that if A = [a jj ′ ] be a symmetric p × p matrix then A ≤ max j ′ p j=1 \|a jj ′ \|. Thus, the result is implied by Pr max j∈S (1) n,ℓ j ′ ∈S (1) n,ℓ \| σ (2) jj ′ ,ℓ − σ jj ′ \| > η ≤ j,j ′ ∈S (1) n,ℓ Pr \| σ (2) jj ′ ,ℓ − σ jj ′ \| > η m max (A.1) where m max = c 1 \|S\|(max j∈S β 2 j )/d 2 0,n . The inequality follows from part (ii) of Lemma 3.2. Let µ k,ℓ = [µ jk : j ∈ S (1) n,ℓ ], Z ijk,ℓ = X ijk,ℓ − µ jk,ℓ , andZ jk,ℓ = n ′ k i=1 X ijk,ℓ /n ′ k , where X ijk,ℓ ∈ D(2) n,ℓ , for i = 1, ..., n ′ k , j = 1, ..., p, k = 1, 2, and ℓ = 1, ..., L, where X ik,ℓ ∼ N p (µ k,ℓ , Σ ℓ ). For the first probability term in (A.1), we have Pr \| σ (2) jj ′ ,ℓ − σ jj ′ \| > η m max ≤ 2 k=1 Pr n ′ k i=1 Z ijk,ℓ Z ij ′ k,ℓ − n ′ kZ jk,ℓZj ′ k,ℓ − (n ′ k − 1)σ jj ′ > (n ′ − 2)η m max ≤ 2 k=1 Pr n ′ k i=1 Z ijk,ℓ Z ij ′ k,ℓ − n ′ k σ jj ′ > (n ′ − 2)η m max + Pr \|n ′ kZ jk,ℓZj ′ k,ℓ − σ jj ′ \| > (n ′ − 2)η m max . Finally, using Lemma A.1, j,j ′ ∈S (2) n,t Pr \| σ (2) jj ′ ,ℓ − σ jj ′ \| > η m max ≤ 2 k=1 C 1 p 2 exp − C 2 (n − 2) 2 η 2 m 2 max n k + C ′ 1 p 2 exp − C ′ 2 (n − 2) 2 η 2 m 2 max , where C 1 , C ′ 1 , C 2 , C ′ 2 are some positive constants. If m max log p/n 1 = o(1) and by taking η = M × m max log p/n 1 , for sufficiently large M > 0, the desired result is obtained. Lemma A.6 Suppose that T has the Student's t-distribution with n > 1 degrees of freedom. Then, for any large constant τ > 0, we have Pr(T > τ ) ≤ c n τ n n − 1 1 + 1 n τ 2 − n−1 2 , where c n = Γ( n+1 2 ) Γ( n 2 ) √ nπ , and Γ(.) is the gamma function. Proof. For any τ > 0, Pr(T > τ ) = ∞ τ c n (1 + x 2 n ) n+1 2 dx < ∞ τ x τ c n (1 + x 2 n ) n+1 2 dx = c n τ n n − 1 1 + 1 n τ 2 − n−1 2 . The result follows from the facts that τ > 0 and τ < x < ∞. B PROOFS OF THE MAIN RESULTS In this Appendix, we provide the proofs of Theorems 2.1-4.2. Proof of Theorem 2.1. Let ǫ ǫ ǫ ik = X ik − µ k , for i = 1, ..., n k , and k = 1, 2, where X ik = (X i \|Y i = k) ∼ N p (µ k , Σ) , and the vectorsǭ ǫ ǫ k = (ǭ 1k ,ǭ 2k , ...,ǭ pk ) ⊤ with entiresǭ jk = 1 n k n k i=1 ǫ ijk . Also, recall ∆ 2 p = µ ⊤ d Σ −1 µ d and µ d = µ 2 − µ 1 . The quantities Ψ lda 1 ( θ θ θ n ), Ψ lda 2 ( θ θ θ n ), and Υ lda ( θ θ θ n ) in (2.6) can be decomposed as Ψ lda 1 ( θ θ θ n ) = (µ 1 − µ µ µ a ) ⊤ Σ −1 ( µ µ µ 2 − µ µ µ 1 ) = 1 2 (−ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 − µ d ) ⊤ Σ −1 (ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 + µ d ) = 1 2 ǭ ǫ ǫ ⊤ 1 Σ −1ǭ ǫ ǫ 1 −ǭ ǫ ǫ ⊤ 2 Σ −1ǭ ǫ ǫ 2 − 2ǭ ǫ ǫ ⊤ 2 Σ −1 µ d − µ ⊤ d Σ −1 µ d = 1 2 I 1 − I 2 − 2I 3 − µ ⊤ d Σ −1 µ d , Ψ lda 2 ( θ θ θ n ) = −(µ 2 − µ µ µ a ) ⊤ Σ −1 ( µ µ µ 2 − µ µ µ 1 ) = − 1 2 (−ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 + µ d ) ⊤ Σ −1 (ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 + µ d ) = − 1 2 −ǭ ǫ ǫ ⊤ 2 Σ −1ǭ ǫ ǫ 2 +ǭ ǫ ǫ ⊤ 1 Σ −1ǭ ǫ ǫ 1 − 2ǭ ǫ ǫ ⊤ 1 Σ −1 µ d + µ ⊤ d Σ −1 µ d = 1 2 I 2 − I 1 + 2I 4 − µ ⊤ d Σ −1 µ d , and and hence, as n 1 → ∞, u(n 1 , n 2 , p − s, τ n ) ∼ p − s τ n e −τ 2 n . Since log(p − s) = o(τ 2 n ), therefore Pr(max j ∈S \|t j \| > τ n ) → 0, and this completes the proof. where the last inequality follows from Lemma A.6, when τ n = O( √ n 2 d 0,n ). Since √ n 2 d 0,n → ∞, log s = o(n 2 d 2 0,n ), and n 2 = o(n 1 ), then as n 1 , n 2 → ∞, we have n n 1 n 2 ∼ 1 √ n 2 , n−2 n−3 ∼ 1 and c n = Γ( n−1 2 ) Γ( n−2 2 ) √ (n−2)π → 1 √ 2π . Therefore, u(n 1 , n 2 , s, d 0,n , τ n ) → 0 and it completes the proof. Proof of Theroem 3.1. (a) The class-specific misclassification rates of Msplit-hr in (3.5) are given by Π Msplit-hr k (D n ) = Φ Ψ Msplit-hr k ( θ θ θ n ) Υ Msplit-hr ( θ θ θ n ) , k = 1, 2, where Ψ Msplit-hr k ( θ θ θ n ) = (−1) k+1 L L ℓ=1 p j=1 r j (µ k ; θ θ θ (2) n,ℓ ) −r n 2 h j ( θ θ θ (1) n,ℓ ), Υ Msplit-hr ( θ θ θ n ) = 1 L 2 L ℓ=1 p j=1 σ 2 j µ (2) dj,ℓ / σ (2),2 j,ℓ 2 h j ( θ θ θ (1) n,ℓ ). By Lemma 3.1, if √ n 2 d 0,n → 0, τ n = O( √ n 2 d 0,n ), log(p−s) = o(τ 2 n ) , and log s = o(n 2 d 0,n ), as n 1 , n 2 → ∞, then max j∈S h j ( θ θ θ (1) n,ℓ ) − 1 p −→ 0 , max j ∈S h j ( θ θ θ (1) n,ℓ ) p −→ 0. Using these results, for any ǫ > 0, we have, for k = 1, 2, Pr j ∈S r j (µ k ; θ θ θ (1 + o p (1)), k = 1, 2. Similarly, we have Υ Msplit-hr ( θ θ θ n ) = 1 L 2 L ℓ=1 j∈S σ 2 j µ (2) dj,ℓ / σ (2),2 j,ℓ 2 (1 + o p (1)). Letǭ (2) jk,ℓ = µ (2) jk,ℓ − µ jk , I k,ℓ = j∈S (ǭ(2) jk,ℓ /σ j ) 2 , for k = 1, 2, and I 3,ℓ = j∈S (ǭ j2,ℓ µ dj /σ 2 j ), for each ℓ = 1, .., L. By the result of Lemma A.4 in the Appendix A, we have j∈S r j (µ j1 , θ θ θ where ∆ 2 p = j∈S (µ 2 dj /σ 2 j ). Now, for η > 0, and k = 1, 2 Pr \|I k,ℓ \| > η ≤ s n k η , (B.3) by taking η = M.s/n k , for sufficiently large M > 0, then I k,ℓ = O p (s/n k ), for k = 1, 2.. By Cauchy-Schwartz inequality, we have I 3,ℓ = O p (∆ p s/n 2 ) and I 4,ℓ = O p (∆ p s/n 1 ). In addition, we have p j=1r n h j ( θ θ θ (2) n,ℓ ) = o p (s/n 2 ). By combining these results in (B.1)-(B.2), we arrive at Ψ Msplit-hr k ( θ θ θ n ) = O p (s/n 2 ) + O p ∆ p s/n 2 − 1 2 ∆ 2 p + O p ∆ 2 p log p/n 1 , (B.4) for k = 1, 2. Let I 5,ℓ = j∈S (ǭ j2,ℓ −ǭ j1,ℓ ) 2 /σ 2 j , I 6,ℓ = j∈S (ǭ j2,ℓ −ǭ j1,ℓ )µ dj /σ 2 j for each ℓ = 1, .., L. Similar to (B.3), we result I 5,ℓ = O p (s/n 2 ) and also I 6,ℓ = O p (∆ p s/n 2 ). Therefore Υ Msplit-hr ( θ θ θ n ) = 1 L 2 L ℓ=1 I 5,ℓ + 2I 6,ℓ + ∆ 2 p = O p (s/n 2 ) + O p ∆ p s/n 2 + ∆ 2 p + O p ∆ 2 p log p/n 1 . (B.5) By combining (B.4) and (B.5), we have, for k = 1, 2, Π Msplit-hr k (D n ) = Φ      O p (s/n 2 ) + O p (∆ p s/n 2 ) − 1 2 ∆ 2 p + O p ∆ 2 p log p n 1 O p (s/n 2 ) + O p (∆ p s/n 2 ) + ∆ 2 p + O p ∆ 2 p log p n 1 1/2      = Φ − 1 2 ∆ p {1 + O p (κ n )} , where κ n = max{∆ −1 p s/n 2 , log p/n 1 }. (b) When ∆ p → ∞, by Lemma A.2, if ∆ 2 p κ n = o(1), then Msplit-hr is asymptoticallystrong optimal and the result follows. The condition ∆ 2 p κ n = o(1) is equivalent to ∆ 2 p log p/n 1 = o(1), and ∆ 2 p s = o(n 2 ). Proof of Lemma 3.2. We follow a similar line of proof as in (Pan et al., 2016, Theorem 1), to show the results of both parts (a) and (b), under the imbalanced setting. (a) It is enough to show that for any ℓ = 1, ..., L, as n 1 , n 2 → ∞, Pr S ⊂ S (1) n,ℓ → 0. Suppose that there exist an index j in S for which j ∈ S (1) n,ℓ . Thus, \|µ dj \| ≥ d 0,n and \| µ (1) dj,ℓ \| < τ n , where d 0,n = min j∈S \|µ dj \|. It results in \| µ (1) dj,ℓ − µ dj \| > d 0,n − τ n . By conditions τ n ≍ d 0,n and λ max (Σ) < c 0 , and for some constants C 1 , C 2 > 0, we have Pr S ⊂ S (1) n,ℓ ≤ p j=1 Pr \| µ (1) dj,ℓ − µ dj \| > d 0,n − τ n ≤ C 1 p d 0,n − τ n n ′ 1 + n ′ 2 n ′ 1 n ′ 2 exp − C 2 n ′ 1 n ′ 2 (d 0,n − τ n ) 2 n ′ 1 + n ′ 2 . The last term tends to zero, since log p = o(n 2 d 2 0,n ) and n 2 = o(n 1 ), and thus the result follows. (b) By condition λ max (Σ) < c 0 , we have µ ⊤ d µ d = µ ⊤ d Σ −1 ΣΣ ⊤ Σ −1 µ d ≤ λ max (ΣΣ ⊤ )β ⊤ β ≤ c 0 × \|S\| × max j∈S β 2 j . (B.6) Let S * = {j : \|µ dj \| > d 0,n /r}, for some constant r > 1. Thus, µ ⊤ d µ d ≥ \|S * \|d 2 0,n /r 2 . This together with (B.6), result in \|S * \| ≤ C 3 \|S\| max j∈S β 2 j /d 2 0,n . = m max , for constant C 3 > 0. The result in part (b) follows by proving that, \|S n,ℓ \| < \|S * \|, with probability tending to one, for any ℓ = 1, ..., L. If there exists an index j in S (1) n,ℓ for which j ∈ S * , thus \| µ (1) dj,ℓ \| > τ n and \|µ dj \| < d 0,n /r and consequently, \| µ (1) dj,ℓ − µ dj \| > τ n − d 0,n /r. Therefore, by condition τ n ≍ d 0,n and for constants C 4 , C 5 > 0 We consider the following decompositioñ µ µ µ ⊤ d,ℓ Σ −1 ℓμ µ µ d,ℓ = (μ µ µ d,ℓ − µ d,ℓ ) ⊤ Σ −1 ℓ (μ µ µ d,ℓ − µ d,ℓ ) + 2(μ µ µ d,ℓ − µ d,ℓ ) ⊤ Σ −1 ℓ µ d,ℓ + µ ⊤ d,ℓ Σ −1 ℓ µ d,ℓ = A 1 + 2A 2 + A 3 Now by Lemma 3.2 and Markov's inequality, also using the Condition (C2) in the Appendix A, we have for a constant C 1 > 0, Pr (μ µ µ d,ℓ − µ d,ℓ ) ⊤ Σ −1 ℓ (μ µ µ d,ℓ − µ d,ℓ ) > η ≤ C 1 η n n 1 n 2 m max . If η = M mmax n 2 , then for large M > 0, A 1 = O p (m max /n 2 ). By Cauchy-Schwartz inequality, A 2 2 ≤ (μ µ µ d,ℓ − µ d,ℓ ) ⊤ Σ −1 ℓ (μ µ µ d,ℓ − µ d,ℓ )A 3 . Hence A 2 = O p ( m max /n 2 )A 1/2 3 . Therefore, by combining these results we havẽ µ µ µ ⊤ d,ℓ Σ −1 ℓμ µ µ d,ℓ = O p (m max /n 2 ) + O p ( m max /n 2 ) A 3 + A 3 (B.8) Now for Ψ Msplit-hr 1 ( θ θ θ n ), we havẽ µ µ µ ⊤ d,ℓΣ Σ Σ −1 n,ℓ (µ 1,ℓ −μ µ µ a,ℓ ) =μ µ µ ⊤ d,ℓ Σ −1 ℓ (µ 1,ℓ −μ µ µ a,ℓ ) 1 + O p (m max log p/n 1 ) (B.9) We decompose it as 2μ µ µ ⊤ d,ℓ Σ −1 ℓ (µ 1,ℓ −μ µ µ a,ℓ ) = (μ µ µ 1,ℓ − µ 1,ℓ ) ⊤ Σ −1 ℓ (μ µ µ 1,ℓ − µ 1,ℓ ) − (μ µ µ 2,ℓ − µ 2,ℓ ) ⊤ Σ −1 ℓ (μ µ µ 2,ℓ − µ 2,ℓ ) = 2(μ µ µ 2,ℓ − µ 2,ℓ ) ⊤ Σ −1 ℓ µ d,ℓ − µ ⊤ d,ℓ Σ −1 ℓ µ d,ℓ = B 1 − B 2 − 2B 3 − A 3 Similar to the proof of A 1 , we have B 1 = O p (m max /n 1 ), and B 2 = O p (m max /n 2 ). Also similar to A 2 , we have B 3 = O p ( m max /n 2 ) √ A 3 . Hence, µ µ µ ⊤ d,ℓ Σ −1 ℓ (µ 1,ℓ −μ µ µ a,ℓ ) = O p ( m max n 1 ) + O p ( m max n 2 ) + O p ( m max /n 2 )A 1/2 3 − 1 2 A 3 (B.10) We recall that ∆ 2 p = µ ⊤ d Σ −1 µ d = β ⊤ µ d and S (1) n,ℓ = {j : \| µ µ µ (1) dj,ℓ \| > τ n }. For each ℓ = 1, .., L, and any η > 0 Pr \|µ ⊤ d,ℓ Σ −1 ℓ µ d,ℓ − ∆ 2 p \| > η = Pr \| j∈S (1) n,ℓ β j µ dj − j ′ ∈S β j ′ µ dj ′ \| > η = Pr j∈S (1) n,ℓ ,j ∈S β j µ dj + j∈S (1) n,ℓ ,j∈S β j µ dj − j ′ ∈S,j ′ ∈S (1) n,ℓ β j ′ µ dj ′ − j ′ ∈S,j ′ ∈S (1) n,ℓ β j ′ µ dj ′ > η = Pr \| j∈S,j ∈S (1) n,ℓ β j µ dj \| > η ≤ p j=1 Pr j ∈ S and j ∈ S (1) n,ℓ By part (i) of Lemma 3.2, the last term tends to zero, as n 1 , n 2 → ∞. Therefore, A 3 = ∆ 2 p + o p (1). By combining this result together with (B.7)-(B.10), also withr n,t = O p (m max /n 2 ), we result Pr \| µ dj − µ dj \| > a n (r − 1) ≥ 1 − 2 p j=1 Φ −a n (r − 1) σ j n/n 1 n 2 ≥ 1 − pc 1 exp − log p n 2α . n 1 n 2 n c 2 . (B.11) Therefore in the denominator of Π slda k (D n ), we havẽ µ µ µ ⊤ d Σ −1 n Σ Σ −1 nμ µ µ d = O p (k n 2 ) + ∆ p O p ( k n 2 ) + ∆ 2 p O p (d n 1 ) = O p k n 2 /∆ 2 p + 1 ∆ 2 p O p (d n 1 ). (B.12) Now, the numerator of Π slda k (D n ) can be decomposed as (−1) kμ µ µ ⊤ d Σ −1 n ( µ µ µ k − µ µ µ k ) − 1 2 µ µ µ ⊤ d Σ −1 nμ µ µ d = (−1) kμ µ µ ⊤ d Σ −1 n ( µ µ µ k − µ µ µ k ) − 1 2 ( µ µ µ d − µ µ µ d ) ⊤ Σ −1 nμ µ µ d − 1 2 (µ µ µ d −μ µ µ d ) ⊤ Σ −1 nμ µ µ d − 1 2μ µ µ ⊤ d Σ −1 nμ µ µ d = J 3 + J 4 + J 5 − 1 2μ µ µ ⊤ d Σ −1 nμ µ µ d = μ µ µ ⊤ d Σ −1 nμ µ µ d O p q n /n k + O p q n C h,p /n k 1 + O p (d n 1 ) + μ µ µ ⊤ d Σ −1 nμ µ µ d O p ( q n /n 2 ) 1 + O p (d n 1 ) + O p (k n 2 ) + ∆ p O p ( k n 2 ) O p (d n 1 ) +μ µ µ ⊤ d Σ −1 nμ µ µ d . (B.13) Again, by condition (4.1) we have J 3 =μ µ µ ⊤ d Σ −1 n ( µ µ µ k − µ k ) = μ µ µ ⊤ d Σ −1 nμ µ µ d O p q n /n k + O p q n C h,p /n k 1 + O p (d n 1 ), and J 4 = ( µ µ µ d − µ d ) ⊤ Σ −1 nμ µ µ d = μ µ µ ⊤ d Σ −1 nμ µ µ d O p ( q n /n 2 ). Also, similar to the expression of J 1 , we have J 5 = (µ µ µ d −μ µ µ d ) ⊤ Σ −1 nμ µ µ d = μ µ µ ⊤ d Σ −1 nμ µ µ d O p ( k n 2 ) 1 + O p (d n 1 ). finally, by combining (B.12) and (B.13) we arrive at E r j (µ 1 ; θ θ θ Π slda k (D n ) = Φ (−1) kμ µ µ ⊤ d Σ −1 n ( µ µ µ k − µ µ µ k ) μ µ µ ⊤ d Σ −1 n Σ Σ −1 nμ µ µ d − 1 2 µ µ µ ⊤ d Σ −1 nμ µ µ d μ µ µ ⊤ d Σ −1 n Σ Σ −1 nμ µ µ d = Φ − 1 2 ∆ p O p ∆ − (2) n,ℓ ) + r j (µ 2 ; θ θ θ (2) n,ℓ ) E h j ( θ θ θ (1) n,ℓ ) , where r j (µ k ; θ θ θ (2) n,ℓ ) = µ (2) dj,ℓ (µ jk,ℓ − µ aj,ℓ )/ σ (2),2 j,ℓ . Hencē r n = E r j (µ 1 ; θ θ θ (2) n,ℓ ) + r j (µ 2 ; θ θ θ (2) n,ℓ ) = ( 1 n ′ 1 − 1 n ′ 2 ) Γ(f n ′ − 1) Γ(f n ′ ) f n ′ , where f n ′ = n ′ /2 − 1. L ℓ=1 E E μ µ µ ⊤ d,ℓ Σ −1 ℓ (µ 1,ℓ −μ µ µ a,ℓ ) −μ µ µ ⊤ d,ℓ Σ −1 ℓ (μ µ µ a,ℓ − µ 2,ℓ ) D (1) n,ℓ = 1 L L ℓ=1 E E μ µ µ ⊤ d,ℓ Σ −1 ℓ (µ 1,ℓ + µ 2,ℓ − 2μ µ µ a,ℓ ) = 1 L L ℓ=1 E{r n,ℓ }. The second equation follows from the independence property of D ). and the result follows. Proposition 3. 1 1The bias B n in (3.3) corresponding to δ Msplit- Figure 1 : 1Effect of the number of sample-splits L on Msplit-hr performance for the Simulation setting (i) and p = 1000. Theorem 3. 1 1Suppose that the conditions of Lemma 3.1 are satisfied. Let κ n = max{∆ −1 p s/n 2 , log p/n 1 }. For any fixed L, (a) the MCRs of Msplit-hr are given by Figure 2 : 2Effect of the number of sample-splits L on Msplit-hr performance for the Simula- Theorem 3. 2 2Suppose that the conditions of Lemma 3.2 are satisfied. Let κ ′ n = max{∆ −1 p m max /n 2 , m max log p/n 1 }. If m max log p/n 1 = o(1), then for any fixed L, (a) the MCRs of Msplit-hr are given by of Shao et al. (2011) appears in b n 2 . To simplify the comparison in this case as inShao et al. (2011), suppose that Σ is a diagonal matrix (C 0,p = 1), and let s be the number of nonzero (active) entries of the mean difference vector µ d . If there are two constant c 1 , c 2 > 0, such that c 1 ≤ \|µ dj \| ≤ c 2 , for the active j's, then we have q n = s. This implies that, by the Conditions (C2) and (C3), ∆ 2 p and D 0,p are of order s. Now, in this case, if s → ∞, according to Theorem 4.1-(b)-ii above, under condition (4.1), ∆ 2 p s c = w c 0 , where w c , w , the active features have different marginal signal values \|µ dj \|/σ j , in each of the settings. The performance measures used to compare different methods are: per-class misclassification rates (MCR 1 , MCR 2 ), and the geometric mean (GM) of the MCRs. The results reported in the tables are average and standard deviations (in parentheses) of the measures over 100 generated samples. We also reported median number of true selected features, denoted by A, and falsely selected features denoted by N, respectively. For the new method Msplit-hr, similar to the stability selection technique of Meinshausen and Bühlmann (2010), the selected features for each simulated sample are those with a relative frequency more than 50%, that is the set S n = {j : f j L ≥ 0.5}, where f j is selection frequency of j-th feature among L splits. contains the classification results under the aforementioned assumptions (a) and (b). Under (a), the results show that Msplit-hr and us-hr outperform the other methods in terms of the error rate in the minority class, MCR 2 . In addition, Msplit-hr outperforms us-hr in terms of the error rate in the majority class, MCR 1 . Under (b), the three methods ba-road, lou-road, and Msplit-hr perform similar in terms of the MCRs. some positive constant M 1 . Then uniformly on U τ (h, c 0 (p), M), and for sufficiently large M 1 , under the Condition (C3) and n 2 = o(n 1 ), as n 1 , n 2 → ∞, then Σ n − Σ = O p c 0 (p) (log p/n 1 ) 1−h 2 : {\|t j \| > τ n } = Pr min j∈S \|t j \| > τ n = 1 − Pr min j∈S \|t j \| ≤ τ n . Let t j = t j − \|t j \| ≥ min j∈S \|µ dj \| σ j n/(n 1 n 2 ) − τ n .Also by Lemma A.4 and under the Condition (C2), min j∈S \|µ dj \| σ j n/(n 1 n 2 ) = d 0,n (1 + o p (= u(n 1 , n 2 , s, d 0,n , τ n ), O p ( log p/n 1 ) , (B.1) j∈S r j (µ j2 , θ θ θ (2)n,ℓ )h j ( θ θ θ 1,ℓ − I 2,ℓ − 2I 4,ℓ + ∆ 2 p 1 + O p ( log p/n 1 ) , (B.2) ℓμ µ µ d,ℓ =μ µ µ ⊤ d,ℓ Σ −1 ℓμ µ µ d,ℓ 1 + O p (m max log p/n 1 ) . (B.7) Π MsplitO p (κ ′ n )) ,We can show the same result for Π Msplit-hr 2 (D n ).(b) When ∆ p → ∞, the result follows from Lemma A.2 by condition ∆ 2 p κ ′ n = o(1). Proof of Lemma 4.1. (a) Recall the sequence a n = M 2 (log p/n) α , with 0 < α < 1/2 and M 2 > 0. Let c 1 , c 2 be some positive constants. Inspired by the proof of Lemma 2 of Shao et al. (2011), we have Pr {j:\|µ dj \|>ran}{\| µ dj \| > a n } ≥ 1 − p j=1 1 p q n C h,p /n k + O p k n 2 /∆ 2 p + 1 + O p (d n 1 ) = Φ − 1 2 ∆ p {1 + O p (b n k )} , k = 1 Proof of Proposition 3. 2 .Π 2The MCRs of δ Msplit-hr in (3.8) are given by Msplit ℓ , for each ℓ. Under normal assumption for the distribution of features, the matrix Σ n ′ − 2, where Σ ℓ is the covariance matrix corresponding to the features included in S Table 1 : 1Classification results for the simulation settings (i)-(ii) with a diagonal Σ and p = 1000.(n1, n2) Setting Methods MCR1% MCR2% GM% A N (25,5) (i) us-bcsvm 48.96(13.99) 47.46(14.38) 46.33(5.99) 2 998 DROAD 2.62(7.45) 93.14(15.91) 5.45(11.37) 2 364 HR 15.96(13.02) 67.3(26.43) 26.25(15.45) 1 2 US-HR 44.34(16.45) 45.14(16.49) 42.73(8.91) 1 143.5 BLDA 14.72(11.05) 70.16(22.47) 28.04(11.75) 1 5 BAI 38.86(15.16) 48.2(17.23) 41.15(10.06) 1 75.5 LOUI 41.46(18.14) 43.9(19.43) 39.05(11.75) 1 20.5 Msplit-HR 42.66(16.97) 40.04(16.65) 39.12(11.01) 1 2 (25,5) (ii) us-bcsvm 46.42(14.04) 41.22(12.27) 41.99(5.75) 9 991 DROAD 5.46(8.63) 58.48(30.27) 9.84(10.24) 6 17 HR 12.38(10.52) 55.78(27.55) 20.80(12.40) 1 2 US-HR 39.34(14.75) 35.48(15.43) 35.13(9.11) 4 124 BLDA 11.06(8.74) 57.48(26.60) 20.85(11.30) 2 3 BAI 30.72(13.94) 35.06(16.32) 30.53(10.12) 3 33 LOUI 29.86(14.54) 31.24(16.37) 27.95(10.83) 3 36.5 Msplit-HR 32.2(15.57) 28.22(15.44) 27.61(9.91) 1 3.5 (50,10) (i) us-bcsvm 47.88(10.19) 44.56(10.78) 45.25(5.71) 2 998 DROAD 6.30(9.10) 75.28(30.25) 11.23(11.61) 2 68 HR 19.36(7.47) 40.82 (20.99) 26.15(7.68) 1 1 US-HR 34.22(14.05) 32.66(14.34) 32.12(10.72) 1 0 BLDA 18.26(8.82) 48.68(21.24) 25.27(8.81) 1 3 BAI 31.94(14.39) 36.92(15.19) 32.71(10.52) 1 11 LOUI 29.28(12.21) 34.12(17.04) 29.99(10.39) 1 8.5 Msplit-HR 30.22(12.66) 26.68(13.42) 26.99(9.75) 1 0 (50,10) (ii) us-bcsvm 41.72(9.03) 38.02(10.21) 38.93(5.23) 9 991 DROAD 5.60(6.04) 30.72(19.67) 9.39(6.22) 7 17.5 HR 11.02(7.04) 25.42(15.69) 14.59(6.84) 2 0 US-HR 22.84(10.14) 19.04(8.49) 19.74(7.23) 1 0 BLDA 11.72(6.70) 24.36(15.71) 14.80(6.13) 2 0 BAI 17.6(8.76) 19.8(11.79) 17.18(8.07) 3 3 LOUI 16.72(8.55) 19.16(10.99) 16.55(7.39) 3 3.5 Msplit-HR 19.22(9.58) 17.82(9.07) 17.08(6.84) 2 0 (100,10) (i) us-bcsvm 47.96(10.08) 44.1(10.50) 45.09(5.42) 2 998 DROAD 2.60(5.25) 85.74(22.67) 6.28(9.67) 2 494 HR 19.96(8.53) 34.82(19.40) 24.31(8.09) 1 0 US-HR 34.08(13.17) 30.52(13.07) 31.14(9.29) 1 0 BLDA 16.84(7.60) 45.48(22.81) 25.08(8.01) 1 2 BAI 28.86(12.15) 33.64(16.62) 29.72(9.85) 1 7 LOUI 26.26(11.03) 32.48(16.76) 27.81(9.26) 1 6 Msplit-HR 27.94(12.09) 24.84(13.11) 24.95(8.93) 1 0 (100,10) (ii) us-bcsvm 41.66(9.67) 37.38(10.88) 38.51(6.02) 9 991 DROAD 3.22(4.23) 37.96(20.38) 6.57(6.07) 8 31.5 HR 10.02(6.20) 22.14(12.49) 13.11(5.77) 3 0 US-HR 20.64(10.65) 18.98(10.47) 18.30(7.76) 1 0 BLDA 10.44(6.26) 22.28(14.29) 12.96(5.83) 3 0 BAI 16.44(9.70) 17.08(10.19) 15.49(7.89) 3.5 2 LOUI 15.02(8.37) 15.94(9.32) 14.13(6.42) 3 2 Msplit-HR 16.56(8.98) 14.38(7.88) 14.04(5.36) 3 0 Table 2 : 2Classification results for Simulation settings (i)-(ii) with a diagonal Σ and p = 3000.(n1, n2) Setting Methods MCR1% MCR2% GM% A N (25,5) (i) us-bcsvm 50.84(13.78) 47.32(13.75) 47.32(5.03) 2 2998 DROAD 3.14(7.66) 94.40(12.91) 6.71(13.44) 1 529 HR 14.92(12.68) 73.64(24.84) 26.10(16.85) 0 1.5 US-HR 46.38(16.21) 46.86(16.10) 44.25(7.23) 1 255 BLDA 13.6(11.63) 76.44(23.64) 26.03(14.53) 1 5 BAI 38.04(15.86) 50.92(17.61) 41.77(9.48) 1 107.5 LOUI 41.02(18.00) 47.04(18.60) 41.16(10.20) 1 42 Msplit-HR 43.06(19.61) 44.08(19.25) 40.09(9.18) 1 3 (25,5) (ii) us-bcsvm 47.12(13.88) 45.32(13.29) 44.37(5.11) 9 2991 DROAD 5.54(8.59) 60.58(28.94) 10.24(11.52) 6 16 HR 14.04(12.40) 62.44(27.65) 23.41(14.78) 1 1 US-HR 43.78(15.90) 41.52(15.33) 40.35(8.22) 3 251.5 BLDA 11.04(10.41) 65.64(27.20) 20.50(13.70) 1 3 BAI 33.04(15.35) 41.68(17.60) 34.84(10.61) 3 79.5 LOUI 31.64(16.28) 41.84(18.91) 33.63(10.90) 3 78 Msplit-HR 37.78(17.62) 36.32(17.68) 34.18(10.36) 1 3 (50,10) (i) us-bcsvm 48.46(11.56) 47.98(11.55) 47.04(5.29) 2 998 DROAD 5.7(9.27) 81.02 (24.78) 11.01(13.54) 2 77 HR 18.68(8.29) 40.88 (24.32) 24.61(9.61) 1 0 US-HR 35.62(13.25) 36.66(14.57) 34.87(10.01) 1 0 BLDA 17.16(8.35) 48.18(24.08) 25.74(8.52) 1 2 BAI 32.08(11.86) 37.42(17.28) 33.32(11.03) 1 12.5 LOUI 31.1(12.33) 34.82(17.41) 31.48(11.18) 1 9.5 Msplit-HR 32.3(12.81) 30.98(16.05) 30.35(11.34) 1 0 (50,10) (ii) us-bcsvm 44.08(10.75) 44.16(10.42) 43.04(4.85) 9 991 DROAD 5.12(5.21) 32.40(17.42) 9.24(6.25) 1 25.50 HR 12.98(8.14) 28.7 (18.25) 17.14(8.18) 2 0 US-HR 26.8(12.13) 24.72(12.17) 24.48(8.81) 1 0 BLDA 12.7(6.88) 29.1(19.32) 16.70(7.20) 2 1 BAI 20.24(10.71) 22.44(13.68) 19.55(9.60) 3 4 LOUI 19.02(10.22) 22.74(13.48) 19.45(9.35) 3 9 Msplit-HR 21.4(11.07) 19.2(10.14) 18.93(7.47) 2 0 (100,10) (i) us-bcsvm 48.3(10.85) 48.58(11.83) 47.32(5.50) 2 998 DROAD 1.80(4.23) 88.66(20.70) 4.49(8.31) 2 861.50 HR 18.42(8.24) 42.38(24.65) 25.08(9.02) 1 0.50 US-HR 36.94(14.20) 37.1(13.93) 35.91(10.47) 1 0 BLDA 15.64(8.56) 50.18(26.11) 24.00(9.47) 1 2 BAI 29.92(10.65) 39.64(16.46) 33.37(10.74) 1 14.5 LOUI 27.04(10.72) 36.04(17.31) 29.95(10.71) 1 7 Msplit-HR 31.46(11.87) 29.1(15.09) 29.14(11.13) 1 0 (100,10) (ii) us-bcsvm 44.52(10.96) 44.74(10.91) 43.52(5.09) 9 991 DROAD 3.28(4.47) 38.90(20.50) 6.74(6.38) 1 31.5 HR 10.18(6.08) 27.96(18.17) 14.07(6.17) 2 0 US-HR 24.28(11.97) 24.48(12.20) 22.97(8.76) 1 0 BLDA 10.06(6.06) 28.26(18.13) 14.25(6.17) 2 1 BAI 17.32(9.01) 20.94(14.02) 17.39(8.37) 3 3 LOUI 16.08(8.97) 21.32(13.90) 17.06(8.32) 3 3 Msplit-HR 18.64(9.41) 18.04(11.09) 16.84(8.28) 2 0 Table 3 : 3Average computational time (in seconds) taken by a method to complete per-sampleresults: Simulation setting (i). (n 1 , n 2 , p) us-bcsvm DROAD HR US-HR BLDA BAI LOUI Msplit-HR (25,5,1000) 2.8 21.73 0.9 4.66 1.05 6 6.39 9.27 (50,10,1000) 5.12 30.77 1.47 19.98 3.53 58 260 92 (100,10,1000) 4.76 35.00 5.43 42.22 11.20 421 365 185 (25,5,3000) 7.5 97.58 1.13 9.05 1.75 14.72 13.83 19.63 (50,10,3000) 12.38 146.17 4.40 62.54 12.24 225 219 282 (100,10,3000) 10.67 141.82 19.90 169.97 29.34 1517 2294 1200 Table 4 : 4Classification results for the simulation settings (iii)-(iv) with a general Σ and p = 200.(n1, n2) Setting Methods MCR1% MCR2% GM% S (25,5) (iii) us-bcsvm 46.26(15.97) 51.22(15.50) 46.20(6.30) 200 FAIR 23.22(9.64) 78.56(10.26) 40.53(8.04) 6.87 SLDAmcr 2 42.04(14.38) 57.38(14.87) 47.02(6.55) 147.07 PSIS 31.56(9.44) 66.98(10.51) 45.02(6.21) 1 ROAD 15.47(8.03) 82.67(8.87) 34.16(6.89) 26.17 BA-ROAD 48.20 (12.51) 48.91(12.36) 46.83(4.55) 56.45 LOU-ROAD 48.07(12.27) 49.03(12.40) 46.97(2.55) 54.09 Msplit-HR 53.58(15.06) 45.76(16.23) 47.12(6.65) 4 (25,5) (iv) us-bcsvm 49.96(15.76) 46(15.44) 45.39(5.62) 200 FAIR 20.96(8.00) 76.38(10.82) 38.83(6.85) 8.07 SLDAmcr 2 37.18(14.30) 60.74(14.84) 45.31(7.84) 124.39 PSIS 30.02(9.15) 62.52(16.16) 42.17(8.64) 1 ROAD 15.77(7.24) 78.93(11.66) 33.94(6.09) 24.38 BA-ROAD 46.52(15.06) 46.63(16.65) 43.92(7.17) 48.53 LOU-ROAD 46.34(15.79) 46.22(17.97) 43.17(7.63) 47.26 Msplit-HR 52.32(18.66) 43.02(18.51) 43.77(7.92) 4.5 (50,10) (iii) us-bcsvm 46.34(11.67) 48.88(12.77) 46.27(5.25) 200 FAIR 28.98(8.96) 69.1(8.95) 43.96(6.76) 6 SLDAmcr 2 44.5(11.65) 55.84(12.40) 48.57(5.37) 195 PSIS 37.96(8.26) 60.72(8.67) 47.47(5.54) 1 ROAD 19.98(8.79) 77.54(9.02) 38.07(7.11) 44 BA-ROAD 48.36(14.40) 48.50(14.15) 45.99(9.29) 53.50 LOU-ROAD 47.38(11.90) 49.14(12.19) 46.99(6.34) 53 Msplit-HR 50.76(13.85) 47.64(12.15) 47.65(6.11) 3 (50,10) (iv) us-bcsvm 47.88(11.98) 48.42(12.55) 46.76(5.53) 200 FAIR 23.98(8.65) 64.5(12.53) 38.33(7.57) 6 SLDAmcr 2 37.5(13.23) 54.58(16.77) 43.61(9.53) 189.5 PSIS 32.16(8.79) 51.04(17.23) 39.64(9.48) 1 ROAD 21.68(9.25) 64.44(18.71) 35.45(7.40) 19.50 BA-ROAD 37.82(13.06) 44.74(15.80) 39.00(10.23) 26 LOU-ROAD 39.74(12.51) 42.46(13.50) 39.80(8.27) 27 Msplit-HR 44.62(13.96) 40.1(14.25) 40.77(8.48) 1 (100,10) (iii) us-bcsvm 46.54(11.31) 48.46(12.50) 46.19(5.04) 200 FAIR 26.08(7.63) 70.62(7.94) 42.27(6.27) 6.68 SLDAmcr 2 47.24(13.75) 54.18(12.48) 49.09(6.46) 169.26 PSIS 36.06(8.41) 63(8.95) 46.52(6.62) 1.01 ROAD 11.16(6.67) 86.04(8.46) 29.49(7.64) 71.70 BA-ROAD 44.50(13.57) 50.94(13.67) 45.40(8.71) 85.41 LOU-ROAD 44.62(10.18) 42.44(9.37) 42.79(6.12) 66.22 Msplit-HR 50.42(15.15) 46.46(14.43) 46.22(6.99) 11.45 (100,10) (iv) us-bcsvm 47.76(12.15) 47.26(12.50) 46.03(5.56) 200 FAIR 22(7.97) 67.54(11.80) 37.63(7.61) 8.03 SLDAmcr 2 34.2(11.71) 50.54(17.12) 40.03(8.67) 96.77 PSIS 31.36(8.13) 47.58(18.24) 37.74(9.75) 1 ROAD 11.16(6.67) 86.04(8.46) 29.49(7.64) 71.70 BA-ROAD 44.50(13.57) 50.96(13.67) 45.39(8.71) 85.41 LOU-ROAD 44.12(12.27) 50.54(11.72) 45.84(5.82) 97 Msplit-HR 45.86(17.28) 37.6(14.92) 39.19(8.68) 9.25 Table 5 : 5Classification results for the simulation settings (iii)-(iv) with a general Σ andp = 500. (n1, n2) Setting Methods MCR1% MCR2% GM% S (25,5) (iii) us-bcsvm 49.18(17.81) 50.46(17.69) 46.39(8.47) 500 FAIR 20.48(8.72) 79(8.91) 38.79(8.29) 8.91 SLDAmcr 2 44.84(15.58) 54.56(16.11) 46.97(5.57) 350.18 PSIS 23.22(9.64) 75.68(10.26) 40.53(8.05) 6 ROAD 12.01(8.27) 87.16(8.70) 30.21(8.08) 30.79 BA-ROAD 44.63(13.74) 53.71(14.30) 45.01(9.74) 57.49 LOU-ROAD 45.82(12.19) 52.32(12.41) 47.40(2.84) 69.18 Msplit-HR 55.96(16.92) 44.58(17.43) 46.77(7.11) 3 (25,5) (iv) us-bcsvm 48.9(15.18) 47.66(13.96) 46.17(5.45) 500 FAIR 15.1(8.91) 84.08(8.87) 33.15(10.97) 13.38 SLDAmcr 2 41.04(15.56) 58.3(15.45) 46.52(7.29) 315.96 PSIS 30.26(9.52) 65.9(13.01) 43.60(7.70) 1 ROAD 12.50(8.10) 85.07(10.66) 30.67(7.51) 27.82 BA-ROAD 48.07(13.10) 48.40(14.23) 46.20(6.71) 60.06 LOU-ROAD 48.47(13.37) 47.48(14.94) 45.97(5.10) 59.19 Msplit-HR 53.06(17.66) 43.16(17.54) 44.50(8.83) 2 (50,10) (iii) us-bcsvm 48.46(12.44) 48.64(14.25) 46.94(5.82) 500 FAIR 26.1(8.65) 72.12(8.35) 42.48(6.57) 9 SLDAmcr 2 43.76(12.28) 55.18(11.98) 47.78(5.51) 493.5 PSIS 35.44(8.02) 63.02(9.33) 46.67(5.37) 1 ROAD 14.10(9.43) 83.70(11.63) 31.80(9.44) 59.50 BA-ROAD 48.32(14.03) 49.66(12.68) 47.15(7.68) 64.50 LOU-ROAD 48.18(13.51) 48.26(12.33) 46.71(6.01) 68 Msplit-HR 50.06(15.67) 48.26(14.52) 46.91(5.78) 1 (50,10) (iv) us-bcsvm 49.04(10.87) 49.28(11.96) 48.01(5.56) 500 FAIR 17.38(6.91) 76.14(10.84) 35.41(7.35) 13.5 SLDAmcr 2 38.58(13.19) 51.14(14.79) 42.98(9.05) 473 PSIS 32.02(8.58) 54.56(17.60) 40.97(9.69) 1 ROAD 15.46(9.32) 73.56(19.75) 30.97(7.64) 38.50 BA-ROAD 42.20(13.08) 44.26(15.04) 41.40(8.97) 38 LOU-ROAD 42.60(13.97) 43.16(14.41) 41.25(8.55) 47 Msplit-HR 46.4(14.65) 40.7(15.10) 41.55(8.81) 1 (100,10) (iii) us-bcsvm 47.96(11.92) 49.98(13.98) 47.47(5.67) 500 FAIR 23.91(8.31) 74.6(7.48) 41.28(7.55) 8.22 SLDAmcr 2 44.28(12.54) 54.34(12.45) 47.61(5.27) 403.99 PSIS 33.32(7.73) 65.48(8.83) 46.24(5.87) 1.01 ROAD 4.60(3.47) 94.60(4.99) 19.02(8.42) 96.49 BA-ROAD 45.40(13.39) 51.16(14.24) 45.90(8.21) 105.05 LOU-ROAD 46.24(8.52) 43.12(10.44) 43.99(6.11) 103.05 Msplit-HR 51.76(14.45) 46(14.62) 46.61(6.55) 5.67 (100,10) (iv) us-bcsvm 48.72(11.81) 48.94(12.06) 47.65(5.53) 500 FAIR 15.1(7.41) 79.42(9.89) 33.09(8.62) 17.18 SLDAmcr 2 36.98(12.42) 53.58(14.94) 43.12(9.01) 226.33 PSIS 30.28(7.57) 54.96(18.29) 39.94(9.12) 1.01 ROAD 4.64(3.47) 94.60(4.99) 19.02(8.42) 96.49 BA-ROAD 45.40(13.39) 51.16(14.24) 45.90(8.21) 105.05 LOU-ROAD 45.64(13.15) 49.18(12.87) 45.79(5.82) 107.80 Msplit-HR 43.52(12.21) 44.02(14.34) 42.38(8.26) 5.74 Table 6 : 6Average computational time (in seconds) taken by a method to complete per-sample results: Simulation setting (iv). (n 1 , n 2 , p) us-bcsvm FAIR SLDA mcr2 PSIS ROAD BA-ROAD LOU-ROAD Msplit-HR (25,5,200) 1.91 1.75 8.75 0.28 50.23 110.34 59.93 11.42 (50,10,200) 1.48 4.73 25.71 4.11 66.00 192.10 189.86 39.31 (100,10,200) 1.86 5.56 66.46 3.06 120.93 468.02 443.23 114.44 (25,5,500) 3.00 29.3 80.05 0.41 204.52 254.64 184.99 16.4 (50,10,500) 3.01 27.09 234.35 3.00 272.15 483.82 477.45 62.23 (100,10,500) 3.58 29.66 451.75 3.61 219.95 974.34 1084.75 176.65 Table 7 : 7Classification results for Breast Cancer data set. S denotes the median number ofselected features. Σ Methods MCR 1 % MCR 2 % GM% S Diagonal DROAD 19.62 (10.20) 46.31 (13.38) 28.72 (7.77) 201.50 HR 16.82 (6.70) 46.72 (10.24) 27.08 (5.98) 26 US-HR 20.67 (7.58) 39.79 (9.78) 27.93 (6.10) 32 BLDA 16.8 (6.14) 45.59 (10.86) 26.78 (5.63) 35 BAI 22.24 (7.09) 37 (10.08) 27.83 (5.40) 99 LOUI 22.65 (7.05) 37.03 (10.82) 28.06 (5.20) 83.5 Msplit-HR 20.96 (7.00) 39.56 (10.74) 27.83 (5.19) 6 General us-bcsvm 19.78 (5.74) 34.79 (10.01) 25.50 (4.24) 1500 FAIR 16.24 (5.54) 45.41 (9.31) 26.43 (4.95) 22 SLDA mcr 2 22.91 (12.32) 47.76 (12.19) 31.58 (9.17) 1500 PSIS 27.47 (14.62) 46.17 (15.30) 33.77 (9.85) 1 ROAD 19.51 (10.03) 47.41 (13.81) 28.96 (7.26) 25 BA-ROAD 22.16 (5.95) 38.83 (9.72) 28.62 (4.24) 51.50 LOU-ROAD 22.16 (5.90) 38.10 (9.84) 28.37 (4.32) 56.50 Msplit-HR 24.11 (8.85) 40.55 (10.76) 30.35 (6.50) 5 Table 8 : 8Classification results for Myeloma Cancer data set. S denotes the median numberof selected features. Σ Methods MCR 1 % MCR 2 % GM% S Diagonal DROAD 26.03 (11.29) 49.33 (10.30) 34.93 (8.58) 5 HR 25.94 (11.60) 57.78 (11.92) 37.43 (8.47) 19 US-HR 41.6 (9.74) 41 (12.75) 40.17 (6.66) 92.5 BLDA 25.58 (9.10) 53.28 (11.23) 35.89 (6.36) 11 BAI 34.31 (10.44) 44.17 (13.20) 37.50 (5.98) 30 LOUI 35.14 (10.54) 44.39 (11.35) 38.26 (5.95) 27.5 Msplit-HR 38.18 (13.68) 41.94 (14.37) 37.89 (7.51) 7 General us-bcsvm 53.78 (27.56) 39.44 (28.32) 46.06 (18.47) 1500 FAIR 27.92 (7.64) 49.56 (11.16) 36.50 (6.15) 14 SLDA mcr 2 28.83 (9.79) 47.22 (10.34) 36.18 (7.59) 13 PSIS 31.42 (15.24) 50.11 (10.33) 38.43 (10.72) 1 ROAD 26.01 (10.27) 53.22 (10.63) 36.47 (8.13) 7.50 BA-ROAD 34.01 (13.75) 43.17 (13.92) 35.52 (9.38) 20 LOU-ROAD 33.74 (9.63) 42.78 (10.67) 38.05 (6.43) 23.50 Msplit-HR 34.74 (11.84) 42.61 (11.66) 37.27 (7.16) 6 High Performance Computing Center (NHPCC) at Isfahan University of Technology for their computational support to conduct our numerical experiments. Arezou Mojiri is grateful to(late) Soroush Alimoradi and also Ali Rejali for their help and constant support during her graduate studies. Abbas Khalili was supported by the Natural Sciences and Engineering Research Council of Canada through Discovery Grants (NSERC RGPIN-2015-03805 and NSERC RGPIN-2020-05011). https://github.com/statcodes/ROAD Both data sets are publicly available from the R package datamicroarray(Ramey, 2016), and are available at https://github.com/. Υ lda ( θ θ θ n ) = ( µ µ µ 2 − µ µ µ 1 ) ⊤ Σ −1 ΣΣ −1 ( µ µ µ 2 − µ µ µ 1 ) = (ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 + µ d ) ⊤ Σ −1 (ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 + µ d ) = (ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 ) ⊤ Σ −1 (ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 ) + 2(ǭ ǫ ǫ 2 −ǭ ǫ ǫ 1 )We first show that I 1 =ǭ ǫ ǫ ⊤ 1 Σ −1ǭ ǫ ǫ 1 = p/n 1 + o p ( p/n 1 ).Note thatǭ ǫ ǫ 1 ∼ N p (0, n −1 1 Σ). By Chebyshev's inequality, for any τ > 0, Pr n 1 p \| I 1 − p n 1 \|> τ ≤ 1 τ 2 V ar{I 1 . n 1 /p}.This together with the fact that V ar{I 1 . n 1 /p} → 0, when n 1 , n 2 → ∞ such that n 2 = o(n 1 ), implies that I 1 = p/n 1 + o p ( p/n 1 ). Similarly, we haveI 4 =ǭ ǫ ǫ ⊤ 1 Σ −1 µ d = O p ∆ 2 p /n 1 ,By combining the above results, we haveSince n 2 p ∆ 2 p = o(1), as long as n 1 , n 2 → ∞, thus we obtainBy Lemma A.6 of the Appendix A, with c n =where n = n 1 + n 2 . The last inequality follows from the upper bound described in Lemma A.6, for the tail of a Student's t-distributed random variable, with n − 2 degrees of freedom.Since n 2 = o(n 1 ) as n 1 , n 2 → ∞, we then obtainThe last term tends to zero, as log p = o(n 2 d 2 0,n ) and √ n 2 d 0,n → ∞.Proof of Theroem 3.2. (a) The misclassification rates of Msplit-hr in (3.8), are given as, k = 1, 2Since (log p/n 2 )(n 1 / log p) 2α = o(1) and n 2 = o(n 1 ), as n 1 , n 2 → ∞, (B.11) tends to 1, and the result of part (a) holds.(b) Similar to part (a), for some positive constants c 1 , c 2 , we haveThis together with (log p/n 2 )(n 1 / log p) 2α = o(1) and n 2 = o(n 1 ), prove that the right hand side of the above inequality tends to 1, as n 1 , n 2 → ∞.(c) The result follows from parts (a) and (b).Proof of Theorem 4.1. (a) The misclassification rates of slda in Class k = 1, 2, are givenFollowing by the proof of Theorem 1 ofShao et al. (2011), we have, andμ µ µ d1 and µ µ µ d1 are two vectors of dimension q, whose elements correspond to those features x j s for which \| µ dj \| > a n . By condition (4.1),, and J 1 = O p (k n 2 ), where k n 2 = max{ qn n 2 , D g,p a 2(1−g) n }. Consequently, by condition (4.1),Proof of Theorem 4.2. The class-specific MCRs of the road in (4.2) are given byThe oracle versions of the MCRs, evaluated at the true parameter values of Σ and µ k , are given bywe have that, for η 1 > 0,Pr \| µ jk − µ jk \| > η 1 ≤ C 1 p exp{−C 2 n k η 2 1 }.Thus, by choosing η 1 = M 1 a n k , for some M 1 > 0, we arrive at µ µ µ k −µ k ∞ = O p ( log p/n k ).Also, by Lemma A.1 in the Appendix A, for η 2 > 0,Pr \|n k µ jk µ lk − σ jl )\| > (n − 2)η 2 /4≤ p 2 C 1 exp{−C 2 (n − 2) 2 η 2 2 /n k } + p 2 C 3 exp{−C 4 (n − 2) 2 η 2 2 }.Thus, by choosing η 2 = M 2 log p/n 1 , for some M 2 > 0, we arrive at Σ n − Σ ∞ = O p ( log p/n 1 ). Using the Lipschitz property of the cumulative distribution function of standard normal, Φ(.), we haveNow,andAccording to the same notations inFan et al. (2012), let f 0 (w) = w ⊤ µ d /(w ⊤ Σw) 1/2 , f 1 (w) = w ⊤ µ µ µ d /(w ⊤ Σw) 1/2 , and f 2 (w) = w ⊤ µ µ µ d /(w ⊤ Σw) 1/2 . By the proof of Theorem 1 ofFan et al. (2012), we havec } log p/n 2 .Therefore, we haveC REMAINING PROOFSIn this Appendix, we provide the proofs of our claim in Remark 2.1, and also the proofs of Propositions 3.1 and 3.2.Proof of the Claim in Remark 2.1. Recall I i , i = 1, ..., 6, defined in Theorem 2.1. When p is fixed with respect to the sample size, and n 2 = o(n 1 ), then as n 1 , n 2 → ∞, we have, V ar(I 1 ) = 2p n 2 1 → 0 , V ar(I 2 ) = 2p n 2 2 → 0 , V ar(I 3 ) = ∆ 2 p n 2 → 0 , V ar(I 4 ) = ∆ 2 p n 1 → 0 , V ar(I 5 ) = n 2 p n 2 1 n 2 2 → 0 , V ar(I 6 ) = ∆ 2 p n 2 → 0.On the other hand, E(I 1 ) = p n 1 , E(I 2 ) = p n 2 , E(I 3 ) = E(I 4 ) = 0, E(I 5 ) = np n 1 n 2 , and E(I 6 ) = 0. Thus, by following the proof of Theorem 2.1, we have Ψ lda 1 ( θ θ θ n ) Υ lda ( θ θ θ n ) = p n 1 − p n 2 + o p (1) − ∆ 2 p 2 np n 1 n 2 + o p (1) + ∆ 2 p 1/2 = − 1 2 ∆ p + o p (1), Ψ lda 2 ( θ θ θ n ) Υ lda ( θ θ θ n ) = − p n 1 + p n 2 + o p (1) − ∆ 2 p 2 np n 1 n 2 + o p (1) + ∆ 2 p 1/2 = − 1 2 ∆ p + o p (1). 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[ "Color Sextet Vector Bosons and Same-Sign Top Quark Pairs at the LHC", "Color Sextet Vector Bosons and Same-Sign Top Quark Pairs at the LHC" ]	[ "Hao Zhang \nEnrico Fermi Institute\nUniversity of Chicago\n60637ChicagoIllinoisU.S.A\n\nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n", "Edmond L Berger \nHigh Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisU.S.A\n", "Qing-Hong Cao \nEnrico Fermi Institute\nUniversity of Chicago\n60637ChicagoIllinoisU.S.A\n\nHigh Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisU.S.A\n", "Chuan-Ren Chen \nInstitute for Physics\nMathematics of the Universe\nUniversity of Tokyo\n277-8568ChibaJapan\n", "Gabe Shaughnessy \nHigh Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisU.S.A\n\nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisU.S.A\n" ]	[ "Enrico Fermi Institute\nUniversity of Chicago\n60637ChicagoIllinoisU.S.A", "Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina", "High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisU.S.A", "Enrico Fermi Institute\nUniversity of Chicago\n60637ChicagoIllinoisU.S.A", "High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisU.S.A", "Institute for Physics\nMathematics of the Universe\nUniversity of Tokyo\n277-8568ChibaJapan", "High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisU.S.A", "Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisU.S.A" ]	[]	We investigate the production of beyond-the-standard-model color-sextet vector bosons at the Large Hadron Collider and their decay into a pair of same-sign top quarks. We demonstrate that the energy of the charged lepton from the top quark semi-leptonic decay serves as a good measure of the top-quark polarization, which, in turn determines the quantum numbers of the boson and distinguishes vector bosons from scalars.	10.1016/j.physletb.2010.12.005	[ "https://arxiv.org/pdf/1009.5379v1.pdf" ]	118,660,288	1009.5379	94cd2188b1b292989dbfda1d3ea321f85c31c9ec	Color Sextet Vector Bosons and Same-Sign Top Quark Pairs at the LHC 27 Sep 2010 Hao Zhang Enrico Fermi Institute University of Chicago 60637ChicagoIllinoisU.S.A Department of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina Edmond L Berger High Energy Physics Division Argonne National Laboratory 60439ArgonneIllinoisU.S.A Qing-Hong Cao Enrico Fermi Institute University of Chicago 60637ChicagoIllinoisU.S.A High Energy Physics Division Argonne National Laboratory 60439ArgonneIllinoisU.S.A Chuan-Ren Chen Institute for Physics Mathematics of the Universe University of Tokyo 277-8568ChibaJapan Gabe Shaughnessy High Energy Physics Division Argonne National Laboratory 60439ArgonneIllinoisU.S.A Department of Physics and Astronomy Northwestern University 60208EvanstonIllinoisU.S.A Color Sextet Vector Bosons and Same-Sign Top Quark Pairs at the LHC 27 Sep 2010 We investigate the production of beyond-the-standard-model color-sextet vector bosons at the Large Hadron Collider and their decay into a pair of same-sign top quarks. We demonstrate that the energy of the charged lepton from the top quark semi-leptonic decay serves as a good measure of the top-quark polarization, which, in turn determines the quantum numbers of the boson and distinguishes vector bosons from scalars. Introduction - The cross section for production of top quarks is relatively high at the energies of the Large Hadron Collider (LHC). While conventional mechanisms produce either a single top-quark or a top-antitop pair, it is important to be alert to the observation of a pair of same-sign top quarks. One consequence would be the appearance of a pair of same-sign leptons with large transverse momentum. In a recent paper, we explore the potential for discovery of an exotic color-sextet scalar in the production of a pair of same-sign top-quarks in early runs of the LHC at 7 TeV [1]. The standard model (SM) backgrounds are small. We demonstrate that one can measure the scalar mass and the top-quark polarization, and confirm the scalar nature of the resonance with 1 fb −1 of integrated luminosity. Moreover, the top-quark polarization distinguishes gauge triplet-and singlet-scalars. In the present manuscript, we address color sextet vector production in same-sign top-quark pair-production, and we show that its discovery, and the determination of its properties, could also be accomplished with early LHC data. Important in our analysis is the recognition that among the products of semi-leptonic top-quark decay, the direction of the charged-lepton is highly correlated with the top-quark spin. The top-quark polarization can be measured from the distribution in cos θ [2], the cosine of helicity angle between the charged-lepton momentum in the top-quark rest frame and the top-quark momentum in the center-of-mass frame of the production process (i.e. the rest frame of the parent scalar or vector). Gauge triplet scalars decay to t L t L , and gauge singlet scalars to t R t R . Here, t L and t R denotes top quarks with lefthanded and right-handed polarization. In either case, both top quarks produce the same angular distribution, either (1 − cos θ)/2 (t L t L )or (1 + cos θ)/2 (t R t R ), allowing unambiguous identification of the scalar [1]. A color sextet vector decays into a t L t R pair. In this case, the observed inclusive angular distribution in the final state would be a flat, a sum of the shapes from the t L and the t R decays. The flat profile would distinguish a vector from a scalar, but it also would admit a mechanism that yields unpolarized top quarks. In this paper we establish that one can separate the angular distributions corresponding to t L and t R from the color sextet vector decay into a t L t R pair. The solution relies of the introduction of asymmetric cuts on the momenta of the leptons from the top-quark decays. The Model -The most general SU (3) C × SU (2) L × U (1) Y effective invariant Lagrangian for color sextet scalars Φ and vectors V µ has the form [3][4][5] L = g 1L q c L iτ 2 q L + g 1R u c R d R Φ 6,1,1/3 + g ′ 1R d c R d R Φ 6,1,−2/3 + g ′′ 1R u c R u R Φ 6,1,4/3 + g 3L q c L iτ 2 τ q L · Φ 6,3,1/3 + g 2 q c L γ µ d R V µ 6,2,−1/6 + g ′ 2 q c L γ µ u R V µ 6,2,5/6 + h.c. ,(1) where q L = (u L , d L ) denotes the left-handed quark doublet, u R and d R are the corresponding right-handed gauge singlet fields, and q c ≡ Cq T is the charge conjugated quark field. For the sake of simplicity, color and generation indices are omitted. The subscripts on the Φ and V fields denote the standard model gauge quantum numbers (SU (3) C , SU (2) L , U (1) Y ) 1 . We are interested in same-sign top-quark pair production via a sextet vector decay. Only the axial-vector part of the coupling contributes while the pure vector coupling vanishes due to the identical quarks. The effective coupling of the color sextet vector to a pair of identical quarks (qq) is L int = g 2 K M abq a γ µ γ 5 q c b V µ M + h.c. ,(2) where the K M ab are Clebsch-Gordan coefficients; a and b are the color indices in the fundamental representation; M is the color index in the sextet representation; and g is the coupling strength. Without loss of generality, we concentrate on real and flavor-conserving couplings in this work. The coupling of the vector to two up-type quarks is largely constrained by the measurement of D 0 −D 0 mix-ing which is affected by the vector at the tree level. The \|∆C = 2\| Hamiltonian induced by V is H ∆C=2 = 2g uu g cc m 2 V C 3 (µ)Q 3 + C 2 (µ)Q 2 .(3) The four-fermion operators Q 2 and Q 3 are Q 2 = (ū Lα γ µ c Lα ) (ū Rβ γ µ c Rβ ) ,(4)Q 3 = (ū Lα c Rα ) (ū Rβ c Lβ ) ,(5) and the Wilson coefficients are C 2 (m V ) = −1, C 3 (m V ) = 2. The vector's contribution to ∆m D ≡ \|m D 0 − mD0 \| is ∆m D = ℜ 2 D 0 \|H\|D 0 2m D = 1 m D 2g uu g cc m 2 V C 2 Q 2 + C 3 Q 3 ,(6) where the hadron matrix elements are [7] Q 2 = − 5 6 f 2 D m 2 D B D , Q 3 = 7 12 f 2 D m 2 D B D ,(7) with f D = 222.6 ± 16.7 +2.3 −2.4 MeV, m D = 1865 MeV, and B D = 0.82 [8]. To be consistent with the measured ∆m D [9], x D = ∆m D Γ D ∼ 8 × 10 −3 , Γ D = 1.6 × 10 −12 GeV, the coupling g qq is stringently constrained: g uu g cc 1.6 × 10 −8(8) for m V ≃ 1 TeV, after an enhancement factor ∼ 2.1 is included from the QCD running of the Wilson coefficients from the scale m V to m c . This strong constraint on the product g uu g cc allows freedom for the production of V at the LHC if the coupling g cc to the second generation quarks is minimized. Alternatively, the first generation may be suppressed while the second generation is not, but such sea-quark initiated processes are relatively suppressed. For the choice g = 1, we show the cross section for V production via the process uu → V at the LHC in Fig. 1(a) and the tt cross section via the process uu → tt in Fig. 1(b). The large cross sections arise from the large parton distribution functions for valence u quarks in the initial state. If the LHC energy is raised from 7 TeV to 14 TeV the cross section is increased by roughly a factor of 3 to 4. Discovery Potential -We focus on the same sign dilepton decay mode in which the W bosons from both t → W b decays lead to a final state containing an electron or muon, W → lν, accounting for about 5% of all tt decays. We concentrate on the clean µ + µ + final state because muon reconstruction has a large average efficiency of 95 − 99% within the pseudorapidity range \|η\| < 2.4 and transverse momentum range 5 GeV ≤ p T ≤ 1 TeV, while the charge mis-assigned fraction for muons with p T = 100 GeV is less than 0.1% [11]. These events are characterized by two high-energy same-sign leptons, two jets from the hadronization of the b-quarks, and large missing energy ( E T ) from two unobserved neutrinos. We generate the dominant backgrounds with ALPGEN [12]: I: Signal and background cross sections (pb) before and after cuts, with g = 1, for six values of mV (GeV). The decay branching ratios of the signal Br(tt) are given in the second column. The "no cut" rates correspond to all lepton and quark decay modes of W -bosons, whereas those "with cut" are obtained after all cuts, the restriction to 2 µ + 's and with tagging efficiencies included. pp → W + (→ ℓ + ν)W + (→ ℓ + ν)jj,(9)pp → tt → bW + (→ ℓ + ν)b(→ ℓ + )W − (→ jj). (10) mV The first process (W W jj) is the SM irreducible background while the second (tt) is a reducible background as it contributes when some tagged particles escape detection, carrying small p T or falling out of the detector rapidity coverage. For example, one of the b-quarks decays into an isolated charged lepton while one of the two jets from the W − boson decay is mis-tagged as a b-jet. Other SM backgrounds, e.g. triple gauge boson production (W W W , ZW W , and W Zg(→ bb)), occur at a negligible rate after kinematic cuts, and are not shown here. At the analysis level, all signal and background events are required to pass the following acceptance cuts: p j T ≥ 50 GeV, \|η j \| ≤ 2.5 p ℓgreater T ≥ 50 GeV, p ℓ lesser T ≥ 20 GeV, \|η ℓ \| ≤ 2.0, ∆R jj,jℓ,ℓℓ > 0.4,(11) where we order the two charged leptons in the final state by their energies and label the more energetic lepton as "greater" and the other one as "lesser". Owing to spin correlations, the charged lepton from right-handed top quark decay is more energetic than the one from the left-handed top quark decay. This difference motivates our asymmetric cut on the p T of the two charged leptons. The separation ∆R in the azimuthal angle (φ)pseudorapidity (η) plane between the objects k and l is ∆R kl ≡ (η k − η l ) 2 + (φ k − φ l ) 2 .(12) We model detector resolution effects by smearing the final state energy according to δE E = A E/GeV ⊕ B,(13) where we take A = 10(50)% and B = 0.7(3)% for leptons(jets). To account for b-jet tagging efficiencies, we demand two b-tagged jets, each with a tagging efficiency of 60%. We also apply a mistagging rate for charm-quarks ǫ c→b = 10% for p T (c) > 50 GeV. The mistag rate for a light jet is ǫ u,d,s,g→b = 0.67% for p T (j) < 100 GeV and 2% for p T (j) > 250 GeV. For 100 GeV < p T (j) < 250 GeV, we linearly interpolate the fake rates given above. After lepton and jet reconstruction, we demand that the two hard leptons are of the same sign, a requirement which greatly reduces the SM background, giving a rejection of order 10 −4 and 10 −3 for the tt and W W jj processes, respectively. After the cuts are imposed, we find a total of 1.0 background event, 0.4 from tt and 0.6 from W W jj for 1 fb −1 of integrated luminosity. After b-tagging and restriction to the µ + µ + mode, 15-30% the signal events survive the analysis cuts depending on the vector mass. Signal and background cross sections are shown in Table I, before and after cuts, for 6 values of m V . Because the decay width of V is narrow, Γ(V → qq) = g 2 m V 24π 1 − 4m 2 q m 2 V 3/2 ,(14) one can factor the process uu → tt into vector production and decay terms, σ(uu → V → tt) = σ 0 (uu → V ) × g 2 Br(tt), = σ 0 (uu → V → tt) × g 2 Br(tt) Br 0 (tt) .(15) The decay branching ratio Br(tt) ≡ Br(V → tt) is Br(tt) = g 2 tt R g 2 uu + g 2 tt R , R = (1 − 4m 2 t /m 2 V ) 3/2 .(16) Subscript "0" denotes the reference value g = 1. We choose to work with the following two parameters in the rest of this paper: the vector mass m V and the product g 2 Br(V → tt). The kinematics of the final state particles are determined by the vector mass, whereas the couplings of the vector to the light and heavy fermions change the overall normalization. In Fig. 2, we show the expected numbers of signal events as a function of m V for a range of values of the coupling g 2 Br(V → tt). We obtain the event rate lines by converting the required cross section into g 2 Br(V → tt) via Eq. 15. Based on Poisson statistics, one needs 8 signal events in order to claim a 5σ discovery significance (equivalent to a 99.999943% confidence level) on top of 1 background event. We plot the 5σ discovery line (solid) in the figure. Rates for other values of g uu and g tt can be obtained from Eq. 15. The search for same-sign top quark pair production in the dilepton mode at the Tevatron imposes an upper limit σ(tt +tt) ≤ 0.7 pb [13][14][15]. The constraint is plotted in the orange shaded region. The CDF collaboration measured the tt invariant mass spectrum in the semi-leptonic decay mode [16]. Since b andb jets from t → W b are not distinguished well, tt pairs lead to the same signature as tt in the semi-leptonic mode. Hence, the m tt spectrum provides an upper limit on σ(tt +tt), shown in the cyan shaded region in Fig. 2. The lower gray shaded region is the region in which V would hadronize before decay, washing out the spin correlation effects we utilize to probe the coupling and spin of the sextet state. Top Quark Polarization -We use the observation of a pair of same sign dileptons as indicative of a same sign top quark pair and to suppress SM backgrounds efficiently. The two missing neutrinos in the final state complicate event reconstruction. Following Ref. [1], we use the MT2 method to select the correct b-µ combinations and to verify whether the final state is consistent with t → W b parentage. Then we make use of the onshell conditions of the two W bosons and two top quarks to solve for the neutrino momenta [17,18]. Once the neutrino momenta are known, the kinematics of the entire final state is fixed, and the vector boson mass is computed from the invariant mass of the two reconstructed top quarks. The next step is to verify that the top-quarks exhibit opposite polarization, accomplished here by making use of the difference in the momentum spectra of decay leptons from left-handed and right-handed top quarks [19]. The energy of a charged lepton in a right-handed topquark decay is harder than the one in a left-handed topquark decay. The differential distribution in the energy of the charged lepton energy is lepton, and z = cos θ where θ is the helicity angle defined in the Introduction. The variables x ′ and z ′ are defined in the top-quark rest frame and are linked to x and z in the laboratory frame through the top quark boost: dΓ dx = zmax zmin dz ∂(x ′ z ′ ) ∂(x, z) dΓ dx ′ dz ′ ,(17)where x = 2E ℓ /E t isx ′ = xγ 2 (1 − zβ), z ′ = z − β 1 − zβ ,(18) where γ = E t /m t and β = 1 − 1/γ 2 . The lower and upper limits of integration are z min = Max 1 β 1 − 1 γ 2 x , −1 ,(19)z max = Min 1 β 1 − B γ 2 x , 1 .(20) The differential cross section dΓ/dx ′ dz ′ in the rest frame of the top quark is [20] dΓ dx ′ dz ′ = α 2 w 32π m t AB x ′ (1 − x ′ ) × ArcTan Ax ′ B − x ′ 1 +ŝ t z ′ 2 ,(21)where A = Γ W /m W , B = m 2 W /m 2 t , ArcTan(x) = arctan(x), for x ≥ 0, π + arctan(x), for x < 0, andŝ t labels the top quark spin direction. Here, m W and Γ W denote the mass and width of the W boson, respectively. In this work we choose the helicity basis to measure the top quark polarization. In this basis, the top quark spin is chosen to be along (against) the direction of motion of the top quark in the center of mass frame of the system. After a boost from the top-quark rest frame to the laboratory frame, we obtain the energy distributions of the charged leptons from left-handed and right-handed topquark decays shown in Fig. 3. The solid curves denote the t L decay while the dashed curves the t R decay. As the charged lepton follows the top quark spin, the lepton from -1 -0.8-0.6 -0.4-0.2 -0 0.2 0.4 0.6 t R decay tends to follow the direction of motion of the top-quark, and is more energetic. The lepton from the t L decay tends to move against the direction of motion of its top quark, and it is therefore less energetic. The difference in the energy spectra becomes more evident with increasing m V . For example, for a 1000 GeV vector, the charged lepton from t R decay peaks near x = 0.5 while the one from t L decay peaks below x = 0.1. We note that both solid and dashed curves have a kink feature. For the left-handed top-quark decay, the kink arises largely from the boost, which generates the lower integration limit z min . The limit yields a scale of x ≃ 1 − β for a heavy vector. On the other hand, the kink in the distribution for right-handed top-quark decays comes mainly from the non-continuity of the ArcTan function in the matrix element, which yields a scale of x ≃ B (1 + β) after the boost. In sextet vector boson decay into a t L t R pair, the charged leptons exhibit a mixture of the energetic and soft spectra. To utilize this feature, we order the energy of the leptons and define the top quark containing the greater energy lepton as t greater and the other top quark as t lesser . The cos θ distributions of the reconstructed t lesser and t greater are displayed in Fig. 4. These results show that one can differentiate the (1 + cos θ) (t R ) and (1 − cos θ) (t L ) shapes. The reconstruction is not perfect, however, owing to imperfect assignment of the charged lepton. As shown in Fig. 3, charged leptons from t L decay can be more energetic than those from t R decay with small probability. With increasing vector mass, the probability of wrong assignment becomes smaller. As a consistency check on this method for determining polarizations, we apply the same reconstruction to -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 FIG. 5: Distribution of cos θ of the reconstructed t lesser and tgreater of a singlet scalar decay into tRtR. For illustration we choose mV = 600 GeV. singlet scalar decay into two right-handed top-quarks. The reconstructed t lesser and t greater distributions for singlet scalar decay are shown in Fig. 5. Both distributions maintain the expected (1 + cos θ) trend, but the shapes are distorted by event reconstruction. Hence, we gain added confidence that the top-quark polarization can be determined and used to discriminate vector bosons from scalar bosons. Summary -We present a study of the search for exotic charge 4/3 color-sextet vector bosons in the production of same-sign top-quark pairs at the LHC at 7 TeV. We examine the final states in which both top-quarks decay semi-leptonically. We show that vector bosons can be distinguished from scalars. The top quarks from vector decay exhibit opposite polarization, one left-handed and one right-handed. The inclusive distribution in the helicity angle cos θ of a charged lepton is flat since it receives contributions from both top quarks. However, we show that an energy selection on the charged leptons can re-store the characteristic shapes that distinguish left-and right-handed top quarks. Correspondingly, we show that LHC data should allow one to observe V → t R t L and distinguish vector from scalar decay. → V → t tFIG. 1: Leading order cross sections (pb) for (a) V production via uu → V and (b) like-sign t-pair production via uu → V → tt at LHC energies: 7 TeV (solid), 10 TeV (blue dashed), and 14 TeV (red dashed) for g = 1. We use the CTEQ6L parton distribution functions[10] and choose the renormalization and factorization scales as mV . FIG. 2 : 2Event number contours as a function of the vector mass and the parameter g 2 Br(V → tt) after all cuts with an integrated luminosity of 1 fb −1 . The shaded regions are excluded, as explained in the text. FIG. 3 : 3the energy fraction of the charged Normalized distributions of the energy of the charged lepton from top-quark decay for mV = 500, 700, 1000 GeV. The solid lines correspond to the left-handed top-quark decay while the dashed lines to the right-handed top-quark decay. 0.8-0.6 -0.4-0.2 -0 0.2 0.4 0.6 0. 0.8 1 FIG 1. 4: (a) Distribution in cos θ for reconstructed top quarks from V → tRtL without energy selection on the decay leptons. Both µ + leptons exhibit similar distributions. (b) Distribution in cos θ for the reconstructed t lesser (black-solid) and tgreater (red-dashed). For illustration we choose mV = 600 GeV. TABLE Br(tt) No cut With cut mV Br(tt) No cut With cut Background No cut With cut500 0.446 10.97 1.71 800 0.483 4.22 1.21 tt 97.62 0.0004 600 0.466 8.22 1.76 900 0.487 3.02 0.92 W W jj 9.38 0.00001 700 0.477 5.89 1.53 1000 0.489 2.22 0.70 W W W/Z 0.03 0.0006 The vector V 6,2,−1/6 is not considered here as it cannot decay into a top-quark pair. Its collider phenomenology will be presented elsewhere[6]. . E L Berger, Q.-H Cao, C.-R Chen, G Shaughnessy, H Zhang, arXiv:1005.2622E. L. Berger, Q.-H. Cao, C.-R. Chen, G. 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[ "Fast bars in SB0 galaxies", "Fast bars in SB0 galaxies" ]	[ "S Ryder ", "D J Pisano ", "M Walker ", "K Freeman ", "Enrico Maria Corsini ", "J A L Aguerri ", "Victor P Debattista ", "\nDipartimento di Astronomia\nInstituto de Astrofísica de Canarias\nInstitut für Astronomie\nUniversità di Padova\nPadova, La LagunaItaly, Spain\n", "\nETH Hönggerberg\nZürichSwitzerland\n" ]	[ "Dipartimento di Astronomia\nInstituto de Astrofísica de Canarias\nInstitut für Astronomie\nUniversità di Padova\nPadova, La LagunaItaly, Spain", "ETH Hönggerberg\nZürichSwitzerland" ]	[ "Dark Matter in Galaxies ASP Conference Series" ]	We measured the bar pattern speed in a sample of 7 SB0 galaxies using the Tremaine-Weinberg method. This represents the largest sample of galaxies for which bar pattern speed has been measured this way. All the observed bars are as rapidly rotating as they can be. We compared this result with recent high-resolution N -body simulations of bars in cosmologically-motivated dark matter halos, and conclude that these bars are not located inside centrally concentrated halos.	10.1017/s0074180900183354	[ "https://arxiv.org/pdf/astro-ph/0311042v1.pdf" ]	16,379,208	astro-ph/0311042	6cd98de94fb8931b56f2d37441bc4e858bdb91ed	Fast bars in SB0 galaxies 2003 S Ryder D J Pisano M Walker K Freeman Enrico Maria Corsini J A L Aguerri Victor P Debattista Dipartimento di Astronomia Instituto de Astrofísica de Canarias Institut für Astronomie Università di Padova Padova, La LagunaItaly, Spain ETH Hönggerberg ZürichSwitzerland Fast bars in SB0 galaxies Dark Matter in Galaxies ASP Conference Series 2003 We measured the bar pattern speed in a sample of 7 SB0 galaxies using the Tremaine-Weinberg method. This represents the largest sample of galaxies for which bar pattern speed has been measured this way. All the observed bars are as rapidly rotating as they can be. We compared this result with recent high-resolution N -body simulations of bars in cosmologically-motivated dark matter halos, and conclude that these bars are not located inside centrally concentrated halos. The pattern speed of a bar, Ω, is its main kinematic observable. When parametrized by the distance-independent ratio R ≡ D L /a B between the corotation radius, D L , and the bar semi-major axis, a B , it permits the classification of bars into fast (1.0 ≤ R ≤ 1.4) and slow (R > 1.4) ones. A model-independent method for measuring Ω directly was obtained by Tremaine & Weinberg (1984, hereafter TW). The TW method is given by the simple expression X Ω sin i = V, where X and V are luminosity-weighted mean position and velocity measured along slits parallel to the line-of-nodes. If a number of slits at different offsets from the major-axis are obtained for a galaxy, then plotting V versus X for the different slits produces a straight line with slope Ω sin i. We defined a sample of 7 SB0 galaxies with intermediate inclination, a bar at intermediate position between the major and minor axes of the disk and no evidence of spirals and patchy dust (NGC 1023: Debattista et al. 2002;ESO 139-G9, IC 874, NGC 1308, NGC 1440, and NGC 3412: Aguerri et al. 2003NGC 2950. For each sample galaxy we obtained I−band imaging and long-slit spectra with slits parallel to disk major axis. We measured Ω with the TW method. The corotation radius, D L ≡ V c /Ω, was derived from the circular velocity, V c , after applying the asymmetric drift correction to the stellar velocities and velocity dispersions. The length of the bar, a B , was derived from the analysis of the surface brightness distribution. For all of the sample galaxy R is consistent with being in the range 1.0 to 1.4, within the errors, i.e. with each having a fast bar (Fig. 1). The unweighted average for the sample is R = 1.1. Some of the values of R are nominally less than unity, this suggests that the large range of R is a result of random errors and/or scatter. The corotation radius, D L , and the bar semi-major axis, a B , for the sample galaxies, NGC 936 (Merrifield & Kuijken 1995), NGC 4596 (Gerssen et al. 1997) and NGC 7079 (Debattista & Williams 2003). Dashed lines corresponding to R = 1 and R = 1.4, separate the fast-bar, slow-bar and forbidden regimes. Debattista & Sellwood (2000) argued that bars this fast can only survive if the disc in which they formed is maximal. Recent high resolution N -body simulations with cosmologically-motivated dark matter halos produce bars with R as large as 1.7 (Valenzuela & Klypin 2003). Even discounting our argument above in favor of a more restricted range of R, Fig. 1 shows that R = 1.7 is possible only for the bars of IC 874, NGC 1440, NGC 3412 and, marginally, NGC 936 (Merrifield & Kuijken 1995), while the bars of ESO 139-G009, NGC 1023, NGC 1308, NGC 2950, NGC 7079 (Debattista & Williams 2003) and NGC 4596 (Gerssen et al. 1997) never reach this value of R. Therefore we conclude that the N -body models of Valenzuela & Klypin (2003) probably produce slower bars than the observed. Figure 1 . 1Figure 1. The corotation radius, D L , and the bar semi-major axis, a B , for the sample galaxies, NGC 936 (Merrifield & Kuijken 1995), NGC 4596 (Gerssen et al. 1997) and NGC 7079 (Debattista & Williams 2003). Dashed lines corresponding to R = 1 and R = 1.4, separate the fast-bar, slow-bar and forbidden regimes. . J A L Aguerri, V P Debattista, E M Corsini, MNRAS. 338465Aguerri, J. A. L., Debattista, V. P., & Corsini, E. M. 2003, MNRAS, 338, 465 . E M Corsini, V P Debattista, J A L Aguerri, ApJ. in press (astroph/0310879Corsini, E. M., Debattista, V. P., & Aguerri, J. A. L. 2003, ApJ, in press (astro- ph/0310879) . V P Debattista, J A Sellwood, ApJ. 543704Debattista, V. P., & Sellwood, J. A. 2000, ApJ, 543, 704 . V P Debattista, T B Williams, ApJ. submittedDebattista, V. P., & Williams, T. B. 2003, ApJ, submitted . V P Debattista, E M Corsini, J A L Aguerri, MNRAS. 33265Debattista, V. P., Corsini, E. M., & Aguerri, J. A. L. 2002, MNRAS, 332, 65 . J Gerssen, K Kuijken, M R Merrifield, MNRAS. 288618Gerssen, J., Kuijken, K., & Merrifield, M. R. 1997, MNRAS, 288, 618 . M R Merrifield, K Kuijken, MNRAS. 274933Merrifield, M. R., & Kuijken, K. 1995, MNRAS, 274, 933 . S Tremaine, M D Weinberg, ApJ. 2825Tremaine, S., & Weinberg, M. D. 1984, ApJ, 282, L5 . O Valenzuela, A Klypin, MNRAS. 345406Valenzuela, O., & Klypin, A. 2003, MNRAS, 345, 406	[]
[ "Probing penguin coefficients with the lifetime ratio τ (", "Probing penguin coefficients with the lifetime ratio τ (" ]	[ "Yong-Yeon Keum [email protected]:[email protected] \nDESY -Theory group\nAPCTP\n207-43 Cheongryangri-dong, Dongdaemun-gu, Notkestrasse 85130-012, D-22607Seoul, HamburgKorea, Germany\n", "Ulrich Nierste \nDESY -Theory group\nAPCTP\n207-43 Cheongryangri-dong, Dongdaemun-gu, Notkestrasse 85130-012, D-22607Seoul, HamburgKorea, Germany\n" ]	[ "DESY -Theory group\nAPCTP\n207-43 Cheongryangri-dong, Dongdaemun-gu, Notkestrasse 85130-012, D-22607Seoul, HamburgKorea, Germany", "DESY -Theory group\nAPCTP\n207-43 Cheongryangri-dong, Dongdaemun-gu, Notkestrasse 85130-012, D-22607Seoul, HamburgKorea, Germany" ]	[]	We calculate penguin contributions to the lifetime splitting between the B s and the B d meson. In the Standard Model the penguin effects are found to be opposite in sign, but of similar magnitude as the contributions of the current-current operators, despite of the smallness of the penguin coefficients. We predict	10.1103/physrevd.57.4282	[ "https://arxiv.org/pdf/hep-ph/9710512v1.pdf" ]	16,377,397	hep-ph/9710512	3cea5ad066380ce10a0fe70a15781424590fec9e	Probing penguin coefficients with the lifetime ratio τ ( 28 Oct 1997 Yong-Yeon Keum [email protected]:[email protected] DESY -Theory group APCTP 207-43 Cheongryangri-dong, Dongdaemun-gu, Notkestrasse 85130-012, D-22607Seoul, HamburgKorea, Germany Ulrich Nierste DESY -Theory group APCTP 207-43 Cheongryangri-dong, Dongdaemun-gu, Notkestrasse 85130-012, D-22607Seoul, HamburgKorea, Germany Probing penguin coefficients with the lifetime ratio τ ( 28 Oct 1997 We calculate penguin contributions to the lifetime splitting between the B s and the B d meson. In the Standard Model the penguin effects are found to be opposite in sign, but of similar magnitude as the contributions of the current-current operators, despite of the smallness of the penguin coefficients. We predict τ (B s ) τ (B d ) − 1 = (−1.2 ± 10.0) · 10 −3 · f Bs 190 MeV 2 , where the error stems from hadronic uncertainties. Since penguin coefficients are sensitive to new physics and poorly tested experimentally, we analyze the possibility to extract them from a future precision measurement of τ (B s ) /τ (B d ). Anticipating progress in the determination of the hadronic parameters ε 1 , ε 2 and f Bs /f B d we find that the coefficient C 4 can be extracted with an uncertainty of order \|∆C 4 \| ≃ 0.1 from the double ratio (τ (B s ) − τ (B d ))/(τ (B + ) − τ (B d )), if \|ε 1 − ε 2 \| is not too small. INTRODUCTION Introduction The theoretical achievement of the Heavy Quark Expansion (HQE) [1] has helped a lot to understand the inclusive properties of B-mesons. The measurements of lifetime differences among the b-flavoured hadrons test the HQE at the order (Λ QCD /m b ) 3 . Today's experimental information on the B-meson lifetimes is in agreement with the expectations from the HQE, but the present theoretical predictions still depend on 4 poorly known hadronic parameters B 1 , B 2 , ε 1 and ε 2 [2,3]. Recently they have been obtained by QCD sum rules [4]. Lattice results are expected soon from the Rome group [5] and will allow for significantly improved theoretical predictions of the lifetime ratios. Weak decays are triggered by a hamiltonian of the form H = G F √ 2   V CKM 2 j=1 C j Q j − V ′ CKM 6 k=3 C j Q j + C 8 Q 8   .(1) Here Q 1 and Q 2 are the familiar current-current operators, Q 3 . . . Q 6 are penguin operators and Q 8 is the chromomagnetic operator. Their precise definition is given below in (3). The factors V CKM and V ′ CKM represent the factors stemming from the Cabibbo-Kobayashi-Maskawa matrix and are specific to the flavour structure of the decay. Feynman diagrams in which the spectator quark participates in the weak decay amplitude induce differences among the various b-flavoured hadrons. Such non-spectator effects have been addressed first by Bigi et al. in [6] evaluating the matrix elements in the factorization approximation in which ε 1 = ε 2 = 0. Then Neubert and Sachrajda [2] have found that even small deviations of ε 1 , ε 2 from zero drastically weaken the prediction of [6] for the lifetime ratio τ (B + )/τ (B d ), which can sizeably differ from 1. On the other hand the deviation of τ (B s )/τ (B d ) from unity has been estimated to be below 1% in [2,6] and the detailed analysis of Beneke, Buchalla and Dunietz [3]. Here τ (B s ) is the average lifetime of the two CP-eigenstates of B s . Experimentally the ratio τ (B s )/τ (B d ) can also be addressed by the measurements of the corresponding semileptonic branching fractions. Since spectator effects in the semileptonic decay rate are negligible, one may use τ (B s )/τ (B d ) = B SL (B s )/B SL (B d ). So far only the effect of Q 1 and Q 2 has been considered in [2,3,6]. Taking into account the present experimental uncertainty and the fact that C 1 and C 2 are much larger than C 3−8 in the Standard Model this is justified. Yet once the lifetime ratio τ (B s )/τ (B d ) is measured to an accuracy of a few permille, the situation will change: The smallness of \|τ (B s )/τ (B d )−1\| is caused by the fact that the weak annihilation contribution of Q 1,2 depicted in Fig. 1 almost yields the same contribution to the decay rates of B s and B d . The difference in the CKM-factors is negligible and the lifetime difference is induced by the small difference of the (c, c) vs. (c, u) phase space and by SU(3) F violations of the hadronic parameters. These effects suppress \|τ (B s )/τ (B d ) − 1\| by roughly an order of magnitude compared to \|τ (B + )/τ (B d ) − 1\|. The contributions stemming from the penguin operators and the chromomagnetic operator, however, do not exhibit such a cancellation. Their contribution to the non-spectator rate of B s comes with the same power of the Wolfenstein parameter λ = 0.22 as the contribution of Q 1,2 . In contrast the effects of Q 3−8 to the non-spectator rate of B d or B + are suppressed by two powers of λ and are therefore negligible. Hence one expects the contributions of Q 3−6 and Q 8 to \|τ (B s )/τ (B d ) − 1\| to be of the same order as those of Q 1 and Q 2 . τ (B + )/τ (B d ) is not modified, so that the phenomenological conclusions drawn from this ratio in [2] are unchanged. Observables sensitive to C 3−8 like τ (B s )/τ (B d ) are phenomenologically highly welcome. The smallness of C 3−8 is a special feature of the helicity structure of the corresponding diagrams in the Standard Model. In many of its extensions the values of these coefficients can easily be much larger. Such an enhancement due to supersymmetric contributions has been discussed in [7]. Up to now the focus of the search for new physics has been on new contributions to C 8 [7]. Yet many interesting possible non-standard effects modify C 3−6 rather than C 8 : New heavy particles mediating FCNC at tree-level or modifications of the b-s-g chromoelectric formfactor affect C 3−6 , but not C 8 . Likewise new heavy coloured particles yield extra contributions to C 3−6 , e.g. in supersymmetry box diagrams with gluinos modify C 3−6 . It is especially difficult to gain experimental information on the numerical values of the penguin coefficients C 3−6 . Even penguin-induced decays to final states solely made of d and s quarks do not provide a clean environment to extract C 3−6 : Any such decay also receives sizeable contributions from Q 2 via CKM-unsuppressed loop contributions [8,9]. In exclusive decay rates these "charming penguins" preclude the clean extraction of the effects of penguin operators [8]. In semi-inclusive decay rates like B → X s Φ the situation is expected to be similar. In inclusive decay rates such as the total charmless b decay rate the effect of "charming penguins" can be reliably calculated in perturbation theory. Yet these rates are much more sensitive to new physics contributions in C 8 rather than in C 3−6 , because Q 8 triggers the two-body decay b → s g, while the effects of Q 3−6 involve an integration over three-body phase space [9]. Notice from Fig. 2 and Fig. 3, however, that this phase space suppression of the terms involving C 3−6 is absent in the non-spectator diagrams inducing the lifetime differences. This work is organized as follows: In the following section we calculate the contributions to τ (B s )/τ (B d ) involving Q 3−6 or Q 8 . Here we also obtain the dominant part of the radiative corrections to order α s . In sect. 3 we discuss the phenomenological consequences within the Standard Model and with respect to a potential enhancement of C 3−8 by new physics. Penguin Contributions For the non-spectator contributions to the B s decay rate we need the \|∆B\| = \|∆S\| = 1hamiltonian: H = G F √ 2 V cb V * cs   6 j=1 C j Q j + C 8 Q 8   (2) with Q 1 = (sc) V −A · (cb) V −A · 1 1, Q 2 = (sc) V −A · (cb) V −A · 1 1 Q 3 = q=u,d,s,c,b (sb) V −A · (qq) V −A · 1 1, Q 4 = q=u,d,s,c,b (sb) V −A · (qq) V −A · 1 1 Q 5 = q=u,d,s,c,b (sb) V −A · (qq) V +A · 1 1, Q 6 = q=u,d,s,c,b (sb) V −A · (qq) V +A · 1 1 Q 8 = − g 8π 2 m bs σ µν (1 + γ 5 ) T a b · G a µν .(3) The colour singlet and non-singlet structure are indicated by 1 1 and 1 1 and V ± A is the Dirac structure. For more details see [9,10]. Fig. 1 has been calculated in [2,3] and yields contributions to the non-spectator part Γ non−spec of the B s decay rate proportional to C 2 2 , C 1 · C 2 and C 2 1 . The result involves four hadronic matrix elements, which are parametrized by the B-factors B 1 , B 2 , ε 1 and ε 2 [2]: In (2) we have set V ub V * us = O(λ 4 ) to zero. The diagram ofB s \|sγ µ (1 − γ 5 ) b bγ µ (1 − γ 5 ) s\| B s = f 2 Bs M 2 Bs B 1 B s \|s (1 + γ 5 ) b b (1 − γ 5 ) s\| B s = f 2 Bs M 2 Bs B 2 B s \|sγ µ (1 − γ 5 ) T a b bγ µ (1 − γ 5 ) T a s\| B s = f 2 Bs M 2 Bs ε 1 B s \|s (1 + γ 5 ) T a b b (1 − γ 5 ) T a s\| B s = f 2 Bs M 2 Bs ε 2 .(4) Here T a is the colour SU(3) generator, M Bs = 5369 ± 2 MeV and f Bs are the mass and decay constant of the B s meson. τ (B s )/τ (B d ) − 1 is proportional to Γ non−spec (B d ) − Γ non−spec (B s ). The main differences between the result of Fig. 1 for these two rates are due to the different mass of u and c and the difference between f B d and f Bs . Hence the current-current parts of τ (B s )/τ (B d ) − 1 proportional to C 2 2 , C 1 · C 2 or C 2 1 are suppressed by a factor of z or ∆ with z = m 2 c m 2 b = 0.085 ± 0.023, ∆ = 1 − f 2 B d M B d f 2 Bs M Bs = 0.23 ± 0.11.(5) The result for ∆ in (5) is the present world average of lattice calculations [11]. There are also SU(3) F violations in the B-factors, but they are expected to be small from the experience with those appearing in B 0 − B 0 -mixing. We want to achieve an accuracy of 2 permille in our prediction for τ (B s )/τ (B d ), which corresponds to an accuracy of 20-30% in τ (B s )/τ (B d ) − 1. In the Standard Model the diagram is of the same order of magnitude as radiative corrections to Fig. 2 and therefore negligible. Yet in models in which quark helicity flips occur in flavour-changing vertices \|C 8 \| can easily be ten times larger than in the Standard Model [7]. The contribution of Q 1 vanishes. moment operator. These effects are suppressed by a factor of m b /(Λ QCD · 16π 2 ) with respect to those discussed above. In [3] they have been estimated from heavy meson spectroscopy to be an effect of order one permille in τ (B s )/τ (B d ). We are now interested in the diagram of Fig. 2 involving one large coefficient C 1,2 and one small penguin coefficient C 3−6 . Diagrams with two insertions of penguin operators yield smaller contributions proportional to C 2 3−6 and are neglected here. To order λ 2 in H we have V ′ CKM = 0 in (1) for the B d system and penguin effects are only relevant in τ (B s ). Hence the penguin contributions to τ (B s )/τ (B d ) − 1 do not suffer from the suppression factors z and ∆. Next we want to evaluate the diagram of Fig. 3 which encodes the interference of Q 1,2 with the chromagnetic operator Q 8 . This part of Γ non−spec already belongs to the order α s and is small in the Standard Model, but it can be sizeable in the new physics scenarios discussed in [7]. We also must discuss radiative corrections to the contributions involving the large coefficients C 1 and C 2 . Dressing the diagram in Fig. 1 with gluons gives contributions to Γ non−spec for both B d and B s and therefore yield small corrections of order C 2 2 ∆α s /π or less. The penguin diagram of Fig. 4, however, contributes only to Γ non−spec (B s ) in the order λ 4 . Hence Fig. 4 yields an unsuppressed contribution of order C 2 2 α s /π to τ (B s )/τ (B d ) − 1 and cannot be neglected. The result of these penguin loop diagrams can easily be absorbed into the penguin coefficients C 3−6 : In the result of the diagram of Fig. 2 one must simply replace C j by C ′ j = C N LO j + α s 4π C 2 Re r 2j 1, √ z, µ/m b , j = 3, . . . , 6.(6) Here r 2j encodes the result of the penguin diagram and can be found in [9] in the NDR scheme. To cancel the scheme dependence of r we must also include the next-to-leading order (NLO) corrections to C j as indicated in (6). More precisely: We must include the NLO mixing of C 2 into C j in C N LO j , j = 3, . . . , 6, but the penguin-penguin mixing only to the LO. The difference between these partial NLO-coefficients, which are tabulated in [9], and the full C N LO j 's has a negligible impact on our result. Here we bypass this technical aspect of scheme independence by tabulating the C ′ j 's in Tab. 1. Our result for the non-spectator part of the B s decay rate reads: Γ non−spec (B s ) = − G 2 F m 2 b 12π \|V cb V cs \| 2 √ 1 − 4zf 2 Bs M Bs [a 1 ε 1 + a 2 ε 2 + b 1 B 1 + b 2 B 2 ](7) with a 1 = 2C 2 2 + 4C 2 C ′ 4 [1 − z] + 12zC 2 C ′ 6 + [1 + 2z] α s π C 2 C 8 a 2 = − [1 + 2z] 2C 2 2 + 4C 2 C ′ 4 + α s π C 2 C 8 b 1 = [C 2 + N c C 1 ] (1 − z) C 2 N c + C 1 + 2C ′ 3 + 2 C ′ 4 N c + 6z C ′ 5 + C ′ 6 N c b 2 = − [1 + 2z] [C 2 + N c C 1 ] 1 N c [C 2 + N c C 1 ] + 2 C ′ 3 + C ′ 4 N c(8) Here N c = 3 is the number of colours. By setting C ′ j , j = 3, . . . , 6, and C 8 in (8) to zero one recovers the result of [2]. 3 The result for the non-spectator contributions to the B d decay rate reads [2]: Γ non−spec (B d ) = G 2 F m 2 b 12π \|V cb V ud \| 2 (1 − z) 2 f 2 Bs M Bs (∆ − 1) a d 1 ε 1 + a d 2 ε 2 + b d 1 B 1 + b d 2 B 2 (9) with 4 a d 1 = 2C 2 2 1 + z 2 , a d 2 = −2C 2 2 (1 + 2z) , b d 1 = 1 N c (C 2 + N c C 1 ) 2 1 + z 2 , b d 2 = − 1 N c (C 2 + N c C 1 ) 2 (1 + 2z) .(10) When we combine (7-10) in order to predict τ (B s )/τ (B d ) − 1: τ (B s ) τ (B d ) − 1 = Γ non−spec (B d ) − Γ non−spec (B s ) Γ total + O(10 −3 ) = K(z) · ∆ 2C 2 2 (ε 1 − ε 2 ) + (C 2 + N c C 1 ) 2 N c (B 1 − B 2 ) (11a) − 3C 2 2 zε 1 − 3 2 (C 2 + N c C 1 ) 2 N c zB 1 (11b) + ∆ z C 2 2 (ε 1 − 4ε 2 ) + (C 2 + N c C 1 ) 2 2N c (B 1 − 4B 2 ) (11c) + 4C 2 C ′ 4 + (1 + 2z) α s π C 2 C 8 (ε 1 − ε 2 ) (11d) + 2 (C 2 + N c C 1 ) C ′ 3 + C ′ 4 N c (B 1 − B 2 ) (11e) − 4z C 2 C ′ 4 (ε 1 + 2ε 2 ) + 12z C 2 C ′ 6 ε 1 +2z (C 2 + N c C 1 ) − C ′ 3 + C ′ 4 N c (B 1 + 2B 2 ) + 3 C ′ 5 + C ′ 6 N c B 1 + O(2 · 10 −3 ) Here K(z) reads K(z) = 16π 2 \|V ud \| 2 B SL m 3 b f 1 (z) [1 + α s (µ)/(2π) h SL ( √ z)] f 2 Bs M Bs [1 − 2z](12)≃ 0.060 1 − 4 ( √ z − 0.3) (1 − 2z) B SL 0.105 4.8 m b 3 f Bs 190 MeV 2 .(13) In (12) we have used the common trick to evaluate the total width Γ total in terms of the semileptonic rate and the measured semileptonic branching ration B SL via Γ total = Γ SL /B SL . f 1 and h SL are the phase space and QCD correction factor of Γ SL calculated in [12]. We use the notation of [9]. The approximation in (13) reproduces K(z) to an accuracy of 3%. The numerical value of h SL entering (13) corresponds to the use of the one-loop pole mass (≃ 4.8 GeV) for m b . For simplicity we have expanded K(z) and the terms in the curly braces in (11) up to the first order in z. The size of the error in (11) is estimated as 2 · 10 −3 . Its main source is the SU(3) Fbreaking in the kinetic energy and chromomagnetic moment matrix elements appearing at order Λ 2 QCD /m 2 b of the HQE, which has been calculated to equal (0-1) · 10 −3 in [3]. Then terms of order 16π 2 Λ 4 QCD /m 4 b can maximally be of the same order of magnitude. Conversely the remaining NLO correction of order C 2 2 ∆ α s /π and the CKM-suppressed contributions are much smaller. Likewise the SU(3) F -breaking in ε 1 , ε 2 , B 1 and B 2 is expected to be at the level of a few percent and therefore smaller than the present uncertainty in ∆. The first three lines (11a-11c) contain the result of the current-current operators calculated in [2,3]. The remaining lines comprise the penguin effects. Note that the terms in (11d-11e) are neither suppressed by ∆ nor by z. For z = 0 the hadronic parameters in (11) only appear in the combinations ε 1 − ε 2 and B 1 − B 2 , both of which are of order 1/N c . The coefficients of B 1 − B 2 suffer from numerical cancellations, e.g. 0.09 ≤ C 2 + 3C 1 ≤ 0.57 (cf. Tab. 1), so that for most values of the input parameters only the terms involving ε 1 and ε 2 in (11a), (11b) and (11d) are important. Finally we discuss a potential systematic uncertainty: The derivation of (11) has assumed quarkhadron duality (QHD) for the sum over the final states. QHD means that inclusive observables are unaffected by the hadronization process of the quarks and gluons in the final state. The new results for inclusive observables in B decays presented at the 1997 summer conferences are consistent with QHD [13]. There are two potential sources of QHD violation in our problem: First it may be possible that the spectator decay rate of the b-quark is affected by the hadronization process. Yet the ballpark of this effect is independent of the flavour of the spectator quark and cancels out in the ratio τ (B s )/τ (B d ). SU(3) F -breaking can only appear in the hadronization of the final state antiquark which picks up the spectator quark and we do not expect the SU(3) Fbreaking in the spectator decay rate to be larger than the SU(3) F -breaking in the (Λ QCD /m b ) 2terms of the HQE. This effect should further not depend on whether the hadron containing the spectator quark recoils against other hadrons or against a lepton pair. Hence one can control the SU(3) F -breaking in the spectator decay rate by comparing the hadron energy in semileptonic B d and B s decays. More serious is a potential violation of QHD in the non-spectator contribution Γ non−spec itself. In a theoretical analysis for the similar case of the width difference ∆Γ Bs of the two B s eigenstates the size of QHD violation has been estimated to be moderate, maximally of order 30%. We can incorporate this into (11) by assigning an additional error of ±0.3 to ∆. In any case the issue of QHD violation in lifetime differences will be experimentally tested in the forthcoming years, when high precision measurements of τ (B + )/τ (B d ) and of ∆Γ Bs are confronted with accurate lattice results for the hadronic parameters. Phenomenology In the following we want to investigate the numerical importance of the penguin contribution. Then we analyze which accuracy is necessary to detect or constrain new physics contributions to C 3−6 by a precision measurement of τ (B s )/τ (B d ). The three main hadronic parameters entering (11) are ∆, f Bs and ε 1 − ε 2 , while B 1 and B 2 come with small coefficients. The canonical sizes of the B-factors are ε i = O(1/N c ) and B i = 1+O(1/N c ). An important constraint on the ε i 's is given by the measured value of τ (B + )/τ (B d ) [2]. The result of [2] for Γ non−spec (B + ) is obtained from (9) by replacing the a d i , b d i 's with a u 1 = −6 C 2 1 + C 2 2 , b u 1 = − 3 N c (C 2 + N c C 1 ) 2 + 3N c C 2 1 , a u 2 = b u 2 = 0. (14) The experimental world average [14] τ leads to the following constraint: ε 1 ≃ (−0.2 ± 0.1) 0.17 GeV f B 2 m b 4.8 GeV 3 + 0.3ε 2 + 0.05.(16) In [4] the ε i 's and B i 's have been calculated with QCD sum rules within the heavy quark effective theory (HQET). The results are ε 1 (µ = m b ) = −0.08 ± 0.02 and ε 2 (µ = m b ) = −0.01 ± 0.03 and B 1,2 = 1 + O(0.01). In view of the smallness of the ε i 's, however, it is conceivable that other neglected effects are numerically relevant.For example a NLO calculation of the matching between HQET and full QCD amplitudes replaces ε i in (7) and (9) by ε i + d i B i , where d i is a coefficient of order α s (m b )/π. Here we will consider the range \|ε 1 \|, \|ε 2 \| ≤ 0.3, and further obey (15). In Tab. 2 we have tabulated τ (B s )/τ (B d ) − 1 for various values of ∆ and ε 1 , ε 2 . We have further split τ (B s )/τ (B d ) − 1 into its current-current part consisting of (11a-11c) and the new penguin part involving C ′ 3−6 , C 8 . These results can be found in Tab. 3. From Tab. 3 we realize that the penguin contributions calculated in this work are comparable in size, but opposite in sign to the current-current part obtained in [3]. This makes the experimental detection of any deviation of τ (B s )/τ (B d ) from 1 even more difficult, if the penguin coefficients are really dominated by Standard Model physics. The results of Tab. 2 can be summarized as τ (B s ) τ (B d ) − 1 = (−1.2 ± 8.0 ± 2.0) · 10 −3 · f Bs 190 MeV 2 4.8 GeV m b 3 . (17) Here the first error stems from the uncertainty in ε 1 and ε 2 and will be reduced once lattice results for the hadronic parameters are available. The second error summarizes the remaining uncertainties. If ∆ and ε 2 simultaneously aquire extreme values, τ (B s )/τ (B d )−1 can be slightly outside the range in (17) (see Tab. 2). Today we have little experimental information on the sizes of the penguin coefficients. Their smallness in the Standard Model allows for the possibility that they are dominated by new physics. The total charmless inclusive branching fraction Br(B → no charm) is a candidate to detect new physics contributions to C 8 [7], but it is much less sensitive to C 3−6 [9]. The decreasing experimental upper bounds on Br(B → no charm) [14] therefore constrain C 8 but leave room for a sizeable enhancement of C 3−6 . Now (11) reveals that τ (B s )/τ (B d ) is a complementary observable mainly sensitive to C 4 , while C 8 is of minor importance. As mentioned in the introduction, many interesting new physics scenarios affect C 3−6 , but not necessarily C 8 . We remark here that we constrain ourself to new physics scenarios, in which the CKM factors of the new contributions are the same as the ones of the Standard Model. This is fulfilled to a good approximation in most interesting models [7]. Now any new physics effect modifies C 3−6 at some high scale of the order of the new particle masses, while the Wilson coefficients entering (11) Observe that ∆C 4 (200 GeV) = −0.05 already increases C ′ 4 (m b ) by more than a factor of two. Clearly the usefulness of τ (B s )/τ (B d ) to probe C 3−6 crucially depends on the size of \|ε 1 − ε 2 \| and f Bs . We now investigate the sensitivity of τ (B s )/τ (B d ) to ∆C 4 (µ = m b ) in a possible future scenario for the hadronic parameters. We assume ε 1 = −0.10 ± 0.05, ε 2 = 0.20 ± 0.05, B 1 , B 2 = 1.0 ± 0.1, f Bs = (190 ± 15) GeV, ∆ = 0.23 ± 0.05, m b = (4.8 ± 0.1) GeV. (18) The assumed accuracy for f Bs will be achieved, once more experimental information on the B s system is obtained, e.g. after the detection of B s − B s -mixing. Also a more precise measurement of f Ds is helpful, because lattice QCD predicts the ratio f Bs /f Ds much better than f Bs [11]. The error bars of the other hadronic parameters likewise appear within reach, if one keeps in mind that information on ε 1 and ε 2 will not only be obtained from the lattice but also from other observables like τ (B + )/τ (B d ). Experimental progress in (15) and a next-to-leading order calculation of the coefficients in (14) and (10) τ (B s ) − τ (B d ) τ (B + ) − τ (B d ) = B SL (B s ) − B SL (B d ) B SL (B + ) − B SL (B d ) ,(19) which depends on ε 1 , ε 2 and ∆, while the dependence on f B and m b cancels. The corresponding plot for the parameter set of (18) can be found in Fig. 6 We find a smaller error band for Fig. 6 shows that e.g. Conclusions We have calculated the contributions of the penguin operators Q 3−6 , of the chromomagnetic operator Q 8 and of penguin diagrams with insertions of Q 2 to the lifetime splitting between the B s and B d meson. In the Standard Model the penguin effects are found to be roughly half as big as the contributions from the current-current operators Q 1 and Q 2 , despite of the smallness of the penguin coefficients. Yet they are opposite in sign, so that any deviation of τ (B s ) − τ (B d ) from zero is even harder to detect experimentally. Assuming a reasonable progress in the determination of the hadronic parameters a precision measurement of τ (B s )/τ (B d ) can be used to probe the coefficient C 4 with an accuracy of \|∆C 4 \| = 0.1. Hence new physics can only be detected, if C 4 is dominated by non-standard contributions. The sensitivity to C 4 depends crucially on the difference of the hadronic parameters ε 1 and ε 2 . For the extraction of C 4 the double ratio (τ (B s )− τ (B d ))/(τ (B + ) − τ (B d )) turns out to be more useful than τ (B s )/τ (B d ). Figure 1 :Figure 2 : 12Non-spectator (weak annihilation) contribution to the B s decay rate involving two current-current operators. The corresponding diagram for the B d decay is obtained by replacing s by d and the upper c by u. Weak annihilation diagram involving one penguin operator Q 3−6 . Penguin contributions to the non-spectator rate of the B d meson are CKM suppressed and therefore negligible. Figure 3 : 3Therefore we use the same B 1 , B 2 , ε 1 and ε 2 in τ (B s ) and τ (B d ). Likewise there is SU(3) Fbreaking in the matrix elements of the b-quark kinetic energy operator and the chromomagnetic Contribution of Q 8 to Γ non−spec (B s ). Figure 4 : 4Penguin diagram contribution to Γ non−spec (B s ). The final state corresponds to a cut through either of the (c, c)-loops. The contributions of Q 1 vanish by colour. This is the only NLO contribution to τ (B s )/τ (B d )−1 involving Q 1,2 without suppression factors of ∆ or z. : The effective Wilson coefficients C ′ j defined in (6) for z = 0.085, α s (M Z ) = 0.118 and m b = 4.8 GeV. Varying z within the range given in (5) affects the C ′ j by 3-4 % and is negligible for our purposes. The C (0) j 's are the LO Wilson coefficients. (B + ) τ (B d ) = 1.07 ± 0.04 (15) Model prediction for 10 3 · [τ (B s )/τ (B d ) − 1] obtained from (11) for f Bs = 190 MeV, µ = m b = 4.8 GeV, z = 0.085, α s (M Z ) = 0.118 and B 1 = B 2 = 1. The entries marked with * are in conflict with the experimental constraint (15), which also implies ε 1 < ∼ 0. There is an overall error of ±2.0 (see 11) for all entries. : The columns labeled with 'peng' list the penguin contribution to 10 3 ·[τ (B s )/τ (B d ) − 1] as a function of ε 1 − ε 2 and f Bs . The other input parameters have little impact on the size of the penguin contribution. The current-current part of 10 3 · [τ (B s )/τ (B d ) − 1] is listed for ε 1 = −0.1 and ∆ = 0.23. For the remaining parameters see Tab. 2. are evaluated at a low scale µ ≈ m b . The renormalization group evolution down to µ ≈ m b mixes the new contributions to C 3−6 . New physics contributions ∆C 3−6 (µ = 200GeV ) affect C 4 (µ = 4.8 GeV) by ∆C 4 (µ = 4.8 GeV) = −0.35 ∆C 3 (200 GeV) + 0.99 ∆C 4 (200 GeV) −0.03 ∆C 5 (200 GeV) − 0.22 ∆C 6 (200 GeV). Figure 5 : 5will significantly improve the constraint in (16). In Fig. 5 we show Dependence of τ (B s )/τ (B d ) − 1 on a new physics contribution ∆C 4 . The shaded area corresponds to a variation of the input parameters within the range of (18). The horizontal lines mark the Standard Model range corresponding to ∆C 4 = 0. the dependence of τ (B s )/τ (B d ) − 1 on ∆C 4 (µ) for the scenario in (18). A cleaner observable is the double ratio (τ (B s ) − τ (B d ))/(τ (B + ) − τ (B d )) than for τ (B s )/τ (B d ) − 1. If ∆C 4 < −0.075 or ∆C 4 > 0.140, we find the allowed range for (τ (B s ) − τ (B d ))/(τ (B + ) − τ (B d )) incompatible with the Standard Model. An experimental lower bound τ (B s )/τ (B d ) > 1.005 would indicate a new physics contribution ∆C 4 < −0.063 in our scenario. Likewise the experimental detection of a sizeable negative lifetime difference τ (B s ) − τ (B d ) may reveal nonstandard contributions to C ′ 4 of similar size as its Standard Model value. Figure 6 : 6Dependence of (τ (B s ) − τ (B d ))/(τ (B + ) − τ (B d )) on ∆C 4 for the parameter set in (18). This double ratio depends on f Bs and f B d only through ∆, and the factor of m −3 b in (11) cancels. the bound τ (B s ) − τ (B d ) < −0.20(τ (B + ) − τ (B d )) would indicate ∆C 4 > 0.051. We conclude that the detection of new physics contributions to C 4 of order 0.1 is possible with precision measurements of τ (B s )/τ (B d ). e-mail:[email protected] 2 e-mail:[email protected] PENGUIN CONTRIBUTIONS Notice that our notation of C 1 and C 2 is opposite to the one in[2].4 In the large N c limit one finds Γ non−spec helicity suppressed in analogy to the leptonic decay rate. This shows that one cannot neglect the O(1/N c ) terms. . I I Bigi, N Uraltsev, A Vainshtein, Phys. Lett. B. 293430I.I. Bigi, N. Uraltsev, and A. Vainshtein, Phys. Lett. B 293, 430 (1992); Erratum ibid. 297477Erratum ibid. 297, 477 (1993). . B Blok, M Shifman, ibid. 399Phys. Rev. D. 459. A. Manohar and M. Wise3991310Nucl. Phys.B. Blok and M. Shifman, Nucl. Phys. B399 (1993) 441; ibid. 399 (1993) 459. A. Manohar and M. Wise, Phys. Rev. D 49, (1994) 1310. . B Blok, L Koyrakh, M Shifman, A I Vainshtein, 3356; Erra- tum ibid. D50Phys. Rev. 493572B. Blok, L. Koyrakh, M. Shifman and A.I. Vainshtein, Phys. 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[ "Using Terminological Knowledge Representation Languages to Manage Linguistic Resources", "Using Terminological Knowledge Representation Languages to Manage Linguistic Resources" ]	[ "Pamela W Jordan \nIntelligent Systems Program\nUniversity of Pittsburgh Pittsburgh PA\n15260\n" ]	[ "Intelligent Systems Program\nUniversity of Pittsburgh Pittsburgh PA\n15260" ]	[]	I examine how terminological languages can be used to manage linguistic data during NL research and development. In particular, I consider the lexical semantics task of characterizing semantic verb classes and show how the language can be extended to flag inconsistencies in verb class definitions, identify the need for new verb classes, and identify appropriate linguistic hypotheses for a new verb's behavior.	10.3115/981863.981917	null	11,255,089	cmp-lg/9605024	d2f4acd95b8b5458d988d9d4b4553c5054dc6ba0	Using Terminological Knowledge Representation Languages to Manage Linguistic Resources Pamela W Jordan Intelligent Systems Program University of Pittsburgh Pittsburgh PA 15260 Using Terminological Knowledge Representation Languages to Manage Linguistic Resources [email protected] I examine how terminological languages can be used to manage linguistic data during NL research and development. In particular, I consider the lexical semantics task of characterizing semantic verb classes and show how the language can be extended to flag inconsistencies in verb class definitions, identify the need for new verb classes, and identify appropriate linguistic hypotheses for a new verb's behavior. Introduction Problems with consistency and completeness can arise when writing a wide-coverage grammar or analyzing lexical data since both tasks involve working with large amounts of data. Since terminological knowledge representation languages have been valuable for managing data in other applications such as a software information system that manages a large knowledge base of plans (Devanbu and Litman, 1991), it is worthwhile considering how these languages can be used in linguistic data management tasks. In addition to inheritance, terminological systems provide a criterial semantics for links and automatic classification which inserts a new concept into a taxonomy so that it directly links to concepts more general than it and more specific than it (Woods and Schmolze, 1992). Terminological languages have been used in NLP applications for lexical representation (Burkert, 1995), and grammar representation (Brachman and Schmolze, 1991), and to assist in the acquisition and maintenance of domain specific lexical semantics knowledge (Ayuso et al., 1987). Here I explore additional linguistic data management tasks. In particular I examine how a terminological language such as Classic can assist a lexical semanticist with the management of verb classes. In conclusion, I discuss ways in which terminological languages can be used during grammar writing. Consider the tasks that confront a lexical semanticist. The regular participation of verbs belonging to a particular semantic class in a limited number of syntactic alternations is crucial in lexical semantics. A popular research direction assumes that the syntactic behavior of a verb is systematically influenced by its meaning (Levin, 1993;Hale and Keyser, 1987) and that any set of verbs whose members pattern together with respect to syntactic alternations should form a semantically coherent class (Levin, 1993). Once such a class is identified, the meaning component that the member verbs share can be identified. This gives further insight into lexical representation for the words in the class (Levin, 1993). Terminological languages can support three important functions in this domain. First, the process of representing the system in a taxonomic logic can serve as a check on the rigor and precision of the original account. Once the account is represented, the terminological system can flag inconsistencies. Second, the classifier can identify an existing verb class that might explain an unassigned verb's behavior. That is, given a set of syntactically analyzed sentences that exemplify the syntactic alternations allowed and disallowed for that verb, the classifter will provide appropriate linguistic hypotheses. Third, the classifier can identify the need for new verb classes by flagging verbs that are not members of any existing, defined verb classes. Together, these functions provide tools for the lexical semanticist that are potentially very useful. The second and third of these three functions can be provided in two steps: (1) classifying each alternation for a particular verb according to the type of semantic mapping allowed for the verb and its arguments; and (2) either identifying the verb class that has the given pattern of classified alternations or using the pattern to form the definition of a new verb class. Sentence Classification The usual practice in investigating the alternation patterning of a verb is to construct example sentences in which simple, illustrative noun phrases are used as arguments of a verb. The sentences in (1) exemplify two familiar alternations of give. (1) a. John gave Mary a book b. John gave a book to Mary. Such sentences exemplify an alternation that belongs to the alternation pattern of their verb. 1 I will call this the alternation type of the test sentence. To determine the alternation type of a test sentence, the sentence must be syntactically analyzed so that its grammatical functions (e.g. subject, object) are marked. Then, given semantic feature information about the words filling those grammatical functions (GFs), and information about the possible argument structures for the verb in the sentence and the semantic feature restrictions on these arguments, it is possible to find the argument structures appropriate to the input sentence. Consider the sentences and descriptions shown below for pour: (2) a. [Mary,,hi] io ~ patient[liquid] pour2: subj --+ agent[volitional] obj ---* patient[l/quid] ppo ---* recipient[volitional] Given the semantic type restrictions and the GFs, pour1 describes (2a) and pourz, (2b). The mapping from the GFs to the appropriate argument structure is similar to lexical rules in the LFG syntactic theory except that here I semantically type the arguments. To indicate the alternation types for these sentences, I call sentence (2a) a benefactive-ditransitive and sentence (2b) a benefactive-transitive. Classifying a sentence by its alternation type requires linguistic and world knowledge. World knowledge is used in the definitions of nouns and verbs in the lexicon and describes high-level entities, such as events, and animate and inanimate objects. Properties (such as LIQUID) are used to define specialized entities. For example, the property NON-CONSUMABLE (SMALL CAPITALS indicate Classic concepts in my implementation) specializes a LIQUID-ENTITY to define PAINT and distinguish it from WATER, which has the property that it is CON-SUMABLE. Specialized EVENT entities are used in the definition of verbs in the lexicon and represent the argument structures for the verbs. The linguistic knowledge needed to support sentence classification includes the definitions of (1) verb types such as intransitive, transitive and alltransitive; (2) verb definitions; and (3) concepts that define the links between the GFs and verb argument structures as represented by events. 1In the examples that I will consider, and in most examples used by linguists to test alternation patterns, there will only be one verb; this is the verb to be tested. Verb types (SUBCATEGORIZATIONS) are defined according to the GFs found in the sentence. For example, (2a) classifies as DITRANSITIVE and (2b) as a specialized TRANSITIVE with a PP. Once the verb type is identified, verb definitions (VERBs) are needed to provide the argument structures. A VERB can have multiple senses which are instances of EVENTs, for example the verb "pour" can have the senses pour or prepare, with the required arguments shown below. 2 Note that pour1 and pour2 in (2) For a sentence to classify as a particular ALTERNA-TION, a legal linking must exist between an EVENT and the SUBCATEGORIZATION. Linking involves restricting the fillers of the GFs in the SUBCATEGO-RIZATION to be the same as the arguments in an EVENT. In Classic, the same-as restriction is limited so that either both attributes must be filled already with the same instance or the concept must already be known as a LEGAL-LINKING. Because of this I created a test (written in LISP) to identify a LEGAL-LINKING. The test inputs are the sentence predicate and GF fillers arranged in the order of the event arguments against which they are to be tested. A linking is legal when at least one of the events associated with the verb can be linked in the indicated way, and all the required arguments are filled. Once a sentence passes the linking test, and classifies as a particular ALTERNATION, a rule associated with the ALTERNATION classifies it as a speciMizalion of the concept. This causes the EVENT arguments to be filled with the appropriate GF fillers from the SUBCATEGORIZATION. A side-effect of the alternation classification is that the EVENT classifies as a specialized EVENT and indicates which sense of the verb is used in the sentence. Semantic Class Classification The semantic class of the verb can be identified once the example sentences are classified by their alternation type. Specialized VERB-CLASSes are defined by their good and bad alternations. Note that VERB defines one verb whereas VERB-CLASS describes a set of verbs (e.g. spray/load class). Which AL-TERNATIONs are associated with a VERB-CLASS is a matter of linguistic evidence; the linguist discovers these associations by testing examples for grammaticality. To assist in this task, I provide two tests, have-instances-of and have-no-instances-of. 2For generality in the implementation, I use argl ... arg, for all event definitions instead of agent ... patient or preparer ... preparee. The have-instances-of test for an ALTERNATION searches a corpus of good sentences or bad sentences and tests whether at least one instance of the specified ALTERNATION, for example a benefactiveditransitive, is present. A bad sentence with all the required verb arguments will classify as an ALTERNATION despite the ungrammatical syntactic realization, while a bad sentence with missing required arguments will only classify as a SUBCATEGORIZATION. The have-no-instances-of test for a SUBCATEGORIZA-TION searches a corpus of bad sentences and tests whether at least one instance of the specified SUBCATEGORIZATION, for example TRANSITIVE, is present as the most specific classification. Discussion The ultimate test of this approach is in how well it will scale up. The linguist may choose to add knowledge as it is needed or may prefer to do this work in batches. To support the batch approach, it may be useful to extract detailed subcategorization information from English learner's dictionaries. Also it will be necessary to decide what semantic features are needed to restrict the fillers of the argument structures. Finally, there is the problem of collecting complete sets of example sentences for a verb. In general, a corpus of tagged sentences is inadequate since it rarely includes negative examples and is not guaranteed to exhibit the full range of alternations. In applications where a domain specific corpus is available (e.g. the Kant MT project (Mitamura et al., 1993)), the full range of relevant alternations is more likely. However, the lack of negative examples still poses a problem and would require the project linguist to create appropriate negative examples or manually adjust the class definitions for further differentiation. While I have focused on a lexical research tool, an area I will explore in future work is how classification could be used in grammar writing. One task for which a terminological language is appropriate is flagging inconsistent rules. When writing and maintaining a large grammar, inconsistent rules is one type of grammar writing bug that occurs. For example, the following three rules are inconsistent since feature1 of NP and feature1 of VP would not unify in rule 1 given the values assigned in 2 and 3. 1) S --. NP VP <NP feature1 > = <VP feature1 > 2) NP ~ det N <N feature1 > = + <NP> = <N> 3) VP --* V <V feature1 > = -<VP> ~ <V> Conclusion I have shown how a terminological language, such as Classic, can be used to manage lexical semantics data during analysis with two minor extensions. First, a test to identify LEGAL-LINKINGs is necessary since this cannot be directly expressed in the language and second, set membership tests, have-instances-of and have-no-instances-of are necessary since this type of expressiveness is not provided in Classic. While the solution of several knowledge acquisition issues would result in a friendlier tool for a linguistics researcher, the tool still performs a useful function. poured [Tinaobj] [a glass of mflkio]. b. [Marys,bj] poured [a glass of milkobj] for [Tinam, o]. poura: subj ~ agent[volitional] obj ~ recipient[voUtional] An environment for acquiring semantic information. Damaris M Ayuso, Varda Shaked, Ralph Weischedel, Proceedings of 25th ACL. 25th ACLDamaris M. Ayuso, Varda Shaked, and Ralph Weischedel. 1987. An environment for acquir- ing semantic information. In Proceedings of 25th ACL, pages 32-40. An overview of the KL-ONE knowledge representation system. J Po3nald, James Brachman, Schmolze, Cognitive Science. 9Po3nald J. Brachman and James Schmolze. 1991. An overview of the KL-ONE knowledge representation system. Cognitive Science, 9:171-216. Living with CLASSIC: When and how to use a EL-ONE-like language. Ronald J Brachman, Deborah L Mcguinness, Peter F Patel-Schneider, Lori A Resnik, Principles of Semantic Networks. John F. SowaRonald J. Brachman, Deborah L. McGuinness, Pe- ter F. Patel-Schneider, and Lori A. Resnik. 1991. Living with CLASSIC: When and how to use a EL-ONE-like language. In John F. Sowa, editor, Principles of Semantic Networks, pages 401-456. . Morgan Kaufmann, San Mateo, CAMorgan Kaufmann, San Mateo, CA. Lexical semantics and terminological knowledge representation. Gerrit Burkert, Computational Lezical Semantics. Patrick Saint-Dizier and Evelyne ViegasCambridge University PressGerrit Burkert. 1995. Lexical semantics and ter- minological knowledge representation. In Patrick Saint-Dizier and Evelyne Viegas, editors, Compu- tational Lezical Semantics. Cambridge University Press. Plan-based terminological reasoning. Premkumar Devanbu, Diane J Litman, James FPremkumar Devanbu and Diane J. Litman. 1991. Plan-based terminological reasoning. In James F. 91: Principles of Knowledge Representation and Reasoning. Richard Allen, Erik Fikes, Sandewall, Editors, Morgan KaufmannSan Mateo, CAAllen, Richard Fikes, and Erik Sandewall, edi- tors, KR '91: Principles of Knowledge Representa- tion and Reasoning, pages 128-138. Morgan Kauf- mann, San Mateo, CA. A view from the middle. K L Hale, S J Keyser, Center for Cognitive Science, MIT. Lexicon Project Working Papers 10. K. L. Hale and S. J. Keyser. 1987. A view from the middle. Center for Cognitive Science, MIT. Lexicon Project Working Papers 10. English verb classes and alternations: a preliminary investigation. B Levin, University of Chicago PressB. Levin. 1993. English verb classes and alterna- tions: a preliminary investigation. University of Chicago Press. Automated corpus analysis and the acquisition of large, multi-lingual knowledge bases for MT. T Mitamura, E Nyberg, J Carbonell, Proceedings of TMI-93. TMI-93T. Mitamura, E. Nyberg, and J. Carbonell. 1993. Automated corpus analysis and the acquisition of large, multi-lingual knowledge bases for MT. In Proceedings of TMI-93. The EL-ONE family. A William, James G Woods, Schmolze, Semantic Networks in Artificial Intelligence. Fritz LehmannOxfordPergamon PressWilliam A. Woods and James G. Schmolze. 1992. The EL-ONE family. In Fritz Lehmann, editor, Se- mantic Networks in Artificial Intelligence, pages 133-177. Pergamon Press, Oxford.	[]
[ "Mass loss and supernova progenitors", "Mass loss and supernova progenitors" ]	[ "John Eldridge \nAstrophysics Research Centre\nPhysics Building\nQueen's University\nBT7 1NNBelfastUK\n" ]	[ "Astrophysics Research Centre\nPhysics Building\nQueen's University\nBT7 1NNBelfastUK" ]	[]	We first discuss the mass range of type IIP SN progenitors and how the upper and lower limits impose interesting constraints on stellar evolution. Then we discuss the possible implications of two SNe, 2002ap and 2006jc, for Wolf-Rayet star mass-loss rates and long Gamma-ray bursts.	10.1063/1.2819001	[ "https://arxiv.org/pdf/0711.2630v1.pdf" ]	14,157,235	0711.2630	fcd83c6d280e268bd99ef65fd21ef83dfbe26600	Mass loss and supernova progenitors 16 Nov 2007 John Eldridge Astrophysics Research Centre Physics Building Queen's University BT7 1NNBelfastUK Mass loss and supernova progenitors 16 Nov 2007stars: evolutionsupernovae: generalsupernovae: 2002ap2006jc We first discuss the mass range of type IIP SN progenitors and how the upper and lower limits impose interesting constraints on stellar evolution. Then we discuss the possible implications of two SNe, 2002ap and 2006jc, for Wolf-Rayet star mass-loss rates and long Gamma-ray bursts. INTRODUCTION Core-collapse supernovae (SNe) are the violent deaths of stars more massive than ≈ 7.5M ⊙ . They occur when nuclear burning reactions or electron degeneracy-pressure can no longer support the core against gravitational collapse. Either a neutron star or black hole is formed from the collapsing iron core and the outer layers of the star explode in a violent display. The nature of this display depends strongly on the final state of the progenitor star and the circumstellar medium; because there are many paths of stellar evolution the are many types of SNe. SNe are classified according to their observed spectra and lightcurves. The first differentiation is made by the absence or presence of hydrogen in a spectrum. If hydrogen emission lines exist then a SN is type II and type I otherwise. For type II SNe there are four subgroups: IIP when there is a plateau to the lightcurve lasting a few months. These are the most common type of SN. Type IIL have a linear decay to their lightcurve, IIn have narrow hydrogen emission lines in their spectrum and IIb initially appear to be type II until after a short time the hydrogen lines disappear and the SN becomes type Ib. The hydrogen free type I SNe have three separate subtypes. Type Ia are thought to be thermonuclear explosions of Chandrasekhar mass white dwarfs and are not considered further here, type Ib have helium lines in emission while type Ic have no helium lines. Single star models predicts that the type II SNe will be the result of stars between about 7 to 27M ⊙ (Heger et al., 2003;Eldridge & Tout, 2004;Poelarends et al., 2007) as these retain their hydrogen envelopes. Stars more massive than this lose all their hydrogen via stellar winds and therefore give rise to type Ib/c SNe. The only way to confirm this mapping is to observe the progenitors of SN. This is achieved by searching telescope archives to discover pre-explosion images. While the progenitors of SNe 1987A and 1993J where detected both were in nearby galaxies and both rare and unusual SNe. It was not until 1999 when the HST archive covered enough galaxies at sufficient depth and resolution that it was possible to start looking for progenitors of SN in a large number of galaxies at distances up to 20 megaparsecs. The first detection for the most common, type IIP, SN was for 2003gd (Smartt et al., 2004). This confirmed that their progenitors were red supergiants. With eight years of observations there are currently 32 SNe with useful pre-explosion images, 18 of these are type IIP (6 detections) and the remainder being types IIb and Ib/c. There are no detections of type Ib/c progenitors and with the growing number of the nondetections there is growing evidence that our standard view of mass loss during the late stages of evolution may be incorrect. In this proceedings we briefly highlight some of the main conclusions the mass range inferred for type IIP progenitors. We then discuss two interesting cases of two type Ib/c SNe, 2002ap and 2006jc that provide very stringent limits on the evolution of the most massive stars. THE POPULATION OF TYPE IIPS. With the sample of 18 type IIP detections and non-detections it is possible to begin to characterise the population of the progenitors Smartt et al. (2007). The main important result for their study is that by using a maximum likelihood analysis it is possible to infer the minimum and maximum initial masses for type IIP SN progenitors. The initial mass range is between 7.5 to 16.5M ⊙ , however the error bars are large and the range could be as large as 6 to 18M ⊙ . The initial mass depends strongly on the mixing and mass-loss scheme of the stellar models use to estimate an initial mass from a final luminosity. To remove this systematic it is better to work out the range of final helium core masses which is approximately 1.9 to 6M ⊙ in the STARS stellar models (Eldridge & Tout, 2004). The minimum mass can be used to constrain models of convection in stellar models. Most models with helium cores at the lower end of this range experience second dredgeup and become AGB stars. In fact the stellar models used in this work do experience second dredge-up and we use the pre-dredge up models. Therefore something is required to prevent these stars from becoming AGB stars. It is possible for the most massive AGB stars to undergo SN, however their observational signature in pre-explosion images would be quite different to the red supergiants observed to date as they are cooler and therefore more luminous at infra-red wavelengths . The maximum mass is due to one of two factors, because stars above this limit have lost most (or all) of their hydrogen and produce another type of SN or it is because the cores are massive enough to form a black hole and this also leads to another type of SN. In reality it is probably a combination of these factors. However the black hole explanation could be favoured as the inferred helium core mass is similar to that which is expected to produce a black hole rather than a neutron star at core-collapse (Heger et al., 2003;Eldridge & Tout, 2004). BUT WHAT ABOUT OTHER TYPES? With only one detection for the other SN types there is little that can be said. If the upper limits that have been obtained on the progenitors' luminosity are compared to the luminosity of observed Wolf-Rayet stars, the suspected progenitors, it is apparent that the pre-explosion in all but one cases are not deep enough to have revealed the progenitors. For a detection it is a waiting game to determine for type type Ib/c SN to occur nearby and have deep pre-explosion images. The culprits for type Ib/c progenitors are Wolf-Rayet (WR) stars. These are evolved massive stars that have completed core hydrogen burning and lost (or in the process of losing) their hydrogen envelopes and are naked helium stars. These stars are also the preferred progenitors of long Gamma-ray bursts (Woosley & Bloom, 2006). Despite this there are two interesting pre-explosion images for type Ib/c SNe. SN 2002ap, because the limit is so low that we were only just unable to detect the progenitor and are able to rule out single stars for the progenitor. While before SN 2006jc a luminous outburst transient was detected two years prior to the SN and this may have interesting consequences for our understanding of stellar-wind mass-loss. 2002ap Crockett et al. (2007) used previously unused deep pre-explosion images to reexamine the progenitor of SN 2002ap. The limit is the most stringent to date on any type Ib/c progenitor. They were able to rule out all standard single-star models are suggest that the most likely progenitor was a binary. Figure 1 shows the luminosity limit derived from the pre-explosion images on a theoretical Hertzsprung-Russell diagram. The grey stellar track is of a 17M ⊙ star that has its hydrogen envelope stripped by interaction via Roche-Lobe Overflow in a binary. The final stellar model has a final mass or 5M ⊙ that agrees with the mass inferred from modelling the SN lightcurve by Mazzali (2002). Also shown on the figure are the tracks of possible companion stars the mass of any companion star can also be limited by the pre-and post-explosion images. The mass of the companion star must have been less than ≈ 14M ⊙ . Any mass transfer must have been quite inefficient otherwise the secondary would have accreted a large amount of mass and be visible in the preexplosion image. However there is a problem with this model, there is a large amount of helium in the progenitor model. While the amount of helium required for a type Ib SN is uncertain, over 0.5M ⊙ should produce the signature of helium lines in the SN spectrum. The binary model has around 1M ⊙ of helium in the envelope. Therefore there is some uncertainty in whether this is a reasonable progenitor model. There are solutions, one is that the star was more massive and underwent more severe stripping during the binary interaction. Another is that we underestimate the strength of stellar winds of Wolf-Rayet stars. Increase the mass-loss rates of a star by a factor of 2 -3 would reduce the final mass of helium and therefore produce the required type Ic SN. A source of increased mass-loss rates could be rapid rotation [REF]. An alternative is that the mass-loss rates of WR stars are underestimated. This would be possible if the mass-loss rates are enhanced for only a short time of evolution. A possible mechanism has been suggested by Petrovic et al. (2006) who discuss how helium star envelopes can become inflated. Inflated helium envelopes are caused by a bump in the opacity which causes the envelopes to become extended and the density profile of the envelope to invert so density increases with radius from the core. Petrovic et al. (2006) FIGURE 1. Theoretical Hertzsprung-Russell diagram. The dashed line is the luminosity limit from the non-detection of any object in the pre-explosion image, the grey box represents the uncertainty in the limit for WR stars . The solid grey line is the evolution for a 17M ⊙ star which has its hydrogen envelope removed in a binary interaction. The solid black line is the evolution of a 11.9M ⊙ binary companion and the dotted line is the evolution of a 15.3M ⊙ binary companion. At the time of the primary SN the latter companion would have been observed while the former would have remained undetected. suggest that this inflation may not be physical and note that by increasing the mass-loss rates it is possible to remove the inflated envelope and density inversion. Figure 2 shows a standard stellar model with the mass-loss rates of Nugis & Lamers (2000) and a model which replaces these with the mass loss rates of Petrovic et al. (2006) when the helium envelope becomes inverted. The figure shows that during the nitrogen rich WN evolution much more mass is lost than in the standard case. Further more most of the lost material is helium. The final model retains only 0.25M ⊙ and the mass again agrees with the analysis of Mazzali (2002). With one non-detection it is not possible to decide if all WR mass-loss rates need to be increased but if the number of non-detections increase then the situation may begin to become more serious and the mass-loss rates will have to be closely examined. The only other scenario is that WR stars produce large amounts of dust a few years before core-collapse. This would reduce their apparent luminosity in pre-explosion images. 2006jc Pastorello (2007) and Foley et al. (2007) describe a most unusual SN. SN 2006jc was discovered on 9th October 2006 in the galaxy, UGC4904. Pastorello (2007) found that it was spatially coincident with a bright optical transient that occurred in 2004. The SN itself was classified as a type Ib-n due to narrow helium lines in the spectra. The current interpretation is that the narrow helium lines are due to helium-rich material ejected by the star in a dramatic mass-loss episode that was observed as the optical transient. Then Theoretical Hertzsprung-Russell diagram of the evolution of initially 50M ⊙ stars. The dashed line is the luminosity limit from the non-detection of a WR star the pre-explosion image, the grey box represents the uncertainty. The solid grey line is for a standard WR model (Eldridge & Vink, 2006) and the solid black line a model using the mass-loss rates of (Petrovic et al., 2006). two years after this episode the star exploded as a type Ic SN and therefore the progenitor star had be stripped of helium, otherwise broad helium emission lines would have been observed in combination with the narrow lines. The problem with this interpretation is that if the progenitor was a single star then our understanding of Wolf-Rayet stars and their winds must be revised. The only single stars known to produce similar bright optical transients are LBV stars (Humphreys & Davidson, 1994;van Genderen, 2001). However these stars tend to retain some hydrogen which would have been observed in the SN spectrum. It is conceivable that there are transition objects between LBVs and WR stars that would lead to the observed evolution for the progenitor of SN2006jc. These objects may be rare to get the right amount of mass-loss to remove all helium just before core-collapse. Although such objects would be more common at lower metallicities. An alternative to the LBV-like WR star is that the transient and mass-loss was due to a pair-production instability causing a dramatic mass-loss episode a few years before a normal core-collapse SN (Langer, 2007). The models of Heger & Woosley (2002) show that massive stars with little mass loss will experience such outbursts prior to corecollapse. The upper metallicity limit is uncertain. It is possible to estimate from stellar models to be below SMC metallicity. There is final a possibility with an observed analogue, the may have been progenitor a binary with a WR star and an evolved LBV star. The outburst was produced by the LBV star while the WR star exploded to produce the type Ic SN. There is one similar system in the SMC that has been observe to undergo outburst and at some point in the future may lead to a similar SN (Koenigsberger et al., 2002). These systems are not as rare as might be first thought. If we assume LBV evolution happens after corehydrogen burning is complete then any star which has a secondary companion that has completed core-hydrogen burning could be a possible LBV-WR system. Figure 3 shows how the hydrogen burning lifetime compares to the total lifetime for stars with different initial masses. It is possible to see that the more massive stars can have a wider range of secondary masses for this kind of evolution. If we assume the mass ratio of binaries (q = M secondary /M primary ) has a flat distribution and is independent of primary mass then 60% of binaries with a 200M ⊙ primary might be LBV-WR systems, while this reduces to only 20% for a primary initially 50M ⊙ . The only way to distinguish between these three plausible models will be the rate of these events and the metallicities of the host galaxies. The binary scenario will be only weakly metallicity dependent, the pair-production outburst will be concentrated to low metallicities while the single star LBV/WR has an unknown metallicity dependence but if it is related to inflation of the WR star then is will be concentrated at higher metallicities. The total rate of type Ib-n SN is < 4% of all type Ib/c SN. The rate of GRBs as a fraction of all Ib/c SNe is between 0.1 to 1% [REF]. It is possible therefore that some GRBs may occur with 2006jc-like SNe. If this is the case then the circumburst medium inferred from the optical afterglow would be a constant density medium due to the changes in mass-loss rate and wind speed. There are a number of GRBs where this has been observed and is another possible solution when the CBM is a constant density rather than that expected of a free-wind density profile (van Marle et al., 2006;Eldridge, 2007). CONCLUSIONS While the progenitors of type IIP SNe are becoming well understood there is still great uncertainty over the progenitors of other SN types. It is becoming apparent that our understanding of WR mass-loss maybe incorrect and one possible reason that we do not observe more WR stars is they lose more mass than currently thought and that some of this mass-loss may be in luminous outburst such as the one that proceeded SN 2006jc. Or they produce copious amount of dust in the last few years before core-collapse. FIGURE 2 . 2FIGURE 2. Theoretical Hertzsprung-Russell diagram of the evolution of initially 50M ⊙ stars. The dashed line is the luminosity limit from the non-detection of a WR star the pre-explosion image, the grey box represents the uncertainty. The solid grey line is for a standard WR model (Eldridge & Vink, 2006) and the solid black line a model using the mass-loss rates of (Petrovic et al., 2006). FIGURE 3 . 3The lifetimes for massive single stars. The solid line is the total lifetime versus initial mass. The dashed line is the time of the end of core hydrogen burning and the dotted-dashed line is the time of the end of core helium burning. The three shaded regions show the range of secondary masses which have completed core hydrogen burning and are LBV candidates by the time the primary (the high mass edge of the region) experiences a SN. 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[ "D-branes on the Quintic", "D-branes on the Quintic" ]	[ "Ilka Brunner \nDepartment of Physics\nAstronomy Rutgers University Piscataway\n08855-0849NJ\n", "Michael R Douglas \nDepartment of Physics\nAstronomy Rutgers University Piscataway\n08855-0849NJ\n\nI.H.E.S\nLe Bois-Marie\nBures-sur-Yvette91440France\n", "Albion Lawrence \nDepartment of Physics\nHarvard University Cambridge\n02138MA\n", "Christian Römelsberger \nDepartment of Physics\nAstronomy Rutgers University Piscataway\n08855-0849NJ\n" ]	[ "Department of Physics\nAstronomy Rutgers University Piscataway\n08855-0849NJ", "Department of Physics\nAstronomy Rutgers University Piscataway\n08855-0849NJ", "I.H.E.S\nLe Bois-Marie\nBures-sur-Yvette91440France", "Department of Physics\nHarvard University Cambridge\n02138MA", "Department of Physics\nAstronomy Rutgers University Piscataway\n08855-0849NJ" ]	[]	We study D-branes on the quintic CY by combining results from several directions: general results on holomorphic curves and vector bundles, stringy geometry and mirror symmetry, and the boundary states in Gepner models recently constructed by Recknagel and Schomerus, to begin sketching a picture of D-branes in the stringy regime. We also make first steps towards computing superpotentials on the D-brane world-volumes.	10.1088/1126-6708/2000/08/015	[ "https://arxiv.org/pdf/hep-th/9906200v3.pdf" ]	14,177,396	hep-th/9906200	7857456813ebd73958359ac42c4ddd62a1ab4b59	D-branes on the Quintic 27 Dec 1999 June 1999 Ilka Brunner Department of Physics Astronomy Rutgers University Piscataway 08855-0849NJ Michael R Douglas Department of Physics Astronomy Rutgers University Piscataway 08855-0849NJ I.H.E.S Le Bois-Marie Bures-sur-Yvette91440France Albion Lawrence Department of Physics Harvard University Cambridge 02138MA Christian Römelsberger Department of Physics Astronomy Rutgers University Piscataway 08855-0849NJ D-branes on the Quintic 27 Dec 1999 June 1999 We study D-branes on the quintic CY by combining results from several directions: general results on holomorphic curves and vector bundles, stringy geometry and mirror symmetry, and the boundary states in Gepner models recently constructed by Recknagel and Schomerus, to begin sketching a picture of D-branes in the stringy regime. We also make first steps towards computing superpotentials on the D-brane world-volumes. Introduction In this work we study D-branes on the quintic Calabi-Yau, historically the first CY to be intensively studied. Our guiding question will be: to classify all supersymmetrypreserving D-branes at each point in CY moduli space, and find their world-volume moduli spaces. As is well known, results of this type are quite relevant for phenomenological applications of M/string theory, because the world-volume theories we will obtain include a wide variety of four-dimensional theories with N = 1 space-time supersymmetry. The problem includes the classification of holomorphic vector bundles (which are ground states for wrapped six-branes); and almost all M/string compactifications which lead to d = 4, N = 1 supersymmetry (such as (0, 2) heterotic string compactifications and F theory constructions) have a choice of bundle as one of the inputs. Thus, many works have addressed this subject explicitly or implicitly. As usual in string compactification this geometric data is only an input and one would really like to answer the same questions with stringy corrections included. The primary question along these lines is: is the effect of stringy corrections just quantitative -affecting masses and couplings in the effective Lagrangian but preserving the spectrum and moduli spaces -or is it qualitative? If the latter, we might imagine that geometric branes undergo radical changes of their moduli space or are even destabilized in the stringy regime, with new branes which were unstable in the large volume limit taking their place. It should be realized that at present very little is known about this question; for example it has not been ruled out that the D0-brane becomes unstable in the stringy regime or has moduli space dimension different from 3. Clearly these questions are of great importance for the string phenomenology mentioned above and were asked long ago in the context of (0, 2) models. No simple answer has been proposed; we will return to this in the conclusions. A concrete framework which allows an exact CFT study of the stringy regime is provided by the Gepner models. The main lesson from the original study of Gepner models for type II and heterotic strings was that these CFT compactifications are continuously connected to CY compactification. Mirror symmetry is manifest in the 2d superconformal field theory, and this connection was one of the earliest arguments for it in the CY context. The first detailed study of D-branes in Gepner models was made by Recknagel and Schomerus [7] who (following the general approach of Cardy) constructed a large set of examples; further work appears in [8,9]. So far no geometric interpretation or contact with the large volume limit has been made. We will do so in this work. The main tool we will use is the (symplectic) intersection form for three-cycles in the large volume limit. This form governs Dirac quantization in the effective d = 4 theory and as such must be invariant under any variation of the moduli. As argued in [10,11] it is given by the index Tr ab (−1) F in open string CFT and thus is easily computed for the Gepner boundary states. The detailed study of Kähler moduli space by Candelas et. al. [12] then allows relating this to the large volume basis for 2p-branes. We can also make the large volume identification for the 3-branes, aided by the large discrete symmetry group. The detailed outline of the paper is as follows. In section 2 we review the quintic, its homology and moduli space, and give a general overview of D-branes on the quintic in the large volume limit. In section 3 we review the stringy geometry of its Kähler moduli space and the monodromy group acting on B branes. In section 4 we review Gepner models and Cardy's theory of boundary states, which will allow us to review the boundary states constructed by Recknagel and Schomerus. We briefly discuss the theory for K3 compactifications, and show that the results agree with geometric expectations; in particular that the dimension of a brane moduli space on K3 is given by the Mukai formula. In section 5 we compute the large volume charges for the quintic boundary states, and compute the number of marginal operators. This will allow us to propose candidate geometric identifications. In section 6 we discuss the computation of world-volume superpotentials. We begin by presenting evidence that the superpotential is "topological" in a sense that we explain. If true, an important consequence would be that the superpotential for B-type branes has relatively trivial Kähler dependence and can thus be computed in the large volume limit. This would imply general agreement between stringy and geometric results, analogous to the case of the prepotential. In section 7 we discuss superpotential computations in the Gepner model and derive selection rules. Besides charge conservation rules similar to those in the closed string sector, additional boundary selection rules appear, and we illustrate these with the examples of the A 1 and A 2 minimal models. The selection rules will allow us to establish that certain branes have non-trivial moduli spaces. The exact superpotentials should be calculable given the solutions of the consistency conditions of boundary CFT [13,14]; this is work in progress. In section 8 we summarize our results and draw conclusions. A point of notation: in labeling a p-brane, we always ignore its Minkowski spacefilling dimensions (for example, a D4 wraps four dimensions of the CY), but we describe its world-volume Lagrangian in d = 4, N = 1 terms (appropriate if the brane filled all 3 + 1 Minkowski dimensions). The conditions for supersymmetric embeddings with nonabelian fields turned on has not been given, but they have been worked out for single D-branes in refs. [3] [15], for which the action of spacetime supersymmetry and worldvolume κ-symmetry is known [16]. A compactification preserving supersymmetry will occur if there are constant spinors η i on M for each of the spacetime SUSYs. These supersymmetries transform the embedding coordinates (and their superpartners) on the D-brane worldvolumes; they are preserved if one can find a κ-symmetry transformation which cancels the SUSY transformation. This condition can be written as (1 − Γ)η i = 0 (2.1) and those η i which are solutions form the unbroken SUSYs. Γ is defined as follows [15]. Let E m µ be the vielbein connecting frame indices m and spacetime indices µ. We can pull this back to the worldvolume, defining E m α = ∂ α X µ E m µ (X) ,(2.2) where α is a worldvolume index for the p-brane. With this we can pull back the 10D γ-matrices Γ m : Γ α = E m α Γ m . (2.3) Define Γ (p+1) = 1 (p + 1)! √ g ǫ α 1 ...α p+1 Γ α 1 ...α p+1 ,(2.4) where g αβ = η mn E m α E n β (2.5) is the induced metric on the Dp-brane. We can now write: Γ = √ g √ g + F ∞ℓ=0 1 2 ℓ ℓ! Γ α 1 β 1 ...α n β n F α 1 β 1 . . . F α n β n Γ n+(p−2)/2 (11) Γ (p+1) (2.6) When F = 0 this can be written in the simpler, more familiar form: Γ = ǫ α 1 ...α p+1 ∂ α 1 X µ 1 . . . ∂ α p+1 X µ p+1 Γ µ 1 . . . Γ µ p+1 (2.7) where Γ µ = E m µ Γ m . The conditions in this latter case have been worked out in some detail, as we will describe below. These conditions match those in refs. [17] [18] for boundary states of BPS D-branes in flat space with constant background fields. Solutions to Eq. (2.1) in the presence of nonzero F have been worked out for flat, intersecting branes in refs. [17][18] [15]. In the case of BPS D-branes in Calabi-Yau 3-fold compactifications the geometric conditions implied by (2.1) (and the analog for boundary states) have been worked out in [3] [6]. These solutions fall into two classes: "A-type" branes wrapping special Lagrangian submanifolds and "B-type" branes wrapping holomorphic cycles. Let us describe each of these in turn. B branes "B-type" BPS branes wrap even-dimensional, holomorphic cycles in the Calabi-Yau [3] [6]. For B (even-dimensional) branes, (2.1) is solved by holomorphically embedded curves (2-branes) and surfaces (4-branes), as well as by 0 and 6-branes with the obvious (trivial) embeddings. We may also have gauge fields on these branes. In general the gauge field may change the definition of a supersymmetric cycle via Eq. (2.1). However, if the brane is wrapped around a holomorphic cycle, we can find conditions for the gauge field to preserve the supersymmetries. In the case of N coincident D6-branes wrapping the entire CY threefold, if we assume that the gauge fields live only in the threefold then the SUSY-preserving gauge field must satisfy the "Hermitian Yang-Mills equations" [19]: F ij = 0 ω 2 ∧ tr F = cω 3 ,(2.8) where (i, j) andī, are holomorphic and antiholomorphic indices, respectively, on the CY. These equations define a "Hermitian-Einstein" connection A with curvature F . The first equation tells us that the vector bundle is holomorphic. The second equation tells us that the vector bundle is "ω-stable"; conversely, ω-stability guarantees a solution to these equations [20] (c.f. chapter 4 of [21] for a discussion and definitions.) For branes wrapped around holomorphic submanifolds of M , these equations must be altered. The gauge fields polarized transverse to the cycle are replaced by "twisted" scalars Φ which are one-forms in the normal bundle to the embedding [4], and Eq. (2.8) becomes a generalization of the Hitchin equations for Φ and F [19]. It is believed that all topological invariants of a D-brane configuration are given by an element of a particular K-theory group on M [22] [23]. When the K-theory group and/or the cohomology of M has torsion the K-theory interpretation is important; one may have objects charged under the torsion. The charge can be written [22] as a generalization of the results of [24] [25]: v(E) = ch(f ! E) Â (M ) (2.9) Here E is a vector bundle on Σ; remember that we must extend the U (1) part of the gauge field F by the NS B-field, so properly the vector bundle E is a polynomial in F . Let π : M → Σ be the projection onto the worldvolume and N be the normal bundle of Σ ֒→ M . There is a K-theory element δ(N ) which is roughly a delta function on the worldvolume and depends on N ; we can thus define f ! (E) = π * E ⊗ δ(N ). The moduli space of D-branes will not just be the moduli space of vector bundles in this K-theory class but rather the moduli space of coherent semistable sheaves in this class [26] [19]. Some advantages of this definition through K-theory and sheaves, besides the fact that it seems to be correct, are that it places configurations with D6-branes (gauge field configurations on M ) on an equal footing with configurations without D6-branes, and that it can describe certain singularities which lead to sensible string compactifications. In examples without torsion, such as the quintic, one may describe the D-brane charge in a less esoteric fashion. Assuming the branes give rise to particles in the macroscopic directions, for a 2n-dimensional worldvolume Σ we can write the D-brane coupling to the RR gauge fields via the "Wess-Zumino term" as [27,24,25]: Σ C ∧ ch(F − B) Â (M ) A(N ) (2.10) where C = C (2n+1) + C (2n−1) + . . . + C (1) is a sum over the (k)-form RR potentials that couple to the 2n-brane. These RR charges reduce to conventional electric and magnetic charges in the four noncompact dimensions. Given two D-branes which reduce to particles, the most basic observable we can study is the Dirac-Schwinger-Zwanziger symplectic inner product on their charges, I(a, b) = Q Ea · Q M b − Q M a · Q Eb . (2.11) We will refer to this as the "intersection form" as it is closely related to the topological intersection form for two-and four-branes. For two six-branes, from the formulas above it is I(a, b) = ch(F a ) ch(−F b )Â(M ) . (2.12) Finally, we quote a general theorem regarding stability (Bogomolov's inequality [28]; c.f. [29,21]): given a variety X of dimension n and ω an ample divisor on X, then a ω-semistable torsion free sheaf E of rank r and Chern classes c i will satisfy S 2r c 2 − (r − 1)c 2 1 ∧ ω n−2 ≥ 0 . (2.13) The parenthesized combination is called the "discriminant" of the sheaf and is equal to c 2 (End(E)). In the special case c 1 (E) = 0 this amounts to requiring c 2 (E) ≥ 0. A branes An "A-type" BPS brane wraps a three-dimensional special Lagrangian submanifold Σ [3]: 1 ω\| Σ = 0 Ree iθ Ω\| Σ = 0 . (2.14) Here Ω is the holomorphic 3-form of the Calabi-Yau and θ is an arbitrary phase. Equivalently to the second equation, we can require that Ω pulls back to a constant multiple of the volume element on Σ. Furthermore the gauge field on this manifold must be flat. A nice introduction to the general theory of these is [31]. It is shown there (and in the references therein) that the moduli space has complex dimension b 1 (Σ). The space of flat U (1) connections has real dimension b 1 (Σ), and ω ij can be used to get an isomorphism between T * Σ and N Σ; thus the deformations of Σ pair up with the Wilson lines to form b 1 (Σ) complex moduli. For three-branes, the DSZ inner product (2.11) is precisely the geometric intersection form. One application of these branes is the Strominger-Yau-Zaslow formulation of mirror symmetry, a precise formulation of the idea that "mirror symmetry is T-duality" [32]. Since mirror symmetry exchanges the sets of A and B branes, an appropriately chosen moduli space of A branes on M will be the moduli space of D0-branes on the mirror W . Clearly b 1 = 3 for such A-branes, and SYZ argue that Σ will be a T 3 in this case. A similar proposal was made for general B branes with bundles in [33]. Another application is the construction of N = 1 gauge theories with the help of brane configurations. Supersymmetric three-cycles have been used to explore the strong coupling limit by lifting the brane configurations to M-theory in [34]. Not too many explicit constructions of special Lagrangian submanifolds are known and it appears (e.g. see [31]) that the problem is of the same order of difficulty as writing explicit Ricci-flat metrics on a CY. A general construction we will use below is as the fixed point set of a real involution. General world-volume considerations Given a system X of A or B D-branes, we can consider the system which is identical except that it extends in the flat 3 +1 dimensions transverse to M . This system will have a d = 4, N = 1 supersymmetric gauge theory as its low-energy world-volume theory, whose data is a gauge group G X ; a complex manifold C X parameterized by chiral superfields φ i ; a Kähler potential K on C X ; an action by holomorphic isometries of G X on C X (linearizing around a solution this corresponds to the usual choice of representation R of the gauge group), and a superpotential W (a holomorphic function on C X invariant under the action of G X ). If G X contains U (1) factors, each of these can have an associated real constant ζ a (the "Fayet-Iliopoulos terms"). In the classical (g s → 0) limit, the moduli space of this theory is the solutions of F i = ∂W/∂φ i = 0 (the "F-terms") and D a = ζ a (the "D-terms") modulo gauge transformations, where D a is the moment map generating the associated gauge transformation (and ζ a ≡ 0 in the non-abelian parts of the gauge group). We review this well-known material for a number of reasons. First, we remind the reader that although some of our later discussion will use other realizations of this D-brane system (for example as particles in 3 + 1 dimensions), the world-volume theories for these other realizations are all obtained by naive dimensional reduction from the 3 + 1 theory (if g s ∼ 0), while the 3 + 1 language makes it easy to impose supersymmetry. Second, it is known that the study of bundles and sheaves on CY three-folds is much more complicated than that for K3; this complication has a direct physical counterpart in the reduced constraints of N = 1 supersymmetry. The most basic example of this is the fact that -unlike the case for K3 -there is no formula for the dimension of the moduli space of E given c(E). The main reason for this is that this dimension is not necessarily constant -the moduli space can have branches of different dimension, and can depend on the moduli of the CY as well. Physically, this corresponds to the possibility of a fairly arbitrary superpotential in the low energy theory. Indeed, the language of superpotentials and N = 1 effective Lagrangians might be the best one for these problems, much as hyperkahler geometry and hyperkahler quotient is for instanton problems in four dimensions. Just as the self-dual Yang-Mills equations can be regarded as an infinite-dimensional hyperkahler quotient, we might pose the problem of rephrasing the YM equations under discussion as the problem of finding the moduli space of an N = 1 effective theory with an infinite number of fields. The basic outlines of part of this treatment are known (see [35], ch. 6 for a very clear discussion of the four-dimensional case). The two equations (2.8) will correspond directly to the F-term (superpotential) constraints and the D-term constraints, respectively. Indeed, the problem of solving F ij = 0 is a purely holomorphic problem, while it is not hard to see that the expression F a ∧ ω n−1 is the moment map generating conventional gauge transformations. The stability condition on the bundle is exactly the infinite-dimensional counterpart of the usual condition in supersymmetric gauge theory for an orbit of the complexified gauge group to contain a solution of the D-flatness conditions (e.g. see [36]). Donaldson's theorem proving the existence of such solutions proceeds exactly by considering the flow generated by i times the moment map to a minimum; the Uhlenbeck-Yau generalization is quite similar (for technical reasons a different equation is used). The other part of the story -translating the problem of finding holomorphic vector bundles into solving constraints on a finite-dimensional configuration space, which can be derived from a superpotential -does not seem to have been addressed in as systematic a manner; clearly this could be useful. In a sense the six-dimensional problem is the "universal" one which also describes the lower-dimensional branes. Not only can their charges be reproduced, but gauge field singularities will correspond to specific lower dimensional branes (e.g. the small instanton). Furthermore, there is a sense in which even the lower-dimensional brane world-volume theories are six-dimensional if we include "winding strings" (by analogy to tori and orbifolds, although this idea has not yet been made precise). Treating a system of N D0's as quantum mechanics requires neglecting these strings, which one expects to be problematic once the separation between branes approaches the size of the space. We now turn from these abstract ideas to our concrete example. D-branes on the quintic Perhaps the best-studied family of Calabi-Yau manifolds is the quintic hypersurfaces in IP 4 . A relatively thorough discussion of these is contained in the classic paper [12]. The moduli space of these manifolds is locally the product of b 2,1 = 101 complex structure deformations and b 1,1 = 1 deformations of the complexified Kähler forms B + iJ (where B is the flux of the NS-NS B-field). We will be particularly interested in the Fermat quintic P = 5 i=1 z 5 i = 0 (2.15) where z i are the homogeneous coordinates on IP 4 . Note that this equation has a S 5 × Z 4 5 discrete symmetry; the Z 5 generators are g i : z i → ωz i and satisfy the relation 5 i=1 g i = 1, while the S 5 permutes the coordinates in the obvious way. B branes on the quintic As we have discussed, D-branes on the quintic can be described by vector bundles or sheaves on this space. Let us denote the charge carried by a single D2p-brane wrapped about a generator of H 2p as Q 2p = 1. Transporting a D-brane configuration about closed, nontrivial cycles of the moduli space of Kähler structures will induce an associated Sp(4, Z) monodromy on the B branes. We will discuss the monodromy more completely in the next section, but there is already one cycle in the moduli space which can be understood in the large volume limit: B → B + 1, where B is the NS 2-form. The action on the charge Q can be seen from Eqs. (2.9),(2.10) [37]. Mathematically this corresponds to the possibility to tensor the vector bundle V 2p with a U (1) bundle of c 1 = 1. This preserves stability and the dimension of the moduli space. Given a bundle V this operation and its inverse can be used to produce a related bundle with −r < c 1 ≤ 0: this is referred to as a "normalized" vector bundle. There is no classification of vector bundles and coherent sheaves on the quintic, but we can write down a few examples in order to orient ourselves when discussing specific boundary states at the Gepner point. BPS D2-branes wrap holomorphic 2-cycles of the Calabi-Yau, the same cycles as appear in worldsheet instanton corrections. Such cycles can have arbitrary genus and arbitrary degree. Degree one rational curves are generically rigid on the quintic [38]. Nontheless for special quintics, families may exist; for example, in the case of the Fermat quintic (2.15), there are 50 one-parameter families essentially identical to the family [39][40]: (z 1 , z 2 , z 3 , z 4 , z 5 ) = (u, −u, av, bv, cv) a 5 + b 5 + c 5 = 0; a, b, c ∈ C ,(2.16) where (u, v) are homogeneous coordinates in IP 1 . Once we perturb away from the Fermat point, these moduli are lifted and a finite number of rational curves remain [39]. This could be described in the world-volume theory by a superpotential of the general form W = φψ 2 where φ are complex structure moduli; φ = 0 is the Fermat point; ψ are curve moduli, and ψ = 0 a curve which exists for generic quintics. The infinitesimal description of deformations of such cycles is as sections of the normal bundle, which by the Calabi-Yau condition will be O(a)⊕O(b) with a+b = −2 for a rational curve. One might think that all one needs to find examples of families is to find examples with a ≥ 0 or b ≥ 0, but this is not true as deformations can be obstructed. The canonical example is given by resolving the singularity in C 4 xy = z 2 − t 2n . (2.17) For n = 1 this is the conifold singularity and the "small" resolution contains a rigid IP 1 , parameterized by x/(z − t) = (z + t)/y. It can be shown [41] that for n > 1 the resolution also contains a IP 1 , now with normal bundle O ⊕ O(−2), but the deformation is obstructed at n'th order, as could be described by the superpotential W = ψ n+1 . (2.18) Intuitively this can be seen by deforming (2.17) by a generic polynomial in t 2 , which splits the singularity into n conifold singularities, each admitting a rigid IP 1 . If we then tune the parameters to make these IP 1 's coincide, a superpotential describing the n vacua will degenerate to (2.18). Such singularities do appear in large families of quintic CY's [38]. 2 2 (Note added in v2): The idea that the moduli space of such a curve can always be described as the critical points W ′ = 0 of a single holomorphic function was apparently not known to mathematicians. We thank S. Katz for a discussion on this point. It turns out that the curves in (2.16) provide another example of obstructed deformations [39]. 3 The normal bundle of these curves is N = O(1) ⊕ O(−3); as dimH 0 (N ) = 2, there must be another obstructed deformation; call it ρ. The correct counting of curves upon deforming away from the Fermat point can be reproduced by a superpotential ρ 3 . The modulus ρ is also connected to the fact that pairs of the 50 families in (2.16) intersect (e.g. take (2.16) and the family (av, bv, u, −u, cv) with c = 0); it describes deformations into the second family. All of this structure can be summarized in the superpotential W (ρ, ψ) = ρ 3 ψ 3 + φF (ρ, ψ) + . . . ; where φF generalizes the φψ 2 term discussed above. Higher genus curves can generically come in families and examples can be found as complete intersections of hypersurfaces in IP 4 with the quintic. A particular example is the intersection of two hyperplanes with the quintic [40]: 5 k=1 a k z k = 5 k=1 b k z k = 0 , a k , b k ∈ C . (2.19) It is easy to see that there are six independent complex parameters after rescaling the equations. The curve is genus 6, and the area of the curve C is C J = 5, where J is the unit normalized Kähler form of IP 4 , i.e. IP 4 J ∧ J ∧ J ∧ J = 1 Quintic J ∧ J ∧ J = 5 (2.20) Thus this brane has Q 2 = 5. 4 There will be six additional complex moduli coming from Wilson lines of the U (1) gauge field around the 12 cycles of the curve. Similarly, four-branes can be obtained as the intersection with another hypersurface in IP 4 . For example, the intersection of the quintic with a single hyperplane k a k z k = 0 3 (Note added in v3): We would like to thank S. Katz for explaining this example, pointing out a mistake in our earlier draft, and suggesting the superpotential discussed here. 4 See ref. [42], chapters 1 and 2 for a nice description of complete intersections in projective spaces, and of techniques for performing the calculations we allude to here. produces a four-parameter family of four-cycles S. Their volume is S J = 5 and so Q 4 = 5. In addition c 2 (T S) = 11J 2 ; so that the coupling of C (1) to p 1 /48 in eqs. (2.9),(2.10) leads to an induced 0-brane charge of 55/24. The four-brane generically may support nontrivial gauge field flux over two-cycles, corresponding to D2-brane charge, or instanton solutions, corresponding to zero-brane charge. Some discussion of the moduli space of four-branes in a Calabi-Yau can be found in [43]. By (2.13), stability of the vector bundle on the four-brane requires Q 0 > 0. Finally we can look at the case of D6-branes wrapping the entire Calabi-Yau manifold. In fact we will find that all of the boundary states we examine at the Gepner point will have non-trivial six-brane charge. A single six-brane by itself will have no moduli. The U (1) gauge field on a single 6-brane can support flux with first Chern class c 1 = n corresponding to Q 4 = n. We can get the relevant bundles by restriction from U (1) bundles on IP 4 . The latter have no moduli, and we will not gain any upon restriction. We can also imagine binding D2-branes to the D6-brane, by analogy to 2 − 6 (or [44,45] in the physics literature, and were previously known as "monads" in the math literature). We consider a complex of holomorphic vector bundles 0 → A → a B → b C → 0 such that ker a = 0, im a is a subbundle of B, im b = C and define our new bundle as E = ker b/im a. For a hypersurface M in IP n , simple bundles to start with are direct sums of the line bundles O(n) restricted to M , as in 0 → ⊕O → ⊕ m i=1 O(q i ) → O( m i=1 q i ) → 0 This data allows computing the Chern classes: c n = ( m i=1 q i ) n − m i=1 q n i . The dimension of the moduli space can also be computed, but this is not as easy. A physical realization of this construction is to start with fields λ i parameterizing sections of B (e.g. the world-sheet fermions of a heterotic string theory), include a superpotential enforcing the constraints b a i λ i = 0, and gauge invariances identifying λ i ∼ λ i +a i . Although it is not the only place this construction appears (e.g. see [46]), the most relevant version for present purposes is in linear (0, 2) models [45]. These constructions have the advantage that they can be studied with conventional world-sheet techniques; a disadvantage is that one requires the anomaly cancellation conditions c 1 = 0 and c 2 (V ) = c 2 (T ) to get a sensible model, so only a subset of possible V can be obtained. The anomaly cancellation conditions also appear in D-brane constructions of the dual type I theories as the consistency condition that the total RR charge vanishes [47]. However in this context we need not consider branes which fill the noncompact dimensions but can instead consider lower dimensional branes, for which these consistency conditions are not required (a point emphasized in [5]). It seems likely that this additional freedom will lead to a simpler theory. Another construction of vector bundles on a CY is the Serre construction. Given a holomorphic curve (satisfying certain conditions), this produces a rank 2 vector bundle with a section having its zeroes on the curve. In [48] this is used to produce an example of a vector bundle with an obstructed deformation (on a different CICY). Finally, to conclude this section, there are a few explicit constructions of bundles on IP 4 in the literature using monads, such as the Horrocks-Mumford bundle (r = 2, c 1 = 5, c 2 = 10) and the bundle of Tango (r = 3, c 1 = 3, c 2 = 5, c 3 = 5) [49], which can be restricted to the hypersurface P = 0 to produce new examples. A branes on the quintic The simplest example of supersymmetric 3-cycles on the quintic are the real surfaces Im ω j z j = 0 with ω 5 j = 1; this was described in [3] for ω = 1. These cycles are determined by the five phases (ω 1 , ω 2 , ω 3 , ω 4 , ω 5 ) up to the diagonal Z 5 action ω i → ωω i (which is just a remnant of the equivalence of homogeneous coordinates under complex multiplication), so they come in a 625-dimensional irrep of the discrete symmetry S 5 × Z 4 5 . The equation (ω j x j ) 5 = 0, where ω i x i ∈ IR, always has a unique solution for x k in terms of the other real coordinates; thus the cycle is the real projective space IRP 3 . The first homotopy group is π 1 (IRP 3 ) = Z 2 ; by the discussion above (c.f. [31]) the wrapped 3-branes cannot have any continuous moduli, but they can support a discrete Z 2 -valued Wilson line. To compare these cycles with Gepner boundary states it will be useful to find their intersection matrix. Let us choose the coordinate system z 1 = 1 on IP 4 , so that ω 1 = 1. Regard the cycle (1, 1, 1, 1, 1) as an embedding of the coordinates x 2 ,x 3 and x 4 into the quintic with positive orientation. The other surfaces are obtained by Z 4 5 rotation from this one, 5 i=1 g k i i (1, 1, 1, 1, 1). Since the intersection matrix must respect the Z 4 5 symmetry, it can be written as a polynomial in the generators g i and is determined by the matrix elements (1, 1, 1, 1, 1)\|(1, ω 2 , ω 3 , ω 4 , ω 5 ) = (1, 1, 1, 1, 1)\|g k 2 2 g k 3 3 g k 4 4 g k 5 5 \|(1, 1, 1, 1, 1) (2.21) where g k i i : z → ω k i z. S 5 symmetry also constrains the problem in an obvious way. There are different possibilities for intersections with the surface (1, 1, 1, 1, 1) in this coordinate system. If ω 2 , ω 3 , ω 4 and ω 5 are all different from 1 there is no intersection in this coordinate patch. If only three of them are different from 1 there is exactly one intersection in this coordinate patch and the intersection has the signature sgn Im ω 2 Im ω 3 Im ω 4 assuming that ω 5 = 1. If the two surfaces intersect on a higher dimensional locus the intersection number has to be calculated by a small deformation of one of the two surfaces. This deformation has to be normal to both surfaces. Because of the special Lagrangian property of the undeformed surfaces this "normal bundle" of the intersection locus can be identified with its tangent space. The intersection number is then given by the number of zeros of a section of the tangent bundle of the intersection locus. For example, in the case that exactly two ω j 's are not 1 the intersection locus is a circle. A circle can have a nowhere vanishing section of its tangent bundle and the intersection number in this coordinate patch is 0. As another example, let precisely one ω j = 1. The intersection locus is then an IRP 2 . A section of its tangent bundle has one zero, as can be seen by modding out the 'hedgehog configuration' of an S 2 by Z 2 . The orientation of this intersection is given by the intersection in the remaining complex dimension, i.e. by Im ω j . In order to compute the full intersection we must look at all possible patches. This can be done by using the constraint 5 i=1 g i = 1 to rewrite (1, ω 2 , ω 3 , ω 4 , ω 5 ) as (ω −1 2 , 1, ω −1 2 ω 3 , ω −1 2 ω 4 , ω −1 2 ω 5 ) and so on. We then add all of the intersection numbers for all of these patches. Thus, although we find that (1, 1, 1, 1, 1)\|(1, ω, ω, ω, ω) = 0 in the z 1 = 1 coordinate patch, the total intersection number -the coefficient of 5 i=2 g i in the intersection matrix -is 1. Another example is the intersection of (1, 1, 1, 1, 1) with (1, ω, ω, ω, 1) which gives a circle in the patch z 2 = 1 and a point in the patch z 1 = 1. A simple general formula that matches all of these results is I IRP 3 = 5 i=1 (g i + g 2 i − g 3 i − g 4 i ). (2.22) Stringy geometry Type IIb string compactification on a general CY threefold M leads to an N = 2, d = 4 supergravity with b 2,1 + 1 vector fields (b 2,1 vector multiplets plus the graviphoton) and b 1,1 + 1 hypermultiplets (including the 4d dilaton); in IIa these identifications are reversed. The most basic physical observables which reflect the structure of M are those described by the special geometry of the vector multiplets. This geometry is determined by a prepotential F K of Kähler deformations in the IIa case, and by the prepotential F c for complex structure deformations in the IIb case. A fundamental result from the study of the worldsheet sigma model is that F c can be determined entirely from classical target space geometry; it receives no worldsheet quantum (α ′ ) corrections. Let us then discuss the complex structure moduli space. Choose a basis for the 3-cycles Σ i ∈ H 3 (M, Z Z) (where i = 0, . . . , b 2,1 , b 2,1 + 1, . . . , 2b 2,1 + 2), so that the intersection form η ij = Σ i · Σ j takes the canonical form η i,j = δ j,i+b 2,1 +1 for i = 0, . . . , b 2,1 (an a cycle with a b cycle). The b 2,1 + 1 vector fields come from reducing the RR potential The primary observables are the periods of the holomorphic three-form, Π i = Σ i Ω. In N = 2 language these are the vevs of the scalar fields in the corresponding vector multiplets. The a-cycle Π i 's can be used as projective coordinates on the moduli space; the b-cycle periods then satisfy the relations Π j = η ij ∂F /∂Π i . If we fix (for example) Π 0 = 1 to pass to inhomogeneous coordinates, the related vector field is the graviphoton. These periods determine the central charge of a three-brane wrapped about the cycle Σ = i Q i [Σ i ]: Z = Σ Ω = Q i Π i . Thus the mass of a BPS three-brane is [50]: m Q = c\|Z\| = c\|Q · Π\| (3.1) where c is independent of Q. If we use four-dimensional Einstein units for m, it is c = 1/g s ( Ω ∧Ω) 1/2 . In contrast to F c , F K receives world-sheet instanton corrections to the classical computation. The exact worldsheet result can be obtained by mirror symmetry: F K for IIa on M is equal to F c for IIb on the mirror W to M . Of course this requires a map between the periods of M and W . This analysis has been carried out for the quintic in [12] (see [51] for a summary) and we will quote the result in this case. The mirror W to the quintic threefold M can be obtained [52] as a Z 3 5 quotient of a special quintic 0 = 5 i=1 z 5 i − 5ψz 1 z 2 z 3 z 4 z 5 . The transformation ψ → αψ with α 5 = 1 can be undone by the coordinate transformation z 1 → α −1 z 1 and thus the complex moduli space of W 's can be parameterized by ψ 5 . This is an "algebraic" coordinate, which although not directly observable, does appear naturally in the world-sheet formulations [53,54]. The moduli space M has three singularities, about which the three-cycles in W will undergo monodromy. Each singularity has physical significance. First, ψ 5 → ∞ is the "large complex structure limit" mirror to the large volume limit. In this limit [51] (5ψ) −5 → e 2πi(B+iJ) , (3.2) where B is the NS B-field flux around the 2-cycle forming a basis of H 2 (M ), and J is the size of that 2-cycle. Next, ψ 5 → 1 is a conifold singularity; here a wrapped threebrane becomes massless [55]. This turns out to be mirror to the "pure" six-brane [56,57]. Each singularity in M gives a noncontractible loop, which is associated with a monodromy on the basis of 3-cycles in W (or even homology in M ) and thus on the periods. We let A be the monodromy induced by ψ → αψ around ψ = 0; clearly A 5 = 1. T will be the monodromy induced by going once around the conifold point, and B will be the monodromy induced by taking ψ → α −1 ψ around infinity. These satisfy the relation B = AT . One may make the physics associated with a given singularity manifest by choosing variables (the periods) for which the associated monodromy is simple. In our case the periods Π i satisfy a Picard-Fuchs differential equation of hypergeometric type. Since b 3 = 4 it is fourth order and quite tractable. There will be four independent solutions and as per the discussion above, we generally want to choose a basis making one of the monodromies simple. Two such bases are particularly natural. The first is the large volume basis which we will denote (Π 6 , Π 4 , Π 2 , Π 0 ) t . Up to an upper triangular transformation this is determined by the asymptotics as ψ 5 → ∞    Π 6 Π 4 Π 2 Π 0    →    − 5 6 (B + iJ) 3 − 5 2 (B + iJ) 2 B + iJ 1    . (3.3) The coefficients correspond to the classical volumes of the cycles. The signs were chosen so that the supersymmetric brane configurations have positive relative charges. We will derive the monodromy below. The other natural basis for us makes the monodromy A simple, and is appropriate for describing the Gepner point. If we choose a solution Π G 0 (ψ) analytic near ψ = 0, the set of solutions Π G i (ψ) = Π G 0 (α i ψ) (3.4) will provide a basis with the single linear relation 0 = 4 i=0 Π G i . It turns out that the 0-brane period Π 0 (the solutionω 0 of [12], equation (3.15)) is analytic near ψ = 0 and thus we can set Π G 0 = Π 0 and define the others using (3.4). We then (as in [12]) choose the period vector ( Π G 2 , Π G 1 , Π G 0 , Π G 4 ) t . In this basis, the three monodromy matrices are 5 A G =    −1 −1 −1 −1 1 0 0 0 0 1 0 0 0 0 1 0    T G =    1 4 −4 0 0 0 1 0 0 −1 2 0 0 4 −4 1    B G =    −1 −7 5 −1 1 4 −4 0 0 0 1 0 0 −1 2 0    (3.5) In [12], the relation between the large volume and Gepner bases proceeds through a third basis which we will call Π 3 , which is naturally described by a particular basis of 3-cycles in W . The intersection form in this basis has the canonical form η 13 = η 24 = −1, and the T monodromy is simple: Π 3 i → Π 3 i + δ i,2 Π 3 4 . Thus Π 3 4 is the vanishing cycle at the conifold and Π 3 2 is its dual. This turns out to be enough information to relate it to the Gepner basis uniquely up to a remaining SL(2, Z) acting on Π 3 1 and Π 3 3 , which we may fix arbitrarily. One then finds a transformation of Π 3 to a basis satisfying (3.3). This is an SL(2, Z) transformation of the type which was unfixed in the previous step; so the Π 3 basis has no significance intrinsic to our problem of relating Π G to the large-volume basis. Thus we will merely quote the final result for this change of basis, which is: Π = M Π G Q = Q G M −1 A = M A G M −1 . . . M = L    0 −1 1 0 − 3 5 − 1 5 21 5 8 5 1 5 2 5 − 2 5 − 1 5 0 0 1 0    (3.6) Here Q and Q G are the charge vectors in the large-radius and Gepner basis respectively. (In the notation of [12], M = KN m: with K a matrix taking the vector (Q 4 , −Q 6 , Q 2 , Q 0 ) of their conventions to our conventions; and N taken with a ′ = b ′ = c ′ = 0.) The matrix 5 There is a typo in table I in [12] as published in Nuclear Physics B. L is an as-yet undetermined Sp(4, Z) ambiguity in the Q 2 and Q 0 charges of the six-and four-branes: [12]). L =    1 0 −b −c 0 1 a b 0 0 1 0 0 0 0 1    with (a, b, c) integers (the (a ′ , b ′ , c ′ ) of Given the classical intersection form η in the large-radius limit, we can now determine the intersection form in the Gepner basis: η g = M −1 η(M −1 ) t =    0 −1 3 −3 1 0 −1 3 −3 1 0 −1 3 −3 1 0    ,(3.7) where η 14 = −η 41 = −η 23 = η 32 = 1 from [12]. 6 L does not enter since it is symplectic, and so preserves η. η g has determinant 25 and thus the Gepner basis is not canonically normalized; this point will not be important for us. We want to better understand the ambiguity L. We can start by comparing the monodromy B with our expectations from the large volume limit. One may define a basis of charges such that Γ RR k is the charge under the RR potential C (k+1) , with the switch in four-and six-brane charge as in (3.3). In this basis the effect of the shift B → B + 1 follows from Eq. (2.10): B L =    1 1 − 5 2 − 5 6 0 1 −5 − 5 2 0 0 1 1 0 0 0 1    . (3.8) The factors 1/2 and 1/6 in this expression come from expanding the exponential (they can also be seen in (3.3)) and indicate that in this basis the charges are not integers. we can see that the charges are modified in precisely this way. The modification due to b comes from theÂ term in (2.12)(as c 2 = 50 for the quintic). a induces a two-brane charge on the four-brane and might come from c 1 of its normal bundle. These effects were referred to in [19] as the "geometric Witten effect". The B monodromy in the Π basis (3.3) is B =    1 1 3 − a −5 − 2b 0 1 −5 −8 + a 0 0 1 1 0 0 0 1    . The most interesting ambiguity comes from c which induces zero-brane charge on the six-brane. In [12] this was attributed to the sigma model four-loop R 4 correction in the bulk Lagrangian. In the D-brane context, one possibility is that this comes from an as yet unknown term at this order in the D-brane world-volume Lagrangians. We should also keep in mind that the intersection form we are computing involves the bulk propagation of the RR fields between the branes, so another possibility is that it comes from a partner to the R 4 term in the bulk Lagrangian which affects the RR kinetic term in a curved background. In [12], the redefinition L was used to make the charge basis integral, but an overall Sp(4, Z) ambiguity was left over. It is in general more useful to have an integer charge basis so we will follow this procedure (this was already done implicitly as we took integer coefficients in the change of basis). We can resolve most of the Sp(4, Z) ambiguity by calling the state which becomes massless at the mirror of the conifold point a "pure" sixbrane with large volume charges (1 0 0 0), following [56,57]. This determines b = c = 0. A geometrical argument for this is that any fluxes on the six-brane would produce additional contributions to its energy. If there is a line from the large volume limit to the conifold point along which the six-brane becomes massless with no marginal stability issues, this argument will presumably be valid. Another argument is that we will find this state as a Gepner model boundary state with no moduli, as is appropriate for a pure six-brane. Finally, this choice simplifies the charge assignments for the other boundary states. We still have the ambiguity in a to fix. As it happens this does not enter into the results we discuss, so we have no principled way to do this. We will simply set it to zero. If the CFT has a chiral symmetry algebra one may simplify the problem by demanding that the boundary conditions are invariant under the symmetry. We can start with the Virasoro algebra which must be preserved (particularly in string theory where the symmetry is gauged). Let the boundary be at z =z in some local coordinates. Reparameterizations should leave the boundary fixed, so we must impose T =T . If the remaining symmetry algebra is generated by chiral currents W (r) with spin s r , then the boundary conditions are Boundary states in W (r) = ΩW (r) Ω † , (4.1) where Ω is an automorphism of the symmetry algebra. We are interested in describing BPS D-branes which preserve N = 1 spacetime SUSY. The closed-string sector will have at least N = (2, 2) worldsheet SUSY and the boundary conditions must preserve a diagonal N = 2 part [60,61]. Eq. (4.1) leads to two classes of boundary conditions [6]: the "A-type" boundary conditions T =T , J = −J, G + = ±Ḡ − ,(4.2) and the "B-type" boundary conditions A CFT on an annulus can also be studied in the closed-string channel where time flows from the one boundary to the other. The boundaries appear as initial and final conditions on the path integral and are described in the operator formalism by "coherent" boundary states [62,63]. The boundary conditions (4.1) can be rewritten in the closed-string channel as operator conditions on these boundary states; for example T =T , J =J, G + = ±Ḡ + .J n =J −n A type J n = −J −n B type. The relative sign change from (4.2),(4.3) can be understood as the result of a π/2 rotation on the components of the spin one current; it means that the A-type states are charged under (c, c) operators and the B-type under (c, a) operators. The solution to these conditions [64,65] are linear combinations of the "Ishibashi states": \|i Ω = N \|i, N ⊗ U Ω\|i, N . (4.4) Here \|i is a highest weight state of the extended chiral algebra; the sum is over all descendants of \|i ; and U is an anti-unitary map with U \|i, 0 = \|i, 0 ⋆ and UW (r) n U † = (−1) s rW (r) n . Modular invariance requires that calculations in either channel have the same result. This gives powerful restrictions on possible boundary states. In particular one requires that a transition amplitude between different boundary states can be written as a sensible open-string partition function, via a modular transformation. For rational CFTs with certain restrictions, Cardy [13] showed that the allowed linear combinations of Ishibashi states (4.4) are: \|I Ω = j B j I \|j Ω = j S j I S j 0 \|j Ω . (4.5) If χ j is a character of the extended chiral algebra, then S j i is the matrix representation of the modular transformation τ → −1/τ . In this notation capital and lower-case letters denote the same representation; we use capital letters to denote this particular linear combination of Ishibashi states. We may also associate a bra state to the representation I ∨ conjugate to I: Ω I ∨ \| = j Ω j\|B j I .Z I ∨ J = k N k IJ χ k . (4.7) The Gepner model in the bulk Gepner models [66,67] (see also [68] for a quick review) are exactly solvable CFTs which correspond to Calabi-Yau compactifications at small radius [53]. They are tensor products of r N = 2 minimal models together with an orbifold-like projection that couples the spin structures and allows only odd-integer U (1) charge. We will review their construction here. For simplicity we will discuss theories with d + r = even, where d is the number of complex, transverse, external dimensions in light cone gauge. Our building blocks are the N = 2 minimal models at level k; these are SCFTs with [69,70,71,72]. The superconformal primaries are labelled by 3 integers, (l, m, s) with central charge c = 3k k+2 < 30 ≤ l ≤ k; \|m − s\| ≤ l; s ∈ {−1, 0, 1}; l + m + s = 0 mod 2 . (4.8) The integers l and m are familiar from the SU (2) k WZW model and can be understood from the parafermionic construction of the minimal models [73,74]. s determines the spin structure: s = 0 in the NS sector; and s = ±1 are the two chiralities in the R sector. 7 The conformal weights and U (1) charges of these primary fields are: h l m,s = l(l + 2) − m 2 4(k + 2) + s 2 8 , q l m,s = m k + 2 − s 2 . (4.9) The N = 2 chiral primaries are clearly (l, ±l, 0) in the NS sector. The related Ramond sector states (l, ±l, ±1) can be reached by spectral flow. The minimal models can also be described by a Landau-Ginzburg model of a single superfield with superpotential X k+2 [75,76,77,78,79]. At the conformal point X l = (l, l, 0) and the Landau-Ginzburg fields provide a simple representation of the chiral ring. The N = 2 characters and their modular properties are described in [80,81,66,67]; we will follow the notation in [66,67]. One extends the s variable to take values in actually defined in the range l ∈ {0, · · · , k}, m ∈ Z 2k+4 and s ∈ Z 4 , where l +m+s = even. They obey the identification χ l m,s = χ k−l m+k+2,s+2 by which the fields can be brought into the range (4.8). Not every c = 9 tensor product of minimal models will give a consistent string compactification with 4d spacetime SUSY. We must find a reasonable GSO projection, and we must project onto states with odd integer U (1) charges [60]. We must then add "twisted" sectors in order to maintain modular invariance. The resulting spectrum is most easily represented by the partition function, for which we require some notation. We will tensor r minimal models at level k j with the CFT of flat spacetime. The latter also has a N = 2 worldsheet SUSY in our case, and we denote the characters by the indices i. The vector λ = (l 1 , · · · , l r ) gives the l j quantum numbers and the vector µ = (m 1 , · · · , m r ; s 1 , · · · , s r ), the charges and spin structures. Now define β j=1,...,r to be the charge vector with a two at the position of s j , and all other entries zero; and define β 0 to be the charge vector with all entries one. The modular invariant partition function in light cone gauge can be written as [66,67]: Z = (i,ī),λ,µ b 0 ,b j δ β (−1) b 0 χ i,λ,µ (q)χī ,λ,µ+b 0 β 0 + j b j β j (q) ,(4.10) Here χ i,λ,µ is the character for the r minimal models specificed by λ, µ and for the character of the flat transverse spacetime coordinates (labelled by i). In the sum, b 0 = 0, · · · , 2K − 1, b j = 0, 1 and K = lcm{2, k j + 2}. δ β is a Kronecker delta function enforcing both odd integral U (1) charge and the condition that all factors of the tensor product have the same spin structure. The kth minimal model has a Z k+2 × Z 2 symmetry [66,82] which acts as: gφ l m,s = e 2πi m k+2 φ l m,s , hφ l m,s = (−1) s φ l m,s . (4.11) With the above projection, all Z 2 symmetries have the same action on a given state and are identified. The remaining Z 2 symmetry acts only on R states by reversing their sign. The Z k+2 symmetry is correlated with the U (1) charge. In particular, the diagonal generator G = j g j is the identity for integral U (1) charges. The Gepner model is an orbifold theory; the orbifold group H is the group generated by G. The remaining discrete symmetry is ⊗ r i=1 Z k r +2 /H. For example, the (k = 3) 5 model is an orbifold by the diagonal Z 5 of (Z 5 ) ⊗5 . Boundary states in the Gepner model It is difficult to construct the most general boundary state for the Gepner model, because the Gepner model is not rational. Following [7], we will consider states which respect the N = 2 world-sheet algebras of each minimal model factor of the Gepner model separately, and can be found by Cardy's techniques. These might be called "rational boundary states." They are labeled according to Cardy's notation by α = (L j , M j , S j ) and an automorphism Ω of the chiral symmetry algebra. In our case there are two choices of Ω giving either A-or B-type boundary conditions; Ω must have the same action on every factor of the tensor product. Recknagel and Schomerus [7] proved the modular invariance of A-and B-type boundary states with internal part: \|α = 1 κ Ω α λ,µ δ β δ Ω B λ,µ α \|λ, µ Ω . (4.12) The coefficients are: B λ,µ α = r j=1 1 √ 2(k j + 2) sin(l j , L j ) k j sin(l j , 0) k j e iπ m j M j k j +2 e −iπ s j S j 2 , (4.13) a result of eq. (4.5) for the minimal models and the extra coefficient κ Ω α described in the appendix. Here (l, l ′ ) k = π (l + 1)(l ′ + 1) k + 2 . δ Ω denotes the constraint that the Ishibashi state \|λ, µ Ω must appear in the closed string partition function (4.10). For A-type boundary states this is no constraint as the Ishibashi states are already built on diagonal primary states and δ β already enforces that total U (1) charge is integral. However, the B-type Ishibashi states have opposite U (1) charge in the holomorphic and antiholomorphic sector, and these only appear as a consequence of the GSO projection; so the δ B constraint requires that all the m j are the same modulo k j + 2. Finally, an integer normalization constant C has to be included in κ Ω α to get the correct normalization for the open-string partition function. It is easy to see from eqs. (4.12),(4.13) that the action of the Z k j +2 (Z 2 ) symmetries is M j → M j + 2 (S j → S j + 2). As a result of the δ β constraint, the two physically inequivalent choices for S j are S = S j = 0, 2 mod 4. The S j = odd case seems to be inconsistent because their RR-charges do not fit into a charge lattice together with the S = even states; thus they will violate the charge quantization conditions 8 . In the end, due to the Z 2 symmetry, it is enough to consider only boundary states with S = 0. A boundary state can be written as g M 1 2 1 · · · g M r 2 r h S 2 \|L 1 · · · L r Ω := \|L 1 · · · L r ; M 1 · · · M r ; S Ω = g M 1 −L 1 2 1 · · · g M r −L r 2 r h S 2 \|L 1 · · · L r ; M ′ 1 = L 1 · · · M ′ r = L r ; S ′ = 0 Ω . For B-type boundary states, the δ β constraint in eq. (4.12) implies in addition that the physically inequivalent choices of M j can be described by the quantity M = j K ′ M j k j + 2 , where K ′ = lcm{k j + 2}. We will be interested in counting the number of moduli for a D-brane state; these will be the massless bosonic with an insertion of (−1) F [80,81]. With this in mind, a calculation similar to that in [7] leads to 9 Z A αα (q) = 1 C NS λ ′ ,µ ′ K−1 ν 0 =0 r j=1 N l ′ j L j ,L j δ (2k j +4) 2ν 0 +M j −M j +m ′ j χ λ ′ µ ′ (q) ,(4.14) and Z B αα (q) = 1 C NS λ ′ ,µ ′ δ (K ′ ) M −M 2 + K ′ 2k j +4 m ′ j r j=1 N l ′ j L j ,L j χ λ ′ µ ′ (q) . (4.15) (Here δ (n) x is one when x = 0 mod n and zero otherwise.) This shows that only a U (1) projection and the SU (2) k fusion rule coefficients constrain the open string spectrum of B-type boundary states; these states are much richer as a consequence. 8 The amplitude between a S = odd boundary state and aS = even boundary state also has interchanged roles of R-and NS-states in the open string sector. 9 N l L,L are the SU (2) k fusion rule coefficients [83]: they are one if \|L −L\| ≤ l ≤ min{L + L, 2k − L −L} and l + L +L = even, and zero otherwise; note that our indices thus differ from those in [83] by a factor of two. The condition that two D-brane boundary states \|α and \|α , with the same external part, preserve the same supersymmetries is [7]: Q(α −α) := − S −S 2 + r j=1 M j −M j k j + 2 = even . (4.16) To explore the charge lattice of the boundary states, and to find the geometric interpretation of given boundary states, we wish to calculate the intersection (2.11)(2.12) of our branes. The CFT quantity which computes this is I Ω = tr R (−1) F in the open string sector [11]. The best way to do this is to start in the closed string sector and to do a modular transformation to the open string sector. In the closed string sector this trace corresponds to the amplitude between the RR parts of the boundary states with a (−1) F L inserted. The calculation is done in the Appendix and the result for A-type boundary states is: I A = 1 C (−1) S−S 2 K−1 ν 0 =0 r j=1 N 2ν 0 +M j −M j L j ,L j . (4.17) For B-type boundary states, I B = 1 C (−1) S−S 2 m ′ j δ (K ′ ) M −M 2 + K ′ 2k j +4 (m ′ j +1) r j=1 N m ′ j −1 L j ,L j . (4.18) The intersection matrix depends only on the differences M −M as was required by the discrete symmetry. We also see that the Z 2 action S → S + 2 changes the orientation of a brane. In the next section we will rewrite these formulas in a more compact notation and use them to identify the charges of the boundary states. D-branes on K3 and the Mukai formula For compactifications with N = 4 worldsheet supersymmetry, the index in the Ramond sector is directly related to the number of marginal operators in the NS sector. We now use this to give a CFT proof of Mukai's formula [84,19] for the dimension of the moduli space of 1/2-BPS D-brane states. K3 compactifications are geometric throughout their moduli space [85]. The BPS D-brane states in these compactifications are described by coherent semistable sheaves E [19] which can be labelled by the Mukai vector [84,19]. In terms of the rank r and Chern classes c i of E, this is v(E) = r, c 1 , 1 2 c 2 1 − c 2 + r ∈ H 0 (M, Z Z) ⊕ H 2 (X, Z Z) ⊕ H 4 (M, Z Z) (4.19) There is a natural inner product on the space of Mukai vectors: (r, s, ℓ), (r ′ , s ′ , ℓ ′ ) = s · s ′ − rℓ ′ − ℓr ′ (4.20) where s · s ′ is defined by the natural intersection pairing of 2-cycles on M . In fact this is just (minus) the intersection form (2.12). Mukai's theorem [84] states that the complex dimension of the moduli space of an irreducible coherent sheaf E is: dimension = v(E), v(E) + 2. (4.21) We now argue that this follows from the relation tr a,a (−1) F = v(E a ), v(E a ) (4.22) and general properties of supersymmetry. First, only two d = 2, N = 4 representations have nonvanishing Witten indices [86,87]. We list them below together with the NS weights related by spectral flow: identity rep. : (h = 0, ℓ = 0) NS −→ (h = 1/4, ℓ = 1/2) R tr(−1) F = −2 "massless ′′ rep. : (h = 1/2, ℓ = 1/2) NS −→ (h = 1/4, ℓ = 0) R tr(−1) F = 1 , (4.23) where ℓ is the SU (2) R isospin. The identity representations lead to world-volume d = 6, N = 1 (or d = 4, N = 2) gauge multiplets, while the massless representations lead to world-volume half-hypermultiplets, so there will be one complex scalar in the open-string sector for each massless multiplet. Let there be N g identity and N m massless multiplets; then the Witten index is tr (−1) F = N m − 2N g . (4.24) Using (4.22) we find that (4.21) will be true if the world-volume theory has a (Higgs branch) moduli space of complex dimension N m −2N g +2. This moduli space is essentially determined by the d = 6, N = 1 world-volume supersymmetry: it is the hyperkähler quotient of the configuration space by the subgroup G of the gauge group which acts nontrivially on the hypermultiplets. The resulting space has complex dimension N m − 2dimG. Now, any brane configuration will have an overall U (1) acting trivially whose partners in the vector multiplet are the center of mass position of the brane; if more U (1)s act trivially we will have more center of mass moduli, so such a configuration must correspond to a reducible bundle. Therefore dimG = N g − 1 for an irreducible bundle and we have proven (4.21). Generalizations Mukai's theorem used the Hirzebruch-Riemann-Roch formula together with special properties of K3 surfaces; these properties allowed one to extract the dimension of the moduli space of a bundle directly from the holomorphic Euler characteristic. We have a similar statement for CY threefolds if we keep track of both chiralities separately. The selfintersection number of a brane on a threefold is of course zero, but we can get non-trivial statements if we consider the intersection of two different branes. For example, consider the index of the Dirac operator on the bundle E. Since the world-volume is Kähler this is ind / D = 3 i=0 (−1) i dimH i (M, E) = χ(E) which is the holomorphic Euler characteristic. By the Hirzebruch-Riemann-Roch formula, χ(E) = M ch(E)Td(T M ) . (4.25) Here ch(E) = r + c 1 (E) + 1 2 c 2 1 (E) − 2c 2 (E) + 1 6 c 3 1 (E) − 3c 1 (E)c 2 (E) + 3c 3 (E) + . . . , Discussion of the 3 5 model Let us apply these results to the example to model (k = 3) 5 , the Gepner point in the moduli space of the quintic. We will consider boundary states labelled by L j ∈ {0, 1}, 0 ≤ M j < (2k + 4) = 10, and S = 0. Let the Z 4 5 symmetry be generated by the operators g j taking M j → M j + 2, and satisfying g 1 · · · g 5 = 1. Note that g 1/2 j which takes M j → M j + 1 is well-defined for these states (using the identifications on LM S, it relates branes to antibranes). We will be particularly interested in computing the intersection forms (4.17) and (4.18), as we will be able to use them to extract the charges and open string spectrum for a given brane. The main advantage of considering these quantities over the charges themselves is that they are canonically normalized, as already noted in [1]. We can consider the intersection form as a matrix I acting on the space of boundary states; since it commutes with Z 4 5 it can be written as a function of the generators g i . The main content of formulae (4.17) and (4.18) is contained in the SU (2) fusion rule coefficients. In these equations the labels M j , M j can be thought of as indices of a matrix acting on the states. The particular fusion coefficients we will need are: 10 N M j − M j 00 → (1 − g 4 j ), N M j − M j 01 → g 1 2 j (1 − g 3 j ) = N 00 g 1 2 j (1 + g 4 j ); N M j − M j 11 → (1 + g j − g 3 j − g 4 j ) = N 01 g 1 2 j (1 + g 4 j ) . (5.1) These various fusion matrices are related by successive multiplication with g 1 2 j (1 + g 4 j ) , so we can express the RR charges of all our boundary states in terms of those for Q(\|00000 Ω ). By eq. (4.16) there are two cases of pairs of branes preserving a common susy. If the total ∆L is even (so integral powers of g appear), a pair with ∆M = ∆S = 0 (brane and brane) will preserve susy. If the total ∆L is odd (powers g 5k+5/2 appear), a pair with ∆M = 5 and ∆S = 2 (brane and anti-brane) will preserve susy. In the case that the two D-branes are both A-type or B-type, the massless open string spectrum can also be expressed in terms of the fusion coefficients. It is easy to see from The SUSY-preserving moduli of the D-branes are constructed from chiral vertex operators. The Witten index counts these operators albeit with a sign depending on their chirality. In our explicit CFT calculation we can remove this sign by hand, and thus the total number of chiral fields can be calculated using (4.17) and (4.18) with the fusion matrices replaced by their absolute values. 11 We can again write this "modified" matrix as a polynomial P Ω (g j ) in the shift matrices g j . For example, the matrix for boundary states \|11111 B is: P B (g) = (1 + g + g 3 + g 4 ) 5 . (5.2) If spacetime supersymmetry is preserved, the chiral fields have integer U (1) charges, and are related to antichiral fields by spectral flow. In particular charge-2 chiral fields in Z Ω αα , are related to charge-−1 antichiral fields in Z Ω αα ; the latter are the hermitian conjugate of 10 The coefficients for m > l are defined in the Appendix. 11 In other words, we define N m LL = +N −m−2 LL , rather than the opposite sign in the Appendix. charge-1 chiral fields in Z Ω αα . Thus k m k in the open-string channel will be a multiple of 5 for marginal, chiral vertex operators. Examination of the fusion coefficients in (4.17) and (4.18) reveals that the number of massless chiral superfields is given by counting terms in 1 2 (P Ω (g j ) − 2) with the total power of g being a multiple of 5 2 . Applying these statements to eq. (5.2) shows that the D-brane described by \|11111 B A boundary states The intersection matrix (4.17) for the A-type boundary states with L j = 0 is I A = (1 − g 4 1 )(1 − g 4 2 )(1 − g 4 3 )(1 − g 4 4 )(1 − g 1 g 2 g 3 g 4 ). (5.3) To determine the rank of the intersection matrix we can count the number of nonzero eigenvalues. The g j can be diagonalized as g j = diag(1, e 1 \|10000 A ) = Q(\|00000 A ) + Q(g 1 \|00000 A ), so these are even farther from an integral basis. The intersection matrix for the \|11111 A states, 5 i=1 (1 + g i − g 3 i − g 4 i ) coincides with the intersection matrix (2.22) for the three-cycles Im ω j z j = 0, and thus we identify these states with the IRP 3 's. This leads to a potential contradiction with the large volume limit in that the L = 1 states have one marginal operator, while the IRP 3 's do not. Although it might be that this is indeed a contradiction, from what we know at present an equally likely resolution is that the L = 1 marginal operator is not strictly marginal; in other words the world-volume theory has a superpotential for the corresponding field ψ, perhaps of the form W = ψ 3 + ψφ where φ is the Kähler modulus (ψ 5 in the notation of section 3). Such a superpotential has two ground states and would also fit the fact that the IRP 3 has a Z 2 Wilson line in the large volume limit. 12 B boundary states As we discussed in the previous section, the B-type boundary states at fixed L j are described by the single integer, M = M j and the g j for different j are identified. The intersection matrix (4.18) for L = 0 states can be written as: I B = (1 − g −1 ) 5 = 5g − 10g 2 + 10g 3 − 5g 4 . (5.4) We want to describe these boundary states in the Gepner basis. The Gepner intersection form (3.7) in the same notation is: I g = −g + 3g 2 − 3g 3 + g 4 . (5.5) A linear change of basis preserving the action of Z 5 can be written as a polynomial in the operator g as well and a transformation of the form I → mIm t will be I → Im(g)m(g −1 ). The relation I B = (1 − g)(1 − g −1 )I g provides this change of basis. The results of section 3 allow us to write these charges in the large volume basis. The Gepner charge vector Q G is related to the large volume charge vector Q as Q = Q G M −1 . Thus Q G = ( 0 1 −1 0 ) becomes Q = ( −1 0 0 0 ) which is a pure (anti)six-brane. The other charges can be found by acting with the operator A L . One can now compute the charges for the L = 0 branes by using the multiplicative relation in (5.1). For example, we have Q(g 5 2 h\|10000 B ) = −Q(g 2 \|00000 B ) − Q(g 3 \|00000 B ). Starting with M = 0 and successively applying this operation produces a subset of branes which preserve the same supersymmetry. This can be checked by computing the central charges using the periods at the Gepner point, which are simply the fifth roots of unity. Thus the central charge for the L'th brane in this series is Z(L) = (2 cos π 5 ) L Z(0). The charges in the Gepner basis charges can written in large volume basis viq eq. Comparison with geometrical results To what extent can we compare these results with the geometrical branes and bundles we discussed in section 2? The only clear match is the six-brane which indeed has no moduli as expected. Our states can plausibly be identified with vector bundles since they obey the stability condition c 2 > 0. We were not able to identify any of them with the explicit constructions we mentioned in section 2. This may just reflect our lack of knowledge of vector bundles on the quintic; thus we might regard our results as predictions of the existence of new vector bundles. We should note that the numbers of marginal operators we obtained are only upper bounds for the dimension of the moduli space as in general these theories will have potentials. The problematic objects are the \|11000 B branes as an object with these charges cannot be a classical line bundle. For reasons explained in section 2 we do not believe it is a quantum bound state either, since we have found it at string tree level. There is a piece of evidence that it is some sort of bound state of the six-brane with the two-brane (2.19): namely, they come in the same multiplet of the discrete symmetries. Like all B branes, the \|11000 B branes are invariant under Z 4 5 , while S 5 acts by permuting the L i labels. The two-brane construction (2.19) also picks out two of the five coordinates and thus comes in the same multiplet. This identification creates a puzzle opposite to the one we faced for the IRIP 3 's: the geometric object appears to have more moduli (12) than the boundary state. Such a mismatch could not be fixed by a superpotential. On the other hand, it could be that the (unknown) mechanism which binds the two-brane to the six-brane removes moduli, so this is not a clear disagreement. One candidate for such a bound state is the instanton in noncommutative U (1) gauge theory [88]. Again by analogy with flat space, (since noncommutative gauge theory has not been formulated on curved spaces, this is all we can say), at generic values of B we might expect the D6-brane gauge theory to be noncommutative [89,90,91]. The center-of-mass position of the instanton would then (presumably) give the moduli of a two-brane and provide at least some of the moduli we observe. A potential problem with this idea is that we can continue to B = 0 in the large volume limit, and there is no sign that this bound state is unstable there. One may ask why the D0-brane does not appear on our list. One possible explanation is that the path from the large volume limit to the Gepner point crosses a line of marginal stability, and the D0 does not exist at the Gepner point. To test this we found the periods for all the branes in (5.6) by numerically integrating the Picard-Fuchs equations along the negative real ψ axis. We found that the D0 is lighter than any brane from the list along the whole trajectory, so we have no evidence for instability. Our favored explanation is simply that all of the B branes by construction are invariant under the Z 4 5 discrete symmetry, while any location we might pick for the D0 would break some of this symmetry. Thus, even if the D0 exists at the Gepner point, it cannot be a rational boundary state, at least in this model. Superpotential and topological sigma models The calculations of the previous section describe the field content of the D-brane world-volumes, but not their dynamics. The primary question in this regard is to find the world-volume potential and true moduli spaces for the brane theories. In CFT language, the marginal boundary operators operators we found might not be strictly marginal. N = 1, d = 4 supersymmetry tells us that the world-volume potential will be a sum of F-terms and Fayet-Iliopoulos D-terms. The D-terms are simply determined by the gauge group and charges of the matter fields. In the case of a single brane or N identical branes we have checked in the models we are studying that the gauge group is U (N ) with all matter uncharged under the diagonal U (1), so there is no possibility for a D-term. More generally we must consider such terms, for example in the case of D0-branes near orbifold points. However, we may expect a non-vanishing superpotential, in general constrained only by holomorphy and the symmetries of the problem. These conditions are often stronger than they might appear, but in general the superpotential must be found by explicit computation. It should eventually be possible to do exact calculations at the Gepner point, as we will discuss in the next section. In this section we will try to make some general statements about the superpotential in these models by showing that they can be calculated as amplitudes in some topologically twisted version of the open string theory. In particular we will use this fact to describe the cubic term in the superpotential, and to discuss to what extent the superpotential couples to the background CY geometry. Known examples of brane superpotentials In order to motivate the search for superpotentials in these theories we will start with a few examples where we know they arise. The most obvious example is N D3-branes in flat space; one may write the N = 4 Lagrangian in N = 1 notation so that there are 3 adjoint complex scalar fields Z i=1,2,3 = Z i a t a with the superpotential Tr Z 1 [Z 2 , Z 3 ] (here t a are adjoint matrices for U (N )). Of course this vanishes for N = 1 but not for N > 1. A plausible generalization of this to weak curvature (still preserving N = 1 worldvolume SUSY) is a function W written as a single trace of the adjoint chiral superfields and with the property that δ δZ i a δ δZ j b δ δZ k c W = f abc Ω ijk (z) (6.1) for variations around the diagonal vevs Z i = z i 1. The assumption of this form of the superpotential is a fairly weak constraint; see [92] for analysis along these lines. In the case of a large Calabi-Yau threefold, we will find below that this assumption is correct, and that Ω is the holomorphic (3, 0) form of the threefold. A well-studied genuinely stringy example is that of D-branes near orbifold singularities, or near resolved orbifolds with string-scale curvature. In these examples a "single" brane (in the orbifold limit they are described by Chan-Paton factors in the regular representation of the orbifold group) can have a superpotential, which furthermore can have non-trivial dependence on the closed string moduli. A "single" brane is described via Chan-Paton factors transforming in the regular representation of the orbifold group [5]. The superpotential takes the general form W = Tr Z 1 [Z 2 , Z 3 ]\| proj + ζ i Tr Z i . (6.2) The spectrum of these models is obtained as a subset of the N = 4 SYM spectrum [5,93,94], and the notation "W \| proj " indicates that the N = 4 superpotential is simply restricted to In the present context we see that we should be wary of arguments that rely on the distinction between configurations involving "one" or "several" branes, or equivalently "one" or "several" distinct world-sheet boundary conditions, as they can be continuously connected. Other examples where this distinction is questionable are a small instanton leaving a Dp-brane as a Dp − 4-brane, or an intersection of 2-branes as described by [96]. Another point we will return to is that the closed string moduli ζ i which appear in the superpotential in this example are complex structure moduli. Of course orbifold resolution also depends on Kähler moduli, but these enter in the Fayet-Iliopoulos D-terms. Our final example is the superpotential on a wrapped two-brane. Recall that a supersymmetric theory arises when we wrap the 2-brane on a holomorphic cycle. The massless fields correspond to infinitesimal deformations of this cycle into a cycle close by in the D-brane moduli space. Witten [97] has argued that an M-theory two-brane (or 5-brane) wrapped around a two-cycle Σ has the superpotential W (Σ) = B Ω (6.3) when Σ is homologically trivial, where B is a three-manifold bounded by Σ. Indeed, this is a holomorphic functional of the embedding coordinates, which is stationary by holomorphic curves. When Σ is in a nontrivial homology class, the superpotential is defined up to an additive constant as: W (Σ) − W (Σ 0 ) = B Ω (6.4) where Σ 0 is an arbitrarily chosen referent holomorphic 2-cycle in the same homology class as Σ, and B has boundary Σ − Σ 0 . Here the additive constant depends both on Σ 0 and the homology class of B. For purely classical, geometric deformations these formulae should hold for D2-branes; there may also be terms arising from the gauge fields on the D2-brane worldvolume. Before discussing the computation in general, we note that in all of our examples, which are of B-type branes, the superpotential depends on closed string moduli only through the complex structure, not through the Kähler structure. Could it be that this is a general statement? We can see some potential problems with the statement by considering the other branes on the list. First of all, we need to describe not just the embedding but also the gauge bundle on the branes. To the extent that this is determined by a choice of a holomorphic vector bundle, this will fit into the same class of problems depending only on complex structure data. However, one might object that general four-and six-brane configurations involve a gauge bundle with c 2 = 0 and such a holomorphic bundle will correspond to a solution of Yang-Mills only if it is stable, a condition which depends on the Kähler class. This condition indeed should enter into the potential but as we discussed in section 2, it is more natural to expect that it appears as a D-term, which would not contradict the decoupling statement. Mirror symmetry for boundary states [6] and the statement we are considering would together imply that the superpotential for A-type branes depends on the Kähler structure of the background, but not the complex structure. But any formula for the potential on the brane analogous to (6.3) will necessarily involve both structures, since the special Lagrangian condition cannot be stated without bringing in Ω. This situation can only be compatible with our decoupling statement if the terms involving Ω are D-terms, an assertion not contradicted by any existing results. 13 One might object that this possibility would require charged matter under a gauge group which is not immediately apparent, but the small instanton and orbifold examples show that such gauge groups can be broken and become invisible in the large volume limit. This would lead to the further interesting possibility that, at special moduli points in the space of a "single" 3-brane, enhanced gauge symmetry could appear. The simplest way this could happen is for the brane to split in two at a self-intersection, leading to U (1) 2 gauge symmetry. We conclude that we have several examples in which decoupling (before taking stringy corrections into account) is clear, and no examples in which it is clearly false. Thus we will consider this decoupling statement further below. CFT computation of superpotential Given the above examples, we have good reason to believe that the Gepner model boundary states we have constructed correspond to D-branes with worldvolume superpotentials. We want to know how to calculate these in the models at hand. 13 Related questions are being considered by G. Tian. We are interested in BPS D-branes in N = 2 compactifications of type II string theory; these lead to N = 1 worldvolume theories. Thus the open-(closed-) string sectors will have N = 2 (N = (2, 2)) worldsheet supersymmetry [60,61]. To fix notation we write out the OPE algebra for the holomorphic piece: T (z)T (w) ∼ 1 2 c (z − w) 4 + 2T (w) (z − w) 2 + ∂T (w) (z − w) + · · · T (z)G ± (w) ∼ 3 2 G ± (w) (z − w) 2 + ∂G ± (w) (z − w) + · · · G ± (z)G ± (w) ∼ · · · G + (z)G − (w) ∼ c 6 (z − w) 3 + 1 2 J(w) (z − w) 2 + 1 2 T (w) + 1 2 ∂J(w) (z − w) + · · · J(z)G ± (w) ∼ ± G ± (w) (z − w) + · · · J(z)J(w) ∼ c 3 (z − w) 2 + · · · . (6.5) For compactifications with c = 9 J(z) can be constructed from the internal part of the spacetime SUSY current [60]. It can be written in terms of a single boson H, J(z) = i √ 3∂H (6.6) and operators with charge q under this U (1) R-symmetry can be written as: O q = e i(qH/ √ 3) O 0 . (6.7) The spacetime SUSY currents can be constructed from the macroscopic spin fields and the internal U (1) current algebra. In the (±1/2) picture, the currents can be written as: Q ± 1 2 ,α (z) = e ±φ/2 S α Σ ± (6.8) where Σ ± = e ±i √ 3H/2 (6.9) is the spectral flow operator of the N = 2 worldsheet algebra, mapping the NS sector to the R sector and vice-versa. φ is the bosonized superconformal ghost. On the 4d noncompact worldvolume, we can have massless chiral superfields Φ i IJ with scalar components φ i , fermionic components ψ i and auxiliary components F i . i will label the (complex) internal moduli of the D-brane configuration on the CY threefold M ; these moduli correspond to marginal boundary operators of the internal CFT. (IJ) label gauge indices, which are described by Chan-Paton factors on the worldsheet. These could be adjoint indices if there are coincident branes, or bifundamental indices if there are several types of (possibly intersecting) branes. More abstractly, the off-diagonal terms are boundary condition-changing operators [13] and the diagonal terms are boundary condition-preserving operators. The superpotential can be written via a holomorphic function W (Φ) and it contributes the following terms to the Lagrangian: d 4 x d 2 θ tr W (Φ) + h.c. = d 4 x ∂ i ∂ j ∂ ∂φ i IJ W (φ)F i IJ − ∂ ∂φ i IJ ∂ ∂φ j KL W (φ)ψ i IJ ψ j KL + h.c. (6.10) where we use the superfield conventions in [98]. We are interested in small fluctuations about a reference D-brane state, so we expand W (φ) in a Taylor series in φ. All of the terms of interest in Eq. (6.10) will be of the form tr w i 1 i 2 i 3 ...i n F i 1 φ i 2 . . . φ i n − 1 2 ψ i 1 ψ i 2 φ i 3 . . . φ i n . The coefficients w may also depend on the closed-string background. We will examine small fluctuations about some reference background so that we can sensibly expand w in a Taylor series in fluctuations of the closed-string background. The worldvolume fermions are represented in the open string theory by dimension 1 boundary Ramond vertex operators constructed from spin fields. In the (−1/2) picture they can be written as: V (−1/2) R,IJ = ζ α i,a e −φ/2 S α Σ i t a IJ ,(6.11) where S α is the spacetime part of the spin field and has dimension 1/4; Σ i is the internal part of the boundary vertex operator and has dimension 3/8; ζ carries polarization and gauge indices; and t a IJ are the Chan-Paton matrices. Since we are interested in computing a potential term we are interested in zero-momentum amplitudes, so we can omit spacetime momentum factors e ik·X . Note that the spacetime directions will have standard Dirichlet or Neumann conditions, so that S α is easily related to its bulk counterpart, for example by the doubling trick. 14 Similarly, the worldvolume scalars are represented by NS vertex operators. In the (−1) picture they can be written as (6.12) where e −φ has dimension 1/2, and ψ is a dimension 1/2 boundary operator arising from the internal sector. To find the 0-picture operator, we find the superpartner of ψ under the (gauged) worldsheet N = 1 SUSY, V (−1) NS,IJ = ζ i,a e −φ ψ i t a IJT F (z)ψ i (w) = 1 2 (z − w) O i (w) , (6.13) where T F = 1 √ 2 (G + +G − )G − (z)ψ i (w) = 1 2 (z − w) O i . The internal part of the vertex operators for the auxiliary fields, in the (0) picture, can be constructed from the internal part of the (−1)-picture scalar vertex operators, via the spectral flow operator mapping the NS sector back to itself [100]: V aux,IJ = lim z→w (z − w)e −i √ 3H V (−1) NS,IJ (6.14) Essentially this is because one gets the auxiliary component by acting on the scalar fields twice with the spacetime SUSY current. For deformations preserving spacetime SUSY, the internal part of the vertex operators should be constructed from the chiral ring of the N = 2 algebra. 15 This is because the marginal (0)-picture operators will have vanishing R-charge; thus they may be added to the worldsheet Lagrangian while maintaining N = 2 worldsheet supersymmetry. The (−1)-picture operators will have charge q = 1; the (−1/2)-picture Ramond operators will have charge q = −1/2; and the auxiliary fields will have charge q = −2. We will also want to include bulk vertex operators in order to measure the effects of the closed-string background. For SUSY-preserving deformations, we will be interested in marginal operators in the (c, c) or (a, c) ring. The vertex operators for massless fields can be constructed from dimension ( 1 2 , 1 2 ) operators ψ(z,z) in the internal sector, with charge (1, 1) if they are in the (c, c) ring, (−1, 1) if they are in the (a, c) ring, and so on. In the (−1) picture these operators are: for (a, c) operators. V (−1) bulk = e −φ(z)−φ(z) ψ(z,z) . Now we wish to calculate the tree-level contribution to the nth order term of the superpotential; we will expand out the coefficients to kth order in the closed string fields. We are particularly interested in the case n > 2, as we are studying putative moduli. We will examine the contribution of this term to the fermion bilinear part of the action. On the disc, we must fix 3 real moduli due to the SL(2, IR) symmetry. In addition, we must absorb the superconformal ghost number violation on the disc. These requirements can be met by using the (−1/2) picture for the two fermionic vertex operators and placing them at opposite sides of the disc, or equivalently at z = 0 and z = ∞ in the upper half plane. Furthermore we will take one of the NS vertex operators to be in the (−1) picture and fix its location between the two R vertex operators, i.e. at z = 1 in the upper half-plane. The remaining open-and closed-string vetex operators are in the (0) and (0, 0) pictures, and are integrated respectively over the boundary and bulk of the worldsheet. The resulting amplitude on the disc is: So far we have not specified which of the four chiral rings the closed-string vertex operators live in. We will discuss below how operators in the different rings may or may not couple to these amplitudes for a given boundary condition. A = lim δ i ,ǫ i →0 e −φ/2 S α Σ i 1 I 1 I 2 (x 3 ) x 3 −δ 2 x 2 +ǫ 2 dy 1 O (i 2 ,(0)) I 2 I 3 (y 1 ) y 2 −δ 3 x 2 +ǫ 3 dy 2 O (i 3 ,(0)) I 3 I 4 (y 2 ) . . . e −φ ψ i k (−1) I k I k+1 (x 2 ) . . . e −φ/2 S β Σ i ℓ I ℓ I ℓ+1 (x 1 ) . . . y n−4 −δ n−3 x 3 +ǫ n−3 dy n−3 O (i n ,(0)) (y n−3 )× D dz 1 . . . dz k O 1(0,0) (z 1 ,z 1 ) . . . O k(0,0) (z k ,z k ) (6.18) where ǫ 2 > ǫ 3 > . . . ǫ k−1 , ǫ k+2 > . . . > ǫ ℓ−1 , ǫ ℓ+1 > . . . > ǫ n We will also be interested in superpotential terms which are linear in the superfields and contain couplings to closed-string moduli, such as the last term in Eq. (6.2). This term will not show up as a fermion bilinear; only the auxiliary field will in fact couple. e −φ−φ ψ (−1,−1) (z,z)V F,II (x) (6.19) All of these prescriptions allow us to perform tree-level calculations for fixed boundary conditions in the Gepner models. In the rest of this section we will discuss these amplitudes in general compactifications as correlators in the topologically twisted version of the internal CFTs. In this language we can revisit our question regarding Kähler decoupling from the superpotential of B-type branes. Topological CFT with boundaries We begin by reviewing and generalizing the discussions in refs. [101,102,103] of topological CFTs with boundary. Topological CFTs can be constructed from N = 2 CFTs via "twisting" the stress tensor with the U (1) current [104]; that is, we define a new stress tensor: T top (z) = T (z) ± 1 2 ∂J(z) . (6.20) Note the sign ambiguity; as we will discuss, the overall sign is physically unimportant but the relative sign between left-and right-moving sectors is physically meaningful. This one finds again that the topological twisting is equivalent to an amplitude with half-units of spectral flow applied to the initial and final states. More generally, to derive the twisted theory on a surface with boundary via the above coupling to the background field, one must take the boundary contribution in Jω + c.c. into account. If we rewrite the disc as a long strip with two caps, the background charge will be concentrated on the boundary of these caps and the result will again be spectral flow applied to the in-and out-states [103]. Care should be taken with any boundary operator insertions on or near this part of the boundary, as they may have contact terms with the charge insertion. The relative sign of the twisting of the holomorphic and anti-holomorphic parts of the stress tensor comes from the relative sign of the background charge. The "A-model" arises from an axial twisting while "B-model" arises from a vector twisting [105]. In the presence of D-branes, the twisting must be compatible with the boundary conditions. We can see easily that One may similarly construct topological operators on the boundary from the chiral primary boundary operators. From these one may also construct "one-form" operators from the commutators or anticommutators of the operator with modes of the spin-2 operators G: 1 √ 2 G − − 1 2 +Ḡ − − 1 2 , O (0) (6.24) In the cases that we can construct the boundary condition and boundary operators via the doubling trick, these can be written as closed-string one-form operators via holomorphic contour integrals, as above. Again, the integral of these operators along the boundary are BRST-invariant, up to potential contact terms with other boundary operators. One may similarly construct BRST-invariant operators in the A-model with A-type boundary conditions. If the CFTs correspond to geometric Calabi-Yau sigma-models, then we can see following refs. [105,6] The amplitude (6.18) factorizes into three pieces. The first is the superghost piece; the second is the two-point function of the spin fields polarized in the spacetime directions. These give essentially universal answers which we can expect from 4d Lorentz invariance. Fig. 1: Computing the disc contribution to the D-brane superpotential. A). CFT contribution. R vertex operators on "caps" can be written as R ground states. B). Representation of R ground state as path integral on half-disc. Spectral flow maps this to NS ground state created by insertion of NS vertex operator. (x ) 3 i(-1/2) Σ R φ j(-1) (x ) 2 NS φ m(0) dx (x ) x 1 2 x + ε − δ k(-1/2) Σ (x ) 1 R k(-1/2) Σ (x ) 1 R B) A) = 3 e -i H/2 k(-1) (x ) 1 φ NS = \| k, R > = \| k, NS > e -i H/2 3 3 The internal CFT amplitude is the interesting part. It is an expectation value of n chiral or antichiral NS boundary operators and k bulk NS-NS operators in one of the four closedstring chiral rings, with two additional half-unit spectral flow operators each mapping NS states to R states. The fixed boundary operators become 0-form observables and the integrated boundary operators, 1-form observables. If the closed-string operators are (c, c) for the B-type twisting, corresponding to complex structure deformations, they become (almost)-invariant topological observables. If they are (a, c) operators, corresponding to Kähler deformations, they are exact with respect to the left-moving BRST current and one might hope that they decouple. We will address this issue below. The result is (up to the caveats above) a correlator of topological operators in the topologically twisted theory. The fixed operators become 0-form observables and the integrated operators become 1-form and 2-form observables. We may bring the techniques of topological field theory to bear on this calculation, and will do so below. Similarly, the computation of Eq. (6.19) is topological (with the same caveats). Here the auxiliary field is related by a full unit of spectral flow to the (0)-form observable of the associated scalar field. The superghost part of the amplitude merely takes care of the relevant zero modes. The internal CFT part is once again an amplitude in the topologically twisted theory of a (0)-form boundary observable and a (0, 0)-form bulk observable. Computations in the geometric sigma model The topological symmetry of these correlators, and the localization properties of the topological path integrals [106,105], make the above calculations relatively straightforward. To see this we will compute the cubic part of the superpotential for a D0-brane in a weakly curved background, and discuss the linear part of the superpotential for generic wrapped B-branes. To begin with we need to construct the relevant topological observables. The closedstring case has been described in ref. [105] and the open-string case for fully Neumann boundary conditions has been described in [101]. We need to generalize these results to arbitrary B-model boundary conditions. In the untwisted sigma-model, the propagating worldsheet fields are 3 complex scalars φ i , and three complex fermions ψ i ± . ψ i has U (1) charge 1 and ψī has charge −1. Thus in the B-twisted theory ψ i have dimension zero and become worldsheet scalars, while ψī have dimension one and become worldsheet one-forms. The BRST currents G + ,Ḡ + give rise to global symmetries parameterized by constant Grassman scalars ǫ,ǭ (since these are scalars such constants are well-defined on any worldsheet). In order to write these transformations in the simplest form, it is convenient to rewrite the fermions as: ξ i = ψ i + + ψ i − θ = g i ψ i − − ψ i + . (6.26) If we integrate out the auxiliary fields on the worldsheet, the BRST transformations become These do not necessarily close off-shell once we have integrated out the auxiliary fields. In the presence of a boundary we must set ǫ =ǭ. 16 Then the transformations simplify: in particular, the important transformations are: δ B φ i = i 2 ǫ ξ i + g i θ δ B φī = 0 δ B ξ i = iΓ i jk ǫψ i + ψ k − +ǭψ j − ψ k + δ B θ = ig i Γ i jk ǫψ j + ψ k − −ǭψ j − ψ k + + g i,k g il (iǫψ i + + iǭψ i − )θl δ B ψ − = −ǭ∂φ δ B ψ + = −ǫ∂φ .δ B φ i = iǫξ i δ B ξ i = 0 δ B θ = 0 . (6.28) Recall that in the Dirichlet directions of the untwisted model, ψ i is fixed and ψ i + = −ψ i − ; in the Neumann directions (when we have turned off the NS 2-form and boundary gauge field), ψ + = ψ − . Thus along Dirichlet directions, ξ vanishes at the boundary; while along Neumann directions, θ vanishes at the boundary. Of course, for curved boundaries, whether a given polarization is "Dirichlet" or "Neumann" will depend on φ; this can be defined by a projection matrix IP i j (φ(C)) : T M −→ T C. The (0)-form topological observables in the bulk were constructed in [105]. They are of the form: Λ i 1 ...i p 1 ... q ξ i 1 . . . ξ i p θ 1 θ q ; (6.29) Λ is a ∂-closed (0, p) form with values in ∧ q T (0,1) M . The boundary observables will live on the appropriate holomorphic submanifold C ⊂ M and will take values in the Chan-Paton algebraĝ. In the fully Neumann case, the boundary observables areĝ-valued (p, 0)-forms, while in the fully Dirichlet case they will be antiholomorphic functions of the position of the boundary, with values in ∧ q T (0,1) M ⊗ĝ. In the 2-brane and 4-brane cases, the observables will be forms on C valued in the normal to see that the BRST operator acts as the holomorphic differential on C. Thus topological observables are ∂-closed, and trivial BRST-exact observables are ∂-exact. The construction of these topological amplitudes makes it clear that we have some anomalous U (1) charge. This essentially counts fermion number in these models. In the closed-string B-model, nonvanishing correlators have fermion number 3 for both θ and ξ corresponding to zero modes for each of these fields. On the disc, the boundary conditions will kill the ξ zero modes polarized along the Dirichlet directions and the θ zero modes polarized along the Neumann directions. Thus for a holomorphic p-cycle, nonvanishing correlators will have ξ fermion number p and θ fermion number 3 − p. The other fact that makes these amplitudes straightforward to calculate is that the topological path integral localizes onto constant maps, restricted to the submanifold defined by the boundary conditions. The correlation functions are then integrals of the appropriate forms over the moduli space of constant maps; in these cases they will be integrals of the pullback of forms on M onto the submanifold C. Let us start by computing the cubic term in the superpotential for a brane sitting at a point in the CY. The topological boundary observables corresponding to the world-volume chiral fields will be O = φī a θīt a (6.30) where t a is a matrix in the adjoint of the Chan-Paton group. The correlation function is: This superpotential is similar to the term coming from fully Neumann boundary conditions. The topological string theory in this latter case has been argued to be a holomorphic six-dimensional version of Chern-Simons theory [101]; the vertex operators which describe chiral fields in the spacetime Lagrangian describe antiholomorphic gauge fieldsĀ on the Calabi-Yau. The low-energy Lagrangian has been argued to be φī 1 a φī 2 b φī 3 c ( θī 1 θī 2 θī 3 + θī 2 θī 1 θī 3 ) tr (t a t b t c ) (6.S = M Ω ∧ Ā ∧ ∂Ā + 2 3Ā ∧Ā ∧Ā (6.33) This second term is the superpotential. Finally, we can look for linear terms in the superpotential coming from a coupling to closed strings. Again, the boundary topological operator will be linear in the worldsheet scalar fermions; for the correlator to have the right fermion number, the closed-string operator must be quadratic. We can analyze these couplings for 0-, 2-, 4-and 6-cycles separately and we find that some of these amplitudes vanish automatically. Indeed, these results should not surprise us. If we change the complex structure of the manifold, the holomorphic 2-and 4-cycles, and the homomorphic bundles on them, will change. We should generically find that the reference cycle is no longer a stable, supersymmetric configuration. On the other hands, the 0-and 6-cycles are holomorphic regardless of the complex structure, so we expect them to be supersymmetric so long as the closed-string background maintains N = 2 spacetime SUSY. V N = A i,a ξ i t a ,(6. Decoupling of non-topological moduli One of the more powerful statements one can make in topological closed string theory is that the Kähler (complex structure) deformations decouple from topological amplitudes in the B (A) model. This is related to the fact that the spacetime of the theory has N = 2 supersymmetry and the vector multiplets and hypermultiplets decouple (away from singular points in the moduli space). One can show in the topologically twisted B (A) models that insertions of integrated (c, a) ((a, c)) operators, which one would get by taking derivatives of the amplitudes with respect to the Kähler (complex structure) moduli, lead to vanishing amplitudes. In the open string case the status of this decoupling is less clear. To start with, the spacetime SUSY is only N = 1 and the Lagrangian is far less constrained. For example, recall the analogous E 8 × E 8 heterotic string, compactified on a CY 3-fold. There the charged multiplets arising from the Kähler and complex structure decouple from each other at finite order in α ′ [44] but couple due to worldsheet instantons [107]. Furthermore, in the generic (0, 2) model it may not make sense to identify deformations with Kähler or complex structure deformations. Actually, a total decoupling is not to be expected, even from geometric considerations. For example, if we consider the theory of D9-branes wrapped on the CY, the four-dimensional action will come with the prefactor V 6 /g s where V 6 is the volume of the CY. On the other hand this is a B-brane so the topological amplitudes naturally depend on complex moduli. Thus the strongest conjecture we could make is that the superpotential (for a B brane) takes the form W = m(φ K )W (φ c , ψ) (6.35) where m(φ K ) is proportional to the brane tension (3.1). . This is consistent with (6.35); furthermore this amplitude also corresponds to a chiral ( d 2 θ) term in the effective action, the one-loop correction to the gauge coupling. A known example which illustrates this is In a system involving several different branes, (6.35) does not even predict a universal multiplicative dependence of the total superpotential on the Kähler moduli. At the very least it will be the sum of several terms of this form but with different m(φ K ). There will also be terms involving strings stretched between different branes. Geometrically these would be expected to come with m(φ K ) for the surface of intersection; it would be quite interesting to make a more general proposal along these lines. In any case, it is preferable to have string world-sheet arguments for decoupling. Thus we proceed to consider the the cubic and quartic terms in the superpotential as computed on the disk, to see if derivatives with respect to Kähler moduli φ K are consistent with (6.35). We will work in the untwisted theory in order to ensure that we are not avoiding any subtleties; our statements can be carried over to the twisted theory. Recall that the cubic term in the superpotential for B-type boundary conditions is calculated via the 3-point disc amplitude. The part arising from the internal CFT is: Σ i 1 IJ O i 2 JK Σ i 3 KI (6.36) plus a sum over any orderings consistent with the Chan-Paton factors. We may fix the ordering by picking suitable Chan-Paton factors, which we will do here. Now the first derivative of this amplitude with respect to some Kähler deformation will lead to the above amplitude with the insertion of an integrated (0, 0)-picture vertex operator constructed from the (a, c) (or (c, a)) ring: 37) and the complete amplitude is show in fig. 2. Conformal invariance allows us to deform the integral of G + out to the boundary. This amounts to using the superconformal Ward identities. Let us concentrate on the case where the doubling trick allows us to describe amplitudes on the upper-half plane via amplitudes on the full complex plane. The contour may be deformed to a sum of integrals of G + around each boundary operator 18 , plus a contour integral around the image of (see fig. 2) V = D d 2 w z→w z→w dzdzG +Ḡ− (z)ψ i (w,w) ,(6.V (0,1) (a,c) = d 2 w dzḠ − (z)ψ i (w,w) (6.38) In the end, the contour integrals around the boundary operators will vanish as the operators are chiral. The contour integral around the image of the bulk operator in the lower halfplane may be expressed in the upper half-plane as an integral z→wḠ + (z) 18 Taking some care with the branch cuts created by the spin fields. Fig. 2: Perturbation of cubic part of superpotential by Kähler deformation. The superconformal Ward identities allow us to pull dzG + (z) to the contour C. This can be deformed to a contour integral around each of the boundary operators and an integral over C of dzḠ + (z) which can be deformed back to the insertion ofḠ − ψ. D G + ψ G - (a,c) I J K φ (x ) j 2 Σ κ 1 (x ) C Σ ι 3 (x ) IJ JK KI The result is an integral over the insertion of∂ w ψ which can be integrated by parts to an integral of ψ over C. around V (0,1) (a,c) . Using the superconformal algebra, this term becomes: (6.39) In this integral over the boundary, we must take some care when the contour passes near one of the boundary operator insertions. The result is the correlation function d 2 w∂ w ψ i = ∂D ψ i .Σ α 1 IJ O α 2 JK Σ α 3 KI C ψ i (6.40) where the contour C is shown in fig. 2. We get two potential contact terms from this correlator. One arises from the operator product of the (a, c) operator with the boundary [14]: Either this is removed by the GSO projection, or the perturbation by ψ i has changed the acceptable boundary conditions -for example by changing the stability condition on vector bundles -so that the original boundary is no longer a stable D-brane. Such a divergence will have to be removed by perturbing the boundary conditions. The second case is a more genuine contact term; it is a dimension-one operator which is integrated over the boundary. In this case the 3-point correlator satisfies: lim y→0 ψ i (x + iy, x − iy) ∼ C I ψ i O α 1 y 1−δ O α O α (x)(6.∂ i − C bound iα ∂ α Σ α 1 IJ O α 2 JK Σ α 3 KI = 0 . (6.42) If O α is a topological operator then the perturbation by ψ i has the fairly simple effect of moving the vev slightly along a flat direction. It should not affect the form of the superpotential, in keeping with our claim. A very similar formula to (6.42) appears in [6]. In that case they find that by defining Higher order amplitudes are more subtle since they have a moduli space of insertions of vertex operators. When applying the Ward identities, we will find integrals of total derivatives with respect to these moduli, leading to contributions from the boundaries of moduli space. 19 We can already illustrate this phenomenon by looking at the contribution to the quartic term of the superpotential from a single derivative with respect to the Kähler moduli. The resulting amplitude is: lim ǫ,δ→0 Σ i,(−1/2) (x 3 ) x 3 −ǫ x 2 +δ dx 1 √ 2 G − +Ḡ − , O j,(−1) (x) O k,(−1) (x 2 )Σ ℓ,(−1/2) (x 1 ) d 2 w z→w z→w dzdzG + (z)Ḡ − (z)ψ (a,c) (6.43) plus a potential sum over orderings. As before we may fix the orderings of the boundary operators via a judicious choice of Chan-Paton factors. Once again, we pull the contour integral of G + off of the bulk operator and apply the superconformal Ward identities. In addition to the terms which we have already argued to vanish, we get a term coming from the contour integral of G + around the integrated NS operator. Again, let us look at the case where we may describe this amplitude via the doubling trick. Then the above anticommutator can be replaced with a contour integral of G − around a point on the real line, and the contour integral of G + around this leads simply to a derivative of ψ j . The result is the difference of contact terms: lim ǫ→0 Σ i (x 3 ) O j (x 3 − ǫ) − O j (x 2 + ǫ) O k (x 2 )Σ ℓ (x 1 ) d 2 w z→w dzḠ − (z)ψ (a,c) (w,w) (6.44) In the twisted theory we might hope that factorization and associativity means that this difference would vanish. This would be true if there was no insertion of ψ (a,c) . With such an insertion, it is not clear that the amplitude will factorize onto topological intermediate states, so we cannot complete this argument at present. The upshot of all of this is that there is a simple world-sheet mechanism which could lead to decoupling. It is very analogous to the known decoupling of bulk Kähler and complex structure deformations: the decoupling operator is a descendant with respect to an operator which annihilates the boundary chiral fields (say for Kähler and B-type, the operator G + ). The situation is better than that for (0, 2) heterotic string models as there are still two N = 2 algebras involved; they are identified only on the boundary. Such world-sheet arguments are valid up to the possible contributions of contact terms and to make them precise, one needs to show that the contact terms either vanish or have simple interpretations (e.g. as connection coefficients on the moduli space). We have interpreted some but not all of these terms and thus can say that we have found further evidence for decoupling but by no means a proof. Correlation functions in minimal models and Gepner models We now turn to the problem of computing correlation functions in the Gepner model. To begin with, let us recall a few properties of Gepner model boundary correlators, which are comparable to properties of bulk correlators. As with correlators in the bulk theory, in the boundary theory there are restrictions due to ghost number conservation. This can easily be seen using the doubling trick and has been discussed in the previous section. In addition, the boundary fields transform under particular representations of the chiral algebra, similar to chiral halves of bulk fields. The chiral algebra is the tensor product of the chiral algebras of the minimal models involved. The fields obey the same fusion rules. Correlators forbidden by the fusion rules therefore vanish. In this section we point out a number of differences with bulk theory computations and interpret their consequences. Ordering effects Correlation functions involving boundary operators require a specification of operator ordering along the boundary (which we will place on the real line): ψ 1 (x 1 )ψ 2 (x 2 ) . . . ψ n (x n ) x 1 > x 2 > . . . > x n This ordering corresponds directly to the ordering of the matrix fields in the world-volume Lagrangian: for example terms tr ψ 1 ψ 2 ψ 3 and tr ψ 1 ψ 3 ψ 2 come from these two orderings of the three-point function. In some particularly simple models (for example, free field theory), correlation functions of boundary operators can be analytically continued to the bulk. In this case it is possible to determine the effect of arbitrary permutations of the fields. This was formalized by Recknagel and Schomerus [111] in a discussion of non-supersymmetric conformal field theories. Two boundary operators ψ 1,2 were called mutually local if ψ 1 (x 1 )ψ 2 (x 2 ) = ψ 2 (x 2 )ψ 1 (x 1 ) (7.1) inside correlators. Here, the left hand side implies x 1 > x 2 and the right hand side x 1 , x 2 . Recknagel and Schomerus then argued that self-local marginal boundary operators are truly marginal. The argument is basically that an o.p.e. ψ(x 1 )ψ(x 2 ) → 1 x 1 − x 2 ψ(x 1 ) + . . . of the form which would spoil marginality is incompatible with (7.1). Free fermion correlators can be continued into the bulk as well, and in section 6 we saw that the superpotentials governing these operators were completely antisymmetric. In particular they vanish in the theory of a single brane. It seems quite plausible that this result applies to all operators which are strictly marginal in the large volume limit; however since we do not know whether an operator we find at the Gepner point is marginal in the large volume limit until we compute the superpotential (and many interesting operators are never strictly marginal), such considerations appear somewhat circular. In general, one does not expect either that boundary correlation functions have a continuation into the bulk or that the boundary operators have such simple exchange relations. By general principles (which we review in the next subsection) boundary correlation functions in minimal models and Gepner models are particular combinations of several chiral conformal blocks, each of which has different exchange relations, chosen to be single-valued on the boundary. To make any statement about ordering effects, we must consider this analysis. Sewing constraints In this section we will briefly discuss sewing constraints on boundary fields. Correlation functions in two-dimensional CFT with boundaries have been studied for rational conformal field theories in [112,14]. In the bulk, the n-point functions on the sphere are As discussed in section 4, for RCFTs the possible boundary conditions preserving all the symmetries are labeled by the primary fields and can be implemented by boundary states carrying these labels, and we have written analogous states for Gepner models. The field content of the theory can be read off from the partition function Z αβ ; thus the propagating fields also carry the labels α, β. In the case that α = β, the field φ αβ is a boundary condition-changing operator. If α = β, it preserves the boundary condition. Let us concentrate on the correlation functions for boundary fields. The boundary OPEs are: φ αβ i (x)φ βγ j (y) = k c αβγ ijk φ αγ k (y)(x − y) h k −h i −h j + . . . y < x . (7.2) The structure constants c αβγ ijk , together with the vacuum amplitude, determine the threepoint functions: φ αβ i (x i )φ βγ j (x j )φ αγ k (x k ) = c αβγ ijk † c αγα k † k1 1 α (x i − x j ) h k −h i −h j (x j − x k ) −2h k . (7.3) We can also evaluate the correlator in the other channel: φ αβ i (x i )φ βγ j (x j )φ αγ k (x k ) = c αβα ii † 1 c βγα jki † 1 α (x j − x k ) h i −h j −h k (x i − x j ) −2h i (7.4) The dependence on the coordinates is dictated by conformal symmetry. As mentioned above, conformal symmetry does not relate three-point functions with different orderings. Comparison of (7.3),(7.4) leads to a consistency condition on the structure constants. In addition to these conditions on the OPE coefficients, we must demand the crossing symmetry of the four-point functions. Nonvanishing correlation functions for boundary fields are of the form In [113] an explicit solution was given for the Virasoro minimal models: In particular, they are completely symmetric in the case that there is only one boundary condition involved. The symmetry of the structure constant is of course a direct consequence of the symmetries of the F -matrices, which are specific for minimal models. (In general, there will be a phase involved [114].) φ αβ 1 φ βγ 2 φ γδ 3 φ δα 4 ,(7.c αβγ ijk = F kβ Another simple example is the U (1) boson. The primary fields are given by tachyon vertex operators e ikX . The vertex operator e ikX connects the boundary conditions \|n to \|n + k . Therefore, the condition (7.5) is fulfilled, whenever momentum conservation holds. The sewing constraints determining the OPEs have not yet been solved for N = 2 minimal models or for Gepner models. We will return to this in future work. Boundary selection rules Given two boundary states, \|α , \|β , the partition function will contain the characters of a particular set of marginal operators, whose insertion changes the boundary conditions from α to β. In general, this set of marginal operators will be a subset of all possible weight one representations, and are determined by the fusion rules [13]. As a consequence, certain correlators remain uncorrected, because the required marginal operator does not propagate with the particular boundary conditions. As a simple application, consider a boundary condition-changing marginal operator φ αβ . All non-vanishing correlators φ γδ 1 φ δǫ 2 φ ǫγ 3 containing only boundary changing operators cannot be corrected by insertions of φ αβ . On the other hand, one can generate a non-zero correlator which vanishes at lower order. Three-point functions in the Gepner models A superpotential for massless fields is computed from n > 2-point functions as we have discussed. Let us briefly discuss the conditions under which a three point function can be non-vanishing. To compute a φφF term we start by picking three vertex operators in the NS sector and we apply spectral flow by one unit to one of them. This is done by splitting the operators in a charged and an uncharged part as in (6.7) and applying the spectral flow e −iH √ 3 . This gives the following correlator: O αβ 0 1 e −iH 2 √ 3 O βγ 0 2 e i H √ 3 O γδ 0 3 e i H √ 3 (7.7) Including the ghost contributions, the result is the product of the OPE coeffient c αβγ 123 for the uncharged operators, with the vacuum expectation amplitude for the boundary condition α. Thus, a cubic term in the superpotential is directly proportional to a structure constant c αβγ 123 . The A 1 model The power of boundary selection rules can be nicely illustrated with the example of the and we have to check whether it is compatible with the boundary conditions. We can determine, which boundary conditions allow for the field φ 1 by using the fusion rules. The field φ αβ 1 exists whenever N 1 αβ does not vanish. This is the case for αβ = 11, −11 and 1 − 1. As a consequence, our candidate is suppressed by boundary selection rules. The A 2 model In the A 2 model, the boundary selection rules do not forbid all correlators. We will give an example of an allowed correlator, where permutation of the operators requires different boundary conditions. The A 2 minimal model can be seen as one real boson χ and one real fermion λ. The central charge is 3 2 . Apart from the identity there are two more chiral primary fields, φ 1 = σe iχ 2 √ 2 , φ 2 = e iχ √ 2 . There are two corresponding anti-chiral fields of opposite charges. φ −1 = µe − iχ 2 √ 2 , φ −2 = e −iχ √ 2 . There is also an uncharged field λ, which is the ordinary fermion, or in minimal model language the field l = 2, m = 0. The spectrum for various boundary conditions can now be determined by fusing the fields labeling the boundary conditions. 1 1 1,1 , 1 1,1 , 1 −1,−1 , 1 2,2 , 1 −2,−2 1 λ,λ φ 1 φ 1,2 1 φ −2,1 1 φ 1,−2 1 φ 2,−1 1 φ −1,λ 1 φ 1 λ, 1 φ 2 φ 1,−1 2 φ −1,1 2 φ 2,λ 2 φ λ,−2 2 λ λ 1,1 λ −2,2 λ 2,−2 (7.8) The boundary conditions for the anti-chiral fields follow from this table: For any chiral field φ αβ q we have an antichiral field φ βα −q . A non-trivial four-point function to compute is φ 2 φ 2 φ 1 φ 1 , where one operator is a one-form and integrated over the boundary. There exists another ordering φ 2 φ 1 φ 2 φ 1 The first ordering requires the following boundary conditions: φ 2,λ 2 φ λ,−2 2 φ −2,1 1 φ 1,2 1 The other ordering requires the boundary conditions: φ 1,−1 2 φ −1,λ 1 φ λ,−2 2 φ −2,1 1 . Evaluation of the two four-point functions leads to structure constants with different boundary conditions (which will in general be not equal). In the final result, the expectation value of the identity is taken with two different boundary conditions. Therefore, the two results are not expected to agree. 7.7. The models (k = 2) 2 and (k = 2) 4 , (k = 1) 3 and (k = 1) 9 There are two Gepner models containing only the (k = 2) model: the model (k = 2) 2 , which corresponds to a 2-torus; and (k = 2) 4 , which corresponds to a K3 surface. We will consider in this section the case of a single boundary condition (i.e.a single brane). In this case it is known that the superpotential vanishes, which we can check using the Gepner model description. The marginal operator in the (k = 2) 2 model is the operator φ (1) 2 φ(2) 2 . This operator corresponds to the complex fermion ψ in sigma-model language. We know that this operator is an anticommuting variable. Therefore, all superpotential terms involving this operator vanish in the absence of Chan-Paton factors, due to the sum over operator ordering. Similarly, as a consequence of the fusion rules, in the (k = 2) 4 case all marginal operators are of the form φ (i) 2 φ (j) 2 . We know from the torus that these operators anticommute. Therefore, all superpotential terms vanish after summing over all permutations. This verifies the result discussed in section 4.4. for the case N g = 1. Similar statements can be made for the models consisting only of (k = 1) models, like the torus (k = 1) 3 and the orbifolded six-torus (k = 1) 9 . The fermion in the twotorus is given by the field φ (1) (1,1,0) φ (2) (1,1,0) φ(3) (1,1,0) . Simlilarly, we can form three complex fermions for the six-torus example. The marginal operators propagate for the L = 1 boundary conditions in these models. Again, we know that the superpotential vanishes by antisymmetry. The quintic Let us now turn to the applications of the boundary selection rules to the (k = 3) 5 Gepner model. We are particularly interested in correlators of marginal operators. For the B-boundary states discussed in section 5, we found that if we impose the same boundary To compute superpotential terms, we start with three (or more) chiral primary fields in the NS sector. One way to relate this to a physical amplitude is to apply a unit of spectral flow, so that from the space time point of view we are computing the F φφ term in the worldvolume Lagrangian. For 4-and higher-point functions, charge conservation requires us to apply the operator G to the additional operators (which changes the label s of the Gepner model fields from 0 to 2). In our example, unit spectral flow is implemented by the operator φ 5 (3,−3,0) . The four marginal operators for the boundary conditions L = {1, 0, 0, 0, 0} are given by ψ i = φ (1) (2,2,0) φ (i) (3,3,0) , where the upper index labels the minimal model, i = 2, 3, 4, 5. Spectral flow relates these operators to φ (1) (1,−1,0) φ (i) (0,0,0) φ 3 (3,−3,0) . There is no non-vanishing three-point function for these operators. However, there are some higher-order terms which are allowed. The four point function ψ 2 ψ 3 ψ 4 ∂Σ Gψ 5 is not suppressed by the selection rules. We can ask about possible corrections to this correlator. The fusion rules tell us that all higher order terms have to be of the form ∂Σ Gψ j 5 , since the fifth powers of φ (2,2,2) and φ (3,3,2) contain the identity in their fusion. Note that there are a lot more correction terms for the corresponding bulk 4-point function. Let us move on to the next most complicated example, the example with 11 marginal operators. The marginal operators are of the form φ (1) (2,2,0) φ (i) (3,3,0) , φ (2) (2,2,0) φ (i) (3,3,0) , φ(1) (1,1,0) φ To conclude this section, let us briefly commment on the A-type boundary conditions. Here, the L = 1 states have one marginal operator, which is the operator φ (1,1,0) . . . φ (1,1,0) . The three-point function between three operators of this type is allowed by the selection rules. Also, higher-order correlators containing G φ (1) (1,1,0) . . . φ (5) (1,1,0) 5 are allowed. Taking the 5th power is required by the fusion rules: φ 5 (1,1,0) contains the identity. For the A-type boundary conditions we have also argued for a coupling of this operator to a bulk field. The closed string observables in the A-model are the (a, c) fields. On the quintic, this is the Kähler deformation ( −1,0) ). Taking this operator to the boundary, we get boundary fields contained in the OPE of (1,1,0) . This certainly makes a bulk-boundary coupling of the desired form possible. 5 i=1 φ (1,1,0) , 5 i=1 φ (1,5 i=1 φ (1,1,0) × 5 i=1 φ Consequences of the selection rules Perhaps the simplest conclusion we can draw from these selection rules is that the B branes with L ≥ 1 have non-trivial moduli spaces. Consider the example of \|10000 B : the superpotential must take the form W = ψ 2 ψ 3 ψ 4 ψ 5 f (ψ 5 2 , ψ 5 3 , ψ 5 4 , ψ 5 5 ). No matter what f is, the subspace ψ 2 = ψ 3 = 0 (or any two ψ's zero) solves W ′ = 0. On the other hand, we found that the branes \|11111 A , which we identified with the large volume IRP 3 s, admitted a superpotential W = ψ 3 f (ψ 5 ) + φψg(ψ 5 ), which would resolve the potential contradiction with the lack of moduli in the large volume limit. A nontrivial f and g would break the ψ → −ψ R symmetry of the leading order superpotential, which has no reason to exist in the large volume limit. However we cannot test the prediction for the number of minima of W at this point. Conclusions and further directions In this work we began a systematic study of D-branes in the stringy regime of the quintic Calabi-Yau. Our main result was the determination of the charges (in the usual large volume conventions) of the explicit Gepner model boundary states constructed by Recknagel and Schomerus. Our tools were the intersection form, and the monodromy and continuation formulas for the CY periods. The techniques clearly generalize to any Calabi-Yau given this data. The primary question we hope to address is whether the spectra and low energy worldvolume theories of branes in the stringy regime are the same (up to renormalizations of couplings) as in the large volume limit or not. We will refer to this as the "geometric This might be considered a relatively mild failure as it is associated with a singularity of the Riemannian geometry or gauge bundle. If all failures of the geometric hypothesis were associated with singularities, conversely it would be true under the mild condition that the geometry stayed non-singular. As we mentioned in the introduction, we can imagine much more drastic failures -a priori, the spectrum of branes at the Gepner point might have satisfied none of the relations we expected from geometry and gauge theory. The results we have presented here are not (yet) inconsistent with the geometric hypothesis. Most of the branes we find could certainly correspond to the appropriate geometric constructions -holomorphic vector bundles and special Lagrangian submanifolds. For example, all of the branes we found satisfied the (mathematical) stability condition on vector bundles. The lack of any classification of these makes it difficult for us to assert that branes which we did not identify actually do not have geometric constructions. The elliptically fibered case may be more promising in this regard. The most problematic case was a brane which would correspond to a rank 1 bundle with c 1 = 0 but c 2 = 0. Although such things do not exist in conventional gauge theory, they are known to exist in modified gauge theories (such as noncommutative gauge theory), so one can imagine that this object has a description in the large volume limit. We also presented a general argument that B-type branes should be described by geometric considerations -namely, that their world-volume potentials are determined by quantities in B-twisted topological models, which are equal to their classical values in the B-twisted topological sigma models, up to hopefully minor effects of Kähler deformations. Besides a formal world-sheet argument we showed that many known cases fit with this idea. By contrast, the superpotentials in A brane theories can depend directly on Kähler moduli and a priori it would seem much more likely that the geometric hypothesis fails. Finally, we made some first steps towards explicit computation of the superpotentials on these branes. These superpotentials appear to be quite non-trivial and it appears that such computations are doable with existing techniques; we will return to this in future work. An exciting possibility is that topological open string theory can be developed to the point where exact superpotentials can be obtained, perhaps with some analog of the special geometry determining the bulk prepotentials. One important direction to develop is to find more direct ways to get the geometric interpretation of these states. The results here suggest that this will be simpler in the A picture -the simplest picture is that each component minimal model has a specific boundary condition for its LG superfield. If we had the D0-brane boundary state, we could apply a probe construction to get the geometrical picture for the B boundary states (indeed we could derive the corresponding (0, 2) models); perhaps larger classes of boundary states containing the D0 can be found. A study of curves of marginal stability is in progress, to decide whether the large volume and Gepner D-branes should be expected to match up, and whether new phenomena appear near the conifold point. Let us close with a brief discussion the physical relevance of our primary question. To the extent that branes in the stringy regime are qualitatively different than geometric branes, all of the work on compactification using branes will have to be reconsidered. On the other hand, to the extent that they are qualitatively the same, these techniques will provide new ways of deriving geometric results, such as the existence and moduli space dimension for vector bundles. Questions of existence of branes are also directly relevant for non-perturbative constructions of M theory. For example, Matrix theory constructions to date use D0-branes as their starting point. Compactifications on some manifold M are believed to be described by D0-branes in a certain scaling limit of type IIA string theory on M [115]. The authors of ref. [116] argued that for Calabi-Yau compactifications this limit was the mirror of the conifold point. If it were to turn out that the D0-brane did not exist in the stringy regime, this construction would have to be reconsidered. In any case we believe there is much to be discovered in this direction. We Appendix A. An explicit calculation for A-type boundary states This appendix shows the explicit calculation of the intersection number of two A-type boundary states. The Witten index tr R (−1) F in the open string sector is obtained from the transition amplitude between the (internal) RR parts of the boundary states with (−1) F L inserted 20 . The first part of this calculation is very close to that in [7]. For the A-type boundary states the δ A constraint is trivial, as we have discussed. where S (λ,µ),(λ ′ ,µ ′ ) is the modular transformation matrix and ev means l j +m j +s j = even. The β-constrained sum together with the charge projection operator can be rewritten as Putting all these equations together and collecting terms we get: tr R (−1) F q H = 1 κ A α κ Ã α 1 K ev λ ′ ,µ ′ ν 0 e iπ d 2 (2ν 0 +1) r j=1 1 2(k j + 2) 2 × × R l j ,m j ,s j sin(l j , L j ) sin(l j ,L j ) sin(l j , l ′ j ) sin(l j , 0) × ×e iπ m j k j +2 (2ν 0 +1+M j − M j +m ′ j ) e iπ s j 2 (−2ν 0 −1−S j + S j −s ′ j ) χ λ ′ µ ′ (q). (A.3) The sums over l j , m j , s j can be evaluated as follows: tr R (−1) F q H = 1 κ A α κ Ã α 1 K 1 2 r (−1) S− S 2 R λ ′ ,µ ′ ν 0 (−1) d 2 (2ν 0 +1) r j=1 N l ′ j L j ,L j δ (2k j +4) 2ν 0 +1+M j − M j +m ′ j × ×(−1) ν 0 + 1+s ′ j 2 χ λ ′ µ ′ (q) . (A.5) This fits with the normalization constant being κ A α = C K2 r , where C is an extra integer constant depending on the specific model. It can be understood from the β constraints. Imposing the same spin structure on all subtheories reduces the number of states by a factor of 1 2 r , the U (1) constraint gives another factor of 1 K . To simplify this result we have to use the fact that the R ground states are given by φ l l+1,1 which are identified with φ k−l −k+l−1,−1 ; only these states will contribute to the Witten index. We continue the upper index of fusion rule coefficients N l L,L with a period of 2k + 4; we identify N −l−2 L,L = −N l L,L ; and we set N −1 L,L = N k+1 L,L = 0. This continuation is natural from the point of view of the Verlinde formula. Neglecting an overall factor of (−1) d 2 we find that: tr R (−1) F q H = 1 C (−1) S− S 2 ν 0 (−1) (d+r)ν 0 r j=1 2k j +3 m j =0 N −m ′ j −1 L j ,L j δ (2k j +4) 2ν 0 +1+M j − M j +m ′ j = = 1 C (−1) S− S 2 ν 0 (−1) (d+r)ν 0 r j=1 N 2ν 0 +M j − M j L j ,L j . (A.6) discussion of D-branes on large volume CY We are interested in BPS states in type II string theory described by collections of Dbranes at points on or wrapping some cycle in a Calabi-Yau manifold M . A configuration for N coincident D-branes with worldvolume Σ wrapped on such a cycle is specified by an embedding X : Σ → M and a U (N ) gauge field A on Σ, with field strength F = dA+[A, A]. The U (1) part of U (N ) appears in combination with the B-field, F = F − X * B, where X * B is the pullback of the NS B-field onto the worldvolume. 0 − 4) configurations in flat space. For Q 6 = 1 and Q 4 = 0 this appears singular; U (1) gauge fields do not support smooth instanton solutions. The brane counterpart to this is that the 2 − 6 strings cannot be given vevs which bind the branes and give mass to the relative U (1)s. This might lead us to predict that such states, if they exist at all, exist only as quantum-mechanical bound states. Such a state should be easily identifiable because it appears at the junction of Coulomb and Higgs branches of the moduli space; a small perturbation should put it on the Coulomb branch and produce two U (1) gauge fields in the macroscopic direction. In the classical considerations of this paper, it should not show up at all. For Q 6 > 1, we require information about vector bundles on the Calabi-Yau. A wellknown example with Q 6 = 3 is deformations of the tangent bundle. This has vanishing c 1 and c 2 (E) = 10J giving us Q 2 = 50. The dimension of the moduli space is 224. This example can be generalized as follows. (Such generalizations are due to for example C ( 4 ) 4on the a cycles, while the b cycles produce their d = 4 electromagnetic duals. Thus a three-brane wrapped about the cycle Σ = i Q i Σ i has (electric,magnetic) charge vector Q i . Note that H 3 (X) forms a nontrivial vector bundle over the moduli space M c of complex structures; a given basis in H 3 (X, Z Z) will have monodromy in Sp(b 3 , Z Z) as it is transported around singularities in M c . Finally, at ψ 5 5= 0 the model obtains an additional Z 5 global symmetry; this is an orbifold singularity of moduli space. The Gepner model (3) 5 lives at this point in Kähler moduli space of M [53]. agree if a = 11/2 and b = −25/12, i.e. if we make a non-integral redefinition of the charge lattice. The explanation of this is that the intersection form in the conventions leading to (3.8) is actually not canonical, because it includes the other terms in (2.10). If we act on the basis (3.3) with the matrix L(a = 11/2, b = −25/12, c), correspond to the open-string channel where the boundary propagates in worldsheet time. For Calabi-Yau compactification at large volume, A-type boundary conditions correspond to D-branes wrapped around middle-dimensional supersymmetric cycles; and B-type boundary conditions to D-branes wrapped around even-dimensional supersymmetric cycles[6]. states are in one-to-one correspondence with open-string boundary conditions which we will label the same way. Cardy argued that the open-string partition function was determined by the fusion rule coefficients. Let worldsheet time and space be labeled by τ and σ respectively; and let the boundary run from σ = 0 to σ = π, and the boundary conditions be I ∨ and J, respectively. Then the number of times that the representation k appears in the open-string spectrum is precisely the fusion rule coefficient N k IJ ; in other words, the open-string partition function will be Z 4 . 4The NS characters are labelled by s = 0, 2 and the different values of s denote opposite Z 2 fermion number. The contribution from the NS primary is in χ l,m,0 . Similarly, in R sector s = ±1 denotes contributions from opposite fermion number: the s = 1(s = 3) character includes the contribution from the s = 1(s = −1) Ramond-sector primary. These characters are (i.e. NS) open-string states. To find their contribution to the open-string partition function, it is enough to examine the NS-NS part of a transition amplitude in the internal dimensions. The reason is that the (open-string) NS characters arising from the modular transformations of the RR part of the transition amplitude come Thus on a threefold, Td(M ) =Â(T M ), and combining eqs. (4.25) and (2.12), we find: ind / D = D6, D(E) = tr D6,D(E) (−1) F , (4.26) where D(E) is the D-brane representation or generalized Mukai vector for E. On the other hand, the Ramond ground states which contribute to the open string index are exactly the fermion zero modes which contribute to the index of / D. In the type I case where E is a gauge bundle with vevs entirely in an SU (3) subgroup and with the gauge connection equal to the spin connection, c 1 (E) = 0; this gives a brane picture case. If we are interested not in the bulk gauge theory on 9-branes in type I but in a gauge theory on a brane B intersecting another brane A, the generalization is that the number of generations (with respect to the B gauge group) associated with the brane A is the intersection form A, B . For B-type branes this follows from eq. (2.12) and the Hirzebruch-Riemann-Roch theorem for the bundle E(A) * ⊗ E(B); for A-type branes eachintersection contributes a chiral multiplet with chirality given by the sign of the intersection[17]. ( 4 . 414) and(4.15) that if the two boundary states are the same, there is exactly one vacuum and one spectral flow operator in the open string channel; if they are not the same, neither state propagates. This means that the unbroken worldvolume gauge group is (the centerof-mass) U (1), and the brane can be viewed as a single object (a priori, it still might be a bound state). has 101 marginal operators. This particular case can also be worked out by checking that the fusion rules lead to all possible L values, so for every operator in the (c, c) ring of the model there is a corresponding chiral open string operator. 5 ). Zero eigenvalues appear if a g j = 1 or if g 1 g 2 g 3 g 4 = 1. The combinatorics leads to 204 nonzero eigenvalues, which is the number of independent 3-cycles on the quintic. Thus, the L j = 0 states provide a basis for the charge lattice. So far as we can tell they do not provide an integral basis of the charge lattice. Furthermore, the charges of the other A-type Gepner boundary states can be obtained from these by successive multiplication by g 12 ( 12Note added in v2): Actually, the two choices of Wilson line are topologically distinct bundles so they would not be continuously connected in the large volume limit. This would suggest that the potential should have a unique minimum. On the other hand, it can be shown that any simply connected six-dimensional manifold X with H * (X) torsion-free (such as the quintic CY) has K(X) ∼ = H * (X), and thus the K theory class distinguishing the two bundles becomes trivial when lifted to the CY. (We thank D. Freed and J. Morgan for explaining this to us.) Thus there is no candidate for a space-time topological charge which could distinguish the two D-branes, and it is not ruled out that transitions between the two choices of bundle are possible in the full string theory. (3. 6 ) 6. Tabulating these results and the numbers of marginal operators, we have (for the Z 5 representatives related to the six-brane) patternQ L+1 = Q L + Q L−1 follows from the identity (−g 2 − g 3 ) 2 = 1 − g 2 − g 3 .It is also easy to compute the number of marginal operators between pairs of distinct boundary states. For example, \|00000 B and \|(1 . . .) L (0 . . .) have (for 1 ≤ L ≤ 5) 4, 3, 3, 4 and 1 (respectively) marginal operators. Each corresponds to a chiral superfield of charge (1, −1) and its charge conjugate (since the mutual intersection numbers are zero, none of these pairs has chiral spectra). The number of operators between two branes of higher L of course depends on which L i are non-zero. this subset. The ζ i are closed string moduli. This intrinsically stringy background illustrates the important lesson that by varying both closed and open string moduli, it is possible to bring down new massless open string states invisible in the weakly-curved geometric limit described above. For example, the C 2 /Z 2 model of a single brane has U (1) gauge symmetry generically; but when both closedand open-string moduli are tuned to the orbifold point, the gauge symmetry is enhanced to U (1) 2 . Furthermore a new branch of moduli space meets this point, where the single brane breaks up into branes wrapping the shrunken cycles of the orbifold[5,95]. This new branch is a transition from Higgs to Coulomb branch and as usual in supersymmetric gauge theory it is almost impossible to predict such transitions starting from the Higgs branch. is the gauged N = 1 part of the N = 2 superconformal currents. Here O i has dimension 1. The vertex operator is simply O i ; if it is exactly marginal its integral over the boundary is a valid conformal deformation of the worldsheet action. Note that if ψ i is a chiral primary, G + (z)ψ i (w) = (non − singular terms) as z → w, and may write picture operators can be constructed via the N = 2 supercurrents which cancel the U (1) charge, i.e. w,w) = dz dzG + (z)G − (z)ψ(w,w)(6.17) − 3 3; this prescription of the limits of integration ammounts to a point-splitting regularization on the boundary. In addition we sum over all orderings consistent with the Chan-Paton indices; amplitudes with adjacent operators φ IJ φ KL are only nonvanishing if K = L. For gauge-invariant amplitudes we would sum over all such indices and thus over all orderings. Such a term can be computed on the disc with a single closed-string insertion and a single open-string insertion. SL(2, IR) invariance allows us to fix the positions of both vertex operators. In addition, if we place the closed-string vertex operator in the (−1, −1) picture and the open-string operator in the (0)-picture we have absorbed the superconformal ghost number violation (the left-and right-moving ghost zero modes will be tied together by the boundary condition.) The relevant tree-level amplitude is thus: twisting may be achieved by adding a charge of ±c/3 at infinity; the change in the stress tensor is simply the shift derived in the Feigin-Fuchs construction. In closed string theories one can see this most simply by adding to the action the coupling of the U (1) current to a background gauge field A = 1 2 ω where ω is the worldsheet spin connection[103]. In the cases we are interested, where J = i integrated by parts to get a coupling of H to the Riemann curvature. For amplitudes on the sphere, one may use conformal invariance to write the sphere as a flat cylinder with two hemispherical caps. The initial and final states are created by the path integral on those caps with any operator insertions one might have there; the curvature on these hemispherical caps means that the above terms in the Lagrangian become the half-unit spectral flow operators applied to the initial and final states.If one constructs the open-string case via the doubling trick on the Riemann sphere, A-type boundary conditions are compatible with the A-model and Btype boundary conditions are compatible with the B-model, as in each case the "twisted" stress tensor satisfies T top =T top and thus satisfies sensible boundary conditions. In these twisted models, the conformal dimensions of N = 2 primary operators are shifted by half their U (1) charge, with the sign depending on the twisting. In the Bmodel, G + andḠ + become dimension-zero "scalar" Grassman operators and suitably define BRST currents. The NS (c, c) operators are annihilated by them and have dimension 0 with respect to T top . These operators are denoted the "topological" operators and their correlators are independent of position, as one can see using the conformal Ward identities in the presence of the background charge. We denote them as the "0-form" operators O i(0) . In the closed string theory, we can also define "(1, 0)-" and "(0, 1)-form" operators z→w dzG − (z)O (0) (w,w) = O (1,0) (w,w) z→w dzḠ − (z)O (0) (w,w) = O (0,1) (w,w) , (6.22) and "2-form" (or "(1, 1)-form") operators: z→w z→w dzdzG − (z)Ḡ − (z)O (0) (w,w) (6.23) We can see that the (0)-form operators are simply the internal parts of the (−1, −1) operators of the untwisted theory, while the (1, 1)-form operators are the internal parts of the (0, 0)-picture operators of the untwisted theory. The operators dzO (1,0) ; dzO (0,1) ; d 2 zO (1,1) , are BRST-invariant on Riemann surfaces without boundary. On surfaces with boundary, the integrated 2-form operator is only BRST-invariant up to an integral of the one-form operators along the boundary, as one can see by integrating by parts; similarly the BRST transformations of the one-form operators pick up boundary terms if the curve of integration ends on a boundary of D. that the open-string A-model describes D-branes wrapped around special Lagrangian submanifolds and the topological closed-string operators are the Kähler deformations of the target space; while the B-model describes D-branes wrapped around holomorphic cycles, and the topological closed-string operators correspond to complex structure deformations of the target space. Note that although refs. [101] discuss only the case of purely Neumann boundary conditions for the B-model, our general discussion shows that we may couple this topological theory to any supersymmetric, even-dimensional brane. (Mirror symmetry requires this, if we are allowed to discuss any supersymmetric 3cycle in the mirror). Note also that in this geometric picture, the almost-BRST-invariance of the integrated 2-form observables makes sense: a change of complex structure (Kähler class) will change the definition of holomorphic (special Lagrangian) submanifolds. Now let us return to the fermion bilinear part of the (n > 2)th order superpotential. By stretching the cylinder out into the capped strip (fig. 1) we may write the amplitude as the expectation value of some set of NS vertex operators between Ramond states; these states are created by applying the Ramond vertex operators to the vacuum. N = 2 worldsheet supersymmetry allows us to write the internal-CFT part of these states as the spectral flow operator applied to NS operators acting on the vacuum; bundle times the gauge group of the internal worldvolume, N C\| M ⊗ĝ. The 0-form boundary observables corresponding to the marginal, chiral primaries of the untwisted model are linear in the worldsheet fermions. For all of these observables, open and closed, it is easy 31 ) 31Note that the expectation value is antisymmetric in the fermions; thus this will vanish if there is only one D-brane, after summing over the ordering. The moduli space of constant maps is a point. Chiral deformations of the location by boundary observables live in T 0,1 M and anti-chiral deformations are valued in T 1,0 M . The latter are BRST-exact in our picture, so the above correlator is ∂-closed as a function of the location of the point. 17 The correlators must be antiholomorphic functions of the location of this point, and they live in ∧ 3 T (0,1) M . Serre duality implies that they are components of a closed antiholomorphic (0, 3) form. On the Calabi-Yau manifold there is only one such form Ω within cohomology. Thus the superpotential is W = Ωī 1ī2ī3 f abc Φī 1 a Φī 2 b Φī 3 c . (6.32) For boundaries living on points, the open-string operator is written in Eq. (6.30). The closed-string operator must be quadratic in θ and have no ξ charge by U (1) charge conservation. This latter operator is an element of H 0 (M, ∧ 2 T (1,0) M ). By Serre duality this group is equivalent to the Dolbeaux cohomology group H (0,1) (M ) which vanishes for Calabi-Yau compactifications. Thus there is no closed-string operator which couples to a single open-string operator on the D0-brane. The argument is almost identical for D6-branes and rests on the fact that H (2,0) (M ) is trivial on a Calabi-Yau manifold.For 2-cycles the story is a bit richer. If the open-string vertex operator is polarized along a Neumann direction, work in a coordinate patch where the tangent-normal split is trivial), then fermion number conservation requires that the closed-string operator be quadratic in θ and there are no such nontrivial operators as we have just argued. But for vertex operators polarized in a Dirichlet direction, the closed-string operator must be bilinear in ξ and θ, making it a one-form valued in T (0,1) M . Serre duality relates this to an element of H (2,1) (M ) and this group is certainly nontrivial, so these open-closed correlators are allowed. Similarly we can have nontrivial linear terms for chiral fields coming from Neumann directions along a 4-cycle. the one-loop topological open string amplitude. For the D6-brane this is the Ray-Singer torsion I(V ) associated to the Chan-Paton bundle V on M [103]. In general I(V ) is not independent of the Kähler moduli, but ratios ln(I(V 1 )/I(V 2 )) are, where V 1,2 are two different bundles on M [108] 41) (here I lables the boundary condition in the region of the contact term); the other from the operator product of ψ i with the boundary operators Σ α , O α . Note that O will have zero U (1) charge. Let us deal with each of these in turn; we will in fact argue that this second contact term is taken care of by the first.The bulk-boundary OPE can be treated as a factorization of the disc amplitude onto an intermediate open-string state. The OPE coefficient C I iα will be proportional to the open-closed disc amplitude ψ i O α . There are in fact two classes of terms to worry about in eq. (6.41): δ O α < 1 and δ O α = 1. In the former case the intermediate state is a tachyon. a suitable connection, the chiral primary part of the boundary state of the B-type brane is covariantly constant with respect to deformations of the Kahler moduli. Our result should be the open-string version of this fact.The second contact term above is between the bulk operator ψ i and boundary operator O β . By using the doubling trick this is described as the coalescence of three operators and, associativity allows us to write this by taking the bulk-boundary OPE (6.41) of ψ first, and then taking the OPE of O α and O β . But this will be included in the limits of integration of the first contact term O α over the boundary. determined by the three-point functions; the higher-point functions can be computed by sewing. The result is independent of the decomposition of the n-point function into threepoint functions, as guaranteed by crossing symmetry for the four-point functions. Similar results hold for the case with boundaries. Here, we have three types of sewing constraints, those involving only boundary fields, those, involving both bulk-and boundary fields and those involving only bulk fields. The structure constants for boundary fields depend on the boundary conditions. Fig. 3 : 3Four-point function as illustrated in fig. 3. In the case of rational symmetric models the factorization conditions can be made explicit using the conformal blocks. Note that for boundary correlators, the four point functions are linear in the conformal blocks. If we want to compare the four point functions for open string operators with different orderings, we have to take into account that the change in the ordering will in general require different boundary conditions for the respective four point function to be non-vanishing. Different boundary conditions will in general change both the structure constants and the expectation values of the identity so that we do not expect the four point functions to agree. In the example of the Virasoro minimal model, where we have an explicit solution, the boundary structure constants satisfy c αβγ ijk = c γβα jik . (7.6) conditions on both ends of the string there are boundary conditions with either 101, 50, 24, 11, 4 or 0 marginal operators propagating. The 101 marginal operators propagating between the boundary states L = {1, 1, 1, 1, 1} are of the same form as the complex structure deformations in the closed string case. These are left-right symmetric fields of charge (1, 1). If we use the doubling trick, the boundary marginal operators look the holomorphic part of these operators. They are of the form: boundary conditions there are no further restrictions on possible correlators from the boundary selection rules. All correlators allowed by U (1) charge conservation and fusion rules will also be allowed by the boundary selection rules. The difference from closed string computations is that these correlators depend on: the one-point function of the identity in the presence of particular boundary conditions; the values of the fusion coefficients c ααα ijk for the boundary conditions; and the different integration domain for the four-and higher point functions. The 4,11, 24 and 50 marginal operators are particular subsets of the 101 operators. Here, restrictions from boundary selection rules are possible: some of the correlators which are present in the closed string case cannot appear as the operators do not propagate with the given boundary conditions. We expect the strongest results for the case with 4 marginal boundary operators. the previous case, this example is already much less restricted. For example, we have in this case two types of three-point functions: those containing each type of operator listed in (7.10) once; and that containing the three operators of the third type listed in (7.10). However, a lot of corrections which would be allowed in the corresponding bulk case are absent for these boundary conditions, because in the factors 3, 4, 5 of the minimal models, φ(3,3,0) is the only chiral primary which propagates.Likewise, we potentially get more three point functions for the cases with 24 and 50 operators, and less corrections are suppressed. Finally, for the case with 101 marginal operators, all allowed bulk correlators have a boundary equivalent. hypothesis." Unlike the previously studied cases, supersymmetry is not sufficient to answer this question. From the bulk point of view, N = 2, d = 4 supersymmetry allows lines of marginal stability for BPS states, while from the brane point of view N = 1, d = 4 supersymmetry allows transitions from Higgs to Coulomb branch which are essentially unpredictable from the large volume point of view. There are a number of considerations which would lead us to expect non-geometric phenomena. Perhaps the simplest is that the B monodromies relate branes of different dimension. Another essentially non-geometric phenomenon is the topology change seen in various Calabi-Yaus; both phenomena make the the geometric interpretation of brane probes ambiguous. The mere existence of these phenomena however does not really contradict the geometric hypothesis as we have stated it, if the different geometric objects related by monodromy and topology change lead to the same low energy theories. What we would be saying is that the same brane theory can have multiple large volume limits, a familiar phenomenon in duality. Small instantons and the C 3 /Z 3 orbifold provide examples which do contradict the geometric hypothesis in its simplest form. At a special point (more generally, in complex codimension one) in moduli space, enhanced gauge symmetry and additional states appear. λ , −μ\|(−1) F Lq L 0 − c 24 \|λ, Q(µ)+d/2 S (λ,µ),(λ ′ ,µ ′ ) χ λ ′ µ ′ (q), (A.1) 20 One has to be careful with the definition of (−1) F L in the RR sector. It should be defined by (−1) F L = (−1) J 0 +d/2 because the charge might be half-integer moded. A 1 model, where all correlators between chiral fields are forbidden. The model contains the φ . As discussed in section 6, from these operators we can derive one-form operators. For the chiral operator, we get the one-form operator φ φ . Alternatively, this operation can be interpreted as picture changing, if the A 1 model is part of a string theory compactification. A candidatechiral operators 1, φ (0) 1 = e i √ 3 φ and the antichiral operator φ (0) −1 = e − i √ 3 (1) 1 = e −2i √ 3 for a non vanishing correlator is (φ (0) 1 ) 3 φ (1) 1 would like to thank Paul Aspinwall, Tom Banks, Michael Bershadsky, Ralph Blumenhagen, Robert Bryant, Emanuel Diaconescu, Robbert Dijkgraaf, Dan Freed, Bartomeu Fiol, Jaume Gomis, Rajesh Gopakumar, Sheldon Katz, Albrecht Klemm, Maxim Kontsevich, Juan Maldacena, John Morgan, Hirosi Ooguri, Arvind Rajaraman, Andreas Recknagel, Moshe Rozali, Volker Schomerus, Shishir Sinha, Andy Strominger, Gang Tian, Cumrun Vafa, Edward Witten and Sasha Zamolodchikov for useful discussions and correspondence. This research was supported by DOE grant DE-FG02-96ER40959. A.L. is also supported in part by NSF grant PHY-9802709 and DOE grant DE-FG02-91ER40654. There is some evidence that the special Lagrangian condition receives α ′ corrections[30]. The signs Σ 6 · Σ 0 = +1 and Σ 4 · Σ 2 = −1 in the large volume intersection form η follow from the definition (2.12). The variable m in[74], in sec. 2.1 of[66], and sec. 4 of[67], is what we are calling m − s. c.f.[99] for a nice discussion of this method. Or antichiral ring. We will fix the overall sign ambiguity of the U (1) charges by demanding that the boundary operators be chiral. This is a sensible thing to do in the closed-string sector as well, as B-model path integrals localize onto constant maps into the target space[105]. For multiple derivatives there is potentially a holomorphic anomaly. This is similar to the fact that insertions of the stress tensor into correlators on higher-genus Riemann surfaces lead not only to transformations of the operators but to derivatives of the amplitude with respect to the moduli of the surface[109]. Indeed such terms are important in deriving the one-loop holomorphic anomaly for topological amplitudes[110]. 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[ "A New Approach for Delta Form Factors", "A New Approach for Delta Form Factors" ]	[ "C Aubin \nDept. of Physics\nCollege of William and Mary\nWilliamsburgVA\n", "K Orginos \nDept. of Physics\nCollege of William and Mary\nWilliamsburgVA\n", "\nDept. of Physics\nFordham University\nBronxNY\n", "\nThomas Jefferson National Accelerator Facility\nNewport NewsVA\n" ]	[ "Dept. of Physics\nCollege of William and Mary\nWilliamsburgVA", "Dept. of Physics\nCollege of William and Mary\nWilliamsburgVA", "Dept. of Physics\nFordham University\nBronxNY", "Thomas Jefferson National Accelerator Facility\nNewport NewsVA" ]	[]	We discuss a new approach to reducing excited state contributions from two-and three-point correlation functions in lattice simulations. For the purposes of this talk, we focus on the ∆(1232) resonance and discuss how this new method reduces excited state contamination from two-point functions and mention how this will be applied to three-point functions to extract hadronic form factors.The calculation of the electromagnetic form factors for mesons and baryons is crucial to understanding the structure of hadronic states in QCD. However, they are notoriously difficult to both measure experimentally as well as calculate theoretically due to the complications that arise from the strong interactions. In the case of the ∆(1232) resonance, the form factors themselves are not currently experimentally accessible, although two of the form factors in the static limit are known (the charge) or measured to some degree (the magnetic dipole moment). For the nucleon, experimental results do exist for the form factors as a function of the momentum transfer. This makes a calculation of the nucleon form factors both a check of methodology as well as of QCD, and allows us to be confident that lattice results for the ∆ form factors are reasonable.The electromagnetic form factors of the ∆ are encoded in the matrix elementwhere u α is a Rarita-Schwinger vector-spinor describing the external ∆, and J µ = ∑ qq γ µ q is the vector current. The Lorentz structure of Γ is given, for example, in [1], and has four form factors F * 1,2,3,4 (Q 2 ) that are functions of Q 2 = −(p ′ − p) 2 alone. These form factors give rise, in the limit Q 2 → 0, to the electric charge, magnetic dipole moment, electric quadrupole moment, and magnetic octupole moment of the ∆.Of these moments, the charge is of course known, and from the PDG [2], we have	10.1063/1.3647217	[ "https://arxiv.org/pdf/1010.0202v1.pdf" ]	119,286,970	1010.0202	d47cc0312055081b7d9adf6313640ba79c14f5f4	A New Approach for Delta Form Factors 1 Oct 2010 C Aubin Dept. of Physics College of William and Mary WilliamsburgVA K Orginos Dept. of Physics College of William and Mary WilliamsburgVA Dept. of Physics Fordham University BronxNY Thomas Jefferson National Accelerator Facility Newport NewsVA A New Approach for Delta Form Factors 1 Oct 20101115Ha1238Gc1340Gp1420Gk We discuss a new approach to reducing excited state contributions from two-and three-point correlation functions in lattice simulations. For the purposes of this talk, we focus on the ∆(1232) resonance and discuss how this new method reduces excited state contamination from two-point functions and mention how this will be applied to three-point functions to extract hadronic form factors.The calculation of the electromagnetic form factors for mesons and baryons is crucial to understanding the structure of hadronic states in QCD. However, they are notoriously difficult to both measure experimentally as well as calculate theoretically due to the complications that arise from the strong interactions. In the case of the ∆(1232) resonance, the form factors themselves are not currently experimentally accessible, although two of the form factors in the static limit are known (the charge) or measured to some degree (the magnetic dipole moment). For the nucleon, experimental results do exist for the form factors as a function of the momentum transfer. This makes a calculation of the nucleon form factors both a check of methodology as well as of QCD, and allows us to be confident that lattice results for the ∆ form factors are reasonable.The electromagnetic form factors of the ∆ are encoded in the matrix elementwhere u α is a Rarita-Schwinger vector-spinor describing the external ∆, and J µ = ∑ qq γ µ q is the vector current. The Lorentz structure of Γ is given, for example, in [1], and has four form factors F * 1,2,3,4 (Q 2 ) that are functions of Q 2 = −(p ′ − p) 2 alone. These form factors give rise, in the limit Q 2 → 0, to the electric charge, magnetic dipole moment, electric quadrupole moment, and magnetic octupole moment of the ∆.Of these moments, the charge is of course known, and from the PDG [2], we have µ ∆ ++ = (5.6 ± 1.9)µ N , (2) µ ∆ + = (2.7 ± 3.5)µ N , where we have added all of the errors (including theoretical) in quadrature just to get an idea for how well these are determined experimentally. Thus, it is essential even for this simple quantity to have a well-determined lattice result. Some unquenched results were obtained using the form factor approach in [3] and using a background field technique [4], but there are several difficulties that arise in these different methods. For this work, we will focus on the difficulties with the form factor approach. These are determined by calculating the 3-point correlator: C 3pt (t i ,t,t f , p i , p f ) = FT 0\|χ(t f , x f )J µ (t, 0)χ(t i , x i )\|0 ,(4) where FT is the Fourier Transform of the correlator, and χ is some appropriate interpolating operator for the ∆. In the large time limit t f ≫ t ≫ t i : C 3pt (t i ,t,t f , p i , p f ) → Z(p i , p f )e −E f (t f −t) e −E i (t−t i ) ∆(p f )\|J µ (0)\|∆(p i ) + · · · .(5) Here we have schematically written this so that Z contains various overlap factors of the form 0\|χ\|∆ as well as other kinematic factors that are known. The dots denote contributions from excited states that are generally ignored. FIGURE 1. On the left is the matrix element G E for the ∆ ++ evaluated at q 2 = 0. While there is significant excited state contamination, there is a noticeable plateau that gives (up to renormalization) the charge of the ∆. On the right is the same matrix element for q 2 = 11(2π/a s L) 2 ≈ 4 GeV 2 , where there is no trustworthy signal. The standard approach then is to note that the overlap factors and kinematics can be canceled by an appropriate ratio with 2-point correlators, and from these ratios we can extract the matrix element of interest. Note that additional complications arise in separating the ∆ state and the N − π state when the pion mass is below 300 MeV. Currently we are not near this regime, so it is not an issue for the current analysis. The problem is that the contamination from excited states can be seen to be large, even for the simplest cases. Take here the ∆ ++ E0 form factor, G E0 (q 2 ), which is a linear combination of the F * i form factors. In the limit q 2 → 0, G E (0) = +2, the electric charge of the ∆ ++ in units of \|e\|. We show on the left of Fig. 1 the appropriate ratio of 3-to-2point correlators to get this quantity using the Hadron Spectrum collaboration lattices of 2+1-flavor anisotropic Clover lattices [5] (here with a volume of 16 3 × 128 and a pion mass of roughly 390 MeV, and a −1 t ≈ 5.5 GeV, a s /a t ≈ 3.5). The source and sink are located at t = 0, 28, and we can see a plateau in the center where we could reliably extract the form factor, but there is significant contamination from excited states, seen from the deviation from the plateau. This problem is more pronounced at higher momenta, and we show the same form factor at q 2 = 11(2π/a s L) 2 ≈ 4 GeV 2 on the right of Fig. 1. The location of a plateau is questionable at best for this plot. We can extract the form factor using this standard approach and one gets reasonable results. For the q 2 = 0 point, we find the charge of the ∆ ++ is precisely twice that of the proton, and we can extract the renormalization factor for the vector current (note it is not conserved here because we are using a local current) from G E0 (0), and we obtain 1 Z V = 1.05(1) (statistical errors only). While this determination is trustworthy, extracting the higher momenta form factors is dangerous, due to the excited state contamination. So we would like to examine a new approach, which makes use of the Variational method to better extract states that contribute to a correlator. For now we will discuss this in the context of 2-point correlators for simplicity. For a given state, there are many interpolating operators that could be used to calculate a two-point function, and one could form a matrix of correlators C i j (t) = 0\|O i (t)Ō j (0)\|0 .(6) From this, by solving the generalized eigenvalue problem (GEVP) C(t)x = λ (t,t 0 )C(t 0 )x ,(7) one can show (see [6] and references therein) that the eigenvalues behave like On the left we show the effective mass for a single ∆ operator with noticeable excited state contamination before reaching a plateau. On the right is the effective masses for the two states extracted using the GEVP and two different local ∆ operators. λ i (t,t 0 ) ∼ e −m i (t−t 0 ) + · · · .(8) The reference time t 0 is empirically chosen so that at that time, all n states (for an n × n system) would contribute to the correlator; no more, no less. For our purposes, we find that the choice of t 0 has little effect on our results, primarily because we are using a very small basis of operators. If we restrict ourselves to local operators, there are two ∆ interpolating fields we can use, and we can see that this has little effect to reduce contamination from excited states in the ground state, as shown by comparing the two plots in Fig. 2. On left is a single operator effective mass, and we can see the excited state contamination before the plateau. On the right is the case with two operators, and solving the GEVP. The ground state is unchanged, and although this allows us to perhaps extract an excited state, this is not what we are currently interested in. So we would like to find a way to kill off the excited state contribution in the ground state. This is where the Generalized Pencil-of-Function (GPoF) method comes in [7,8]. In a quantum mechanical system, the important point is that if O ∆ (t) is an interpolating operator for the ∆, then so is O τ ∆ (t) ≡ e Hτ O ∆ (t)e −Hτ = O ∆ (t + τ) ,(9) and this new operator is linearly independent from the original operator. So if we use O ∆ (t), O τ ∆ (t) as our two operators, we can construct a correlator matrix using only a single correlator. This matrix is C(t) = O ∆ (t)Ō ∆ (0) O τ ∆ (t)Ō ∆ (0) O ∆ (t)Ō τ ∆ (0) O τ ∆ (t)Ō τ ∆ (0) = O ∆ (t)Ō ∆ (0) O ∆ (t + τ)Ō ∆ (0) O ∆ (t)Ō ∆ (−τ) O ∆ (t + τ)Ō ∆ (−τ) = C(t) C(t + τ) C(t + τ) C(t + 2τ) .(10) We can replicate this and use a set of operators O τ,n ∆ (t) = O ∆ (t + nτ), and make this correlator as large as we wish. It turns out that using multiple shifts does not give us any additional information, so we will ultimately use just a single shift and a τ = 4. This would be something that is determined for each correlator one is interested in. Once this matrix is created then we follow through the same procedure as before with the GEVP. Note that when the correlator basis grows, there is more of a likelihood of zero singular values, so we perform an SVD cut on the correlator matrix to exclude states that have significantly smaller singular values than the largest by some cutoff, here chosen to be ≈ 10 −3 . In Fig. 3 we show the same operator(s) as before, but now using the GPoF method (again, τ = 4). On the left is one operator with one shift and the right uses both local operators and a single shift. We see the effective mass comes to a plateau much earlier than before, immediately after t 0 (which is also set to 4). Of course, the excited state(s) is(are) much noisier and we could not reliably extract information from that, but if one is only interested in the ground state, this is not an issue. However, one could apply the GPoF method to a large correlator basis to perhaps get cleaner signals for the lower lying states. On the right we perform the shift to the two operators, creating a 4 × 4 basis, but FIGURE 3. This is the same as in Fig. 2, but with a single shift of τ = 4 for the operator(s) included. after the SVD cut, this is reduced to three operators. In either case, we see that the ground state is unchanged, and a plateau is reached far sooner than without the use of the GPoF method. For the unshifted operator if we perform a correlated fit to the correlator using an exponential in the range t ∈ {29, 40}, we get a t m ∆ = 0.2770(50) with a confidence level of 78%. For the single shift of τ = 4, we use the larger (and earlier) range t ∈ {5, 28}, and get a t m ∆ = 0.2825(17) with a confidence level of 84%. Adding more shifts does not change this result, but it is amazing how well the signal improves with just this single shift. It can be seen what is happening qualitatively, since the correlator C(t + τ) comes from the correlation with an operator that is τ time steps away from the other operator, and thus excited states are not going to contributed much to that, and even less to C(t + 2τ). The ground state, however, is going to still contributed significantly to this correlator, so its signal, in a sense, is effectively enhanced. In the three-point case, working through the correlator matrix, we would get C 3−pt (t i ,t,t f ) = C 3−pt (t i ,t,t f ) C 3−pt (t i ,t,t f + τ) C 3−pt (t i ,t + τ,t f + τ) C 3−pt (t i ,t + τ,t f + 2τ) .(11) So unlike the two-point function, we have to actually use three different sink locations, and thus generate a factor of three in the computational cost. However, if this allows a better determination of our ground state signal, this should be worth the additional propagator generation. Once the two-point correlator matrix is diagonalized with the vectors x, we form a matrix V i j = (x j ) i and then use that to diagonalize the three-point correlator C 3−pt diag (t i ,t,t f ) = V −1 C 3−pt (t i ,t,t f )V .(12) Currently, the analysis using this approach is under way and we hope to see a noticeable improvement as we did in the two-point case. This work was partially supported by the US Department of Energy, under contract nos. DE-AC05-06OR23177 (JSA), DE-FG02-07ER41527, and DE-FG02-04ER41302; and by the Jeffress Memorial Trust, grant J-813. FIGURE 2 . 2FIGURE 2. On the left we show the effective mass for a single ∆ operator with noticeable excited state contamination before reaching a plateau. On the right is the effective masses for the two states extracted using the GEVP and two different local ∆ operators. . V Pascalutsa, M Vanderhaeghen, S N Yang, Phys. Rept. 437V. Pascalutsa, M. Vanderhaeghen, and S. N. Yang, Phys. Rept. 437, 125-232 (2007). . K Nakamura, J. Phys. 3775021K. Nakamura, J. Phys. G37, 075021 (2010). . C Alexandrou, Phys. Rev. 79C. Alexandrou, et al., Phys. Rev. D79, 014507 (2009), 0810.3976. . C Aubin, K Orginos, V Pascalutsa, M Vanderhaeghen, Phys. Rev. 7951502C. Aubin, K. Orginos, V. Pascalutsa, and M. Vanderhaeghen, Phys. Rev. D79, 051502 (2009). . H.-W Lin, Phys. Rev. 7934502H.-W. Lin, et al., Phys. Rev. D79, 034502 (2009). . B Blossier, M Della Morte, G Hippel, T Mendes, R Sommer, JHEP. 0494B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, and R. Sommer, JHEP 04, 094 (2009). . Y Hua, T Sarkar, 37Y. Hua, and T. Sarkar, IEEE transactions on antennas and propagation 37, 229-234 (1989). . T Sarkar, O Pereira, IEEE Antennas and Propagation Magazine. 37T. Sarkar, and O. Pereira, IEEE Antennas and Propagation Magazine 37, 48-55 (1995).	[]
[ "Potentially semi-stable deformations of specified Hodge-Tate type and Galois type", "Potentially semi-stable deformations of specified Hodge-Tate type and Galois type" ]	[ "Yong Suk Moon " ]	[]	[]	Let k be a perfect field of characteristic p > 2, and let K be a finite totally ramified extension of W (k)[ 1 p ]. We prove that the locus of potentially semi-stable Gal(K/K)-representations of a given Hodge-Tate type and Galois type is a closed subspace of the universal deformation ring, generalizing the result of Kisin(2007)where k is assumed to be finite.	10.1016/j.jnt.2017.05.018	[ "https://arxiv.org/pdf/1609.08570v2.pdf" ]	119,147,146	1609.08570	334831180e3fec2bb5860b432c78ec6e8c83fe14	Potentially semi-stable deformations of specified Hodge-Tate type and Galois type Yong Suk Moon Potentially semi-stable deformations of specified Hodge-Tate type and Galois type Let k be a perfect field of characteristic p > 2, and let K be a finite totally ramified extension of W (k)[ 1 p ]. We prove that the locus of potentially semi-stable Gal(K/K)-representations of a given Hodge-Tate type and Galois type is a closed subspace of the universal deformation ring, generalizing the result of Kisin(2007)where k is assumed to be finite. that the natural map O E → A/m A is surjective and the map from A to the projective limit of its discrete artinian quotients is a topological isomorphism. If V 0 is absolutely irreducible, then there exists a universal deformation ring R ∈ C with a deformation V R which parametrizes the isomorphism classes of deformations of V 0 ( [SL97]). Note that R is not necessarily noetherian in general when k is not finite. In this paper, we study the geometry of the locus of potentially semi-stable representations with a specified Hodge-Tate type v and Galois type τ . We show that such a locus cuts out a closed subspace in the following sense: Theorem A. There exists a closed ideal a v,τ ⊂ R such that the following holds: for any finite flat O E -algebra A and a continuous O E -algebra homomorphism f : R → A (where we equip A with the (p)-adic topology), the induced representation A[ 1 p ] ⊗ f,R V R is potentially semi-stable of Hodge-Tate type v and Galois type τ if and only if f factors through the quotient R/a v,τ . When the residue field k is finite, Kisin proved the corresponding result in [Kis07, Theorem 2.7.6]. One of the main steps in [Kis07] is the construction of the projective scheme which parametrizes representations of E(u)-height ≤ r for a fixed positive integer r (cf. [Kis07, Section 1.2]). It is obtained as a closed subscheme of the affine Grassmannian for the restriction of scalars Res W (k)/Zp GL d . But this construction does not make sense in general when k is infinite. The main difficulty is that we do not know how to analyze whether the restriction of scalars Res W (k)/Zp for a non-affine scheme over W (k) is representable by an Ind-scheme when k is infinite, even for simple examples such as P 1 W (k) . Another approach to studying the locus cut out by certain p-adic Hodge theoretic conditions, motivated by Fontaine's conjecture in [Fon97], is to analyze torsion representations given as the subquotients of Galois stable lattices satisfying the given conditions. For semistable (or crystalline) representations having Hodge-Tate weights in [0, r], this is carried out by Liu in [Liu07]. And in the case k is finite, Liu proved the corresponding result for semi-stable representations of a given Hodge-Tate type in [Liu15]. We use the functor given in [Liu12] from the category of representations semi-stable over a totally ramified Galois extension K /K to the category of lattices in filtered modules equipped with Frobenius, monodromy, and Gal(K /K)-action, in order to study the refined structure of a Hodge-Tate type and Galois type of torsion representations. We first study semi-stable representations of a given Hodge-Tate type and show that the locus of such representations is p-adically closed (cf. Theorem 3.3). This generalizes the corresponding result in [Liu15] to the case k is not necessarily finite. The proof in [Liu15] is based on reducing to the situation when the coefficient field E contains the Galois closure of K, thereby requiring k to be finite. We remove such a restriction using a different argument. Then, we study potentially semi-stable representations of a given Galois type, and prove that such representations cut out a p-adically closed locus (cf. Theorem 3.17). Acknowledgments I would like to express my sincere gratitude to Mark Kisin for suggesting me to work on this topic and making many helpful comments. This paper is based on a part of author's Ph.D. thesis under his supervision. I also wish to thank Brian Conrad and Tong Liu for helpful discussions on this work. I thank the referee for a careful reading of the paper and making helpful suggestions to improve it. I would like to thank the department of Mathematics at Harvard University and Purdue University for their cordial environment. This work was partly supported by Samsung Scholarship Foundation, South Korea. Torsion Representation and Construction of M st We keep the notations as in the introduction. Let K ⊂K be a finite totally ramified Galois extension of K. In this section, we will first explain the construction of the functor given in [Liu12] from the category of representations semi-stable over K to the category of lattices in filtered modules equipped with Frobenius, monodromy, and Gal(K /K)-action. Then, we will explain the result proved in [Liu12] and [Liu15] that one can associate a Hodge-Tate type and Galois type to a torsion representation up to some constant depending only on K . Potentially Semi-stable Representation and Filtered (ϕ, N, Γ)module For a G K -representation V over Q p , we say V is potentially semi-stable if there exists a finite extension L ⊂K of K such that V restricted to G L := Gal(K/L) is semi-stable. This means precisely that dim Qp V = dim L 0 (B st ⊗ Qp V ∨ ) G L where L 0 is the maximal unramified subextension of L/K 0 . Let e = [K : K 0 ]. We fix a uniformizer π of K , and let F (u) be the Eisenstein polynomial for π over K 0 . Denote by Rep pst,K Qp the category of G K -representations over Q p which become semi-stable over K (i.e., semi-stable as Gal(K/K )-representations). Let Γ = Gal(K /K) and G K = Gal(K/K ). Note that K 0 is equipped with the natural Frobenius endomorphism ϕ. We consider the category of filtered (ϕ, N, Γ)-modules whose objects are finite dimensional K 0 -vector spaces D equipped with: • a Frobenius semi-linear injection ϕ : D → D, • W (k)-linear map N : D → D such that N ϕ = pϕN , • decreasing filtration Fil i D K on D K := K ⊗ K 0 D by K -sub-vector spaces such that Fil i D K = D K for i 0 and Fil i D K = 0 for i 0, and • K 0 -linear action by Γ on D which commutes with ϕ and N . If we extend Γ-action semi-linearly to D K , then for any γ ∈ Γ, γ( Fil i D K ) ⊂ Fil i D K . Morphisms between filtered (ϕ, N, Γ)-modules are K 0 -linear maps compatible with all structures. The functor D K st : V → (B st ⊗ Qp V ∨ ) G K is an equivalence between Rep pst,K Qp and the category of weakly admissible filtered (ϕ, N, Γ)-modules (cf. [CF00], [Fon94]). We define an integral structure of a filtered (ϕ, N, Γ)-module. Definition 2.1. Let D be a filtered (ϕ, N, Γ)-module. A lattice M in D is a finite free W (k)-submodule of D such that M [ 1 p ] := M ⊗ Zp Q p ∼ = D, and ϕ(M ) ⊂ M , N (M ) ⊂ M , and γ(M ) ⊂ M for all γ ∈ Γ. For a lattice M ⊂ D, we equip M K := O K ⊗ W (k) M with the natural filtration by O K -submodules, given by Fil i M K = M K ∩ Fil i D K . If M 1 , M 2 are lattices in filtered (ϕ, N, Γ)-modules D 1 , D 2 respectively, then a morphism f : M 1 → M 2 is a W (k)-linear map such that f ⊗ Zp Q p : D 1 → D 2 is a morphism of filtered (ϕ, N, Γ)-modules. Note that for a lattice M in a filtered (ϕ, N, Γ)-module, the associated graded O Kmodules gr i M K = Fil i M K /Fil i+1 M K is torsion free by the definition of the filtration. Let r be a positive integer. Denote by L r (ϕ, N, Γ) the category of lattices in filtered (ϕ, N, Γ)-modules D satisfying Fil 0 D K = D K and Fil r+1 D K = 0. . Let A cris be the p-adic completion of the divided power-envelope of W (O Cp ) with respect to ker(θ). We fix a compatible system of p n -th roots π n ∈ OK of π for non-negative integers n, and let π := (π n ) ∈ O Cp . We have an [u]. We also fix a compatible system of primitive p n -th roots of unity ζ p n ∈ OK for non-negative integers n, and let : = (ζ p n ) ∈ O Cp . Let t = log[ ] ∈ B + dR . Note that we also have t ∈ A cris . Let B dR = B + dR [ 1 t ], B cris = B + cris [ 1 t ], and B st = B + st [ 1 t ]. We denote by O E the p-adic completion of S[ 1 u ], and let E = Frac(O E ). LetÊ ur be the p-adic completion of the maximal unramified subextension of E in W (Frac(O Cp ))[ 1 p ], and OÊ ur its ring of integers. We let S ur = OÊ ur ∩ W (O Cp ). We let K ∞ = ∞ n=1 K (π n ) and K p ∞ = ∞ n=1 K (ζ p n ). Let K c = K ∞ K p ∞ , which is the Galois closure of K ∞ over K . LetĜ = Gal(K c /K ), G ∞ = Gal(K/K ∞ ), and H K = Gal(K c /K ∞ ). Write t {i} = t i p q(i) q(i)! where q(i) is defined by i = q(i)(p − 1) + r(i) with 0 ≤ r(i) < p − 1. We define R K 0 := { ∞ i=0 a i t {i} \| a i ∈ S K 0 , a i → 0 p-adically as i → ∞}. We have a natural map ν : W (O Cp ) → W (k) induced by the projection O Cp →k, which can be seen to extend uniquely to ν : B + cris → W (k)[ 1 p ]. For any subring A ⊂ B + cris , write I + A := A ∩ ker(ν). We have I + S = uS and I + S = { ∞ i=1 b i i e ! u i \| b i ∈ W (k), b i → 0 p-adically as i → ∞}. DefineR = W (O Cp ) ∩ R K 0 and I + = I +R . The following lemma is proved in [Liu10]. 1.R (resp. R K 0 ) is a ϕ-stable S-algebra as a subring in W (O Cp ) (resp. B + cris ). 2.R and I + (resp. R K 0 and I + R K 0 ) are G K -stable. The G K -actions onR and I + (resp. R K 0 and I + R K 0 ) factor throughĜ. •Ĝ M denotes aR-semi-linearĜ-action onM which commutes with ϕM and induces a trivial action onM/I +M . 3. R K 0 /I + R K 0 ∼ = K 0 andR/I + ∼ = S /I + S ∼ = S/uS ∼ = W (k). • Considering M as a ϕ(S)-submodule ofM, we have M ⊂M H K . A morphism between two (ϕ,Ĝ)-modules M 1 , M 2 of height r is a morphism in Mod r S (ϕ) which commutes with theĜ-actions. We denote by Mod r S (ϕ,Ĝ) the category of (ϕ,Ĝ)modules of height r. ForM ∈ Mod r S (ϕ,Ĝ), we associate a Z p [G K ]-moduleT ∨ (M) := HomR ,ϕ (M, W (O Cp )) with G K -action given by g(f )(x) = g(f (g −1 (x))) for g ∈ G K , f ∈ T ∨ (M). Here, G K -action onM is given byĜ-action onM. Moreover, for M ∈ Mod r S (ϕ), we associate a Z p [G ∞ ]-module T ∨ S (M) := Hom S,ϕ (M, S ur ) similarly. The main result proved in [Liu10] is the following. There exists a natural isomorphism T ∨ S (M) ∼ = →T ∨ (M) of Z p [G ∞ ]-modules. To construct the functor M st , we establish a connection between (ϕ,Ĝ)-modules and filtered (ϕ, N, Γ)-modules. Let V ∈ Rep pst,K ,r ι : B + st ⊗ K 0 D K st (V ) → B + st ⊗ Qp V ∨ such that ι is compatible with Frobenius, monodromy, filtration, and G K -action on both sides. The following is proved in [Liu12]. Proposition 2.6. (cf. [Liu12, Proposition 2.6, Corollary 2.7, 2.8]) There exists a unique K 0 -linear isomorphism i : D K st (V ) → D such that i is compatible with the Frobenius morphisms on both sides and makes the following diagram commutative: D K st (V ) B + st ⊗ Zp T ∨ D B + cris ⊗ Zp T ∨ i mod u Furthermore, such i is functorial. Note that M/uM ∼ = ϕ * (M)/uϕ * (M) ⊂ D/I + S K 0 D = D. We set M st (T ) ⊂ D K st (V ) to be the inverse image of ϕ * (M)/uϕ * (M) under the isomorphism i : D K st (V ) → D given in Proposition 2.6. M st (T ) is a finite free W (k)-lattice in D K st (V ) stable under Frobenius. Furthermore, it is proved in [Liu12, Corollary 2.12, Proposition 2.15] that M st (T ) is stable under G K -action and N on D K st (V ). Thus, M st (T ) is a lattice of the filtered (ϕ, N, Γ)-module D K st (V ). And the association M st (·) is a contravariant functor from Rep pst,K ,r Zp to L r (ϕ, N, Γ) since the isomorphism i in Proposition 2.6 is functorial. Potentially Semi-stable Torsion Representations We now associate torsion filtered (ϕ, N, Γ)-modules to potentially semi-stable torsion representations. Denote by Rep pst,K ,r tor the category of torsion representations L semi-stable over K and of height r, in a sense that there exist lattices L 1 , L 2 ∈ Rep pst,K ,r Zp with a G K -equivariant injection j : L 1 → L 2 such that L ∼ = L 2 /j(L 1 ) as Z p [G K ]-modules, and L is killed by some power of p. Morphisms between two torsion representations in Rep pst,K ,r tor are morphisms of Z p [G K ]-modules. We call such (L 1 , L 2 , j) a lift of L. We will sometimes denote simply by j a lift of L. Note that a lift of L ∈ Rep pst,K ,r tor is not unique. Let L, L ∈ Rep pst,K ,r tor with lifts (L 1 , L 2 , j), (L 1 , L 2 , j ) respectively. If f : L → L is a morphism in Rep pst,K ,r tor , we say a morphismf : L 2 → L 2 in Rep pst,K ,r Zp is a lift of f if f (j(L 1 )) ⊂ j (L 1 ) andf induces f . We denote by M fil,r tor (ϕ, N, Γ) the category whose objects are finite W (k)-modules M killed by some power of p and endowed with the following structures: • a Frobenius semilinear morphism ϕ : M → M , • W (k)-linear map N : M → M satisfying N ϕ = pϕN , • W (k)-linear Γ-action on M which commutes with ϕ and N , and • M K := O K ⊗ W (k) M has decreasing filtration by O K -submodules such that Fil 0 M K = M K and Fil r+1 M K = 0. Also, γ(Fil i M K ) ⊂ Fil i M K for any γ ∈ Γ. Morphisms in M fil,r tor (ϕ, N, Γ) are W (k)-linear maps compatible with above structures. For L ∈ Rep pst,K ,r tor with a lift j : L 1 → L 2 , we can associate an object M st,j (L) ∈ M fil,r tor (ϕ, N, Γ) as follows. By Theorem 2.2, we have the morphism M st (j) : M st (L 2 ) → M st (L 1 ) in L r (ϕ, N, Γ) corresponding to j, and M st (j) is injective by [Liu12, Corollary 3.8]. We set M st,j (L) = M st (L 1 )/M st (j)(M st (L 2 ) ). Then M st,j (L) has natural endomorphisms ϕ and N , and Γ-action induced from M st (L 1 ). Furthermore, tensoring by O K on M st (j) gives the following exact sequence: 0 → O K ⊗ W (k) M st (L 2 ) → O K ⊗ W (k) M st (L 1 ) q → O K ⊗ W (k) M st,j (L) → 0. We define the filtration on M st,j (L) K by Fil i M st,j (L) K := q(Fil i M st (L 1 ) K ). This gives M st,j (L) a structure as an object in M fil,r tor (ϕ, N, Γ). By the snake lemma, we further have the following exact sequence of the associated graded modules: 0 → gr i (M st (L 2 ) K ) → gr i (M st (L 1 ) K ) → gr i (M st,j (L) K ) → 0. If f : L → L is a morphism in Rep pst,K ,r tor with a liftf : (L 1 , L 2 , j) → (L 1 , L 2 , j ), then it induces a morphism M st,f (f ) : M st,j (L ) → M st,j (L) in M fil,r tor (ϕ, N, Γ) . Note that the above construction depends on the choice of the lift of L. However, the following theorem, which can be deduced directly from [Liu15] and [Liu12], shows that the construction depends on lifts only up to a constant. Theorem 2.7. There exists a constant c depending only on F (u) and r such that the following statement holds: for any morphism f : L → L in Rep pst,K ,r tor with lifts j, j of L, L respectively, there exists a morphismh : M st,j (L ) → M st,j (L) in M fil,r tor (ϕ, N, Γ) such that • if there exists a morphism of liftsf : j → j which lifts f , thenh = p c M st,f (f ), • let f : L → L be a morphism in Rep pst,K ,r tor , j a lift of L , andh : M st,j (L ) → M st,j (L ) the morphism in M fil,r tor (ϕ, N, Γ) associated to f , j , and j . If there exists a morphism of liftsg : j → j which lifts f • f , thenh •h = p 2c M st,g (f • f ). Proof. It follows directly from [Liu12, Theorem 3.1] and [Liu15, Theorem 2.1.3, Remark 2.1.5]. The following corollary is immediate. Representation with Coefficient Let A be a Z p -algebra, and denote by Rep pst,K ,r 3 Hodge-Tate Type and Galois Type Hodge-Tate Type Let E be a finite extension of Q p , and let B be a finite E-algebra. Let V B be a free Bmodule of rank d equipped with a continuous G K -action. Suppose that as a representation of G K , V B is semi-stable over K , i.e., V B ∈ Rep pst,K Qp . Then V B is de Rham over K, and we set D K dR (V B ) = (B dR ⊗ Qp V ∨ B ) G K . For any E-algebra A, we write A K := A ⊗ Qp K. Lemma 3.1. (cf. [Liu15, Lemma 4.1.2]) 1. Let B be a finite B-algebra, and write V B = B ⊗ B V B . Then D K dR (V B ) ∼ = B ⊗ B D K dR (V B ), and gr i (D K dR (V B )) ∼ = B ⊗ B gr i (D K dR (V B )). 2. D K dR (V B ) is a free B K -module of rank d. Proof. (1) is proved in [Liu15, Lemma 4.1.2]. For (2), since D K dR (V B ) = K ⊗ K 0 D K st (V B ) , it suffices to prove that D K st (V B ) is a free B ⊗ Qp K 0 -module of rank d. For any finite B-algebra B , we can show similarly as in (1) that D K st (V B ) ∼ = B ⊗ B D K st (V B ). Let B red = B/N (B) where N (B) denotes the nilradical of B. B red is a reduced Artinian ring, so there exists a ring isomorphism B red ∼ = m j=1 E j for some field E j finite over E. E j ⊗ Qp K 0 is isomorphic to a finite direct product of fields, so D K st (V E j ) ∼ = E j ⊗ B D K st (V B ) is finite projective as an E j ⊗ Qp K 0 -module. Note that the Frobenius morphism on K 0 extends E j -linearly to E j ⊗ Qp K 0 , and the extended Frobenius permutes the maximal ideals of E j ⊗ Qp K 0 transitively. Therefore, D K st (V E j ) is a free E j ⊗ Qp K 0 -module of rank d, and D K st (V B red ) = B red ⊗ B D K st (V B ) is a free B red ⊗ Qp K 0 -module of rank d. Let {e 1 , . . . , e d } be a B red ⊗ Qp K 0 -basis of D K st (V B red ), and choose a liftê i ∈ D K st (V B ) of e i . By Nakayama's lemma, {ê 1 , . . . ,ê d } generate D K st (V B ) as a B ⊗ Qp K 0 -module. Thus, we have a surjection of B ⊗ Qp K 0 -modules f : d i=1 (B ⊗ Qp K 0 ) ·ê i D K st (V B ). As a K 0 -vector space, dim K 0 D K st (V B ) = d · dim Qp B. Thus, f is an isomorphism, and D K st (V B ) is a free B ⊗ Qp K 0 -module of rank d. Let D E be a finite E-vector space such that D E,K := D E ⊗ Qp K is equipped with a de- creasing filtration Fil i D E,K of E ⊗ Qp K-modules and {i \| gr i D E,K = 0} ⊂ {0, . . . , r}. We denote v = (D E,K , {Fil i D E,K } i=0,...,r ). We say that V B has Hodge-Tate type v if gr i D K dR (V B ) ∼ = B ⊗ E gr i D E,K as B K -modules for all i. Remark. We can consider a Hodge-Tate type as a conjugacy class of Hodge-Tate cocharacter in the following way. The Hodge-Tate period ring is given by B HT = C p [X, X −1 ] where G K acts on X via the cyclotomic character χ. When V B is de Rham and thus Hodge-Tate, we have the isomorphism α HT : D K HT (V B ) ⊗ K B HT ∼ = −→ V B ⊗ Qp B HT , and D K HT (V B ) ∼ = i gr i D K dR (V B ) as graded B ⊗ Qp K-modules. By base change of α HT via the map B HT → C p given by X → 1, we have the isomorphism D K HT (V B ) ⊗ K C p ∼ = V B ⊗ Qp C p . This gives the grading on V B ⊗ Qp C p whose graded pieces are B ⊗ Qp K-modules. Therefore, we get the induced map G m → Res B/Qp (GL B (V B )) over C p , and we can think of a Hodge-Tate type as a conjugacy class of Hodge-Tate cocharacter. When K/Q p is finite and E contains the Galois closure of K, then we can also consider a Hodge-Tate type as induced from embeddings of K into E, but this point of view does not work in our case where K/Q p is allowed to be infinite. Note also that Hodge-Tate type does not make sense if we allow both B and K to be infinite over Q p , so we always assume B/Q p is finite. Lemma 3.2. For a finite B-algebra B , V B has Hodge-Tate type v if V B has Hodge-Tate type v. Proof. It follows immediately from Lemma 3.1. The goal of this subsection is to prove the following theorem: I such that A/I ⊗ A ρ ∼ = A/I ⊗ β,A ρ as A[G K ]-modules. Let V be the free A[ 1 p ]-module of rank d equipped with the G K -action corresponding to ρ ⊗ Zp Q p , and similarly let V be corresponding to ρ ⊗ Zp Q p . If I ⊂ p c 1 A and V has Hodge-Tate type v, then V also has Hodge-Tate type v. When k is further assumed to be finite, Theorem 3.3 is proved in [Liu15, Theorem 4.3.4]. The proof in [Liu15] is based on reducing to the case when E contains the Galois closure of K , and thus require k to be finite. We remove such a restriction in the following. Since E is a finite extension of Q p , we have a ring isomorphism E K = E ⊗ Qp K ∼ = s j=1 H j for some fields H j finite over K. Note that each H j is an E K -algebra via E K ∼ = s i=1 H i q j → H j where q j is the natural projection onto the j-th factor. For any E K -module M , we write M j := M ⊗ E K H j . Then M ∼ = ⊕ s j=1 M j . For a filtered E K -module D K , we denote (Fil i D K ) j and (gr i D K ) j by Fil i j D K and gr i j D K respectively. Since any finite E K -module is projective, we have gr i j D K ∼ = Fil i j D K /Fil i+1 j D K . Write B H j := B ⊗ E H j .i j D K dR (V B ) is B H j -free and rank B H j gr i j D K dR (V B ) = dim H j gr i j D E,K for all j = 1, . . . , s and i ∈ Z. Proof. This follows from the same argument as in the proof of [Liu15, Lemma 4.1.4]. Proof. This follows from the same argument as in the proof of [Liu15, Proposition 4.1.5]. B red = B/N (B) is a reduced Artinian E-algebra, so B red ∼ = m l=1 E l for some field E l finite over E. We set V E l = E l ⊗ B V B . The following lemma is useful when we consider an extension of the coefficient field E. Lemma 3.6. Let H be a field, and let C be a field (possibly of an infinite degree) over H. Let H be a finite extension of H, and let R and T be finite Proof. We have an induced injection of finite E-algebras B red → B red . By Lemma 3.5, we can reduce to the case when B and B are fields. Then it follows from Lemma 3.4 and Lemma 3.6. extensions of H . If M is a C ⊗ H R-module such that M ⊗ H T is a finite free C ⊗ H R ⊗ H T -module, then M is finite free over C ⊗ H R. As we will apply the functor M st to G K -representations semi-stable over K , we need to consider D K dR (V B ) := (B dR ⊗ Qp V ∨ B ) G K . Note that D K dR (V B ) = D K dR (V B ) ⊗ K K . Thus, by essentially the same argument as in the proof of Lemma 3.7, we see that V B has Hodge-Tate type v if and only if gr i D K dR (V B ) ∼ = B ⊗ E gr i D E,K as B K -modules for all i. Here, D E,K := D E ⊗ Qp K = D E,K ⊗ K K which has the induced filtration from D E,K . Let K 1 ⊂ E be the maximal unramified subextension over Q p . Then K 1 = W (k 1 )[ 1 p ] for some finite field k 1 , and E/K 1 is totally ramified. Choose a uniformizer E ∈ E, and letF (u) be its Eisenstein polynomial over K 1 . Let G(u) be a monic irreducible polynomial in Q p [u] such that K 1 ∼ = Q p [u]/G(u)Q p [u], and let G(u) = m j=1 G j (u) be the decomposition into monic irreducible polynomials in K 0 [u]. Note that G j (u) ∈ W (k) [u]. Denote byḠ j (u) ∈ k[u] the reduction of G j (u) mod p. ThenḠ j (u) is irreducible in k[u] and k[u]/Ḡ j (u)k[u] ∼ = l j for a finite extension l j /k. By the Chinese remainder theorem, W (k 1 ) ⊗ Zp W (k) ∼ = m j=1 W (l j ). SinceF (u) is irreducible over W (l j )[ 1 p ] for each j, we have E ⊗ Qp K 0 ∼ = m j=1 L j and O E ⊗ Zp W (k) ∼ = m j=1 O L j where L j := (W (l j )[ 1 p ])( E ) . For each j = 1, . . . , m, let F (u) = t j s=1 F js (u) be the decomposition of F (u) into monic irreducible polynomials in L j [u], and choose a root js of F js (u) for each s. Then L j ⊗ K 0 K ∼ = t j s=1 L j [u]/F js (u)L j [u] ∼ = t j s=1 T js where T js := L j ( js ). Thus, we have ring isomorphisms Repeating this argument for all s = 1, . . . , w and considering all possible decompositions of F (u) into irreducible factors over some finite field over K 0 , we see that there exists a positive integer c depending only on K 0 and F (u) such that for any L finite over K 0 , if we write L ⊗ K 0 K ∼ = Let C be a finite flat O E -algebra, and let Λ ∈ Rep pst,K ,r C such that Λ is a finite free C-module of rank d and Λ[ 1 p ] has Hodge-Tate type v. Since O E is henselian, C ∼ = n j=1 C j where each C j is a finite flat local O E -algebra. We say C is good if for each j = 1, . . . , n, there exists a prime ideal E ⊗ Qp K ∼ = (E ⊗ Qp K 0 ) ⊗ K 0 K ∼ = m j=1 t j s=1 T js .p j ⊂ C j such that C j /p j ∼ = O F j for some finite extension F j /Q p . Let L K := M st (Λ) K . Lemma 3.11. Suppose C is good. Then L K is finite free over C K := C ⊗ Zp O K of rank d. Proof. By Theorem 2.5, there exists a unique Kisin module M ∈ Mod r S (ϕ,Ĝ) such that T ∨ (M) = Λ. Write S C := C ⊗ Zp S. By the construction of the functor M st in Section 2.2, it suffices to show that M is a finite free S C -module of rank d. The Kisin module corresponding to C j ⊗ C Λ is C j ⊗ C M, so we may assume without loss of generality that n = 1 and so that C is a local ring. The Kisin module corresponding to By Nakayama's lemma, we have a surjection f : O F 1 ⊗ C Λ (via C/p 1 ∼ = O F 1 ) is M := O F 1 ⊗ C M.d i=1 S C · e i M of S C -modules. Λ is a free Z p -module of rank [C : Z p ]d, so M is free over S of rank [C : Z p ]d. Thus, f is an isomorphism. Suppose that C is good and that there exists an ideal J ⊂ C such that Proof of Theorem 3.3. Given above results, Theorem 3.3 follows from essentially the same argument as in the proof of [Liu15,Theorem 4.3.4], except that we do not reduce to the case where E contains the Galois closure of K . We also remark that the proof of [Liu15] reduces to the case A is local. But A is not necessarily finite over O E after such reduction, which has been overlooked in [Liu15]. This is a very minor gap, and we remedy it by only reducing to the case A is good. C/J ∼ = O E /p b O E for some positive integer b. For s = 1, . . . , t, we set C[ 1 p ] s := (C[ 1 p ] ⊗ Qp K ) ⊗ E K , We first reduce to the case where A = O E and A is good. For this, let B := A ⊗ Zp Q p and B := A ⊗ Zp Q p . We have B red = B/N (B) ∼ = j E j and B red ∼ = E j for some E j , E j finite over E. Let L be a finite Galois extension of E containing all Galois closures of E j , E j . Denote O L ⊗ O E ( * ) by ( * ) O L for ( * ) being A, A , ρ, ρ , I, and β. Note that A l /I l . Thus, by replacing E by L and A by A O L , we can assume that A = O E and that A is good. (A O L [ 1 p ]) red = L ⊗ E B red = L ⊗ E E j ∼ = i L with E j embedding into L differently, Let T denote the torsion representation A/I ⊗ A ρ ∼ = A /I ⊗ A ρ ∈ Rep pst,K ,r tor,O E where I = ker(β). We denote by j and j the two lifts ρ and ρ of T respectively. Write and Lemma 3.9 respectively. Assume I ⊂ pcA = pcO E . We claim that if gr 0 s (D K dR (V )) = 0, then gr 0 s (D K dR (V )) = 0. Suppose gr 0 s (D K dR (V )) = 0. By Corollary 3.10, gr 0 s M K is killed by p c . But by Lemma 3.12, there exists x ∈ gr 0 s M K such that p c +2c x = 0. We have a contradiction since p c +2c x = g 0 s (p c h 0 s (x)). On the other hand, let B = A [ 1 p ], and denote d 0 = dim Ts gr 0 s (D K dR (V )). We claim (assuming I ⊂ pcO E ) that d 0 ≤ dim Ts gr 0 s (D K dR (V )). For this, note that as an O Ts -module, gr 0 s L K = N tor ⊕ N where N tor is the torsion submodule of gr 0 s L K and N is a finite free O Ts -module of rank d 0 . By Corollary 3.10, dim Ts gr i s (D K dR (V )) = rank B Ts gr i s (D K dR (V )). Suppose that the above equality fails for some i, and let i * be the smallest such number. gr 0 s M K ∼ = N tor ⊕ d 0 i=1 O Ts /IOWrite d i = dim Ts gr i s (D K dR (V )) and d i = rank B Ts gr i s (D K dR (V )). Suppose first d i * > d i * . We set t 1 = i≤i * d i and t 2 = i≤i * id i . Letĩ = max{i \| j≤i d j ≤ t 1 } and t = i≤ĩ d i . Then i * ≤ĩ and t ≤ t 1 . Let t = ( i≤ĩ id i ) + (t 1 − t )(ĩ + 1). We have t 2 < t . Moreover, t 2 (resp. t ) is the smallest i such that gr i s (D K dR ( t 1 V )) (resp. gr i s (D K dR ( t 1 V ))) is nontrivial. Let χ be a crystalline character such that gr i s (D K dR (χ)) = 0 only when i = −t 2 . Then gr 0 s (D K dR (χ t 1 V )) is nontrivial. From the above result applied to χ t 1 V and χ t 1 V , we see that gr 0 s (D K dR (χ t 1 V )) is also nontrivial, leading to a contradiction. By switching the roles of V and V , it follows similarly that we cannot have d i * < d i * . This completes the proof. Galois Type We now study the Galois types of potentially semi-stable representations. As in Section 3.1, let E be a finite field over Q p , and let B be a finite E-algebra. Let V B be a free B-module of rank d equipped with a potentially semi-stable continuous G K -action. Let D pst (V B ) = lim −→ K⊂K (B st ⊗ Qp V ∨ B ) G K , where the limit goes over finite extensions K of K contained inK. Denote by K ur 0 the union of finite unramified extensions of K 0 contained inK. We have dim K ur 0 D pst (V B ) = dim Qp V B . Lemma 3.13. Let B be a finite B-algebra, and write V B = B ⊗ B V B . Then V B is potentially semi-stable as a G K -representation, and D pst (V B ) ∼ = B ⊗ B D pst (V B ). If V B becomes semi-stable over L ⊃ K, then so does V B . Furthermore, D pst (V B ) is a free B ⊗ Qp K ur 0 -module of rank d. Proof. It follows from essentially the same proof as for Lemma 3.1. D pst (V B ) is equipped with a K ur 0 -semilinear action of G K , and thus a K ur 0 -linear action of the inertia group I K . The Frobenius action commutes with the I K -action, so tr(σ\|D pst (V B )) ∈ B for all σ ∈ I K . Let D E be an E-vector space of dimension d, and let D E,K = D E ⊗ Qp K equipped with a filtration giving a Hodge-Tate type v. Fix a representation τ : I K → End E (D E ) with an open kernel. Note that there exists an I K -stable O E -lattice in D E , so tr(τ (σ)) ∈ O E for all σ ∈ I K . We say V B has Galois type τ if the I K -representation D pst (V B ) is equivalent to τ , i.e., tr(σ\|D pst (V B )) = tr(τ (σ)) for all σ ∈ I K . Let L/K be a finite Galois extension contained inK such that I L ⊂ ker(τ ). Here, I L denotes the inertia subgroup of G L . D L st (V B ) = (B st ⊗ Qp V ∨ B ) G L is an L 0 -vector space where L 0 is the maximal unramified subextension of K 0 contained in L. If V B is semi-stable over L, then D pst (V B ) ∼ = K ur 0 ⊗ L 0 D L st (V B ) . Therefore, V B has Galois type τ if and only if V B is semi-stable over L and tr(σ\|D L st (V B )) = tr(τ (σ)) for all σ ∈ I L/K , where I L/K is the inertia subgroup of Gal(L/K). Lemma 3.14. Let α : B → B be an E-algebra morphism between finite E-algebras. Suppose V is semi-stable over L. Then for all σ ∈ I L/K , we have tr(σ\|D L st (V B )) = α(tr(σ\|D L st (V B ))). In particular, if V B has Galois type τ , then so does V B . If α is injective, then the converse is also true, i.e., V B has Galois type τ if and only if V B has Galois type τ . Proof. D pst (V B ) ∼ = B ⊗ B D pst (V B ) by Lemma 3.13, so tr(σ\|D L st (V B )) = α(tr(σ\|D L st (V B ))) for all σ ∈ I L/K . The remaining statements follow immediately. Consider the case when B is local. If E is its residue field, then E is finite over E and B is naturally an E -algebra. Note that the I K -action on D pst (V B ) has an open kernel. Since the cohomology of a finite group with coefficients in E ⊗ Qp K ur 0 is trivial in all positive degrees, it follows from the deformation theory that the representation D pst (V B ) arises from a representation over E ⊗ Qp K ur 0 . Thus, V B has Galois type τ if and only if V E = E ⊗ B V B has Galois type τ . For a general finite E-algebra B, we have isomorphisms B ∼ = n i=1 B m i and B red ∼ = n i=1 E i , where m 1 , . . . , m n are the maximal ideals of B and E i = B m i /m i B m i . Let V E i = E i ⊗ B V B . The following lemmas are analogous to Lemma 3.5 and 3.7. Lemma 3.15. V B has Galois type τ if and only if V E i has Galois type τ for each i = 1, . . . , n. Proof. It follows directly from Lemma 3.14. Proof. Since the natural map of E-algebras B → B E is injective, it follows from Lemma 3.14. The following theorem is essential in studying the locus of representations with a given Galois type. Theorem 3.17. Let τ be a Galois type, and let L/K be a finite Galois extension inK over which τ becomes trivial. Let A be a finite flat O E -algebra and ρ : G K → GL d (A) be a Galois representation such that ρ ⊗ Zp Q p is semi-stable over L having Hodge-Tate weights in [0, r]. Suppose that for each positive integer n, there exist a finite flat O E -algebra A n , a Galois representation ρ n : G K → GL d (A n ), and an O E -linear surjection β n : A n → A/p n A such that A/p n A ⊗ A ρ ∼ = A/p n A ⊗ βn,An ρ n as A[G K ]-modules, and that ρ n ⊗ Zp Q p is semi-stable over L having Hodge-Tate weights in [0, r] and Galois type τ . Then ρ ⊗ Zp Q p also has Galois type τ . Since this holds for all positive integers n, we have tr(σ\|M st (ρ)) = tr(τ (σ)). Galois Deformation Ring We now construct the quotient of the universal deformation ring which corresponds to the locus of potentially semi-stable representations of a given Hodge-Tate type and Galois type. Let E/Q p be a finite extension with residue field F. Denote by C the category of topological local O E -algebras A satisfying the following conditions: • The natural map O E → A/m A is surjective. • The map from A to the projective limit of its discrete artinian quotients is a topological isomorphism. Note that the first condition implies F is also the residue field of A. The second condition is equivalent to the condition that A is complete and its topology can be given by a collection of open ideals a for which A/a is artinian. Morphisms in C are continuous O E -algebra homomorphism. 4]) Suppose A is a noetherian ring in C. Then the topology on A is equal to the m A -adic topology, and A is m A -adically complete. Furthermore, every O E -algebra homomorphism A → A with A in C is continuous. Let V 0 be a continuous F-representation of G K having rank d. For A ∈ C, a deformation of V 0 in A is an isomorphism class of continuous A-representations V of G K satisfying F ⊗ A V ∼ = V 0 as F[G K ]-modules. We denote by Def(V 0 , A) the set of such deformations. A morphism A → A in C induces a map f * : Def(V 0 , A) → Def(V 0 , A ) sending the class of a representation V over A to the class of A ⊗ f,A V . Assume V 0 is absolutely irreducible. Then, the following is proved in [SL97]. Proposition 4.2. (cf. [SL97, Theorem 2.3]) There exists a universal deformation ring R ∈ C and a deformation V R ∈ Def(V 0 , R) such that for all rings A ∈ C, we have a bijection Hom C (R, A) ∼ = → Def(V 0 , A) (4.1) given by f → f * (V R ). The ring R is noetherian if and only if dim F H 1 (G K , End F (V 0 )) is finite. Note that if K/Q p is not finite, then R is not necessarily noetherian in general. We fix a Hodge-Tate type v and Galois type τ , and let L/K be a finite Galois extension over which τ becomes trivial. Let C 0 be the full subcategory of C consisting of artinian rings. Abusing the notation, we write V ∈ Def(V 0 , A) for a continuous A-representation V to mean that F ⊗ A V ∼ = V 0 . For A ∈ C 0 and a G K -representation V A ∈ Def(V 0 , A), we say V A is potentially semi-stable of type (v, τ ) if there exist a finite flat O E -algebra B, a surjection g : B → A of O E -algebras, and a continuous B-representation V B of G K such that V B ⊗ Zp Q p is potentially semi-stable having Hodge-Tate type v and Galois type τ , and A ⊗ g,B V B ∼ = V A as A[G K ]-modules. For A ∈ C, denote by S v,τ (A) the subset of Def(V 0 , A) consisting of the isomorphism classes of representations V A such that A/a ⊗ A V A is potentially semi-stable of type (v, τ ) for all open ideals a A. Proposition 4.3. For any C-morphism f : A → A , we have f * (S v,τ (A)) ⊂ S v,τ (A ). There exists a closed ideal a v,τ of the universal deformation ring R such that the map (4.1) induces a bijection Hom C (R/a v,τ , A) ∼ = → S v,τ (A). Proof. We check the conditions in [SL97,Section 6]. Let f : A → A be an inclusion of artinian rings in C, and let V A ∈ Def(V 0 , A) be a representation. We first claim that Note that B is a finite flat O E -algebra. Let V B = B ⊗ B V B . Then V B ⊗ Zp Q p is semistable over L. By Lemma 3.2 and 3.14, it has Hodge-Tate type v and Galois type τ . A ⊗ g ,B V B ∼ = V A , so V A ∈ S v,τ (A ). Conversely, suppose V A ∈ S v,τ (A ). Then there exist a finite flat O E -algebra B , a surjection g : B → A , and a B -representation V B such that V B ⊗ Zp Q p is potentially semi-stable having Hodge-Tate type v and Galois type τ , and we have an isomorphism h : A ⊗ g ,B V B ∼ = −→ V A of A -representations. Since O E is henselian, B is a finite product of local rings flat over O E . By Lemma 3.2 and Lemma 3.14, we can take B to be a local ring, since A is local. Thus, we can lift the isomorphism h to an isomorphism of B -modules V A ∈ S v,τ (A) if and only if V A := A ⊗ f,A V A ∈ S v,τ (A ). Suppose that V A ∈ S v,h 1 : V B ∼ = −→ V B such that the composite A ⊗ g ,B V B id⊗h −1 1 −→ A ⊗ g ,B V B h −→ V A is the identity map of A -modules. Let B be the kernel of the composite of morphisms B g → A → A /f (A). Then B is a finite flat O E -algebra, and we have the surjection g : B A of O E -algebras induced from g . Let V B be the kernel of the following composite of morphisms V B h −1 1 −→ V B → A ⊗ g ,B V B h −→ V A → A /f (A) ⊗ A V A . Then V B is a continous B-representation of G K such that B ⊗ B V B ∼ = V B and A ⊗ g,B V B ∼ = V A , since V A = A ⊗ f,A V A . By the main theorem for semi-stable representations in [Liu07], V B ⊗ Zp Q p is semi-stable over L. It has Hodge-Tate type v and Galois type τ by Lemma 3.8 and 3.14, and therefore V A ∈ S v,τ (A). Now, for A ∈ C and a representation V A ∈ Def(V 0 , A), suppose a 1 , a 2 A are open ideals such that A/a i ⊗ A V A ∈ S v,τ (A/a i ) for i = 1, 2. We claim that A/(a 1 ∩ a 2 ) ⊗ A V A ∈ S v,τ (A/(a 1 ∩ a 2 )). There exist a finite flat O E -algebra B i , a surjection g i : B i A/a i , and a B i -representation V B i such that V B i ⊗ Zp Q p is potentially semi-stable having Hodge-Tate type v and Galois type τ , and that A/a i ⊗ g i ,B i V B i ∼ = A/a i ⊗ A V A . Let V B 1 ×B 2 be the (B 1 ×B 2 )-representation corresponding to V B 1 ⊕V B 2 . Note that V B 1 ×B 2 ⊗ Zp Q p is potentially semi-stable having Hodge-Tate type v and Galois type τ . Consider the natural inclusion A/(a 1 ∩ a 2 ) ⊂ A/a 1 × A/a 2 . Let B be the kernel of the composite of morphisms B 1 × B 2 g 1 ×g 2 −→ A/a 1 × A/a 2 → (A/a 1 × A/a 2 )/(A/(a 1 ∩ a 2 )). Then B is a finite flat O E -algebra, and we have the surjection g : B → A/(a 1 ∩ a 2 ) induced from g 1 × g 2 . Let V B be the kernel of the composite of morphisms V B 1 ×B 2 → (A/a 1 × A/a 2 ) ⊗ g 1 ×g 2 ,B 1 ×B 2 V B 1 ×B 2 ∼ = (A/a 1 × A/a 2 ) ⊗ A V A and (A/a 1 × A/a 2 ) ⊗ A V A → (A/a 1 × A/a 2 )/(A/(a 1 ∩ a 2 )) ⊗ A V A . Then V B is a continuous B-representation of G K such that (B 1 × B 2 ) ⊗ B V B ∼ = V B 1 ×B 2 and A/(a 1 ∩ a 2 ) ⊗ g,B V B ∼ = A/(a 1 ∩ a 2 ) ⊗ A V A . By the main theorem for semi-stable representations in [Liu07], V B ⊗ Zp Q p is semi-stable over L. It has Hodge-Tate type v and Galois type τ by Lemma 3.8 and 3.14. Thus, A/(a 1 ∩ a 2 ) ⊗ A V A ∈ S v,τ (A/(a 1 ∩ a 2 )). The result then follows by [SL97, Proposition 6.1]. Finally, we prove the main theorem. Proof. Let A 1 = f (R) ⊂ A. Then A 1 is a finite flat O E -algebra and local. We equip A 1 with the (p)-adic topology. Then A 1 ∈ C, and the map f : A → A 1 is a morphism in C. Let V A 1 = A 1 ⊗ f,R V R and V A = A ⊗ f,R V R ∼ = A ⊗ A 1 V A 1 . Suppose that V A ⊗ Zp Q p is potentially semi-stable of Hodge-Tate type v and Galois type τ . By the main theorem for semi-stable representations in [Liu07], V A 1 ⊗ Zp Q p is semi-stable over L. By Lemma 3.8 and 3.14, V A 1 ⊗ Zp Q p has Hodge-Tate type v and Galois type τ . Thus, V A 1 ∈ S v,τ (A 1 ), and f factors through R/a v,τ by Proposition 4.3. Conversely, suppose f factors through R/a v,τ . Then V A 1 ∈ S v,τ (A 1 ) by Proposition 4.3, so A 1 /p n ⊗ A 1 V A 1 is potentially semi-stable of type (v, τ ) for each n ≥ 1. By the main theorem for semi-stable representations in [Liu07], V A 1 ⊗ Zp Q p is semi-stable over L. And by Theorem 3.3 and 3.17, V A 1 ⊗ Zp Q p has Hodge-Tate type v and Galois type τ . Thus, V A ⊗ Zp Q p is potentially semi-stable of Hodge-Tate type v and Galois type τ . Let r be a positive integer. A Kisin module of height r is a pair (M, ϕ M ) where M is a finite free S-module, and ϕ M : M → M is a ϕ-semi-linear map such that the cokernel of the induced map 1 ⊗ ϕ M : ϕ * (M) → M is killed by F (u) r . A morphism between two Kisin modules M 1 , M 2 is a morphism as S-modules compatible with ϕ M i . Let Mod r S (ϕ) denote the category of Kisin modules of height r. For (M, ϕ M ) ∈ Mod r S (ϕ), we writê M =R ⊗ ϕ,S M. The Frobenius ϕ M on M naturally extends toM by ϕM(a ⊗ m) = ϕR(a) ⊗ ϕ M (m). Definition 2.4. A (ϕ,Ĝ)-module of height r is a triple (M, ϕ M ,Ĝ M ) satisfying the following: • (M, ϕ M ) is Kisin module of height r. T ∨ induces an anti-equivalence between Mod r S (ϕ,Ĝ) and the category of G K -stable Z p -lattices in semi-stable representations of G K having Hodge-Tate weights in [0, r]. 2.T ∨ induces a natural W (O Cp )-linear injection ι : W (O Cp ) ⊗RM → W (O Cp ) ⊗ ZpT (M)such thatι is compatible with Frobenius maps and G K -actions on both sides. Here, T (M) := Hom Zp (T ∨ (M), Z p ). Qp, and let T ⊂ V be a G K -stable Z p -lattice. By Theorem 2.5, there exists a unique M ∈ Mod r S (ϕ,Ĝ) such thatT ∨ (M) = T as Z p [G K ]modules. Let D := S K 0 ⊗ ϕ,S M equipped with the Frobenius endomorphism given byϕ D = ϕ S K 0 ⊗ϕ M . Let D = D/(I + S K 0 )D,which is a finite K 0 -vector space equipped with the Frobenius induced from ϕ D . By [Bre97, Proposition 6.2.1.1], there exists a unique section s : D → D compatible with the Frobenius morphisms on both sides. Thus, D = S K 0 ⊗ K 0 D if we identify D with s(D). So B + cris ⊗RM ∼ = B + cris ⊗ K 0 D, and the mapι given in Theorem 2.5 (2) induces a natural injection D → B + cris ⊗ Zp T ∨ where T ∨ := Hom Zp (T, Z p ). On the other hand, the functor D K st induces an injection With notations as in Theorem 2.7, assume that f : L → L is an isomorphism with the inverse f −1 : L → L. Leth 1 : M st,j (L) → M st,j (L ) be the morphism as in Theorem 2.7 associated to f −1 , j, and j . Thenh •h 1 = p 2c Id on M st,j (L) andh 1 •h = p 2c Id on M st,j (L ). Furthermore, the similar statement holds for the induced morphisms on gr i (M st,j (L) K ) and gr i (M st,j (L ) K ). A-modules such that G K -actions are A-linear. Morphisms in Rep pst,K ,r A are morphisms of A[G K ]-modules. Let Rep pst,K ,r tor,A be the subcategory of Rep pst,K ,r tor whose objects have lifts in Rep pst,K ,r A , and the morphisms are A[G K ]-module morphisms. For L ∈ Rep pst,K ,r tor,A having a lift j : L 1 → L 2 in Rep pst,K ,r A , note that M st (L 1 ) and M st (L 2 ) are naturally A ⊗ Zp W (k)-modules, and thus so is M st,j (L). Proposition 2.9. Let f : L → L be a morphism in Rep pst,K ,r tor,A , and let j and j be lifts in Rep pst,K ,r A of L and L respectively. Then, the associated morphismh : M st,j (L ) → M st,j (L) in M fil,r tor (ϕ, N, Γ) as in Theorem 2.7 is a morphism of A ⊗ Zp W (k)-modules. Proof. It follows immediately from [Liu12, Proposition 3.13] and [Liu15, Lemma 4.2.4] . Theorem 3.3. (cf. [Liu15, Theorem 4.3.4]) There exists a constant c 1 depending only on K , r, and d such that the following statement holds: Let A and A be finite flat O E -algebras and let ρ : G K → GL d (A) and ρ : G K → GL d (A ) be representations such that ρ ∈ Rep pst,K ,r A and ρ ∈ Rep pst,K ,r A . Suppose that there exist an ideal I ⊂ A such that A/I is killed by a power of p and an O E -linear surjection β : A A/ Lemma 3. 4 . 4(cf. [Liu15, Lemma 4.1.4]) With notations as above, V B has Hodge-Tate type v if and only if gr Lemma 3.5. (cf. [Liu15, Proposition 4.1.5]) V B has Hodge-Tate type v if and only if V E l has Hodge-Tate type v for each l = 1, . . . , m. Proof. M is a finite projective C ⊗ H R-module, and there exists a surjection f :M ⊗ H T M of C ⊗ H R-modules having a section. Let {e 1 , . . . , e n } be a basis of M ⊗ H T over C ⊗ H R ⊗ H T . Let N := ⊕ n i=1 (C ⊗ H R) · f (e i ). Then the natural map N → M of C ⊗ H R-modules is an injection since {e 1 , . . . , e n } is a basis of M ⊗ H T over C ⊗ H R ⊗ H T . Furthermore, dim C N = dim C M , so it is bijective. Let L be a finite extension of E, and write B L := L ⊗ E B. Given v as above, let v = (D L := L ⊗ E D, {Fil i D L,K = L ⊗ E Fil i D E,K } i=0,...,r ). Lemma 3.7. (cf. [Liu15, Lemma 4.1.6]) With notations as above, V B has Hodge-Tate type v if and only if V B L := B L ⊗ B V B has Hodge-Tate type v . Proof. Given Lemma 3.6, it follows from the same argument as in the proof of [Liu15, Lemma 4.1.6]. Lemma 3 . 8 . 38Suppose we have an injection B → B of finite E-algebras. If V B = B ⊗ B V Bhas Hodge-Tate type v, then also V B has Hodge-Tate type v. Let t = m j=1 t j . After re-indexing the fields T js , we have E ⊗ Qp K ∼ =t s=1 T j , and the statement analogous to Lemma 3.4 holds for D K dR (V B ). Let O E,K := O E ⊗ Zp O K . The projection q s : E K → T s induces the map O E,K → O Ts , and we have the natural map q : O E,K → t s=1 O Ts . Denote by v p the p-adic valuation normalized by v p (p) = 1. Lemma 3 . 9 . 39There exists a positive integer c depending only on K 0 and F (u) such thatp c ( t s=1 O Ts ) ⊂ q(O E,K ). Proof.For a field L finite over K 0 , let F (u) = w s=1 F s (u) be the decomposition of F (u) into monic irreducible polynomials in L[u], and choose a root s of F s (u) for each s. We haveL ⊗ K 0 K ∼ L s := L( s ). Let q s : L ⊗ K 0 K → Ls be the composition of the above isomorphism followed by the projection onto the s-th factor. Then q s induces a surjection O L ⊗ Zp O K O L s , and we have the natural map O L ⊗ Zp O K → w s=1 O L s . Under this map, h =s F h (π) maps to (0, . . . , 0, h =s F h ( s ), 0, . . . , 0) whose components are 0 except the sth component. Write v p ( h =s F h ( s )) = a b for some relatively prime positive integers a, b. Then ( h =s F h ( s )) b = p a x for some x ∈ O × Ls with v p (x) = 0. Thus, (0, . . . , 0, p a , 0, . . . , 0), whose components are 0 except the s-th component, lies in the image of O L ⊗ Zp O K under the above map. s ) as above, then for each s, (0, . . . , 0, p c , 0, . . . , 0) whose components are 0 except the s-th component lies in the image of O L ⊗ Zp O K . Applying this for each L j , we get the result. Corollary 3 . 10 . 310Let M be a torsion free O E,K -module. Then the torsion part of M s := M ⊗ O E,K ,qs O Ts is killed by p c , where c is the constant given in Lemma 3.9. Proof. Let M = ⊕ t s=1 M s . By Lemma 3.9, there exist morphisms of O E,K -modules q M : M → M and s M : M → M such that q M • s M = p c Id\| M . Let x be a torsion element in M . Then s M (x) = 0, so p c x = q M (s M (x)) = 0. Since M is finite free over S, M /uM is p-torsion free. Thus, M /uM is a projectiveO F 1 ⊗ Zp W (k)-module. Since (M /uM )[ 1 p ]is isomorphic to its pullback by ϕ and ϕ permutes the maximal ideals of O F 1 ⊗ Zp W (k) transitively, M /uM is a free O F 1 ⊗ Zp W (k)-module of rank d. Thus, M is a free O F 1 ⊗ Zp S-module of rank d. qs T s , and define d s := rank C[ 1 p ]s (gr 0 s (D K dR (Λ[ 1 p ]))). Denote Fil i s L K := Fil i L K ⊗ O E,K ,qs O Ts , and similarly for the graded modules. By Lemma 3.11, Fil 0 s L K is free over C s := C K ⊗ O E,K ,qs O Ts of rank d. Lemma 3 . 312. (cf. [Liu15, Lemma 4.2.7]) Suppose that d s = 0. Let l be a positive integer satisfying b ≥ ld + 1. Then there exists x ∈ gr 0 s L K /Jgr 0 s L K such that p l x = 0.Proof. This follows from essentially the same argument as in the proof of [Liu15, Lemma 4.2.7]. For any C-module M , denote M/JM by M/J. We have the following right exact sequence:Fil 1 s L K → Fil 0 s L K → gr 0 s L K → 0. LetFil 1 s L K be the image of Fil 1 s L K in Fil 0 s L Kunder the first map in the above sequence. We then obtain the following right exact sequencẽFil 1 s L K /J → Fil 0 s L K /J → gr 0 s L K /J → 0.DenoteM := Fil 0 s L K /J and letN ⊂M be the submodule given by the image ofFil1 s L K /J. ThenM /N = gr 0 s L K /J. Suppose that p l annihilatesM /N . By Lemma 3.11,M is a finite free O Ts /p b O Ts -module of rank d. Letπ s be a uniformizer of O Ts . Then there exists an O Ts /p b O Ts -basisē 1 , . . . Ts /p b O Ts ) · (π a i sē i ) for some nonnegative integers a i . We haveπ a i s \| p l for all i = 1, . . . , d. Let e 1 , . . . , e d be a C s -basis of Fil 0 s L K which liftsē 1 , . . . ,ē d . For i = 1, . . . , d, let y i ∈Fil 1 s L K which liftsπ a i sē i . If X denotes the d × d-matrix such that (y 1 , . . . , y d ) = (e 1 , . . . , e d )X, then det(X) =π a s + j with a = d i=1 a i and j ∈ J. Since b ≥ ld + 1, we haveπ a s = 0 in C s /J, and thus det(X) = 0 in C s . On the other hand, letz 1 , . . . ,z ds be a C[ 1 p ] s -basis of gr 0 s (D K dR (Λ[ 1 p ])). We have det(X)(e 1 , . . . , e d ) ⊂ Fil 1 s (D K dR (Λ[ 1 p ])), and therefore det(X)z i = 0. This gives a contradiction. and similarly for (A O L [ 1 p ]) red . This induces the natural map ψ l : A O L → (A O L )[ 1 p ] L to the l-th factor of i L. By Lemma 3.5 and 3.7, it suffices to show (assuming I ⊂ p c 1 A for a suitable constant c 1 ) that L ⊗ ψ l ,A O L (ρ) O L has Hodge-Tate type v. Let A l = ψ l (A O L ) and I l = ψ l (I O L ). ψ l : A O L A l ⊂ L is a morphism of O L -algebras, so A l = O L (and analogously for A O L ), and we have a natural projection γ l : A O L /I O L L K := M st (ρ) K , L K := M st (ρ ) K , M K := M st,j (T ) K , and M K := M st,j (T ) K . We have gr i s M K ∼ = gr i s L K /Igr i s L K and gr i s M K ∼ = gr i s L K /I gr i s L K for s = 1, . . . , t. By Corollary 2.8 and Proposition 2.9, there exist morphisms of O Ts -modules g i s : gr i s M K → gr i s M K and h i s : gr i s M K → gr i s M K such that g i s • h i s = p 2c Id\| gr i s M K and h i s • g i s = p 2c Id\| gr i s M K . Now, we setc =c(K , r, d) := (2c + c )d + 1 where c and c are given as in Theorem 2.7 O Ts /IO Ts . Then p c gr 0 s M K =N , again by Corollary 3.10, and therefore h 0 s (g 0 s (N )) ∼ = d 0 i=1 p 2c+c O Ts /IO Ts . Since p c gr 0 s L K surjects onto h 0 s (p c gr 0 s M K ) and g 0 s (N ) ⊂ p c gr 0 s M K , we have by Corollary 3.10 that the O Ts -rank of p c gr 0 s L K is at least d 0 . Thus, the O Ts -rank of gr 0 s L K is at least d 0 , and dim Ts gr 0 s (D K dR (V )) ≥ d 0 . Hence, assuming I ⊂ pcO E , we have gr 0 s (D K dR (V )) = 0 if and only if gr 0 s (D K dR (V )) = 0. For the last step, we set c 1 =c(K , dr, d) and assume I ⊂ p c 1 O E . It suffices to show that for each i, Lemma 3 . 16 . 316Let E be a finite extension of E, and let B E = E ⊗ E B and V B E = B E ⊗ B V B . Then V B has Galois type τ if and only if V B E has Galois type τ . Proposition 4.1. ([SL97, Proposition 2. τ (A). Then there exist a finite flat O E -algebra B, a surjection g : B → A, and a B-representation V B such that V B ⊗ Zp Q p is potentially semi-stable having Hodge-Tate type v and Galois type τ , andA ⊗ g,B V B ∼ = V A . There exists a surjection f : A[x 1 , . . . , x n ] A of O Ealgebras extending f such that f (x i ) ∈ m A for each i. Let I m,A ⊂ A[x 1 ,. . . , x n ] denote the ideal generated by the m-th degree homogeneous polynomials with coefficients in A. Since A is artinian, f (I m,A ) = 0 for a sufficiently large m, and f induces a surjection A[x 1 , . . . , x n ]/I m,A A for such m. Thus, we have surjective homomorphisms of O Ealgebras g : B := B[x 1 , . . . , x n ]/I m,B A[x 1 , . . . , x n ]/I m,A A . Theorem 4. 4 . 4Let A be a finite flat O E -algebra, and let f : R → A be a continuous O Ealgebra homomorphism (where we equip A with the (p)-adic topology). Then the induced representation A[ 1 p ] ⊗ f,R V R is potentially semi-stable of Hodge-Tate type v and Galois type τ if and only if f factors through the quotient R/a v,τ . Let Rep pst,K ,r Qp be the full subcategory of Rep pst,K Qp whose objects have Hodge-Tate weights in [0, r], and let Rep pst,K ,r Zp be the category of G K -stable Z p -lattices of representations in Rep pst,K ,r Qp . The following theorem is proved in [Liu12]: Theorem 2.2. (cf. [Liu12, Theorem 2.3]) There exists a faithful contravariant functor M st from Rep pst,K ,r Zp to L r (ϕ, N, Γ). If we denote by M st ⊗ Zp Q p the functor M st associated to the isogeny categories, then there exists a natural isomorphism of functors between M st ⊗ Zp Q p and D K st . We now explain briefly the construction in [Liu12] of the functor M st in Theorem 2.2. We first recall the definitions of period rings in p-adic Hodge theory. Let S be the p-adic completion of the divided power-envelope of S = W (k)[[u]] with respect to the ideal (F (u)). Denote S K 0 := S [ 1 p ]. Let C p be the p-adic completion ofK, and let O Cp be its ring of integers. We define O Cp := lim ← − x →x p O Cp /p. By the universal property of the ring of Witt vectors W (O Cp ), there exists a unique surjection θ : W (O Cp ) → O Cp , which lifts the projection O Cp → O Cp /p onto the first factor of the inverse limit.2.2 Construction of M st We denote by B + dR the ker(θ)-adic completion of W (O Cp )[ 1 p ] embedding S → W (O Cp ) mapping u to [π], and hence the embeddings S → S → A cris compatible with Frobenius endomorphisms. Let B + cris = A cris [ 1 p ]. Let u = log[π], and B + st = B + cris We have B red ∼ = i E i for some finite extensions E i /E. Let H/E be a finite Galois extension containing the Galois closures of E i for all i. We writeand replace β n and ρ n accordingly.Denote by L 0 the maximal unramified extension of K 0 contained in L. Note that I L/K ∼ = I L/KL 0 . Applying the results of Section 2 with (L 0 , KL 0 , L) in place of (K 0 , K, K ), we get the associated lattice Représentations p-adiques semi-stables et transversalité de Griffiths. Christophe Breuil, Math. Ann. 3072Christophe Breuil, Représentations p-adiques semi-stables et transversalité de Grif- fiths, Math. Ann. 307 (1997), no. 2, 191-224. Construction des représentations padiques semi-stables. Pierre Colmez, Jean-Marc Fontaine, Invent. Math. 1401Pierre Colmez and Jean-Marc Fontaine, Construction des représentations p- adiques semi-stables, Invent. Math. 140 (2000), no. 1, 1-43. Représentations l-adiques potentiellement semi-stables. Jean-Marc Fontaine, Astérisque. 223Jean-Marc Fontaine, Représentations l-adiques potentiellement semi-stables, Astérisque 223 (1994), 321-347. Deforming semistable Galois representations. Proc. Nat. Acad. Sci. U.S.A. 94, Deforming semistable Galois representations, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 11138-11141. Potentially semi-stable deformation rings. Mark Kisin, J. Amer. Math. Soc. 212Mark Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2007), no. 2, 513-547. Tong Liu, Torsion p-adic Galois representations and a conjecture of Fontaine. 40Tong Liu, Torsion p-adic Galois representations and a conjecture of Fontaine, Ann. Sci.École Norm. Sup. 40 (2007), no. 4, 633-674. A note on lattices in semi-stable representations. Math. Ann. 3461, A note on lattices in semi-stable representations, Math. Ann. 346 (2010), no. 1, 117-138. Lattices in filtered (φ, N )-modules. J. Inst. of Math. of Jussieu. 113, Lattices in filtered (φ, N )-modules, J. Inst. of Math. of Jussieu 11 (2012), no. 3, 659-693. Filtration associated to torsion semi-stable representations. Ann. Inst. Fourier. 655, Filtration associated to torsion semi-stable representations, Ann. Inst. Fourier 65 (2015), no. 5, 1999-2035. Explicit construction of universal deformation rings, Modular Forms and Fermat's Last Theorem. Bart De Smit, Hendrick W Lenstra, Springer-VerlagNew YorkBart De Smit and Hendrick W. Lenstra, Explicit construction of universal defor- mation rings, Modular Forms and Fermat's Last Theorem (New York), Springer- Verlag, 1997, pp. 313-326.	[]
[ "Dirac Cohomology on Manifolds with Boundary and Spectral Lower Bounds", "Dirac Cohomology on Manifolds with Boundary and Spectral Lower Bounds" ]	[ "Simone Farinelli " ]	[]	[]	Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to the Dirac Laplacian. A general method allowing to estimate Dirac spectral lower bounds for the Dirac spectrum of a compact Riemannian manifold in terms of the Dirac eigenvalues for a cover of 0-codimensional submanifolds is developed. Two applications are provided for the Atiyah-Singer operator. First, we prove the existence on compact connected spin manifolds of Riemannian metrics of unit volume with arbitrarily large first non zero eigenvalue. Second, we prove that on a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the first positive non zero eigenvalue is bounded from below by a positive uniform constant. *	null	[ "https://arxiv.org/pdf/1405.7162v2.pdf" ]	119,632,237	1405.7162	3b1aeced28fde45562e97c1df2981eda10381a80	Dirac Cohomology on Manifolds with Boundary and Spectral Lower Bounds 28 May 2014 May 29, 2014 Simone Farinelli Dirac Cohomology on Manifolds with Boundary and Spectral Lower Bounds 28 May 2014 May 29, 2014 Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to the Dirac Laplacian. A general method allowing to estimate Dirac spectral lower bounds for the Dirac spectrum of a compact Riemannian manifold in terms of the Dirac eigenvalues for a cover of 0-codimensional submanifolds is developed. Two applications are provided for the Atiyah-Singer operator. First, we prove the existence on compact connected spin manifolds of Riemannian metrics of unit volume with arbitrarily large first non zero eigenvalue. Second, we prove that on a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the first positive non zero eigenvalue is bounded from below by a positive uniform constant. * Introduction When dealing with direct and indirect spectral theory on Riemannian manifolds, the following question naturally arises. Given a formally selfadjoint operator of Laplace type (or Laplacian for short) over a compact Riemannian manifold, consider a finite open cover or a decomposition into 0-codimensional submanifolds with boundary and add an appropriate elliptic boundary condition. Is there a general principle allowing to find lower bounds of the spectrum of the manifold in terms of the spectra of the pieces? The first answer to this question is a dissection principle which was originally formulated for the Laplacian on functions on domains in R m by Courant and Hilbert ([CH93] and [Ch84]). The remarkable fact is that it still holds for any formally selfadjoint operator of Laplace type under Neumann boundary conditions, as conjectured by Bär [Bä91] for the Dirac Laplacian. where M 1 , . . . , M r are finitely many 0-codimensional submanifolds with (smooth) boundaries ∂M k and pairwise disjoint interiors. We require for a boundary ∂M k intersecting ∂M that it agrees with the corresponding boundary connected component of ∂M . The operators P BN and (P \| M k ) BN for the Neumann boundary condition are selfadjoint and have a discrete spectrum. We denote by (λ j ) j≥0 := spec(P BN ) the Neumann eigenvalues for the operator P on M , by (µ k j ) j≥0 := spec((P \| M k ) BN ) the Neumann eigenvalues of the operator P on M k and by (µ j ) j≥0 := r k=1 (µ k j ) j≥0 . All the sequences are ordered non decreasingly and all the eigenvalues are repeated according to their multiplicities. Then, the spectrum on the whole manifold can be bounded from below by all the spectra on the submanifolds as λ j ≥ µ j for all j ≥ 0.(2) The key ingredient in the proof is the density of the smooth sections satisfying the Neumann boundary condition in the Sobolev space H 1 (M, V ) with respect to the Sobolev H 1 norm, as it has been explicitly carried in out in [Fa98]. The main contribution of this paper is a technique allowing to estimate the lower spectral part of a general Dirac operator in terms of the spectra of a finite cover under the appropriate boundary conditions. The original idea in the case of differential forms is due to Cheeger but unpublished, based on the Meyer-Vietoris scheme, was carried out in [Go93]. In order to extend it to the set up of Dirac bundles, a Dirac complex as in non commutative differential geometry is introduced, as well as appropriate elliptic local boundary conditions for both Dirac and Dirac Laplacian. Concepts like derivation and coderivation and boundary conditions like absolute and relative ones can be extended from the context of differential forms to that of Dirac sections. If (V, ·, · , ∇, γ) is a Dirac bundle over the Riemannian manifold (M, g) with normalized orientation S, and if there exists a bundle isomorphism T on V such that γ := iT γ anticommutes with γ and with the Dirac operator Q. The tuple (V, ·, · , ∇, γ, γ) defines a (1, 1)-Dirac bundle structure with corresponding Dirac operators Q and Q. The operators d := 1 2 (Q − iQ) = 1 + T 2 Q and δ := 1 2 (Q + iQ) = 1 − T 2 Q(3) can be seen derivative and coderivative on M, while the zero-order boundary operators B ± := 1 ∓ T γ(ν) 2(4) play the role of the absolute (B − ) and relative (B + ) boundary conditions on ∂M for the Dirac operator Q. The Dirac Laplacian can be decomposed as Q 2 = dδ + δd(5) and the corresponding first order boundary operators read as: B − ⊕ B − d (absolute) B + ⊕ B + δ (relative).(6) Theorem 1.2 (Orthogonal Decomposition of Dirac Sections). Let (V, ·, · , ∇, γ) be a Dirac bundle over the Riemannian manifold (M, g), C ∞ (M, V ) denote the Dirac spinors, i.e. the differentiable sections of V , H B± (M, V ) the harmonic, Ω d B± (M, V ) the exact and Ω δ B∓ (M, V ) the coexact Dirac sections satisfying the absolute (B − ) and the relative (B + ) boundary conditions. Then, the following orthogonal decomposition holds: C ∞ (M, V ) = H B± (M, V ) ⊕ Ω d B± (M, V ) ⊕ Ω δ B∓ (M, V ).(7) This theorem generalizes Morrey's Theorem (cf. [Mo56] and [Sc95]) for differential forms on manifold under the relative or absolute boundary condition. By using this Hodge-De Rham-like decomposition theorem a variational characterization of the Dirac spectrum in terms of the Dirac spectrum on exact Dirac sections can be derived. This is the technical result needed to prove the following: Theorem 1.3 (Spectral Lower Bounds by Dissection). Let (V, ·, · , ∇, γ) be a Dirac bundle over the Riemannian manifold (M, g) with Dirac operator Q. Assume the existence of a a bundle isomorphism T on V anticommuting with γ and with Q. Let (U j ) K j=0 be a collection of closed sets whose interiors cover M . Choose and fix (ρ j ) K j=1 a subordinate partition of unity and set U α0,α1,...,α k := . This is the generalization Cheeger's technique for the Laplacian on differential forms (cf. [Go93]). The lower spectral bound method found can be applied to prove the new results introduced by the following two subsections. Large First Eigenvalues Let (M, g) be a compact, connected n dimensional Riemannian manifold. and λ 1 (∆ g p )) the smallest positive eigenvalue of the Laplacian on p forms. Hersch ( [He70]) proved, that for functions on the sphere S 2 we have λ 1 (∆ g 0 )Vol(S 2 , g) ≤ 8π (10) for every Riemannian metric g. In connection with this result, Berger ([Be73]) asked whether there exists a constant k(M ) such that λ 1 (∆ g 0 )Vol(M, g) 2 n ≤ k(M )(11) for any Riemannian metric g on M . Yang and Yau [YY80] proved that the inequality above holds for a compact surface S of genus Γ with k(S) = 8π(Γ+1). Later, Bleecker ([Bl83]), Urakawa ([Ur79]) and others constructed examples of manifolds of dimension n > 3 for which the inequality (11) is false. Xu ([Xu92]), and Colbois and Dodziuk [CD94] showed that inequality (11) is false for every Riemannian manifold of dimension n > 3. Tanno ([Ta83] posed the analogous question for forms of degree p, if there exist a constant k(M ) such that λ 1 (∆ g p )Vol(M, g) 2 n ≤ k(M )(12) for any Riemannian metric g on M . Pagliara and Gentile ( [GP95]) showed that inequality (12) is false for n > 4 and 2 < p < n − 2. We can adapt now their proof to show Theorem 1.4. Every compact connected spin manifold M admits for a given spin structure s metrics g of volume one with arbitrarily large first non zero Atiyah-Singer operator eigenvalue λ 1 (D (M,g) s ). Lower Spectrum of Degenerating Hyperbolic Three Manifolds According to Thurston's cusp closing Theorem (cf. [Th79]), every complete, non compact, hyperbolic, three manifold M of finite volume is the limit in the sense of pointed Lipschitz of a sequence of compact, hyperbolic, three manifolds (M j ) j≥0 . The Laplace-Beltrami operator on p-forms is selfadjoint and non negative. Its spectrum is contained in [0, ∞[ and can be seen as the disjoint union of pure point spectrum i.e. eigenvalues and continuous spectrum i.e. approximate eigenvalues or, alternatively, as the disjoint union of non essential spectrum i. e. isolated eigenvalues of finite multiplicity and essential spectrum i. e. cluster points of the spectrum and eigenvalues of infinite multiplicity. On the basis of Thurston's Theorem, we expect the eigenvalues of ∆ p on M j to accumulate at points of the spectrum of ∆ p on M . In three dimensions the spectra of functions and coexact 1-forms fully determines the spectra of forms in all degree. In the case of functions, the results of Donnely ([Do80]) implied ess spec(∆ 0 ) = [1, ∞[ and a sharp estimate for the number of eigenvalues of M j in any interval [1, 1 + x 2 ] was given by Chavel and Dodziuk [CD93]. In the case of 1-forms, Mazzeo Theorem 1.5 (Dodziuk, Mc Gowan ). On a degenerating sequence of hyperbolic three manifolds (M j , g j ) j≥0 the lower eigenvalues of the Laplace-Beltrami operator acting on 1−forms accumulate near zero as the inverse of the square of the diameter. More precisely, there exists an integer N 0 ∈ N 0 such that λ N0 (∆ (Mj ,gj ) 1 ) = O(1) diam 2 (M j , g j ) (j → ∞).(13) Recall that c j = O(1) (j → ∞) if and only if (c j ) j≥0 is a bounded sequence and that for a degenerating sequence of hyperbolic manifolds diam(M j , g j ) ↑ +∞ (j → ∞). An explicit lower bound for the first eigenvalue with respect to the diameter has been recently provided by Jammes (cf. [Ja12]). Theorem 1.6 (Jammes). For any real V > 0, there exists a constant c(V ) > 0 such that, if M is a compact three dimensional hyperbolic manifold of volume smaller than V , whose thin part has k components, then λ 1 (∆ (M,g) 1 ) ≥ c(V ) diam 3 (M, g) exp 2k diam 3 (M, g ) λ k+1 (∆ (M,g) 1 ) ≥ c(V ) diam 2 (M, g) .(14) Theorem 1.7 (Jammes). For every non compact three dimensional hyperbolic manifold M of bounded volume, there exits a constant c > 0 and a degenerating sequence of hyperbolic three manifolds (M j , g j ) j≥0 converging to M such that for all j ≥ 0 λ 1 (∆ (Mj ,gj ) 1 ) ≥ c diam 2 (M j , g j ) .(15) In two dimensions the spectrum of ∆ 0 fully determines the spectra of forms of all degree. The analogous questions for surfaces were studied by Wolpert ([Wo87]), Hejahl ( [He90]) and Ji ([Ji93]) and a sharp estimate for the accumulation rate was obtained by Ji and Zworski ( [JZ93]). In addition Colbois and Courtois ( [CC89], [CC89bis]) proved that the eigenvalues below the bottom of the essential spectrum are limits of eigenvalues of M j for both Riemann surfaces and hyperbolic three manifolds. Problems of this kind don't arise in dimensions greater than or equal to four (cf. [Gro79]), because the number of complete hyperbolic manifolds of volume less than or equal to a given constant is finite in this case. In the case of the classical Dirac operator Bär (cf. [Bä00]) proved: Theorem 1.8 (Bär). On a degenerating sequence of oriented, hyperbolic, three manifolds (M j , g j ) j≥0 for any spin structure (s j ) j≥0 on M j the lower eigenvalues of the Atiyah-Singer operator D (Mj ,gj ) sj do not accumulate. More precisely, there exists an integer N 0 ∈ N 0 such that λ N0 (D (Mj , gj ) sj ) = O(1) (j → ∞).(16) The different behaviour of spin Laplacian and Laplacian on forms is due to topological reasons. We can improve Theorem 1.8 proving, by means of Theorem 1.3, that in Theorem 1.9 N 0 = 1 can be chosen and providing an explicit lower bound for the first non zero eigenvalue of the Dirac operator. Theorem 1.9. On a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the lower eigenvalues of the Atiyah-Singer operator do not accumulate and the first positive non zero eigenvalue is bounded from below by a positive uniform constant c > 0 λ 1 (D (Mj , gj ) sj ) ≥ c.(17) Dirac Bundles The purpose of this chapter is to recall some basic definitions concerning the theory of Dirac operators, establishing the necessary self contained notation and introducing the standard examples. The general references are [LM89], [BW93], [BGV96] and [Bä91]. Dirac Bundle Definition. (Dirac Bundle) The quadruple (V, ·, · , ∇, γ), where (i) V is a complex (real) vector bundle over the Riemannian manifold (M, g) with Hermitian (Riemannian) structure ·, · , (ii) ∇ : C ∞ (M, V ) → C ∞ (M, T * M ⊗ V ) is a connection on M , (iii) γ : Cl(M, g) → Hom(V ) is a real algebra bundle homomorphism from the Clifford bundle over M to the real bundle of complex (real) endomorphisms of V , i.e. V is a bundle of Clifford modules, is said to be a Dirac bundle, if the following conditions are satisfied: (iv) γ(v) * = −γ(v), ∀v ∈ T M i.e. the Clifford multiplication by tangent vectors is fiberwise skew-adjoint with respect to the Hermitian (Riemannian) structure ·, · . (v) ∇ ·, · = 0 i.e. the connection is Leibnizian (Riemannian). In other words it satisfies the product rule: d ϕ, ψ = ∇ϕ, ψ + ϕ, ∇ψ , ∀ϕ, ψ ∈ C ∞ (M, V ).(18) (vi) ∇γ = 0 i.e. the connection is a module derivation. In other words it satisfies the product rule: ∇(γ(w)ϕ) = γ(∇ g w)ϕ + γ(w)∇ϕ, ∀ϕ, ψ ∈ C ∞ (M, V ), ∀w ∈ C ∞ (M, Cl(M, g)).(19) Among the different geometric structures on Riemanniann Manifolds satisfying the definition of a Dirac bundle (cf. [Gil95]) the canonical example is the spinor bundle. γ : ·, · := g Λ(T 0,1 M) * ⊗ ·, · W . T M −→ Hom(V ) v −→ γ(v), where γ(v)ϕ := v · ϕ (· is the Clifford product) We identified T M with SO(M ) × α R m (α is the standard representation of R m ) and ΣM with s × ρ C l . Since γ 2 (v) = −g(v, v)1,T M −→ Hom(V ) v −→ γ(v) := ext(v) − int(v) where ext(v)ϕ := v ♭ ∧ ϕ and int(v)ϕ := ϕ(v, ·). Since γ 2 (v) = −g(v, v)1,∇ := ∇ g Λ(T 0,1 M ) * ⊗ ∇ W . γ : T M = T M 1,0 ⊕ T M 0,1 −→ Hom(V ) v = v 1,0 ⊕ v 0,1 −→ γ(v) := √ 2(ext(v 1,0 ) − int(v 0,1 )) where ext(v)ϕ := v ♭ ∧ ϕ and int(v)ϕ := ϕ(v, ·). Since γ 2 (v) = −g(v, v)1, by the universal property, the map γ extends uniquely to a real algebra bundle endomorphism γ : Cl(M, g) −→ Hom(V ). Remark 2.1. Let (M, g, s) be a spin manifold. The spin connection ∇ Σ is locally given by a spinor endomorphism valued 1-form. More exactly, let U ⊂ M be any contractible open subset of M , {e 1 , . . . , e m } a local orthonormal frame for T M \| U with corresponding Christoffel symbols Γ k ij , ( i.e. ∇ ei e j = Γ k ij e k ), and {s 1 , . . . , s l } a local frame for ΣM \| U . We write ∇ Σ \| U = d + ω Σ,U(21) where ω Σ,U ∈ C ∞ (U, T * M ⊗ ΣM \| U ) meaning by this: if for any ϕ ∈ C ∞ (M, ΣM ), we decompose ϕ\| U = f j s j where f j ∈ C ∞ (M, C), then ∇ Σ (ϕ\| U ) = df j ⊗ s j + f j ω Σ,U ⊗ s j .(22) Being ∇ Σ the lift of the Levi-Civita connection, we find (vii) ω Σ,U = 1 4 Γ k ij γ(e j )γ(e k )(e i ) ♭ .(23) One can ask, how general the definition of Dirac bundle is, in particular allowing for formula (7). The answer is as follows: it is always possible to make a bundle of Clifford modules into a Dirac bundle, such that the connection is an extension of the Levi-Civita connection. However, this extension procedure is only locally but not globally unique. In fact the following statement is implicitely contained in chapter 2 of [BW93]. Theorem 2.1. Let (V, γ) be a bundle of Clifford modules over the Riemannian manifold (M, g) like in (iii). Then there exist: • a Hermitian (Riemannian) structure ·, · on V , making the Clifford multiplication γ by tangent vectors v ∈ T M fibrewise skewadjoint like in (iv) • a connection ∇ on V , which is Leibnizian (Riemannian), a module derivation like in (v) − (vi) and an extension of the Levi-Civita connection as in (vii) such that (V, ·, · , ∇, γ) is a Dirac bundle. Dirac Operator and Dirac Laplacian Definition. Let (V, ·, · , ∇, γ) be a Dirac bundle over the Riemannian manifold (M, g). The Dirac operator Q : C ∞ (M, V ) → C ∞ (M, V ) is defined by C ∞ (M, V ) ∇ − −−− → C ∞ (M, T * M ⊗ V ) Q:=γ•(♯⊗1)•∇     ♯⊗1 C ∞ (M, V ) γ ← −−− − C ∞ (M, T M ⊗ V )(24) The square of the Dirac operator P := Q 2 : C ∞ (M, V ) → C ∞ (M, V ) is called the Dirac Laplacian. Remark 2.3. The Dirac operator Q depends on the Riemannian metric g and on the homomorphism γ. If different metrics or homomorphisms are considered, then the notation Q = Q g γ = Q g = Q γ is utilized to avoid ambiguities. Proposition 2.2. The Dirac operator is a first order differential operator over M . Its leading symbol is given by the Clifford multiplication: σ L (Q)(x, ξ) = ı γ(ξ ♯ )(25) where ı := √ −1. The Dirac operator has the following local representation: Q(ϕ\| U ) = γ(e j )∇ ej (ϕ\| U )(26) for a local orthonormal frame {e 1 , . . . , e m } for T M \| U and a section ϕ ∈ C ∞ (M, V ). The Dirac Laplacian is a second order partial differential operator over M . Its leading symbol is given by the Riemannian metric: σ L (Q 2 )(x, ξ) = g x (ξ ♯ , ξ ♯ )1 Vx ∀x ∈ M, ξ ∈ T * x M.(27) Example 2.4 (Atiyah-Singer Operator and Spin Laplacian). The Dirac operator in the case of spin manifolds (M, g, S) is the Atiyah-Singer operator D g γ on the sections of the spinor bundle ΣM . The Dirac Laplacian ∆ g γ := (D g γ ) 2 is the spin Laplacian. Dirac Complexes Definition (Normalized Orientation). Let (V, ·, · , ∇, γ) be a Dirac bundle over the oriented Riemannian manifold (M, g). We consider a positively oriented local orthonormal frame {e 1 , . . . , e m } for T M . Then the product S := ı [ m+1 2 ] γ(e 1 ) · · · γ(e m ) ∈ Hom(V )(28) is called the normalized orientation of the Dirac bundle. Proposition 2.3. The normalized orientation S is well defined and independent of the choice of the positively oriented local orthonormal frame. Moreover, it has the following properties: 1. S 2 = 1 2. ∇S = 0 3. QS = (−1) m−1 SQ. Definition. (Dirac Complex) Let Q be an operator of Dirac type for the vector bundle V over the Riemannian manifold (M, g) and T ∈ Hom(V ). (Q, T ) is called a complex of Dirac type if and only if 1. T 2 = 1 2. QT = −T Q. Notation. By Proposition (2.3) any Dirac bundle over an even dimensional manifold can be made into a complex of Dirac type by means of the normalized orientation. In odd dimensions this is not possible, because normalized orientation and Dirac operator commute. Π ± := 1 ∓ T 2 V ± := Π ± (V ) Q ± := Q\| C ∞ (M,V±) . (29) Proposition 2.4. 1. Q ± : C ∞ (M, V ± ) −→ C ∞ (M, V ∓ ) 2. Q = 0 Q − Q + 0 : C ∞ (M, V + ⊕ V − V ) −→ C ∞ (M, V + ⊕ V − V ) 0 − −−− → C ∞ (M, V + ) Q+ − −−− → C ∞ (M, V − ) Q− − −−− → C ∞ (M, V + ) − −−− → 0 (30) is a complex i.e. Q − Q + = 0. 3. (Π ± Q) 2 = 0 Π + Q + Π − Q = Q Π + QΠ − Q + Π − QΠ + Q = Q 2 . The restriction of a Dirac bundle to a one codimensiional submanifold is again a Dirac bundle, as following theorem (cf. [Gil93] and [Bä96]) shows. Theorem 2.5. Let (V, ·, · , ∇, γ) be a Dirac bundle over the Riemannian manifold (M, g) and let N ⊂ M be a one codimensional submanifold with normal vector filed ν. Then (N, g\| N ) inherits a Dirac bundle structure by restriction. We mean by this that the bundle V \| N , the connection ∇\| C ∞ (N,V \|N ) , the real algebra bundle homomorphism γ N := −γ(ν)γ\| Cl(N,g\|N ) , and the Hermitian (Riemannian) structure ·, · \| N satisfy the defining properties (iv)-(vi). The quadruple (V \| N , ·, · \| N , ∇\| C ∞ (N,V \|N ) , γ N ) is called the Dirac bundle structure induced on N by the Dirac bundle (V, ·, · , ∇, γ) on M . For a spin manifold of arbitrary dimension we will now construct a vector bundle isomorphism T which anticommutes with the Atiyah-Singer operator Q making a generic spin bundle to a complex of Dirac type (Q, T ). Inspired by [BGM05], we embed a given manifold into a cylinder. It can easily proved (cf. [BGM05], Chapter 5) that Proposition 2.6. The original spin manifold and the generalized cylinder satisfy following properties: 1. Spin(M ) = π −1 (i(SO(M ))). γ M and γ Z (ν) anticommute. In fact, for all v ∈ T M γ M (v)γ Z (ν) = −γ Z (ν)γ M (v). (32) 3. (Q M , γ Z (ν)) is a complex of Dirac type. 4. ∇ M γ Z (ν) = 0. Spectral Properties of the Dirac Operator We consider Dirac bundles over compact manifolds, possibly with boundary. The existence of a regular discrete spectral resolution for both Dirac and Dirac Laplacian operators under the appropriate boundary conditions is a special case of the standard elliptic boundary problems theory developed by Seeley ([Sl66], [Sl69]) and Greiner ([Gre70], [Gre71]). The general references are [Gru96] and [Hö85]. See [BW93] and [Gil95] for the specific case of the Dirac and Dirac Lapacian operators. Dirac and Dirac Laplacian Spectra on manifolds without boundary The Dirac operator Q and the Dirac Laplacian P for a Dirac bundle V over a compact Riemannian manifold without boundary are easily seen by Green's formula to be symmetric operators for the C ∞ -sections of Dirac bundle. Taking the completion of the differentiable sections of V in the Sobolev H 1 − and respectively H 2 -topology, leads to two selfadjoint operators in L 2 (V ). Theorem 3.1. The Dirac Q and the Dirac Laplacian P operators of a Dirac bundle over a compact Riemannian manifold M without boundary have a regular discrete spectral resolution with the same eigenspaces. It exists a sequence (ϕ j , λ j ) j∈Z * such that (ϕ j ) j∈Z * is an orthonormal basis of L 2 (V ) and that for every j ∈ Z * it must hold Qϕ j = λ j ϕ j P ϕ j = λ 2 j ϕ j and ϕ j ∈ C ∞ (V ). The eigenvalues of the Dirac operator (λ j ) j∈Z * are a monotone increasing real sequence converging to ±∞ for j → ±∞. The eigenvalues of the Dirac Laplacian are the squares of the eigenvalues of the Dirac operator and hence not negative. Therefore, for Dirac bundle over a manifold without boundary the knowledge of the spectrum for the Dirac operator and the Dirac Laplacian are equivalent. Moreover, in the case of a Dirac complex the spectrum of the Dirac operator is symmetric with respect to the origin. Proposition 3.2. If there is an isomorphism T for the Dirac bundle V anticommuting with the Dirac operator Q, then the discrete spectral resolution of Theorem 3.1 can be chosen such that the equalities λ −j = −λ +j and ϕ −j = T ϕ +j hold for every j ∈ N * . In particular, the dimension of the space of harmonic sections is always even. Remark 3.1. An interesting consequence of Proposition 3.2 and of Proposition 2.6 is that the spectrum of the classical Dirac operator is symmetric with respect to the origin in any dimension. This was known in even dimension and is in line with all the Dirac spectra which can be computed explicitly (cf. [Bä91] Chapter II and [Gin09] Chapter 2). Dirac and Dirac Laplacian Spectra on Manifolds with Boundary The case of manifolds with boundary is more complex and the spectra of the Dirac and Dirac Laplacians are no more equivalent as they are in the boundaryless case. Moreover, while for the Dirac Laplacian it is always possible to find local elliptic boundary conditions allowing for a discrete spectral resolution, this is not always true for the Dirac operator. The Dirac Laplacian P for a Dirac bundle V over a compact Riemannian manifold with boundary is easily seen by Green's formula to be a symmetric operator for the C ∞ -sections of Dirac bundle if we impose the Dirichlet boundary condition B D ϕ := ϕ\| ∂M = 0 or the Neumann boundary condition B N ϕ = ∇ ν ϕ\| ∂M = 0. Taking the completion of the differentiable sections of V satisfying the boundary conditions in the Sobolev H 2 -topology, leads to a selfadjoint operator in L 2 (V ). Theorem 3.3. The Dirac Laplacian P of a Dirac bundle over a compact Riemannian manifold M with boundary under the Neumann or the Dirichlet condition has a regular discrete spectral resolution (ϕ j , λ j ) j≥0 . This means that (ϕ j ) j≥0 is an orthonormal basis of L 2 (V ) and that for every j ≥ 0 it must hold P ϕ j = λ j ϕ j , ϕ j ∈ C ∞ (V ), and Bϕ j = 0 for either B = B D or B = B N . The eigenvalues (λ j ) j≥0 are a monotone increasing real sequence bounded from below and converging to infinity. The Dirichlet eigenvalues are all strictly positive. The Neumann eigenvalues are all but for a finite number strictly positive. The situation for the Dirac operator is more subtle. Altough it is -again by Green's formula-a symmetric operator under the Dirichlet boundary condition, it is not selfadjoint. As a matter of fact the Dirichlet boundary condition is elliptic for the Dirac Laplacian but not for the Dirac operator. If we are looking for local elliptic boundary conditions for the Dirac operator, we need to introduce the following Definition. Let (V, ·, · , ∇, γ) be a Dirac bundle over a manifold M with boundary ∂M . The isomorphism χ ∈ Hom(V \| ∂M ) is called boundary chirality operator for the Dirac bundle if satisfies χ 2 = 1 and anticommutes with the Clifford multiplication, i.e. χγ(v) + γ(v)χ = 0 for any v ∈ T M \| ∂M . The corresponding boundary condition operator is given by B ± := 1 2 (1 ∓ χγ(ν)). In the even dimensional case one can always find boundary chirality operators for any Dirac bundle: it suffices to choose χ := S\| ∂M , where S denotes the normalized orientation. For the special case of the exterior algebra bundle in any dimension the choice χ := ext(ν) + int(ν) leads to the absolute and relative boundary conditions for differential forms which are ellipitic for the Euler operator d + δ. In the odd dimensional case there are obstructions to the existence of local boundary chirality operators for Dirac bundles. As a matter of fact, if there exist a local elliptic boundary condition for the Dirac operator, then tr(S) = 0. The non vanishing of the trace of the normalized orientation, is therefore the topological obstruction, termed the Atiyah-Bott obstruction, for the existence of local elliptic boundary conditions for the full Dirac Operator. In even dimension this obstruction always vanishes because the full Dirac operator and the normalized orientation always anticommute. In odd dimensions the obstruction for the full Dirac operator can or cannot vanish. For the chiral Dirac operator, defined on the sections of the eigenbundles of the normalized orientation the situation is complementary. In even dimension the obstruction vanishes, while in odd ones it does not, see [Gi84] page 248 and [Gil95] page 102. An elliptic boundary condition for both full and chiral Dirac operator always exists in any dimension, but it is defined by mean of a zero order pseudodifferential operator, the spectral projections of the Dirac operator on the boundary. This is the famous Atiyah-Patodi-Singer boundary condition (see [Sl66], [BW93])). In a neighbourhood of the boundary ∂M it is possible to decompose the Dirac operator as Q = γ(ν)(∇ ν + A).(33) Remark that it is not necessary to assume that the geometric structures are a product on this neighbourhood. The operator A\| ∂M is an operator of Dirac type for V \| ∂M over the boundaryless manifold ∂M . The operator A APS := A\| ∂M + 1 2 H1, where H denotes the mean curvature of the boundary, is an operator of Dirac type for ∂M and, by Theorem 3.1, it has a discrete regular spectral resolution (ψ j , µ j ) j≥0 . The subspace of L 2 (V ) defined by E µ (A APS ) := ker(A APS − µ1) is the eigenspace of A APS if µ is in the spectrum of A APS and the zero subspace otherwise. Definition. Let (V, ·, · , ∇, γ) be a Dirac bundle over the oriented Riemannian manifold (M, g) with Dirac operator Q and normalized orientation S. The generalized Atiyah-Patodi-Singer boundary condition for Q is given by B APS (ϕ\| ∂M ) = 0, where B APS denotes the orthogonal projection in L 2 (V \| ∂M ) onto µ<0 E µ (A APS ) ⊕ 1 2 (1 − S)(E 0 (A APS )).(34) The Dirac operator Q for a Dirac bundle V over a compact Riemannian manifold with boundary is easily seen by Green's formula to be a symmetric operator for the C ∞ -sections of Dirac bundle if we impose the boundary conditions B ± induced by a boundary chirality operator or by the generalized APS boundary condition. Taking the completion of the differentiable sections of V satisfying the boundary condition B in the Sobolev H 1 -topology, leads to a selfadjoint operator in L 2 (V ). Of course, the associated first order boundary conditions for the Dirac Laplacian are elliptic as well and lead to a self adjoint operator with pure point spectrum if we define the domain of P as the completion of the differentiable sections of V satisfying the boundary conditions B ⊕ BQ in the Sobolev H 2 -topology. In [FS98] it is given an elementary proof (with no reference to the calculus of elliptic pseudodifferential operators as in [Hö85] or [BdM71]) of the following result: Theorem 3.4. The Dirac Q and the Dirac Laplacian P operators of a Dirac bundle over a compact Riemannian manifold M with boundary have under the boundary conditions B and B ⊕ BQ respectively, for either B = B ± , (if a boundary chirality operator exists), or B = B APS a regular discrete spectral resolution with the same eigenspaces. It exists a sequence (ϕ j , λ j ) j∈Z * such that (ϕ j ) j∈Z * is an orthonormal basis of L 2 (V ) and that for every j ∈ Z * it must hold Qϕ j = λ j ϕ j P ϕ j = λ 2 j ϕ j , ϕ j ∈ C ∞ (V ), B(ϕ j \| ∂M ) = 0 and B((Qϕ j )\| ∂M ) = 0. The eigenvalues of the Dirac operator (λ j ) j∈Z * under the boundary condition B are a monotone increasing real sequence converging to ±∞ for j → ±∞. The eigenvalues of the Dirac Laplacian are the squares of the eigenvalues of the Dirac operator and hence not negative. Remark that the Dirac operator under the complementary APS-boundary condition, that, is the orthogonal projection from L 2 (V ) onto µ>0 E µ (A APS ) ⊕ 1 2 (1 + S)(E 0 (A APS )),(35) is only symmetric but not selfadjoint and thus must not have a discrete real spectrum. The complementary APS-boundary condition is not elliptic. Extending the result in the boundaryless case, for a Dirac complex (Q, T ) preserving the boundary condition B, that is, where T anticommutes with B, the spectrum of the Dirac operator is symmetric with respect to the origin. Dirac Cohomology and Hodge Theory under Boundary Conditions In this section we will prove Theorem 4.3. We will have to introduce for Dirac bundles concepts which mimick the situation for differential forms like derivation, coderivation, absolute and relative boundary conditions. 2. The coderivative defines a complex: δ 2 = 0. 3. The Dirac operator can be decomposed as Q = d + δ. 4. The Dirac Laplacian can be decomposed as P := Q 2 = dδ + δd. The zero-order boundary operators B ± := 1 ∓ T γ(ν) 2 (37) define the absolute B − and relative B + boundary conditions on ∂M for the Dirac operator Q and have following properties 1. B + ⊕ B − = 1. 2. B 2 + = B + = B * + . 3. B 2 − = B − = B * − . 4. γ(ν)B ± = B ∓ γ(ν) and γ(ν) : ker(B + ) ⊕ ker(B − ) → ker(B − ) ⊕ ker(B + ). The following Green's formula holds for all smooth sections ϕ, ψ of the Dirac bundle (dϕ, ψ) − (ϕ, δψ) = − ∂M dvol ∂M γ(ν)B − ϕ, ψ = = − ∂M dvol ∂M γ(ν)ϕ, B + ψ .(38) For the Dirac Laplacian the corresponding first order boundary operators are C − := B − ⊕ B − d (absolute boundary condition) and C + := B + ⊕ B + δ (relative boundary condition). In fact 1. The absolute boundary condition is preserved by the derivative operator: B − ϕ\| ∂M = 0 ⇒ B − dϕ\| ∂M = 0. The relative boundary condition is preserved by the coderivative operator B + ϕ\| ∂M = 0 ⇒ B + δϕ\| ∂M = 0. Proof. The properties of derivative and coderivative are a direct consequence of their definition where an isomorphism T such that (Q, T ) is a Dirac complex was utilized. The properties of the boundary conditions follows from the fact that (iγ(ν)γ(ν)) 2 = 1. The Green's formula (38) follows from the corresponding Green's formulae for the Dirac operators Q and Q. To prove the preservation of the absolute boundary condition by the derivative operator, we note that, by Green's formula (dϕ, ψ) = (ϕ, δψ), for a ϕ satisfying B − ϕ\| ∂M = 0 and any ψ. Applying Green's formula to dϕ and ψ we obtain (ddϕ, ψ) − (dϕ, δψ) = − ∂M dvol ∂M γ(ν)B − dϕ, ψ(40) The left hand side of (40) vanishes because of (39) and the fact that d 2 = 0. Thus, the boundary integral vanishes for all ψ and so does B − dϕ\| ∂M . The proof of the preservation of the relative boundary condition under the coderivative operator reads analogously. After having introduced operators and boundary condition we would like to study the spectrum. Then, the following orthogonal decomposition holds: 1. d ⊂ δ * 0 , d ⊂ δ * 0 , Q 0 ⊂ Q * 0 and P 0 ⊂ P * 0 . 2. d * B± = δ B∓ and δ * B± = d B∓ .C ∞ (M, V ) = H B± (M, V ) ⊕ Ω d B± (M, V ) ⊕ Ω δ B∓ (M, V ).(41) Proof. The proof is based on standard elliptic operator theory and the fact that derivative and coderivative operators preserve the absolute and the relative, respectively, boundary condition. Definition (Dirac Cohomology). The group H B± (M, V ) := {ω ∈ Ω B± (M, V )\|dω = 0}/dΩ d B± (M, V )(42) is called absolute, respectively, relative Dirac cohomology of the Dirac bundle. Since we will not need it going forward, we mention without proof the following result To motivate the terminology introduced so far, we prove that in the case of the Euler operator, for a particular choice of the bundle isomorphism T for the exterior algebra bundle, the derivative and coderivative operators are the classical exterior and interior differentiation for forms, the Dirac Cohomologies are the De Rham cohomologies under the absolute and relative boundary conditions and Theorem 41 the classical Hodge decomposition theorem for differential forms on a manifold with boundary. Proof. This can be verified by a direct computation. Meyer-Vietoris's Scheme and Generalization of Cheeger's Spectral Estimate In this section we will prove Theorem 1. Then: 1. E B± (λ) = E d B± (λ) ⊕ E δ B± (λ) 2. d : E δ B± (λ) −→ E d B± (λ) and δ : E d B± (λ) −→ E δ B± (λ) are isomorphisms between finite dimensional subspaces of L 2 (M, V ). 3. E d B± (λ) = dE δ B± (λ) and E δ B± (λ) = δE d B± (λ) . This lemma has an important consequence. The knowledge of the spectrum of the Dirac Laplacian on all exact (or coexact) Dirac sections implies the knowledge of the spectrum of the Dirac Laplacian on all sections namely. Corollary 5.2. The spectrum of the Dirac Laplacian can be decomposed as spec(P B± ) = {0} ∪ spec(P B± Ω d (M,V ) ) ∪ spec(P B± Ω δ (M,V ) )(44) The multiplicity of zero is the dimension of the absolute or relative, respectively, Dirac Cohomology. The multiplicity of an eigenvalue λ > 0 is the sum of its multiplicities as exact and coexact eigenvalue. Thus, to study the Dirac Laplacian and hence the Dirac spectrum, it suffices to study the spectrum of exact Dirac sections, whose eigenvalues allow for the following minimax characterization. λ d i = inf L sup η∈L η =0 (η, η) (ϕ, ϕ) ϕ ∈ Ω B− (M, V ), dϕ = η(45) where L varies over all i-dimensional subspaces of Ω B− (M, V ). Proof. We take any ϕ ∈ Ω B− (M, V ) such that dϕ = η. By Theorem 4.3, any Dirac section ϕ splits into the orthogonal sum ϕ = h ⊕ dα ⊕ δβ, where h is an harmonic section, and α, β Dirac sections. Set ψ := δβ ∈ Ω δ B− (M, V ). By the orthogonality of the decomposition (ϕ, ϕ) ≥ (ψ, ψ) and, by Green's formula and the coexactness of ψ: (dϕ, dϕ) = (dψ, dψ) = (δdψ, ψ) = (P ψ, ψ). So, (η, η) (ϕ, ϕ) = (dϕ, dϕ) (ϕ, ϕ) ≤ (P ψ, ψ) (ψ, ψ) and inf L sup η∈L η =0 (η, η) (ϕ, ϕ) = inf R sup ψ∈R ψ =0 (P ψ, ψ) (ψ, ψ) where L varies over all i dimensional subspaces of Ω B− (M, V ) and R over all i dimensional subspaces of Ω δ B− (M, V ). The right hand side of this equation is the standard minimax characterization of λ δ i , the i-th eigenvalue of coexact Dirac sections, which by Lemma 5.1 (ii) is equal to the i-th eigenvalue λ d i of exact sections. After having proved the variational characterization of Dirac Laplacian eigenvalues on exact sections satisfying the absolute boundary conditions, we possess now the technical tools to prove Proposition 5.4. Let (V, ·, · , ∇, γ) be a Dirac bundle over a compact Riemannian manifold (M, g). We assume the existence of an isomorphism T anticommuting with with γ and with the Dirac operator Q. Let µ(U ) be the smallest postive eigenvalue of the Dirac Laplacian P on exact Dirac sections satisfying the absolute boundary condition on U . Moreover, for an an open cover of M denoted by {U i } i=0,...,K we introduce the following notation: • U α0,α1,...,α k := i∈{α0,...,α k } U i . • m i := \|{j = i\| U j ∩ U i = ∅}\|. • {ρ i } i=0,...,K : a partition of unity subordinate to the open cover. • C ρ := 1 2 max i∈{0,1,...,K} sup x∈Ui \|∇ρ i (x)\| 2 . • N 1 := K i,j=0 dim H B− (U i,j , V ). • N 2 := K i,j,k=0 dim H B− (U i,j,k , V ). • N := N 1 + N 2 + 1. The N -th eigenvalue of the Dirac Laplacian satisfies the following lower inequality µ N (M ) ≥ 1 K i=0 1 µ(Ui) + 4 mi j=0 Cρ µ(Ui,j ) + 1 1 µ(Ui) + 1 µ(Uj )(46) Proof. Let {Φ i } i≥0 be an orthonormal basis of exact Dirac section in C ∞ (M, V ), where, for all i ≥ 0 Φ i = dχ i and χ i is coexact and thus unique. Therefore: (Φ N , Φ N ) (χ N , χ N ) = (dχ N , dχ N ) (χ N , χ N ) = (δdχ N , χ N ) (χ N , χ N ) = = ((dδ + δd)χ N , χ N ) (χ N , χ N ) = (P χ N , χ N ) (χ N , χ N ) = µ N .µ N = (Φ N , Φ N ) (χ N , χ N ) ≥ (Φ, Φ) (χ, χ) ≥ (Φ, Φ) (ψ, ψ) ,(48) for all ψ such that dψ = Φ ∈ Span({Φ i } i=0,...,N ). As a matter of fact, by Theorem 4.3 ψ = h ⊕ dα ⊕ δβ. By denoting χ := δβ, we have that dχ = Φ and (ψ, ψ) = (h, h)+(dα, dα)+(χ, χ) ≤ (χ, χ) and, hence, inequality (48). Therefore, a lower bound on (Φ,Φ) (ψ,ψ) for any par of Φ, ψ with dψ = Φ ∈ Span({Φ i } i=0,...,N ) will give a lower bound on µ N . We will construct a Dirac section ψ satisfying dψ = Φ in such a way that the L 2 -norm of ψ is controlled in terms of the L 2 -norm of Φ. In order to do this we will be forced at two points during the proof to make specific choices for the coefficients a i 's. Let us consider the following diagram . . . . . . . . . 0 / / Ω(M ) r / / d O O Π i Ω B− (U i ) s / / d O O Π i,j Ω B− (U i,j ) s / / d O O . . . 0 / / Ω(M ) r / / d O O Π i Ω B− (U i ) s / / d O O Π i,j Ω B− (U i,j ) s / / d O O . . . . . . d O O . . . d O O . . . d O O(49) Thereby, we utilize the notation: where α 0 , . . . ,α i , . . . , α p means that the index α i has been dropped from the index sequence α 0 , . . . , α p . The rows in diagram (49) are exact but the columns are not (in general). Since we are interested in lower bounds for exact Dirac sections, we will pick Φ ∈ Span({Φ i } i=0,...,N ), the first N exact eigensections. We restrict now Φ by means of r to get {Φ i } i ∈ Π i Ω(U i ). Since Φ is exact, we can choose {ψ i } i ∈ Π i Ω(U i ) so that dψ i = Φ i . Now, we can use the fact that we have a lower eigenvalue bound for exact sections on U i for all i to choose ψ i 's with bounded L 2 norm. We will then piece together these ψ i 's into a section defined on all M . It is in general not true that ψ i = ψ j on U i,j , i.e. that s{ψ i } = 0. Therefore, we set {ω i,j } = s{ψ i }, where ω i,j := ψ i − ψ j on U i,j . Notice that dω i,j = dψ i − dψ j = Φ − Φ = 0,(52) so that, by Theorem 4.3, we can write ω i,j = h i,j ⊕ dη i,j ,(53) where h i,j is harmonic. We can choose appropriate coefficients a i 's for Φ = N i=0 a i Φ i = 0 so that h i,j = 0. The dimension of the space of such Φ's will be at least N −N 1 = N 2 +1. We pick the unique coexact η i,j such that ω i,j = dη i,j . Therefore, by Proposition 5.3 (dη i,j , dη i,j ) (η i,j , η i,j ) ≥ µ(U i,j )(54) Next, let us consider {ν i,j,k } = s{η i,j } = {(η j,k − η i,k + η i,j )\| U i,j,k } for which dν i,j,k = dη j,k − dη i,k + dη i,j = ω j,k − ω i,k + ω i,j = = ψ k − ψ j − ψ k + ψ i + ψ j − ψ i = 0,(55) and therefore {Φ i } s / / {0} {ψ i } d O O s / / {ω i,j } d O O s / / {0} {τ i } d O O s / / {η i,j,k } s / / d O O {ν i,j,k } d O O(56) We want to replace the ψ i 's with some ψ i 's which are restrictions of a globally defined section on M and such that on U i dψ i = dψ i = Φ i .(57) the exactness of the rows of diagram (49) would allow us, if all the ν i,j,k s were zero, to find {τ i } ∈ Π i Ω(U i ) so that s{τ i } = {η i,j } = {τ j − τ i \| Ui,j }. An explicit choice is given by τ i := K j=1 ρ j η i,j ,(58) where {ρ j } j=0,...,K is the partition of unity subordinate to the cover {U j } j=0,...,K . However, so far we can only claim that dν i,j,k = 0, i.e. that ν i,j,k is closed. On the other hand ν i,j,k is coexact, i.e. ν i,j,k = δα i,j,k . The mapping Φ → ψ i (Φ) → ω i,k (Φ) → ν i,j,k (Φ)(59) is linear in Φ, which is in a space of dimension at least N 2 + 1. Therefore, we can choose Φ = N i=0 a i Φ i = 0 such that ν i,j,k (Φ) = 0 for all i, j, k.(60) As a matter of fact condition (60) represents N 2 linear equations in N 2 + 1 unknowns. So, ds{τ i } = sd{τ i } = {ω i,j },(61) and, if we take ψ i := ψ i − dτ i , then s{ψ i } = {ψ j − ψ i − d(τ j − τ i )} = {ψ j − ψ i − ω i,j } = {0}(62) Therefore, ψ i = ψ\| Ui , where ψ is a globally defined section. Notice that dψ i = dψ i = Φ i on U i . Since (ψ, ψ) ≤ i (ψ i , ψ i ),(63) it follows (Φ, Φ) i (ψ i , ψ i ) ≤ (Φ, Φ) (ψ, ψ) .(64) A lower bound on the left hand side of inequality (64) will give a lower eigenvalue bound for exact sections on M . Note that all norms are L 2 -norms unless otherwise indicated, and are computed on the appropriate open sets. Being Φ i the restriction of Φ to U i , the variational characterization of the eigenvalues in Proposition 5.3 implies Φ 2 ψ i 2 ≥ Φ i 2 ψ i 2 ≥ µ(U i ),(65) so that ψ i 2 ≤ Φ 2 µ(U i ) .(66) Both operators Q andQ satisfy the product rule for all smooth functions f and sections ϕ Q(f ϕ) = γ(gradf )ϕ + f Qϕ andQ(f ϕ) =γ(gradf )ϕ + fQϕ,(67) so that the operator d : = 1 2 (Q − iQ) d(f ϕ) = γ − iγ 2 (gradf )ϕ + f dϕ.(68) This formula allows to estimate dτ i : dτ i 2 = d( j ρ j η i,j ) 2 = j γ − iγ 2 (gradη i,j + ρ j dη i,j 2 ≤ ≤ 2 j γ − iγ 2 (gradη i,j 2 + ρ j dη i,j 2 ≤ ≤ 2 j (C j η i,j 2 + dη i,j 2 ).(69) Since η i,j fullfilling the condition (54) was chosen, we have η i,j 2 ≤ dη i,j 2 µ(U i,j ) = ψ i − ψ j 2 µ(U i,j ) ≤ 2( ψ i 2 − ψ j 2 ) µ(U i,j ) .(70) Assembling the inequalities (70), (69) and (66) into the definition of ψ i , we obtain ψ i 2 ≤ ψ i 2 + dτ i 2 ≤ ≤ ψ i 2 + d( j ρ j η i,j ) 2 ≤ ≤ Φ 2 µ(U i ) + 4 j C ρ ψ i 2 + ψ j 2 µ(U i,j ) + ψ i 2 + ψ j 2 ≤ ≤ Φ 2 µ(U i ) + 4 j   C ρ Φ 2 µ(Ui) + Φ 2 µ(Uj ) µ(U i,j ) + Φ 2 µ(U i ) + Φ 2 µ(U j )   ,(71) and therefore ψ i 2 Φ 2 ≤ 1 µ(U i ) 4 j   C ρ 1 µ(Ui) + 1 µ(Uj ) µ(U i,j ) + 1 µ(U i ) + 1 µ(U j )   .(72) Because of inequality (63) we finally obtain Φ 2 ψ 2 ≤ 1 K i=0 1 µ(Ui) 4 mi j=0 Cρ 1 µ(U i ) + 1 µ(U j ) µ(Ui,j ) + 1 µ(Ui) + 1 µ(Uj ) ,(73) which completes the proof. Definition (Quasi-Isometry). Two Riemannian metrics h and g on a manifold M are said to be quasi-isometric if and only if there are positive constants σ, τ such that σ h ≤ g ≤ τ h.(74) This is an abbreviation of the condition ∀x ∈ M, ∀ξ ∈ T x M σ h x (ξ, ξ) ≤ g x (ξ, ξ) ≤ τ h x (ξ, ξ).(75) A diffeomorphism f : (M, h) → (M ′ , g) is called quasi-isometry, if and only if h and f * g are quasi-isometric. σ and τ are called the constants of the quasiisometry. Two Riemannian manifolds are said to be quasi-isometric, if and only if they admit a quasi-isometry. Using the standard eigenvalue characterization for Dirac Laplacians, we can prove a perturbation result about the spectrum under quasi-isometry, similar to what can be found in cf. [Do82] for differential forms. Lemma 5.5. Let (M, g, s) be a spin manifold without boundary and h another quasi-isometric Riemannian structure, i.e. there exist positive constants σ, τ such that σh ≤ g ≤ τ h. Let (λ g i ) i≥0 := spec(∆ g s ) and (λ h i ) i≥0 := spec(∆ h s be the eigenvalues of the spin Laplacians for the same spin structure and the two metrics on exact sections and assume that the sequences are ordered non decreasingly and that the eigenvalues are repeated according to their multiplicities. Then, the following inequalities hold for all i ≥ 0 σ τ m 2 λ h i ≤ λ g i ≤ τ σ m 2 λ h i .(76) Proof. Following Chapter II in [BG92] we introduce, given a representation µ of O(m) on R m , the vector bundle isomorphism b g h : T M = O g × µ R m −→ T M = O h × µ R m [f, u] −→ [b g h (f ), u],(77) where b g h := H × σ C l [Φ, v] −→ [β γ η (Φ), u],(78) A short computation shows that vector and spinor bundle isomorphism satisfy β γ η (γ(v)ψ) = η(b g h (v))β γ η (ψ), ∇ η v β γ η (ψ) = β γ η ∇ γ v (ψ).(79) for all sections v of T M and all sections ψ of Σ γ M . Therefore, we conclude that Q γ β γ η = β γ η Q γ .(80) Now we can proceed to prove the lemma. First of all, being (ϕ, ϕ) g = M d m x det g ϕ, ϕ ,(81) it follows for all φ σ m 2 (β γ η ϕ, β γ η ϕ) h ≤ (ϕ, ϕ) g ≤ τ m 2 (β γ η ϕ, β γ η ϕ) h ,(82) and σ m 2 (β γ η Q η ϕ, β γ η Q η ϕ) h ≤ (Q γ ϕ, Q γ ϕ) g ≤ τ m 2 (β γ η Q η ϕ, β γ η Q η ϕ),(83) Inserting the commutation relation (80) and dividing by inequality (82) we obtain σ τ m m (β γ η Q η ϕ, β γ η Q η ϕ) h (β γ η ϕ, β γ η ϕ) h ≤ (Q γ ϕ, Q γ ϕ) g (ϕ, ϕ) g ≤ τ σ m 2 (β γ η Q η ϕ, β γ η Q η ϕ) h (β γ η ϕ, β γ η ϕ) h(84) Being β an isometry, the Lemma follows from the variational characterization of the eigenvalues. Remark 5.1. Lemma 5.5 still holds for compact manifolds with boundary, if we impose the absolute or relative boundary condition. Large First Dirac Eigenvalue: Proof of the Result Proof of Theorem 1.4. We take a topological sphere S m and choose a metric g 0 on it, such that S n looks like a cigar, where the middle part has length 3. In particular this middle part is a product for the metric g 0 , i.e. a cylinder I × S m−1 . We then remove the half-sphere H 2 at one end of the cigar and form a connected sum with M . The resulting manifold is diffeomorphic to M and has a submanifold N , with smooth boundary, naturally identified with S m \ H 2 . Let g 1 be an arbitrary metric on M whose restriction to N is equal to g 0 \| N . The manifold N contains an open cylinder of length 3. We subdivide this cylinder into 3 cylinders Z 1 , Z 2 , Z 3 of length 1. Let g t be a metric on M such that g t \| M\Z2 = g 1 \| M\Z2 and such that Z 2 = I × S m−1 becomes a cylinder of length t. This is accomplished by replacing the unit interval by the interval [0, t] and using the product metric on Z 2 . Now Vol(M, g t ) = a + bt, where a and b are positive real constants. We take the following open cover of M : 1. U 1 = H 1 ∪ Z 1 , 2. U 2 = M \ H 1 ∪ Z 1 ∪ Z 2 , 3. U 3 = Z 1 ∪ Z 2 ∪ Z 3 , which has the property that U 1 ∩ U 2 = ∅, U 1 ∩ U 3 = Z 1 , U 2 ∩ U 3 = Z 3 and U 1 ∩ U 2 ∩ U 3 = ∅. Let µ 1 (M t ) be the first positive eigenvalue of the Dirac Laplacian on exact sections on M t = (M, g t ) for the given spin structure. To estimate µ 1 (M ) we apply Theorem 1.3 to M t and the cover {U 1 , U 2 , U 3 }. The eigenvalues µ(U 1 ), µ(U 2 ), µ(U 1,3 ) and µ(U 2,3 ) are independent of t. Let λ k (O) be the k-th eigenvalue of the Dirac Laplacian on O under the absolute boundary condition By using the Künneth's formula, we get the following inequality for µ(U 3 ): µ(U 3 ) ≥ λ 1 (U 3 ) = λ 1 (I × S m−1 ) ≥ min i,j {λ i (I) + λ j (S n−1 )} =: C,(85) where C is a constant independent of t. If m = 3, then S 2 has no harmonic spinors (cf. [Bä91], [Bä92]). In other dimensions if the Riemannian metric on S m−1 allows for non trivial harmonic spinors, a small perturbation of the metric reduces the harmonic spinors to the zero section (cf. [BG92]). Therefore, it is always possible to find a Riemannian metric for which λ 1 (S m−1 ) > 0. Therefore, the constant C is strictly positive. From Theorem 1.3 we get that µ 1 (M t ) ≥ ǫ > 0(86) for an ǫ independent of t. The volume of M t is given by Vo1(M, g t ) = a+bt with constants a, b > 0. Set h t = (a + bt) 7 Lower Dirac Eigenvalues on Degenerating Hyperbolic Three Dimensional Manifolds The Geometry of Three Hyperbolic Manifolds A very readable survey of the geometry of compact, hyperbolic, three manifolds and their degenerations is contained in Gromov [Gro79]. A very thorough discussion of this topic can be found in Thurston [Th79] or in Benedetti and Petronio [BP91]. The Kazhdan-Margulis decomposition gives a simple insight of the geometrical structure of hyperbolic three manifolds, particularly where the injectivity radius is small. There exists a universal (i.e. depending only on the dimension) positive constant µ, called the Kazhdan-Margulis constant, for which the following construction can always be carried out. Any hyperbolic manifold M of finite volume splits into two parts: M = M ]0,µ] ∪ M ]µ,∞[ .(88) M ]0,µ] is called the thin part and contains all points of M , whose injectivity radius is smaller than or equal to µ. The thin part is found to be a finite union of tubes and cusps. A tube T is a tubular neighbourhood of a closed geodesic. A cusp C is the warped product [0, +∞[×F , equipped with the metric du 2 + e −2u ds 2 , where F is a 2-dimensional torus and ds 2 a flat metric on F . The points of M , where the injectivity radius is bigger than µ, form the so-called thick part M ]µ,∞[ . The thick part is non empty and connected. The following theorem, due to Thurston (cf. [BP91] page 197), states that any complete hyperbolic three manifold of finite volume, observed from its thick part, looks on its bounded part like a compact hyperbolic three manifold. Theorem 7.1 (Thurston). Let M be a complete, hyperbolic, three manifold with p cusps, p ≥ 1, and of finite volume vol(M ). Then, there is a sequence (M j ) j≥0 of compact, hyperbolic, three manifolds having p simple closed geodesics, whose lengths go to zero as j → ∞, such that (M j , x j ) converges to (M, x) x =y d(f (x), f (y)) d(x, y) ∈ [0, +∞].(89) The Lipschitz distance between M and N is the defined as d L (M, N ) := inf{\| log dil(f )\| + \| log dil(f −1 )\|} (90) where the infimum is taken over all Lipschitz homeomorphisms f : M → N . The sequence (M j , x j ) j≥0 of metric spaces M j with distinct points x j ∈ M j is said to converge to (M, x) in the sense of pointed Lipschitz, if and only if the following condition is satisfied: for every r > 0 there exists a sequence (ε j ) j≥0 of positive real numbers ε j → 0 + (j → +∞), such that d L B Mj (x j , r + ε j ), B M (x, r) −→ 0 (j → +∞)(91) where B M (x, r) denotes the ball of radius r in M centered at x. As a matter of fact, Thurston shows that the compact manifolds M j , obtained by closing the cusps of an hyperbolic, complete, non compact manifold M using Dehn's surgery, support for all but for a finite number of exceptions an hyperbolic metric and approximate M . Definition. If the limit manifold M is non compact, then the sequence (M j ) j≥0 described above is called a degenerating family of hyperbolic three manifolds. A brief review of Riemannian metrics on tubes and cusps is needed for the following. We refer to [BP91] for more details. To keep the notation simple, the manifold M in Thurston's Theorem is assumed without loss of generality to have only one cusp. There is a positive R j for which the component of the thin part (M j ) ]0,µ] of M j containing the closed simple geodesic γ j , whose length ε j → 0 as j → ∞, is the solid torus T j := {x ∈ M j \| dist(x, γ j ) ≤ R j } .(92) This torus is the quotient of a solid hyperbolic cylinderT j in the universal cover H 3 of M j by the action of an infinite cyclic group of isometries generated by an hyperbolic twist of length ε j and angle ρ j ∈ [o, π[. Some non trivial facts about hyperbolic geometry accounted for example in Colbois and Courtois ( [CC89]) or in Dodziuk and McGowan ([DG95]) force the distinguished constants R j , ε j , ρ j to satisfy the following inequalities: D 1 e −2Rj ≤ ε j ≤ D 2 e −2Rj E 1 e −Rj ≤ ρ j ≤ E 2 e −Rj where D j and E j (j = 1, 2) are positive constants. In terms of Fermi coordinates (r, t, θ) with respect to the geodesicγ j , the lift of γ j in H 3 , we can write the twist as A γj : (r, t, θ) → (r, t + ε j , θ + ρ j )(93) and the metric onT j as g j = dr 2 + cosh 2 r dt 2 + sinh 2 r dθ 2 , where r ∈]0, R j ], t ∈ [0, ε j ] and θ ∈ [0, 2π]. If we change the radial coordinate by u := R j − r ∈ [0, R j [ and introduce the following auxiliary functions ϕ j (u) := 1 4 (e 2u − 1)cosh −2 R j (e −2Rj (1 + e 2u ) + 2) ψ j (u) := 1 4 (e 2u − 1)sinh −2 R j (e −2Rj (1 + e 2u ) − 2),(95) the metric onT j becomes in the new coordinates g j = du 2 + e −2u (1 + ϕ j (u))cosh 2 R j dt 2 + (1 + ψ j (u))sin h 2 R j dθ 2 , (96) from which the similarity with the warped product metric g ′ j = du 2 + e −2u cosh 2 R j dt 2 + sinh 2 R j dθ 2(97) is evident. As a matter of fact ϕ j = o(1) and ψ j = o(1) pointwise on [0, R j ] and, in view of Thurston's Theorem, T j is expected to become a cusp in the limit j → +∞. This idea can be made more precise by means of quasi-isometries. As a matter of fact Colbois and Courtouis ( [CC89]) proved the following result. Proposition 7.2. For any L > 0 denote T j (L) := x ∈ (M j ) ]0,µ] \| dist(x, ∂M j ]0,µ] ) ≤ L , C(L) := x ∈ M ]0,µ] \| dist(x, ∂M ]0,µ] ) ≤ L(98) a piece of the tube and, respectively, a piece of cusp both of length L. Then, T j (R j − log R j ) and C(R j − log R j ) are quasi-isometric and the constants of the quasi-isometries converge to 1 as j → ∞. As a matter of fact it can be proved (cf. [Fa98]) that T j (R j −δ) and C(R j −δ) are quasi-isometric for δ = 1 2 . We conclude by some observations about the fibers of the tubes and the cusp. The warped product metric on T j writes as g ′ j = du 2 + e −2u ds 2 j where (F j , ds 2 j ) is a flat torus. More exactly :F j = R 2 and F j =F j / ∼ w.r.t. the identifications in polar coordinates (t, θ) ∼ (t + ε j , θ + ρ j ) and (t, θ) ∼ (t, θ + 2π) for all (t, θ) and the metric in the universal cover R 2 is given by ds 2 j = cosh 2 R j dt 2 + sinh 2 R j dθ 2 . Colbois-Courtois [CC89] proved: Proposition 7.3. A subsequence of (F j , ds 2 j ) j≥0 converges in the sense of Lipschitz to the flat torus (F, ds 2 ), where C = [0, +∞[×F is the cusp in the limit manifold (M, g). Warped Products We want to compute the eigenvalues of the Dirac Laplacian under the absolute boundary condition for a warped product. We consider a cylindrical manifold Z := [t 0 , t 1 ] × N , where (N, dσ 2 ) is a m − 1 dimensional Riemannian manifold, and t > 0 is a given parameter. A Riemannian metric for Z is defined by g := du 2 + ρ 2 (u)dσ 2 , for a given smooth function ρ. We assume that Z carries a Dirac bundle structure, which by Theorem 2.5, induces on each 1-codimensional submanifold N u := {u} × N with the metric ρ(u)dσ 2 a Dirac bundle structure. Lemma 7.4. Let Q N := Q Nu denote the Dirac operator on N u , P N := (Q Nu ) 2 the Dirac Laplacian on N u and H = − ρ ′ ρ the mean curvature of N u in Z. The Dirac Laplacian P Z on Z Laplacian can be written as P Z σ =P N s σ + [Q N , ∇ Z ∂u ]σ + m − 1 2 H ′ − m − 1 2 2 H 2 σ+ + (m − 1)H∇ Z ∂u σ − (∇ Z ∂u ) 2 σ,(99) for any smooth section σ of the Dirac bundle. Proof. According to Bär [Bä96] the Dirac operator on the warped product writes as Q Z σ = γ(∂ u )Q N σ − m − 1 2 Hγ(∂ u )σ + γ(∂ u )∇ Z ∂u σ.(100) So, for the Dirac Laplacian we have P Z σ = (Q Z ) 2 σ = = Q Z γ(∂ u )Q N σ − m − 1 2 Hγ(∂ u )σ + γ(∂ u )∇ Z ∂u σ = = γ(∂ u )Q N γ(∂ u )Q N σ − m − 1 2 Hγ(∂ u )σ + γ(∂ u )∇ Z ∂u σ + − m − 1 2 Hγ(∂ u )(γ(∂ u )Q N σ − m − 1 2 Hγ(∂ u )σ + γ(∂ u )∇ Z ∂u σ)+ + γ(∂ u )∇ Z ∂u γ(∂ u )Q N σ − m − 1 2 Hγ(∂ u )σ + γ(∂ u )∇ Z ∂u σ = = (Q N ) 2 σ + (Q N ∇ Z ∂u − ∇ Z ∂u Q N )σ − (∇ Z ∂u ) 2 σ+ + m − 1 2 ∂ u H − m − 1 2 2 H 2 σ + 2(m − 1)H∇ Z ∂u σ(101) which is the assertion of the lemma. Thereby we used that ∇ Z ∂u ∂ u = 0 (102) and that − γ(∂ u )∇ Z ∂u (Hγ(∂ u )σ) = ∂ u Hσ + H∇ Z ∂u σ.(103) The proof is completed. Proposition 7.5. The parallel transport Π Z in the Dirac bundle along the u lines of the warped product Z satisfies the following equation: Q Nu Π u0→u Z = ρ(u 0 ) ρ(u) Π u0→u Z Q Nu 0 ,(104) where u 0 , u ∈ [t 0 , t 1 ]. Proof. Let ∂ 1 := ∂ u , ∂ 2 , . . . , ∂ m be an orthogonal frame for T U , where U is an open set in Z. Equation (104) follows from the formulae ∇ Nu 1 ρ(u) ∂i Π u0→u Z = ρ(u 0 ) ρ(u) Π u0→u Z ∇ Nu 0 1 ρ(u 0 ) ∂i(u0)(105) and Π u0→u Z γ 1 ρ(u 0 ) ∂ i (u 0 ) = γ 1 ρ(u) ∂ i Π u0→u Z(106) which can be verified by a computation. Let {σ j (u 0 )} j≥0 be a L 2 -o.n. eigenbasis of Q Nu 0 over N u0 . Since γ(∂ u ) anticommutes with Q Nu , the section γ(∂ u0 )σ j (u 0 ) is an eigenvector for the eigenvalue −µ j , if we assume that σ j is an eigenvector for the eigenvalue µ j . Hence, we may assume σ j+1 = γ(∂ u )σ j and µ j+1 = −µ j . Parallel transport along the u-lines generates sections σ j = σ j (u, ·). By Proposition 7.5, when restricted to N u for u fixed, (σ j (u, ·)) j≥0 is a L 2 -o.n. eigenbasis of Q Nu with eigenvalues µ j (u) = ρ(u0) ρ(u) µ j (u 0 ). Since γ(∂ u ) is parallel along the u-lines, the equality σ j+1 (u, ·) = γ(∂ u )σ j (u, ·) holds for all u. Let now σ be a smooth section over Z. Restriction to N u for a u fixed yields a smooth section over N u which can be expressed in the basis (σ j (u, ·)) j≥0 as σ(u, ·) = j≥0 a j (u)σ j (u, ·).(107) By virtue of this representation we can rephrase eigenvalue equation and boundary condition for the warped product Z. Proposition 7.6. The eigenvalue equation for the Dirac Laplacian P Z σ = λσ on Z is equivalent to the ordinary differential equation on [t 0 , t 1 ] − a ′′ j + 2a ′ j + µ 2 j − µ ′ j − 2 − λ a j = 0 (j ≥ 0).(108) The absolute boundary condition B − σ\| ∂Z = B − Qσ\| ∂Z = 0 is equivalent to the boundary conditions a j (t 0 ) = a j (t 1 ) = 0 (j ≥ 0) (109) or a ′ j (t 0 ) = a ′ j (t 1 ) = 0 (j ≥ 0).(110) If we choose on N one of the three spin structures which exclude harmonic spinors, that is if µ 0 > 0 then the eigenvalues of the Dirac Laplacian on Z under the absolute boundary condition are uniformly bounded away from zero λ ≥ 1,(111) as soon as the cylinder satisfies following conditions t 0 > t min 0 := log(2µ 0 − 2), t 1 > t 0 + arctanh   √ µ 2 0 +exp(2t0)−µ0 exp(t0)−1 exp µ 2 0 2 (exp(2t0)−1)−µ0(exp(t0)−1)−t0   µ 2 0 + exp(2t 0 ) − µ 0 exp(t 0 ) − 1 .(112) Remark 7.1. It is possible to solve the ordinary differential equation (108) explicitly by means of a closed formed expression involving Laguerre polynomials, Bessel and confluent hypergeometric functions. However, after having inserted this expression into the boundary conditions (109) and (120), one obtains equations for λ which are not treatable analytically. Therefore, we have to look just for a rough estimate of the eigenvalues. Proof of Proposition 7.6. Let us first consider the boundary condition. Since σ j is a section of the spinor bundle over Z, whose rank is 2, it can be decomposed as σ j = 2 k=1 σ k j s k ,(113) where s 1 , s 2 are two sections of the spinor bundle satisfying B − s 1 = 0 B + s 1 = s 1 B − s 2 = s 2 B + s 2 = 0.(114) The absolute boundary condition for the Dirac Laplacian on Z reads B − σ \| ∂Z = 0 B − Qσ \| ∂Z = 0,(115) and becomes after having inserted the decomposition (107) j≥0 a j (u)σ 2 j (u, ·)s 2 \| ∂Z = 0 j≥0 a ′ j (u)σ 1 j (u, ·)s 1 \| ∂Z = 0(116) It corresponds therefore to the boundary condition in (7.6). For the differential equation we insert the decomposition (107) into the equation (P Z − λ)σ = 0 where we represent P Z using Lemma 7.4. Since ∇ Z ∂u σ(u, ·) = 0, being σ j parallel along the u-lines, we obtain, after having inserted m = 3 and H ≡ 1 j≥0 −a ′′ j + 2a ′ j + µ 2 j − µ ′ j − 1 − λ a j σ j = 0.(117) This equation is satisfied if and only if all coefficients vanish and the equation (108) follows. To solve it we introduce the substitution a j (u) := e − u 0 ds(µ 2 j (s)−µ ′ j (s)−1) a j (u),(118) under which equation (108) becomes −ā ′′ j + µ 2 j − µ ′ j − λ ā j = 0,(119) and the boundary conditions (109) and (110) become a j (t 0 ) = 0 a j (t 1 ) = 0.(120) for all j ≥ 0 and, respectively,        a ′ j (t 0 ) + µ 2 j e 2t 0 −µj e t 0 −1 exp( 1 2 µ 2 j (exp(2t0)−1)−µj (exp(t0)−1)−t0) a j (t 0 ) = 0 a ′ j (t 1 ) + µ 2 j e 2t 1 −µj e t 1 −1 exp( 1 2 µ 2 j (exp(2t1)−1)−µj (exp(t1)−1)−t1) a j (t 1 ) = 0.(121) We apply Proposition A.2 to the boundary value problem given by the equation (119) and the boundary condition (120) and Proposition A.1 to the boundary value problem given by equation (119) and the boundary condition (121). For the first BVP we set q j (u) := µ 2 j e 2u − µ j e u − λ ≥ k 2 j + (1 − λ), k 2 j := µ 2 j e 2t0 − µ j e t0 − 1(122) If λ < 1 then q j > k 2 j and by Proposition A.2 the boundary condition cannot not be satisfied for any t 1 > 0. Therefore, the eigenvalue must be bounded away from zero: λ ≥ 1. For the second BVP we set q j (u) := µ 2 j e 2u − µ j e u − λ ≥ k 2 j + (1 − λ), k 2 j := µ 2 j e 2t0 − µ j e t0 − 1, α j := µ 2 j e 2t0 − µ j e t0 − 1 exp 1 2 µ 2 j (exp(2t 0 ) − 1) − µ j (exp(t 0 ) − 1) − t 0 < k j .(123) If λ < 1 then q j > k 2 j . By Proposition A.1 the boundary condition cannot not be satisfied for any t 1 > t 0 + 1 kj arctanh αj kj . Therefore, the eigenvalue must be bounded away from zero: λ ≥ 1. We now analyze the bounds t 0 and t 1 a little closer. The function ∆ = ∆(µ, t) := arctanh √ µ 2 +exp(2t)−µ exp(t)−1 exp µ 2 2 (exp(2t)−1)−µ(exp(t)−1)−t µ 2 + exp(2t) − µ exp(t) − 1 (124) is well-defined if t > t 0 min := log(2µ − 2)(125) and satisfies for all j ≥ 0 ∆(µ j , t min 0 ) = 1 k j arctanh α j k j , ∆(µ j , t min 0 ) ≤ ∆(µ 0 , t min 0 ),(126) which can be verified by a direct computation using the first derivatives of ∆. The proof is completed. Proof of the Lower Bound Inequality We first sketch the structure of the proof of Theorems 1.9. We can assume without loss of generality that M has only one cusp. 1. For every j ≥ 0, we cover its approximating manifold M j with three 0codimensional submanifolds with boundary: (a) W j ⊃ (M j ) ]µ,∞[ ∪ x ∈ (M j ) ]0,µ] \| R j ≥ dist(x, γ j ) ≥ R j − r 0 : a compact neighborhood of the thick part of M j . (b) U j ⊃ x ∈ (M j ) ]0,µ] \| R j − r 0 ≥ dist(x, γ j ) ≥ 1 : a relevant piece of the tube (a solid annular torus). (c) V j ⊃ x ∈ (M j ) ]0,µ] \| 1 ≥ dist(x, γ j ) : a tubular neighborhood of the closed geodesic (a solid torus). The submanifolds can be chosen as the closure of a ε neighbourhood (for a fixed small ε ) of the sets specified on the right hand side. The constant r 0 > 0 will be appropriately chosen during the proof. 2. We change the metric on U j to that of the warped product (97). We control the spectrum from below by Lemma 5.5. 3. We compute the spectral bound given by Theorem 1.3. 4. We control the spectra of the bounded parts W j and V j under the absolute boundary conditions using spectral perturbation theory and Lemma 5.5 again. 5. Since the metric of the tube converge to the metric on the cusp, the lower eigenvalues of P on the piece of cusp for the absolute boundary conditions converge to the the lower eigenvalues of P on U j under the absolute boundary condition. Proof of Theorem 1.9. Following the steps above we apply Theorem 1.3 for the cover of the manifold, for which we have N 1 = N 2 = 0, N = 0 to obtain λ 2 1 (P ) ≥ C 1 1 µ(W j ) + 1 µ(U j ) + 1 µ(V j ) + + 4 C j µ(W j ∩ U j ) + 1 1 µ(W j ) + 1 µ(U j ) + +4 C j µ(U j ∩ V j ) + 1 1 µ(U j ) + 1 µ(V j ) −1 ,(127) for a C 1 > 0 and constants C j > 0 depending on the C 1 norm of a partition of unity subordinate to the chosen cover (cf. Theorem 1.3). The constants C j are bounded from above by a constant C 2 > 0. Now, we examine the different eigenvalues involved: • The eigenvalues µ(W j ) and µ(W j ∩ U j ) are bounded from below by a positive constant independent of j because W j converges to a closed ε neighbourhood of the thick part M thick , which is compact. • By Proposition 7.6 the eigenvalue µ(U j ) ≥ 1 as soon as r 0 > log(2µ 0 − 2). By Theorem 3 in [Bä00] (page 478) the spin structure is non trivial along all tubes and hence its restriction on the flat torus T 2 must be non trivial. Therefore µ 0 (T 2 ) ≥ π 2 and r 0 can be chosen accordingly. • The eigenvalues µ(V j ) and µ(U j ∩V j ) are bounded from below by a positive constant independent of j, because the metric there is quasi-isometric to that of the warped product (97) and Lemma 5.5. The lowest eigenvalues are bounded away from zero uniformly in j, because the spin structure is non trivial. We conclude that there exist a positive constant c > 0 such that λ 2 1 (P ) ≥ c,(128) where q ∈ C ∞ ([m 0 , m 1 ]) is a smooth function satisfying q > k 2 for constants k, α ∈ R such that k > 0 and α ≤ k. Then, for the unique solution v of the initial value problem      −v ′′ + k 2 v = 0 v(m 0 ) = a(m 0 ) v ′ (m 0 ) = a ′ (m 0 )(130) the following inequality holds on [m 0 , m 1 ]: a ′ a ≥ v ′ v .(131) In particular lim inf m1→+∞ a ′ (m 1 ) a(m 1 ) ≥ 1 2 k and a ′ (u) a(u) > 0 as soon as u > m 0 + 1 k arctanh α k . (132) Proof. Without loss of generality we can prove the inequality on [m 0 , m 1 [ and choose m 0 := 0 and m 1 = +∞. We need to distinguish several cases: case 0: a(0) = 0 never occurs. In fact, both cases a(0) = 0 and a ′ (0) = 0 are excluded by the assumption on the non triviality of a and by the existence and uniqueness theorem for the solutions of ordinary differential equations. case 1: a(0) > 0. Since a ′ (0) = −αa(0) > −ka(0), we obtain v(u) = a(0) cosh(ku) + a ′ (0) k sinh(ku) > 0 ∀u ∈ [0, +∞[. With w := a ′ v − av ′ it follows w ′ = (q − k 2 )av, w(0) = 0 and w ′ (0) = (q(0) − k 2 )v 2 (0) > 0. So, ε 1 := sup {u ∈]0, +∞[ \| w ′ > 0 on ]0, u[} must belong to ]0, +∞]. If ε 1 < +∞, then by continuity w ′ (ε 1 ) = 0. Analogously, since a(0) > 0, ε 2 := sup {u ∈]0, +∞[ \| a > 0 on ]0, u[} must be in ]0, +∞]. If ε 2 < +∞, then by continuity a(ε 2 ) = 0. Set ε := min{ε 1 , ε 2 }. On [0, ε[ one has w ≥ 0, i.e. a ′ a ≥ v ′ v , being a and v positive. Integrating both sides of this inequality , one gets a ≥ v on [0, ε[. So, on this interval one has w ′ (u) = (q − k 2 )a(u)v(u) ≥ (q(u) − k 2 )v 2 (u) = (q(u) − k 2 )a 2 (0) cosh 2 (ku) and a(u) ≥ v = a(0) cosh(ku). Assume now that ε < ∞. There are two possibilities: if ε = ε 1 , then by continuity w ′ (ε 1 ) = (q(ε 1 ) − k 2 )a 2 (0) cosh 2 (kε 1 ) > 0; if ε = ε 2 , again by continuity a(ε 2 ) ≥ a(0) cosh(kε 2 ) > 0. In both cases there is a contradiction, so it must be ε = ∞. We therefore come to the conclusion that a ′ a ≥ v ′ v on [0, +∞[. Since v(u) = 0 for u ∈ [m 0 , m 1 ] we can write: v ′ (u) v(u) = k a(m 0 ) tanh(k(u − m 0 )) + 1 k a ′ (m 0 ) a(m 0 ) + 1 k a ′ (m 0 ) tanh(k(u − m 0 )) . We insert the boundary condition a ′ (m 0 )+αa(m 0 ) = 0 and simplify by a(m 0 ) = 0: v ′ (u) v(u) = k tanh(k(u − m 0 )) − α k 1 − α k tanh(k(u − m 0 )) .(135) Since k > α, we obtain lim u→+∞ v ′ (u) v(u) = k(136) and the inequalities (132) follow from the estimate (132). Proposition A.2. Let the function a = a(u) be a non trivial solution of the linear second order boundary value problem −a ′′ + qa = 0 a(m 0 ) = 0, where q ∈ C ∞ ([m 0 , m 1 ]) is a smooth function satisfying q > k 2 for a constant k > 0. Then, for the unique solution v of the initial value problem      −v ′′ + k 2 v = 0 v(m 0 ) = 0 v ′ (m 0 ) = a ′ (m 0 )(138) the following inequality holds on ]m 0 , m 1 ]: a ′ a ≥ v ′ v .(139) There exist δ > m 0 such that a(u) ≥ a(δ)e case 1: a ′ (0) > 0. There exist a δ > 0 small enough such that a ′ (δ) > 0 and a(δ) > 0. Note that α := − a ′ (δ) a(δ) < k. We can continue by applying Proposition A.1 and obtain the result stated. case 2: a(0) < 0. Analogously to case 2 in the proof of Proposition A.1. Theorem 1. 1 . 1Let us consider a compact, oriented, Riemannian manifold M with (smooth) boundary ∂M . Let (V, ., . ) be a Hermitian (Riemannian) vector bundle over M and denote by P a formally selfadjoint operator of Laplace type. H B− (U i,j,k , V ) N := N 1 + N 2 + 1 m i := \|{j = i\| U j ∩ U i = ∅}\|For any closed set U ⊂ M let λ(U ) denote the smallest positive eigenvalue of the Dirac operator on exact Dirac sections satisfying the absolute boundary condition B − on ∂U . Then, the N -th positive eigenvalue of the Dirac operator over M has the following positive lower bound: and Phillips ([MP90]) proved spec(∆ 1 ) = [0, ∞[ and the accumulation rate near 0 was estimated by McGowan ([Go93]). Later on, these results were extended by Dodziuk and Mc-Gowan ([DG95]), who gave an asymptotic formula for the number of 1-form eigenvalues in an arbitrary interval [0, x]. Definition. (Spin Manifold) (M, g, s) is called a spin manifold if and only if 1. (M, g) is a m-dimensional oriented Riemannian manifold. 2. s is a spin structure for M , i.e. for m ≥ 3 s is a Spin(m) principal fibre bundle over M , admitting a double covering map π : s → SO(M ) such that the following diagram commutes: (M ) denotes the SO(m) principal fiber bundle of the oriented basis of the tangential spaces, and Θ : Spin(m) → SO(m) the canonical double covering. The maps s × Spin(m) → s and SO(M ) × SO(m) → SO(M ) describe the right action of the structure groups Spin(m) and SO(m) on the principal fibre bundles s and SO(M ) respectively. When m = 2 a spin structure on M is defined analogously with Spin(m) replaced by SO(2) and Θ : SO(2) → SO(2) the connected two-sheet covering. When m = 1 SO(M ) ∼ = M and a spin structure is simply defined to be a two-fold covering of M . The vector bundle over M associated to s w.r.t the spin representation ρ i.e. ΣM := s × ρ C l l := 2 [ m 2 ] is called spinor bundle over M , see [Bä91] page 18. Example 2.1. (Spinor bundle as a Dirac bundle) Let (M, g, s) be a spin manifold of dimension m. We can make the spinor bundle into a Dirac bundle by the following choices: V := ΣM : spinor bundle, rank(V ) = l ·, · : Riemannian structure induced by the standard Hermitian product in C l (which is Spin(m)-invariant) and by the representation ρ. ∇ = ∇ Σ : spin connection = lift of the Levi-Civita connection to the spinor bundle. by the universal property, the map γ extends uniquely to a real algebra bundle endomorphism γ : Cl(M, g) −→ Hom(V ). Example 2.2. (Exterior algebra bundle as a Dirac Bundle) Let (M, g) be a C ∞ Riemannian manifold of dimension m. The tangent and the cotangent bundles are identified by the ♭-map defined by v ♭ (w) := g(v, w). Its inverse is denoted by ♯.The exterior algebra can be seen as a Dirac bundle after the following choices: V := Λ(T * M ) = m j=0 Λ j (T * M ) : exterior algebra over M ·, · : Riemannian structure induced by g ∇ : (lift of the) Levi Civita connection γ : by the universal property, the map γ extends uniquely to a real algebra bundle endomorphism γ : Cl(M, g) −→ Hom(V ).Example 2.3. (Antiholomorphic Bundle as a Dirac Bundle) Let (M, g, Ω, J) be a Kähler manifold of even dimension m with Riemannian metric g, closed Kähler 2-form Ω and almost complex structure J ∈ Hom(T M ) satisfying J 2 = −1. Let W be an holomorphic hermitian bundle over M with hermitian structure ·, · W and canonical connection ∇ W . The antiholomorphic bundle can be seen as a Dirac bundle after the following choices: V := Λ(T 0,1 M ) * ⊗ W : antiholomorphic bundle over M . Remark 2. 2 . 2By formula (7) the connection ∇ is locally well and uniquely defined over contractible open subsets of M . One obtains then a global well defined ∇ by patching together the local connections, using a partition of unity: a convex linear combination of compatible connections is compatible. However, in general different choices of a cover of contractible open subsets of M and of a subordinate partition of unity give rise to different compatible connections. Example 2 . 5 ( 25Euler and Laplace-Beltrami Operators). The Dirac operator in the case of the exterior algebra bundle over Riemannian manifolds (M, g) is the Euler operator d + δ on forms on M . The Dirac Laplacian ∆ := (d + δ) 2 = dδ + δd is the Laplace-Beltrami operator. Example 2 . 6 ( 26Clifford operator and Complex Laplacian). The Dirac operator in the case of antiholomorphic bundles over Kähler manifolds (M, g, Ω, J) is the Clifford operator √ 2(∂ + ∂ * ), while the Dirac Laplacian is the complex Laplacian 2(∂∂ * + ∂ * ∂). Example 2. 7 ( 7Exterior Algebra in Even Dimensions). T := S: normalized orientation. (d + δ, S): signature complex.Example 2.8 (Exterior Algebra in any Dimensions). T defined as T \| Λ k (T * M) := (−1) k 1 Λ k (T * M) (d + δ, T ): (rolled up) De Rham complex. Example 2.9 (Spinor Bundle in even Dimension). T := S: normalized orientation. (D, S): spin complex. Definition (Generalized Cylinder). Let (M, g, Spin(M )) be a spin manifold of dimension n, Riemannian metric g and spin structure Spin(M ). The manifold Z := I × M , where I denotes an interval of the real line, equipped with the Riemannian metric g Z (u, x) := du 2 + g(x) and with the spin structure Spin(Z) := Spin(I) × Spin(M ), with double covering map π : Spin(Z) = Spin(I) × Spin(M ) → SO(Z) = SO(I) × SO(M ), π := (π\| Spin(I) , π\| Spin(M) ) (31) is a spin manifold (Z, g Z , Spin(Z)) termed generalized cylinder, and i : SO(M ) → SO(Z), (e 1 , . . . , e n ) → (ν, e 1 , . . . , e n ) denotes the canonical embedding. Proposition 3. 5 . 5If there is an isomorphism T for the Dirac bundle V anticommuting with the Dirac operator Q and commuting with the boundary condition B for either B = B + or B = B − or B = B APS , then the discrete spectral resolution of Theorem 3.4 can be chosen such that the equalities λ −j = −λ +j and ϕ −j = T ϕ +j hold for every j ∈ N * . In particular, the dimension of the space of harmonic sections satisfying the boundary condition B is always even.The local boundary conditions B ± defined by mean of a boundary chirality operator χ ∈ Hom(V \| ∂M ) are preserved by the Dirac complex (Q, T ) on V if and only if γ(ν)χ and T \| ∂M commute. This is always the case for a Dirac bundle in even dimensions, if we choose T := S and χ := S\| ∂M , where S is the normalized orientation of the Dirac bundle. A special case, where Proposition 3.5 in any dimension for local elliptic boundary conditions, is the De Rham complex with either the absolute or relative boundary conditions. The global boundary condition B APS defined by mean of the projection onto the eigenspaces of the non positive eigenvalues of A APS are preserved by the Dirac complex (Q, T ) on V if and only if A APS and T \| ∂M commute. This is always the case for a Dirac bundle in any dimension, if we choose T := S, where S is the normalized orientation of the Dirac bundle. Proposition 4. 1 . 1Let (V, ·, · , ∇, γ) be a Dirac bundle over the Riemannian manifold (M, g) with a bundle isomorphism T on V such that γ := iT γ anticommutes with γ and with the Dirac operator Q. The tuple (V, ·, · , ∇, γ, γ) defines a (1, 1)-Dirac bundle structure with corresponding Dirac operators Q and Q. The operatorsd := 1 2 (Q − iQ) = 1 + T 2 Q and δ := 1 2 (Q + iQ) = 1 − T 2 Q(36)are called derivative and coderivative operators on M and have following properties 1. The derivative defines a complex: d 2 = 0. Proposition 4. 2 . 2Let H 1 (M, V ), H 1 0 (M, V ) and H 1 B± (M, V ) the domain of definitions of d, d 0 , d B± and δ, δ 0 , δ B± and Q, Q 0 , Q B± , respectively. Let H 2 (M, V ), H 2 0 (M, V ), H 2 C± (M, V ) the domain of definitions of P , P 0 and P B± . They satisfy following properties: 3 . 3(Q, B ± ) are elliptic boundary value problems and Q * B± = Q B± are selfadjoint operators. If M is compact, the operators Q B± have pure point spectra and the corresponding eigensections are smooth. 4. (P, C ± ) are elliptic boundary value problems and P * B± = P B± are selfadjoint operators. If M is compact, the operators P B± have non negative pure point spectra and the corresponding eigensections are smooth.Proof. The proof is based on the Green's formula (38) and standard elliptic operator theory. Theorem 4. 3 ( 3Orthogonal Decomposition of Dirac Sections). Let (V, ·, · , ∇, γ) be a Dirac bundle over the compact Riemannian manifold (M, g), and • Ω(M, V ) := C ∞ (M, V ) be the smooth sections of the Dirac bundle on M , • H B± (M, V ) be the harmonic sections of the Dirac bundle on M satisfying the absolute or relative, respectively, boundary condition, • Ω d B± (M, V ) := ϕ ∈ Ω B± (M, V ) ∃ψ ∈ Ω(M, V ) : dψ = ϕ be the smooth exact Dirac sections on M satisfying the absolute or relative, respectively, boundary condition, • Ω δ B± (M, V ) := ϕ ∈ Ω p B± (M ) ∃ψ ∈ Ω(M, V ) : δψ = ϕ be the smooth coexact Dirac sections on M satisfying the absolute or relative, respectively, boundary condition. Theorem 4. 4 . 4The mappingsI ± : H B± (M, V ) → H B± (M, V ), ω → I(ω) := [ω](43)are a natural isomorphisms between harmonic Dirac sections and Dirac cohomologies.Remark 4.1. Of course decomposition (41) is a variation of the famous Hodge's Theorem and the isomorphims (43) provide a similar result to De Rham's Theorem. The Dirac Cohomology is a Riemannian but not a topological invariant. Proposition 4 . 5 . 45Let (M, g) be an m dimensional Riemannian manifold and {e i } i=1,...,m be a local orthonormal field of T M . Let T i := int(e i ) ext(e i ) − ext(e i ) int(e i ), and T := m i=1 T i P i , where the operator P i be the orthogonal projection onto W i := {ext(e i )ϕ\| ϕ is a local section of Λ(T * M )}. The operator T can be extended to M by a partition of unit argument and satisfies the following 1+T 2 (d + δ) = δ, 4. T (d + δ) = −(d + δ)T , 5. Absolute boundary condition: int(ν)(ϕ)\| ∂M = 0 ⇔ B + (ϕ)\| ∂M = 0, 6. Relative boundary condition ext(ν)(ϕ)\| ∂M = 0 ⇔ B − (ϕ)\| ∂M = 0, where B ± := 1−T γ(ν) 2 for γ(v) := ext(v) − int(v). 3 . 3We first have to introduce several technicalities. Let M be a compact manifolds with boundary. If we impose the absolute boundary condition B − φ\| ∂M = 0 on all Dirac eigensections considered, Theorem 4.3 and the preservation of the first order absolute boundary condition under the derivative d will allow for a special variational characterization of the spectra for Dirac and Dirac Laplacian. Inspired by results for Laplace-Beltrami operator on forms (cf. [DG95]) and using Theorem 4.3, one can prove Lemma 5.1. Let λ ∈ spec(P C± ) be a non zero eigenvalue of the Dirac Laplacian under absolute or relative boundary conditions, and • E B± (λ) := ϕ ∈ Ω B± (M, V ) P ϕ = λϕ be Dirac eigensections with eigenvalue λ, • E d B± (λ) := E B± (λ) ∩ Ω d B± (M, V ) be exact Dirac eigensections with eigenvalue λ, • E δ B± (λ) := E B± (λ)∩Ω δ B± (M, V ) be coexact Dirac eigensections with eigenvalue λ. Proposition 5. 3 . 3If (λ d i ) i≥0 := spec(P Ω d B − (M,V )) are the eigenvalues of the Dirac Laplacian on exact sections, then ( 47 ) 47Then, for every Φ ∈ Span({Φ i } i=0,...,N ), i.e. Φ = N i=0 a i Φ i , there exists a unique χ ∈ Span({χ i } i=0,...,N ), namely χ = N i=0 a i χ i , such that dχ = Φ. The uniqueness follows from the vanishing of a section which is at the same time exact and coexact. Moreover, • r: the restriction operator, which restricts global Dirac sections on M to each open set of the cover according to r(ω) := {ω\| Ui } i . (50) • s: the difference operator, which maps ω ∈ Π α0,...,αp Ω(U α0,...,αp ) with components ω α0,...,αp ∈ Ω(U α0,...,αp ) is defined as (sω) α0,...,αp := p+1 i=0 (−1) i ω α0,...,αi,...,αp , of eigenvalues like those presented in Proposition 5.3 lead to another interesting results concerning the stability of the spectrum of the classical Dirac operator under quasi-isometric perturbations of the Riemannian structure. H g := g −1 h satisfies h(u, v) = g(H g (u), v) for all u, v ∈ T M .Let γ be an element of the bundle Spin(M ) covering g in the bundle SO(M ).Given a representation σ of Spin(m) on C l for l := 2 [ m 2 ] we denote Σ γ := Spin γ (M ) × σ C l and we introduce the spinor bundle isomorphism β γ η : Σ γ M := Spin γ (M ) × σ C l −→ Σ η M := Spin η (M ) 2 m) 2. For (M, h t ) we have that Vol(M, h t → +∞ as t → +∞. The proof is completed. in the sense of pointed Lipschitz, for appropriate x j and x belonging to the thick part of M j and M , respectively. In particular, vol(M j ) ↑ vol(M ), diam(M j thick ) → diam(M thick ) and if M is non compact, then diam(M j ) ↑ ∞. Definition (Pointed Lipschitz Convergence). 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The Spectrum of the Hodge Laplacian for a Degenerating Family of Hyperbolic Three Manifolds. J Dodziuk, J Mc, Gowan, Trans. Amer. Math. Soc. 3476J. DODZIUK and J. MC GOWAN, The Spectrum of the Hodge Lapla- cian for a Degenerating Family of Hyperbolic Three Manifolds, Trans. Amer. Math. Soc. 347, no. 6, 1985-1995, 1995. On the Essential Spectrum of a Complete Riemannian Manifold. H Donnely, Topology. 20H. DONNELY, On the Essential Spectrum of a Complete Riemannian Manifold, Topology 20, (1-14), 1981. S Farinelli, Spectra of Dirac Operators on a Family of Degenerating Hyperbolic Three Manifolds, Diss. Math. Wiss. ETH Zrich, Nr. 12690, Ref.: E. Zehnder ; Korref. und Supervisor: B. Colbois. S. FARINELLI Spectra of Dirac Operators on a Family of Degenerating Hyperbolic Three Manifolds, Diss. Math. Wiss. ETH Zrich, Nr. 12690, Ref.: E. Zehnder ; Korref. und Supervisor: B. Colbois, 1998. On the Spectrum of the Dirac Operator under Bounday Conditions. S Farinelli, G Schwarz, J. Geom. Phys. S. 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[ "Field test of entanglement swapping over 100-km optical fiber with independent 1-GHz-clock sequential time-bin entangled photon-pair sources", "Field test of entanglement swapping over 100-km optical fiber with independent 1-GHz-clock sequential time-bin entangled photon-pair sources" ]	[ "Qi-Chao Sun \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nSchool of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina\n", "Yang-Fan Jiang \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Ya-Li Mao \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Li-Xing You \nShanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina\n", "Wei Zhang \nTsinghua National Laboratory for Information Science and Technology\nDepartment of Electronic Engineering\nTsinghua University\n100084BeijingChina\n", "Wei-Jun Zhang \nShanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina\n", "Xiao Jiang \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Teng-Yun Chen \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Hao Li \nShanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina\n", "Yi-Dong Huang \nTsinghua National Laboratory for Information Science and Technology\nDepartment of Electronic Engineering\nTsinghua University\n100084BeijingChina\n", "Xian-Feng Chen \nSchool of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina\n", "Zhen Wang \nShanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina\n", "Jingyun Fan \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Qiang Zhang \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n", "Jian-Wei Pan \nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n" ]	[ "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina", "Tsinghua National Laboratory for Information Science and Technology\nDepartment of Electronic Engineering\nTsinghua University\n100084BeijingChina", "Shanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina", "Tsinghua National Laboratory for Information Science and Technology\nDepartment of Electronic Engineering\nTsinghua University\n100084BeijingChina", "School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina", "Shanghai Institute of Microsystem and Information Technology\nState Key Laboratory of Functional Materials for Informatics\nChinese Academy of Sciences\n200050ShanghaiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nShanghai Branch\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina" ]	[]	Realizing long distance entanglement swapping with independent sources in the real-world condition is important for both future quantum network and fundamental study of quantum theory. Currently, demonstration over a few of tens kilometer underground optical fiber has been achieved. However, future applications demand entanglement swapping over longer distance with more complicated environment. We exploit two independent 1-GHz-clock sequential time-bin entangled photon-pair sources, develop several automatic stability controls, and successfully implement a field test of entanglement swapping over 1 arXiv:1704.03960v1 [quant-ph]	10.6084/m9.figshare.c.3874669.v1	[ "https://arxiv.org/pdf/1704.03960v1.pdf" ]	118,842,598	1704.03960	382113e1680f37978c3e5cf0b9b0308ab0d858d1	Field test of entanglement swapping over 100-km optical fiber with independent 1-GHz-clock sequential time-bin entangled photon-pair sources April 14, 2017 13 Apr 2017 Qi-Chao Sun National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina School of Physics and Astronomy Shanghai Jiao Tong University 200240ShanghaiChina Yang-Fan Jiang National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Ya-Li Mao National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Li-Xing You Shanghai Institute of Microsystem and Information Technology State Key Laboratory of Functional Materials for Informatics Chinese Academy of Sciences 200050ShanghaiChina Wei Zhang Tsinghua National Laboratory for Information Science and Technology Department of Electronic Engineering Tsinghua University 100084BeijingChina Wei-Jun Zhang Shanghai Institute of Microsystem and Information Technology State Key Laboratory of Functional Materials for Informatics Chinese Academy of Sciences 200050ShanghaiChina Xiao Jiang National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Teng-Yun Chen National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Hao Li Shanghai Institute of Microsystem and Information Technology State Key Laboratory of Functional Materials for Informatics Chinese Academy of Sciences 200050ShanghaiChina Yi-Dong Huang Tsinghua National Laboratory for Information Science and Technology Department of Electronic Engineering Tsinghua University 100084BeijingChina Xian-Feng Chen School of Physics and Astronomy Shanghai Jiao Tong University 200240ShanghaiChina Zhen Wang Shanghai Institute of Microsystem and Information Technology State Key Laboratory of Functional Materials for Informatics Chinese Academy of Sciences 200050ShanghaiChina Jingyun Fan National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Qiang Zhang National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Jian-Wei Pan National Laboratory for Physical Sciences at Microscale and Department of Modern Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics Shanghai Branch University of Science and Technology of China 230026HefeiAnhuiChina Field test of entanglement swapping over 100-km optical fiber with independent 1-GHz-clock sequential time-bin entangled photon-pair sources April 14, 2017 13 Apr 2017* These authors contributed equally to this work Realizing long distance entanglement swapping with independent sources in the real-world condition is important for both future quantum network and fundamental study of quantum theory. Currently, demonstration over a few of tens kilometer underground optical fiber has been achieved. However, future applications demand entanglement swapping over longer distance with more complicated environment. We exploit two independent 1-GHz-clock sequential time-bin entangled photon-pair sources, develop several automatic stability controls, and successfully implement a field test of entanglement swapping over 1 arXiv:1704.03960v1 [quant-ph] more than 100-km optical fiber link including coiled, underground and suspended optical fibers. Our result verifies the feasibility of such technologies for long distance quantum network and for many interesting quantum information experiments. Entanglement swapping [1,2] is a unique feature of quantum physics. By entangling two independent parties that have never interacted before, entanglement swapping has been used in the study of physics foundations such as nonlocality [1,3] and wave-particle duality [4]. It is also a central element in quantum newtwork [5,6], appearing in the form of quantum relay [7][8][9] and quantum repeater [10,11]. The integrity of an experimental realization of entanglemnet swapping is ensured only by satisfying these criteria: proper causal disconnection between relevant events [12,13], and independent quantum sources without common past [3]. Driven by the application of future quantum network, there has been a significant progress in experimental entanglement swapping since its first experimental demonstration [2]. Quantun interference with independent sources was addressed in a number of experimental settings [14][15][16][17][18].Quantum relay was simulated with coiled optical fibers in a laboratory environment [19]. Recently, entanglement swapping and quantum teleportation were realized in both free space and optical fiber link with about 100-km distance [20][21][22][23], in which the quantum sources shared a common past, the Bell state measurement (BSM) were performed locally, and the swapped (teleported) photonic qubits were sent afterwards over long distance for analysis. More recently, several teams succeeded in entanglement swapping and teleportation with high integrity over a fiber network (about 10 km) in the real world, in which they overcomed the challenge in removing the distinguishability between photons from separate quantum sources by defeating the noise in the real world [13,[24][25][26]. To date, there is no report on entanglement swapping or quantum teleportation with suspended optical fiber, which is more susceptible to environment but unavoidable for applications in optical fiber network. Here, we present an implementation of entanglement swapping in an inter-city quantum network, which is composed of about 77-km optical fiber inside the lab, 25-km optical fiber outside of the lab but kept underground, and 1-km optical fiber suspended in the air outside of the lab to account for various types of noise mechanisms in the real world. The schematic of the sequential time-bin entangled photon-pair source [27][28][29] is depicted in Fig.1(a). We carve the continuous wave output of a distributed feedback laser (with coherence time τ c = 300µs) periodically into pulses at a rate of 1 GHz with an electro-optical modulator (EOM). The sequential laser pulses differ with τ = 1 ns and a phase difference of θ = 2πντ . After amplification with an erbium doped fiber amplifier and spectral filtering with dense wavelength division multiplexing filters 15 (DWDM), the laser pulses are fed into a 300-m dispersion shifted fiber immersed the liquid nitrogen to produce photon-pairs via spontaneous fourwave mixing. The quantum state of a produced photon-pair is given by \|Θ = 1 √ n n−1 k=0 e 2ikθ \|t k s \|t k i , where t k = kτ represents time bin k, and s and i are for idler (1,555.73 nm)/signal (1,549.36 nm) photons. We single out the signal/idler photons with cascaded DWDM filters with pump photons suppressed by 115 dB. The schematic of the entanglement swapping experiment is shown in Fig.1 (b), with(out) gray shaded areas indicating indoor (outdoor) environment. The two quantum sources are placed 12.5 km apart at nodes Alice and Bob, respectively, and BSM is performed at the third node (Charlie) between them, as shown in Fig.1 (e). Alice and Bob hold idler photons with 26-km coiled fibers while sending signal photons to Charlie. We specifically keep 1-km optical fiber cable between Bob and Charlie suspended in the air and exposed to sunlight and wind to face inclement weather conditions. The rest deployed fibers are kept underground. The total length of the optical fiber is 103 km with a total transmission loss of ∼ 29 dB. Upon detection, both idler and signal photons are passed through 4-GHz fiber Bragg gratings, which is about half of the bandwidth of the pump pulses and is therefore sufficient to eliminate the frequency correlation between the twin photons [24]. To synchronize the operation of each node in the quantum network, Charlie keeps a master clock which sends pulses to Alice and Bob at 1 GHz, which is used to drive the EOM to synchronize independent quantum sources (see Fig.1 (c,d)). In the BSM, it is critical that the signal photons sent by Alice and Bob arrive at the 50:50 beam splitter (BS) simultaneously. However, the arrival time changes drastically due to the fluctuation of effective length of the optical fiber link which is subjected to the influence in the real world. As shown in Fig. 2, the typical peak-to-peak delays between arrival times of photons from Alice and Bob changes are 200 ps, 500 ps, and 1000 ps in rainy days, cloudy days, and sunny days, respectively, which are much larger than the coherent time of signal photons (∼ 110 ps). We use the difference between the arrival times of signal photons from Alice and Bob as error signals and feed them into a delay line to suppress the relative delay to 6 ps under all weather conditions, which is ∼ 110 ps to ensure high interference visibility. In addition, for each channel, an electronic-controlled polarization controller is used to compensate the polarization fluctuation caused by optical fiber. Before performing entanglement swapping, we characterize the quantum source of Alice (Bob) after entanglement distribution using the Fransontype interferometer. At nodes Alice (Bob) and Charlie, the idler and signal photon are fed into an unbalanced Mach-Zehnder interferometer (MZI) with an arm difference of 1-ns. The interference term in one output of the MZI can be written as n−1 k=1 e 2ikθ (1 + e i(θs+θ i −2θ) ) \|t k s \|t k i , where θ s and θ i are relative phases induced by MZIs for single and idler photons, respectively. The outputs of MZIs are detected by superconducting nanowire single photon detectors (SNSPD) and the detection results are recorded by time-todigital converters (TDC) with a 4-ps resolution and analyzed in real-time. The TDCs are synchronized with 10 MHz clocks at Charlie's node. Fig.3 shows that the measured two-fold coincidence counts between Alice (Bob) and Charlie change sinusoidally as a function of phase (which is dialed by sweeping the temperature of MZI), with a visibility of (89.8 ± 0.5)% for Alice's source and (82.9 ± 1.2)% for Bob's source, respectively. In the entanglement swapping experiment, Charlie performs BSM with signal photons sent by Alice and Bob by interfering them on a BS. By detecting them with two SNSPDs and within a delay of 1 ns (for the same time-bin or adjacent time-bins), the overall state of the two entangled photon-pairs, \|Γ 1,2 = 1 n ( n−1 k=0 e 2ikθ \|t k 1s \|t k 1i ) ⊗ ( n−1 l=0 e 2ilθ \|t l 2s \|t l 2i ), can be cast into \|Γ 1,2 → 1 n { n−1 k=0 e 4ikθ √ 2n \|t k 1i \|t k 2i (\|Φ + s,k + \|Φ − s,k )+ n−2 k=0 e i(4k+2)θ [ 1 √ 2 (\|t k 1i \|t k+1 2i + \|t k+1 1i \|t k 2i ) \|Ψ + s,k 1 √ 2 (\|t k 1i \|t k+1 2i − \|t k+1 1i \|t k 2i ) \|Ψ − s,k ]}, where the four Bell state sets are given by {\|Ψ ± s,k = 1 √ 2 (\|t k 1s \|t k+1 2s ± \|t k+1 1s \|t k 2s )} and {\|Φ ± s,k = 1 √ 2 (\|t k 1s \|t k 2s ± \|t k+1 1s \|t k+1 2s )}. The Bell states \|Φ ± s,k correspond to cases that the two signal photons are output from the same port of the BS and in the same time bin, which can not be discriminated using linear optics. The Bell states \|Ψ + s,k and \|Ψ − s,k correspond to cases that the two signal photons are output in time bins with 1-ns delay, while from the same port and different ports of the BS, respectively. In our experiment, the recovering time of the SNSPD is about 40 ns, much longer than the time delay of two consecutive time bins. So only the Bell states {\|Ψ − k s,k } are discriminated in BSM. As a result, their twin photons are projected to entangled quantum state \|Ψ − i,k = 1 √ 2 (\|t k 1i \|t k+1 2i − \|t k+1 1i \|t k 2i ). To verify the entanglement swapping, both Alice and Bob use a MZI with 1-ns path difference followed by SNSPDs and implement a conditioned Franson-type measurment for time-bin entangled state. The experimental result is shown in Fig. 4. The four-fold coincidence counts show a clear interference fringe and the average visibility of the fitted curves is (73.2 ± 5.6)%. If we assume that the two photons are in a Werner state, we can show that the lower bound of visibility to demonstrate entanglement is 1/3 [30]. The visibility achieved in our experiment clearly exceeds this bound. In our experiment, the total transmission loss of the optical fiber link is about 10 to 20 dB higher than previous field tests with independent sources. By setting the interval between adjacent coherent pump laser pulses equal to that of conventional time-bin entangled photon-pair source, the event rate of experiment with sequential time-bin entangled photon pair source can be increased by 3 times. We achieve a four-fold count rate in energy basis of ∼ 3/h with 1-GHz pump laser pulses. The relative low visibility of the entanglement created in entanglement swapping is mainly attributed to the imperfect sequential time-bin entangled photon-pair sources, which upper bounds the visibility to ∼ 74% ( product of the visibility of two entanglement sources). In our experiment, the average photon-pair number per time bin is µ ≈ 0.023 for both sources, which can decrease the visibility to ∼ 96% (V d ≈ 1/(1 + 2µ)). The wavelength of the CW laser is controlled with stability of 0.18 pm, resulting in a fluctuation of the relative phase in the entangled state. Assuming the temperature fluctuation follows a Gaussian distribution, it can decrease the visibility to ∼ 96%. The distortion of driven signals can also decrease the visibility. The main reason of distortion is the limited bandwidth of PD, which is 45 GHz and 10 GHz for Alice and Bob, respectively. So by employing a 45-GHz PD in Bob's source, it is possible that the visibility of the swapped entanglement can be increased to ∼ 80% so that it can be used to demonstrate quantum key distribution [26]. In summary, we have demonstrated entanglement swapping with two independent sources 12.5 km apart using 103-km optical fiber. Compared with previous experiments with independent sources, we have increased the length of optical fiber from metropolitan distance to inter-city distance. The transmission loss and stability of the optical fiber channel in our experiment is enough to match those of more than 100-km typical underground deployed optical fiber [31]. So, our results show that realizing entanglement swapping between two city is technically feasible. Moreover, the configuration of our experiment allows the space-like separation between any two measurements of those performed in the three nodes, and various of time-space relation can be achieved by combining both coiled optical fiber and deployed optical fiber. This distinguish feature together with the independent sources make our setup a promising platform for many interesting fundamental tests. Funding Information National Fundamental Research Program (under Grants No. 2013CB336800); National Natural Science Foundation of China; Chinese Academy of Science; 10000-Plan of Shandong Province; Quantum Ctek Co., Ltd. Figure 1 : 1Scheme of the entanglement swapping experiment. (a) Setup of sequential time-bin entangled photon-pair source. DFB, distributed feedback laser; EOM, electro-optic modulator; EDFA, erbium doped fiber amplifier; DWDM, dense wavelength division multiplexing filter; DSF, dispersion shifted fiber. (b) Experimental realization. Two sequential time-bin entangled photon-pair sources, S1 and S2, are placed in nodes Alice and Bob, respectively. Each of them keeps the idler photons in coiled fibers and distributed the signal photons to Charlie through deployed optical fiber. The yellow dash lines represent deployed fibers and the circles represent the coiled fibers. The total transmission loss of the 77-km coiled optical fiber is about 16 dB, while that of the deployed optical fiber in Alice-Charlie and Bob-Charlie links are 6 dB and 7 dB, respectively. To synchronize the two sources, Charlie prepares 1-GHz laser pulses (Sync) and sends them to Alice and Bob through deployed optical fiber (represented by the purple dash lines) to generate the driven signal for their EOMs. The setup MZI, unbalanced Mach-Zehnder interferometer; EPC, electronic-controlled polarization controller; FBG, fiber Bragg grating; SNSPD, superconducting nanowire single photon detector. (c) Setup of synchronization signal generation. Charlie first generates laser pulses with repetition rate of 500 MHz, and then doubles them using an MZI with 1-ns path difference. MG, microwave generator; PPG, pulse pattern generator; MWA, microwave amplifier. (d) Setup to generate the driven signal for EOM in (a). VDL, variable delay line; PD, photodiode. (e), satellite image of the experimental nodes. Figure 2 : 2Typical delay compensation (blue line) and relative delay between arrival time of photons from Alice and Bob (red line) under different weather conditions. Measured by a TDC with time resolution of 4 ps, the standard deviations of the relative delay in (a)-(d) are 6.1 ps, 6.5 ps, 6.0 ps, and 6.7 ps, respectively. Figure 3 : 3Two-fold coincidence counts of the sequential time-bin entangled photon-pairs distributed by Alice (a) and Bob (b) as functions of temperature of the MZIs. The measurement results are represented by squares and circles corresponding to Charlie's MZI at temperature of 20.425 • C and 20.6 • C, respectively. The error bars indicate one standard deviation calculated from measured counts assuming Poissonian detection statistics. The visibility of the fitted sinusoidal curves for the squares (circles) is (89.8 ± 0.6)% ( (89.5 ± 1.2)%) and (83.5 ± 1.5)% ((82.3 ± 0.8)%) for (a) and (b), respectively. 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[ "Chapter 4 From Lacaille to Lalande: French work on lunar distances, nautical ephemerides and lunar tables, 1742-85 [FINAL DRAFT]", "Chapter 4 From Lacaille to Lalande: French work on lunar distances, nautical ephemerides and lunar tables, 1742-85 [FINAL DRAFT]" ]	[ "R Dunn & R ", "Higgitt " ]	[]	[ "Navigational Enterprises in Europe and its Empires" ]	In the 1630s, when Galileo Galilei sought a longitude reward from the Dutch States for his curious floating telescopic device for observing eclipses of Jupiter's satellites, lunar distances were under intense discussion in France between Cardinal Richelieu and the French savant Jean-Baptiste Morin. 1 Drawing on the work of sixteenth-century authors including Gemma Frisius, Johannes Werner and Peter Apian, Morin set down 13 propositions outlining astronomical and computational methods for finding longitude from the Moon, including lunar distances, lunar altitudes, meridian transits and hour angles. 2 Morin also described the 'clearing' of observations for refraction and parallax. As Parès has explained, while the true angular distance would be the main objective of computations in the mid-eighteenth century, for Morin it was just one step in the process of defining the Moon's coordinates. From Morin's point of view, longitude difference had to be obtained by comparing the estimated coordinates of the Moon, deduced from lunar altitude and distance observations, with those computed from astronomical tables and tabulated in almanacs.In the mid-seventeenth century, however, stellar positions were not precisely known; the lunar motions had not been solved (predictions still being subject to errors of 30 to 50 arcminutes);	10.1057/9781137520647_4	[ "https://arxiv.org/pdf/2104.01916v1.pdf" ]	132,040,427	2104.01916	587e1182f1812a4d63e7659aa149b1e8fd6b8b69	Chapter 4 From Lacaille to Lalande: French work on lunar distances, nautical ephemerides and lunar tables, 1742-85 [FINAL DRAFT] Palgrave/McMillanCopyright Palgrave/McMillan2015 R Dunn & R Higgitt Chapter 4 From Lacaille to Lalande: French work on lunar distances, nautical ephemerides and lunar tables, 1742-85 [FINAL DRAFT] Navigational Enterprises in Europe and its Empires BasingstokePalgrave/McMillan20151 In the 1630s, when Galileo Galilei sought a longitude reward from the Dutch States for his curious floating telescopic device for observing eclipses of Jupiter's satellites, lunar distances were under intense discussion in France between Cardinal Richelieu and the French savant Jean-Baptiste Morin. 1 Drawing on the work of sixteenth-century authors including Gemma Frisius, Johannes Werner and Peter Apian, Morin set down 13 propositions outlining astronomical and computational methods for finding longitude from the Moon, including lunar distances, lunar altitudes, meridian transits and hour angles. 2 Morin also described the 'clearing' of observations for refraction and parallax. As Parès has explained, while the true angular distance would be the main objective of computations in the mid-eighteenth century, for Morin it was just one step in the process of defining the Moon's coordinates. From Morin's point of view, longitude difference had to be obtained by comparing the estimated coordinates of the Moon, deduced from lunar altitude and distance observations, with those computed from astronomical tables and tabulated in almanacs.In the mid-seventeenth century, however, stellar positions were not precisely known; the lunar motions had not been solved (predictions still being subject to errors of 30 to 50 arcminutes); 4 expedition', the second geodetic investigation of the shape of the Earth, which was sponsored by the Académie (1733- 44). Bouguer was then the best expert the Ministre de la Marine could find for institgating improvements to the navy. As early as 1726, however, he had signalled his rejection of mechanical timekeeping for longitude determination during his travels to the Equator, and maintained this stance in his reports as expert adviser between 1749 and 1758. 8 Such an authoritative rejection of timekeepers was to hold sway at the Académie until news of John Harrison's sea watch (H4) reached France in the 1760s. Following Bouguer's death in 1758, his position was split between the mathematician Alexis-Claude Clairaut and the astronomer Pierre-Charles Lemonnier, both experts on lunar tables and their nautical uses yet with differing views and methods. 9 Jérôme Lalande replaced Clairaut after his death in May 1765. 10 Given responsibility for improving the navy, these savants worked under the control of the Ministre de la Marine, without interference from their peers or the Académie. Improving nautical astronomy was considered a task not for naval officers but for the scientific elite: royal astronomers and members of the Académie. Many of the books by these savants, mathematicians and astronomers -Maupertuis, Bouguer, Clairaut, Le Monnier and Lalandeshould, therefore, be read in the context of their role as 'official responsible for the improvement of navigation', giving a more consistent and deeper view of their scientific activity. Examining their works in this light, it is possible to draw some conclusions about the nature of their expertise and influence. With some hindsight, for example, three of the five might be said to have had a negative influence on the development of scientific navigation in France. • Maupertuis failed to answer the Ministre de la Marine's demands and, for a while, halted Lacaille's attempts to develop nautical astronomy and lunar methods; 11 • Bouguer discouraged timekeeping research during the 1750s, despite knowing skilled clockmakers (the Lepaute and Leroy brothers) with relevant expertise; 12 • Lemonnier disseminated old astronomical and nautical methods; he was not able (or did not wish) to follow new developments in Newtonian celestial mechanics, something that overshadowed his quarrels with Lacaille, Clairaut and Lalande (his expupil Tables with the corrections of lunar tables from observations of a Saros, would also play an important part in Lemonnier's and Pingré's longitude developments, discussed below. 15 In 1742, Nicolas-Louis de Lacaille, 'adjoint astronome' of the Académie, took charge of the computation and publication of the Ephémérides des mouvements célestes. These were computed for a period of ten years and gave the astronomical data needed to compute calendars. Lacaille was aware of the recent requests for a renewal of the methods of nautical astronomy and of Halley's 1731 paper. In 1742, while working on the fourth volume of the Ephémérides for 1745-55, Lacaille thought of adding considerations of new ways of finding longitude at sea from lunar distances. Hearing that Maupertuis, his superior at the Académie and the official 'responsible for the improvement of the navigation', was shortly to publish the Astronomie nautique (1743), however, he shelved his plan, assuming that Maupertuis' book would deal with longitude at sea. As it turned out, it did not. A new opportunity arose, however, when Lacaille met Jean-Baptiste d'Après de Mannevillette, an officer of the Compagnie des Indes who was based in the port of Lorient in Brittany and had good relationships with instrument makers in Paris. In June 1749, having made improvements to the octant, Mannevillette was the first naval officer to apply the lunar-distance method at sea, near Cape Verde, using an octant of Caleb Smith's design. 16 Well trained in mathematics, Mannevillette later said that he was able to determine longitude with an accuracy of between five 7 and 15 'lieues marines' or marine leagues (25 to 45 km); in other words, to an accuracy greater than that required by the 1714 Longitude Act. 17 His results were published much later, in 1775, in the Neptune François collection of sea charts. 18 Mannevillette's voyage to the Cape of Good Hope (1750-54) with Nicolas-Louis de Lacaille helped rekindle practical interest in the lunar-distance method. Both used the method to determine the longitude of Santiago in the Cape Verde Islands with considerable accuracy in November 1750. Several determinations of longitude differences were also made by lunar distance (from Antares) in Rio de Janeiro in January 1751. Given his skill in determining stellar positions, improving tables of atmospheric refraction and correcting tables for solar motions, not to mention his familiarity with Clairaut's work on lunar theory, it is no surprise that Lacaille was able to deploy and correct the tables required for carrying out the lunar-distance method. After completing his work on lunar parallax, geodesy and stellar cataloguing in Cape Town, Lacaille developed his ideas on the voyage back to France. In 1754, he sent a memorial on his new method to the Académie. 19 Noting that most seafarers lacked the scientific training to carry out the lunar method, he argued that it should be adapted and put 'within the reach of ordinary sailors': During this sea voyage, I occupied myself in making trials of the method of observing longitude at sea by means of the distances of the Moon from some zodiacal fixed star. Following my departure from France, I made numerous investigations to facilitate the practice of the method proposed by Mister Halley. I recognized that it was useless to look for another way of using the Moon for the longitude; that it was solely a question of making the calculation easy for ordinary sailors. 20 8 Lacaille also proposed computing the predicted lunar distance from the Sun and other key stars every three hours in a 'nautical almanac', the model Maskelyne would apply ten years later. Nevertheless, there was no consensus within the Académie. Lacaille's main rival, the astronomer Pierre-Charles Lemonnier, was attempting to publish the first nautical almanac entirely devoted to the lunar altitude method (as well as that of hour angles). 21 The computations were performed by Alexandre-Guy Pingré. The resulting almanac, the État du ciel, was published four times between 1754 and 1757, but does not seem to have been used by seafarers owing to its complex computations. Nonetheless, with this publication in progress and with Lacaille at sea near Isle de France, Lemonnier was able to stop Lacaille's project. Lacaille's graphical method After his return to France in 1755, Lacaille was able to defend his proposal. He read a memorial to the Académie in 1759, which set out his plan for a pre-calculated table of lunar distances and added a graphical method to avoid the long and difficult calculations normally required by the lunar-distance method. His ideas were later promoted by Jérôme Lalande, who was elected in 1759 to take charge of the computation and publication of the Connaissance des temps (hereafter CDT). Lalande had some original views on the CDT and its contents, notably adding new scientific matter that can often now only be found there. The volume for 1761, for instance, included Lacaille's procedures and methods for lunar distances. In 1755, he also published the Ephémérides des mouvements célestes, astronomical tables for ten years (1755-65), in which he gave further consideration to lunar distances and his longitude method. In 1760 Lacaille also published a revised edition of Bouguer's Nouveau traité de navigation, which expanded on his graphical method for lunar distance, a method Bouguer had previously overlooked. 9 Lacaille's graphical method derived from an idea proposed in 1692 by Father Paul Hoste, S.J., professor of hydrography for the navy in Brest, and explained more carefully by Bouguer in the first edition of the Nouveau traité de navigation (1753). The method was also taught in Saint-Malo by a navigation teacher called Griffon, for whose 1748 memorial to the Académie Lacaille was the academic referee. 22 Griffon proposed a developed version of Hoste's method. The basis was to draw a circle representing the celestial sphere with the observer at the centre, then plot the pole and the equator at right angles to each other. To determine local time, it was easy to plot the Sun's path, the only data required being solar declination, which mariners could look up in almanacs. Following Bouguer's elaboration, with worked examples, in the Nouveau traité de navigation, Lacaille extended it to the determination of the apparent angular distance between the Moon and Sun. In doing so, he transformed the computations of spherical trignonometry into graphical operations of simple geometry, using only ruler, compass and the four basic operations, something easily within the grasp of the common seafarer. 23 Lacaille did not, however, specify the elements needed to clear or correct the angles for lunar parallax. best to have three observers measuring the two altitudes and the angular distance simultaneously, thus avoiding the need to calculate by interpolation the small but significant horary motion of the Moon, which would be a source of error. 24 As Sadler has explained, Maskelyne was unable to use the graphical method because of an error Lacaille made in the example calculations. 25 In fact, the error came from Lalande in the CDTs for 1761 and 1762, which had mistakenly swapped the figures for Regulus and Aldebaran in the examples for 8 July 1761. 26 10 Like the Chevalier Jean-Charles de Borda after him, Maskelyne thought that Lacaille was too pessimistic about the accuracy with which one could measure the angular distance between the Moon and a star. 27 Nonetheless, Lacaille was the first astronomer to study the propagation of errors, drawing on Roger Cotes's Harmonia Mensurarum (edited by Robert Smith, 1722). Lacaille believed an accuracy of about four minutes of arc was possible; Maskelyne and Borda gave one minute (of arc) for the angular distance, and preferred to develop the lunar-distance method without simplified techniques. In fact, Borda's opposition to Lacaille's methods went deeper. As already noted, Lacaille had argued that lunar distances had to be be adapted to seafarers' use, a view echoed in Alexis Rochon's and Lalande's beliefs that the astronomer's task was to simplify and popularize: We have simplified through tables all the other parts of the longitude calculations […] This part, however, greatly lengthens the longitude method and prevents many seafarers from engaging in these studies: if they [seafarers] continue to neglect these observations at the risk of their fortunes and lives, it is the astronomers' duty to lessen the difficulties and to remind them of the vital matters at stake. 28 In the same year as Rochon's memorial, however, Borda condemned the use of graphical methods, which had 'the drawback of having men, only too inclined to it, becoming used to a process in some way automatic' 29 . Borda concluded elsewhere that the best way for navigators to avoid the difficulties and inconveniences of calculation was to be properly taught how to calculate. Borda's expectations were high, his opinion of seafarers low: 'It is about time that seafarers ceased looking at the mathematical and physical sciences as having no practical use in navigation and its progress. Without the help of science, the Navy would still be in its infancy'. 30 In the eyes of many mathematicians, examiners and savant-officers of the French navy, it was up to seafarers to rise to the demands required by the new navigational methods, rather than for mathematicians to simplify solutions and contrivances to circumvent direct calculation. 31 French and Portugese attempts to publish nautical almanacs The development of lunar methods for determining longitude at sea was directly connected to the establishment of nautical almanacs, not only in France, but also in Portugal. In France, the process began in the early 1750s with a dispute over the contents of the CDT. The Abbé André- 12 Likewise, in 1758 the Portuguese Jesuit Eusebio da Veiga published the Planetario Lusitano, a type of nautical almanac, a year before the dispersal of the Jesuit order. 34 This latter event was to disrupt maritime scientific education for several years in France, and more in Portugal, since maritime education in both countries was mainly conducted by Jesuits. During the 1750s, therefore, there was significant activity around the problem of improving existing ephemerides in France and Portugal and of encouraging astronomers in the belief that they should produce the necessary tools for finding longitude at sea. Lalande, Jeaurat, Maskelyne and lunar distances in the Connaissance des temps Elected as the new director of the Connaissance des temps in 1759, Lalande worked over the next deacde or so to develop lunar-distance methods, despite considerable resistance from the Académie. In 1772, for example, the Académie de marine in Brest proposed translating the British Nautical Almanac and publishing an edition of Dunthorne's and Witchell's formulae for clearing lunar distances. Initially, however, the Ministre de la Marine refused to grant the privilege for printing and forbade any translation, considering it an allégence to France's maritime rival; there was also a question of rivalry between the two royal academies. Only after several exchanges with the Ministre did Lalande, who was a member of both academies (Paris and Brest), gain authorization to add lunar-distance tables to the CDT. As Jim Bennett has shown, there was only a weak notion of state secrecy regarding longitude discoveries. 35 There were no secrets regarding astronomical and nautical computations, with such information circulating easily and quickly between Maskelyne and French astronomers at the end of the eighteenth century. 36 Lalande and his pupils Edme-Sébastien Jeaurat and Pierre Méchain used the work of Maskelyne's computers to complete French ephemerides between 1772 and 1785, as well as 13 lunar-distance tables reduced to the Greenwich meridian, before beginning to compute the same tables, reduced to the Paris meridian after 1790, with the help of the first ever full-time lunardistance computer, Louis-Robert Cornelier-Lémery. 37 The need for purely French nautical ephemerides became even more pressing when Lalande discovered in 1803 that Maskelyne had made errors of five to six seconds in the positions of the stars needed for lunar-distance and other tables: We are occupied these days in recalculating from observations Maskelyne's 34 stars which we've used with complete confidence, and I find it is necessary to add 5 or 6 seconds to the right ascensions. So we'll have to correct all our catalogues, all our tables and all our longitudes of the observed planets! This old pen pusher, lazy drunkard, miser, has usurped our trust. He's very rich, he should have got himself a computer and checked, more than once, this important result. 38 Lunar tables for longitude: Mayer versus Clairaut, or empiricism versus theory? To understand how theoretical knowledge circulated within the network of European astronomers and géomètres (mathematicians), it is necessary to examine the ways in which astronomical tables of the Moon's motions were developed between 1750 and 1770 as the basis for computations of the lunar-distance elements of nautical almanacs and ephemerides. In this context, it is important to remember that in 1765 a reward was given posthumously to Tobias Mayer for his lunar tables. To achieve an accuracy of half a degree of longitude, as specified by the Longitude Act, the tables had to be able to give the ecliptic longitude of the Moon to within one arcminute. 14 Less well known than the award to Mayer's widow is a letter (in English) of 11 April 1765 from Alexis Clairaut to John Bevis, claiming an equal part of the reward. 39 Clairaut was not generally known as a mathematician involved in the development of nautical astronomy, nor for his commitment to the improvement of the navigation, yet his letter claimed that his lunar tables were superior to Mayer's. There was some history to this work. Despite repeated efforts, Newton had failed to formulate a complete theory of the Moon's motions, leaving to later mathematicians and astronomers the task of solving by approximation the three-body problem; that is, the Keplerian problem of the motions of two celestial bodies, but also taking into account the Bouguer, the correction of theoretical terms by means of observations could only be a short-term solution and was intellectually unsatisfactory. Clairaut followed the same line, which he repeated in his letter to Bevis, writing that: as I have done it by the meer theory, it is to hope that their agreement with the observations will hold more constantly than that which is grounded upon an empirick method, which may be good for a time not very distant from the observations made use of in the confection of the tables, and disagree afterwards. 41 For Clairaut and d'Alembert, the correction of lunar tables by means of observations was not the right way to develop lunar theory: We can even observe that in the equations M. Mayer uses for his tables, the values of the coefficients are not exactly the same as those he extracted from theory; from which it would appear that the tables of M. Mayer were partly drawn up from observations, by a sort of trial-and-error method, combined with the principal results derived from theory. 42 Leonhard Euler wisely noted in 1765 that the three-body problem was highly complex and that mathematicians needed time to complete the project; it was premature to argue about it. Even when Euler's son, Johan Albrecht, wrote in 1766 that the lunar tables produced by Clairaut and Mayer were sufficiently accurate for calculating lunar distances, he added that the problem was still not really solved; the necessary approximations and theory remained incomplete. 45 Lalande promoted Lacaille's graphical method in the Connaissance des temps. Not fully explained, either by Lacaille or by Lalande, this method was tried and discussed by Maskelyne during his sea trials, but was abandoned. Nonetheless, the need expressed by Lacaille to develop simplified methods for 'ordinary sailors' began to be met by Jérôme Lalande and Abbott Alexis Rochon at the end of the eighteenth century. Helena between in 1761 offers some evidence. On page after page, his work shows knowledge of celestial mechanics, mainly from Clairaut's early works and Mayer's 1755 tables, and draws from them every element concerned with the problem of determining longitude at sea: angular distance, clearing for the effects of parallax and refraction. 46 Maskelyne had clearly read Lacaille and Lalande in the Connaissance des temps, as well as Bouguer's Nouveau traité de navigation as revised and expanded by Lacaille in 1760. The Nautical Almanac appears, therefore, to be an adaptation of the French ephemeris and a realization of Lacaille's earlier proposals for a nautical almanac. Nonetheless, the full extent of what Maskelyne owed to Lacaille has yet to be fully explored. For his voyage to St Helena in 1761, Nevil Maskelyne took the CDT for 1761 and Lacaille's ephemeris. In a letter published in the Philosophical Transactions the following year, he discussed Lacaille's method, in particular the graphical method, agreeing with Lacaille on one point: the practical dispositions needed for the lunar distance method. Like Lacaille, he felt it François Brancas de Villeneuve proposed modifying the CDT to transform it into a nautical almanac, and published Éphemérides cosmographiques between 1750 and 1755. In April 1755 he sent a memorial to the Ministre de la Marine and the astronomer Joseph-Nicolas Delisleone of Lalande's former teachersin which he castigated the astronomers and exorted them to concentrate on their real work: producing tables for longitude determinations. 32 Brancas added that such a nautical almanac should be published two or three years in advance. While his other proposals were sometimes idiosyncratic, Brancas set out all the principles, known from Delisle and Lalande, for creating a nautical almanac. 33 At the same time, Lacaille and Lemonnier were working on proposals for new nautical ephemerides to help seafarers. Nor were these the only attempts to publish nautical almanacs in France. In several ports, small nautical almanacs existed, called (with local variations) Étrennes maritimes et curieuses, Étrennes nautiques, Étrennes nantaises, and similar. These gave the times of rising and setting and the declinations of the Sun and Moon, information needed to determine local time from the altitude of either body; in other words, the basic astronomical elements used in Hoste's and Bouguer's graphical methods. perturbations caused by a third body. In fact, this problem has no analytical and exact solution and can be solved only by successive approximations, the theory of perturbations. The Moon's ecliptic longitude is obtained by the addition of terms which appear as smaller and smaller corrections to the elliptical Keplerian orbit. From 1743 Clairaut, Euler and d'Alembert developed such a theory in an atmosphere of intellectual competition and rivalry over the motions of both the Moon and comets. 40 But how did these geomètres construct the tables from which astronomers might compute navigational terms and, most importantly, the lunar distances published in nautical ephemerides? And how did astronomers correct or adjust the theoretical computations of the Moon's position against observed lunar positions? Eighteenth-century geomètres, mathematicians and/or astronomers had very different, and often unambiguous, opinions on these questions. On one hand, some argued, tables had to be obtained only from theory, a pure theory. On the other, it seemed necessary to make corrections of the Moon's position by reference to practical observations: the theory was modified, for example, with a term obtained from the computation of the mean of O -C (observed minus computed). In other words, the result could be seen as 15 empirical. Clairaut developed his tables with 22 terms obtained theoretically; Mayer developed his with 26 terms corrected from observations.The divergence between theoretical and empiricial developments in celestial mechanics seems to have begun when Bouguer wrote of the empiricism of Mayer's lunar tables in 1754.For A number of French astronomers helped Clairaut with his calculations; apparently both he and d'Alembert disliked number crunching. Delisle performed the calculations for the lunar tables published in 1754; in 1763-64, Bailly, Jeaurat and Pingré did the same for the second edition of Clairaut's theory of the Moon's motions (Saint-Petersburg, 1765) and for the 1764 annular eclipse of the Sun. Clairaut's and Mayer's lunar tables were also tested in 1764 based on their predictions of an annular eclipse of the Moon due to occur on 1 April; Mayer's tables suggested that the eclipse would not be seen in Paris, Clairaut's that it would. Clairaut won this test, since the eclipse was indeed observed in Paris. The astronomers Cassini III, Bailly and Pingré subsequently recommended that the Académie compute the lunar elements published by Lalande in the CDT on the basis of Clairaut's tables, considered as 'pure' theoretical tables. The rejection of Mayer's tables by some French astronomers, Lacaille and Lalande aside, can also be understood in the light of his failure to answer Lacaille's (and d'Alembert's) challenge of explaining the fundamentals of his theory of lunar motion. 43 Most French astronomers defended and used Clairaut's lunar tables until the late 1780s. Edme-Sébastien Jeaurat began comparisons between Clairaut's and Mayer's tables in June 1759, completing major studies for 1759, 1764, 1776-79, 1780 and 1781-82, the last two also performed by Cornelier-Lémery. In the first three studies, Clairaut's lunar tables came out well: the discrepancies of the errors (O -C) were similar to those of Mayer's tables. Both sets of tables proved to be accurate, with a mean error about one minute of arc for the ecliptic longitude of the Moon. At the beginning of the 1780s, however, Jeaurat and Lémery pointed out that the discrepancies in Clairaut's lunar tables were increasing because, unlike his rivals, Clairaut had not included the secular acceleration of the Moon. Subsequently, Lémery mainly used Euler's tables, computed from his second theory of the lunar motions, until the beginning of the nineteenth century, when they were superseded by the tables of Bouvard, Bürckhardt, Damoiseau and Plana, all based on Laplace's celestial mechanics. 44 Conclusion The development of theories and practices for finding longitude at sea by lunar methods followed different courses in Britain and France in the mid-eighteenth century. British astronomers mainly focused on Halley's methods for lunar distances until the publication of the Nautical Almanac in 1767. French astronomers, by contrast, were aware of Jean-Baptiste Morin's methods and explored them in the 1740s and 1750s. From the 1750s, there were also attempts in France to develop and adapt ephemerides and nautical almanacs for the needs of seafarers. Lacaille played an important and significant role in this. He developed the lunardistance method and gave a model for a nautical almanac containing pre-computed angular distances between the Moon and a bright star every three hours. His works were a source of inspiration for Maskelyne in Britain. Nevil Maskelyne had certainly read Lacaille and Lalande by the start of the 1760s, but what did he know of Lemonnier and Pingré's attempts to produce the État du Ciel ephemeris? What did he know of French debates on longitude? And what influence did they have on Maskelyne's own thinking about these problems and their solutions? Maskelyne's journal of his voyage to St ) . )There were deep disagreements and controversies within and between the main scientific academic organizations too: within the Académie (between factions, as well as individually between d'Alembert and Clairaut, Lemonnier and Lacaille, Lalande and Lemonnier, Lalande and Cassini III, and others); within the Académie de Marine in Brest (Lemonnier was excluded in 1771, for example); and between successive Secrétaires d'état de la Marine because of an overly mercurial and hesitant French naval policy (in particular under Maurepas, Rouillé, Choiseul, Praslin, de Boynes and de Sartines). 13 The first observations of lunar distance at sea: Abbé Lacaille and Jean-Baptiste d'Après de Mannevillette, 1749-51 After the passage of the Longitude Act in 1714 and the partial development of lunar theory by Isaac Newton, it seemed clear to astronomers that the Moon was the only natural clock that could be used regularly at sea. In 'A proposal of a method for finding the Longitude at sea within a degree' in the Philosophical Transactions for 1731, Edmond Halley offered a method based on observations of occultations of a star by the Moon for correcting lunar tables and calculating the ecliptic longitude of the Moon to within two arcminutes. In principle, the method was sufficiently precise for lunar methods at sea, and Halley's paper showed how the observation of a single angular distance between the Moon and a fixed star could help the seafarer determine longitude. Years later, Lacaille and the Jesuit professor of hydrography and astronomer in Marseille, Father Esprit Pezenas, would write that Halley's method was merely a variant of Morin's lunar altitude method, in which the lunar distance was only an intermediate step towards calculating longitude, not the endpoint. 14 Halley's paper, and the publication in 1742 of his6 Astronomical Jean Parès, 'Jean-Baptiste Morin (1583-1654) et la querelle des longitudes de 1634 à 1647' (Thèse de Doctorat de 3ième cycle, Paris, E.H.E.S.S., Université Paris-I, 1976). 2 Jean-Baptiste Morin, La science des longitudes de Jean-Baptiste Morin (Paris: aux dépens de l'Autheur, 1647), pp. 21-40, for Morin's 13 propositions. 3 Royal warrant, quoted in Derek Howse, Greenwich Time and the Longitude (London: Philip Wilson/National Maritime Museum, 1997), p. 42. 4 Jean Picard et les débuts de l'astronomie de précision au XVII e siècle, ed. by Guy Picolet (Paris, CNRS Éditions, 1987). See also, J. D. North, 'The Satellites of Jupiter, from Galileo to Bradley', in The Universal Frame. Historical Essays in Astronomy, Natural Philosophy and Scientific Method (London: Hambledon Press, 1989), pp. 185-214. 5 Guy Boistel, 'L'astronomie nautique au XVIII e siècle en France: tables de la Lune et longitudes en mer' (Thèse de doctorat en épistémologie, histoire des sciences et des techniques, Centre François Viète, Université de Nantes, 2001), part I, pp. 30-75; Guy Boistel, 'Pierre-Louis Moreau de Maupertuis: un inattendu préposé au perfectionnement de la navigation (1739- Annales 2003 de la Société d'histoire et d'archéologie de l'arrondissement. de Saint-Malo', in Annales 2003 de la Société d'histoire et d'archéologie de l'arrondissement de Saint- Malo (2004), 241-61. French title: 'préposé au perfectionnement de la navigation or de la Marine sous toutes ses formes'; see Boistel, 'L'astronomie nautique' and Boistel. Pierre-Louis Moreau de Maupertuis'French title: 'préposé au perfectionnement de la navigation or de la Marine sous toutes ses formes'; see Boistel, 'L'astronomie nautique' and Boistel, 'Pierre-Louis Moreau de Maupertuis'. . 'pierre-Louis Boistel, Moreau De Maupertuis, Boistel, 'Pierre-Louis Moreau de Maupertuis'. commissaire pour la marine et expert pour les longitudes: un opposant au développement de l'horlogerie de marine au XVIII e siècle?' in Pierre Bouguer (1698-1758), un savant et la marine dans la première moitié du XVIII e siècle. Guy Boistel, Pierre Bouguer, Rev. Hist. Sc. 63Guy Boistel, 'Pierre Bouguer, commissaire pour la marine et expert pour les longitudes: un opposant au développement de l'horlogerie de marine au XVIII e siècle?' in Pierre Bouguer (1698-1758), un savant et la marine dans la première moitié du XVIII e siècle (= Rev. Hist. Sc., 63 (2010)), 121-59. . ' Boistel, Boistel, 'L'astronomie nautique'; La Lune au secours des marins: la déconvenue d'Alexis Clairaut', Les génies de la science. Guy Boistel, 23Guy Boistel, 'La Lune au secours des marins: la déconvenue d'Alexis Clairaut', Les génies de la science, 23 (2005), 28-33; Au-delà du problème des trois corps: Alexis Clairaut et ses tables de la Lune à vocation nautique. Guy Boistel, 1751Guy Boistel, 'Au-delà du problème des trois corps: Alexis Clairaut et ses tables de la Lune à vocation nautique (1751- (= Cahiers d'histoire et de philosophie des sciences. Actes du congrès d'histoire des sciences et des techniques. A. Bonnefoy & B. Jolys du congrès d'histoire des sciences et des techniques', in Actes du congrès d'histoire des sciences et des techniques, Poitiers 20-22 mai 2004, ed. by A. Bonnefoy & B. Joly (= Cahiers d'histoire et de philosophie des sciences, hors-série (2006)), 20-9. . ' Boistel, Boistel, 'L'astronomie nautique'; Lalande et la Marine: un engagement sans faille mais non désintéressé. Guy Boistel, une trajectoire scientifique. Guy Boistel, Jérôme Lamy and Colette Le LayRennesPresses Universitaires de RennesJérôme LalandeGuy Boistel, 'Lalande et la Marine: un engagement sans faille mais non désintéressé', in Jérôme Lalande (1732-1807), une trajectoire scientifique, ed. by Guy Boistel, Jérôme Lamy and Colette Le Lay (Rennes: Presses Universitaires de Rennes, 2010), pp. 67-80. . ' Boistel, Boistel, 'L'astronomie nautique'; . 'pierre-Louis Boistel, Moreau De Maupertuis, Boistel, 'Pierre-Louis Moreau de Maupertuis'. . 'pierre Boistel, Bouguer, Boistel, 'Pierre Bouguer'. . ' Boistel, Boistel, 'L'astronomie nautique'; . ' Boistel, Lalande, Marine La, Boistel, 'Lalande et la Marine'. L'observatoire des jésuites de Marseille sous la direction du Père Esprit Pezenas (1728-1763)', in Observatoires et patrimoine astronomiques français. Guy Boistel, Guy Boistel20Guy Boistel, 'L'observatoire des jésuites de Marseille sous la direction du Père Esprit Pezenas (1728-1763)', in Observatoires et patrimoine astronomiques français, ed. by Guy Boistel (= 20 Cahiers d'histoire et de philosophie des sciences. 54Cahiers d'histoire et de philosophie des sciences, 54 (2005)), 27-45; . ' Boistel, ' L'astronomie Nautique, Boistel, 'L'astronomie nautique', part III. After each Saros cycle, the irregularities of Moon's motions are supposed to be the same. This cycle was used to compute eclipses of the Sun by the Moon. For the origins of the word saros and the uses of the cycle for the correction of lunar tables by Halley, Legentil de la Galaisière, Lemonnier, Pingré and d'Alembert in particular, see Boistel. The name was given by Halley from the Chaldean cycle of 223 lunar periods. part III. and part IVThe name was given by Halley from the Chaldean cycle of 223 lunar periods. After each Saros cycle, the irregularities of Moon's motions are supposed to be the same. This cycle was used to compute eclipses of the Sun by the Moon. For the origins of the word saros and the uses of the cycle for the correction of lunar tables by Halley, Legentil de la Galaisière, Lemonnier, Pingré and d'Alembert in particular, see Boistel, 'L'astronomie nautique', part III, pp. 299-302 and part IV, pp. 508-70. A Caleb Smith quadrant appears in 'Portrait of a Merchant Captain. Robert WilloughbyGreenwich, BHC3130National Maritime Museumoil on canvas, 1805A Caleb Smith quadrant appears in 'Portrait of a Merchant Captain', by Robert Willoughby, oil on canvas, 1805, National Maritime Museum, Greenwich, BHC3130. Principes du calcul astronomique, provenant du cabinet de d'Après de Mannevillette' (c.1754), Fonds du Dépôt des Cartes et Plans, Service Historique de la Défense. Jean-Baptiste D'après De Mannevillette, Paris, Vincennes, SH 53Jean-Baptiste d'Après de Mannevillette, 'Principes du calcul astronomique, provenant du cabinet de d'Après de Mannevillette' (c.1754), Fonds du Dépôt des Cartes et Plans, Service Historique de la Défense, Paris, Vincennes, SH 53. See Boistel, ' , for a study of Lacaille's and. part IIISee Boistel, 'L'astronomie nautique', part III, pp. 281-500 for a study of Lacaille's and Mannevillette's works, longitude determinations at sea and methods; see also Autour de d'Après de Mannevillette: savant navigateur havrais du siècle des Lumières. Eric SaunierLe HavreCentre Havrais de Recherche HistoriqueMannevillette's works, longitude determinations at sea and methods; see also Autour de d'Après de Mannevillette: savant navigateur havrais du siècle des Lumières, ed. by Eric Saunier (Le Havre: Centre Havrais de Recherche Historique, 2008); . Manonmani Filliozat, D'Après deManonmani Filliozat, 'D'Après de . Capitaine Mannevillette, La De, Compagnie des Indes. Thèse de l'École des ChartesMannevillette, capitaine et hydrographe de la Compagnie des Indes (1707-1780)' (Thèse de l'École des Chartes, Paris, 1993). Projet pour rendre la méthode des longitudes sur mer. Nicolas-Louis De Lacaille, 16pratiquable au commun des navigateurs', 1754, A.N., Marine, 2 JJ 69 (J.-N. Delisle's papersNicolas-Louis de Lacaille, 'Projet pour rendre la méthode des longitudes sur mer pratiquable au commun des navigateurs', 1754, A.N., Marine, 2 JJ 69 (J.-N. Delisle's papers), item 16; Instruction détaillée pour l'observation et le calcul des longitudes sur mer par la distance de la Lune aux étoiles ou au Soleil. Nicolas-Louis De Lacaille, 1754, A.N., Marine3Nicolas-Louis de Lacaille, 'Instruction détaillée pour l'observation et le calcul des longitudes sur mer par la distance de la Lune aux étoiles ou au Soleil', 1754, A.N., Marine, 3 JJ 13, items 3 and 9. 21 . Nicolas-Louis De Lacaille, Relation abrégée du Voyage fait par Ordre du Roi au cap deNicolas-Louis de Lacaille, 'Relation abrégée du Voyage fait par Ordre du Roi au cap de M Bonne-Espérance, De La Caille, Histoire et Mémoires de l'Académie Royale des Sciences… pour l'année 1751. Paris13: Imprimerie royale, 1755), Mémoires, read on 13. see also Procès-verbaux des séances de l'Académie Royale des SciencesBonne-Espérance, par M. l'abbé de la Caille ', Histoire et Mémoires de l'Académie Royale des Sciences… pour l'année 1751 (Paris: Imprimerie royale, 1755), Mémoires, read on 13 November 1754, pp. 519-36 ; see also Procès-verbaux des séances de l'Académie Royale des Sciences, 13 Je m'étais beaucoup exercé à ces sortes d'observations, et j'avais reconnu qu'il étoit inutile d'avoir recours à une autre façon d'employer la Lune pour les longitudes; qu'il ne s'agissait uniquement que d. 73Tomeen rendre le calcul pratiquable au commun des marinsNovember 1754, Tome 73, fols 475-93 (fol. 491): 'Je m'étais beaucoup exercé à ces sortes d'observations, et j'avais reconnu qu'il étoit inutile d'avoir recours à une autre façon d'employer la Lune pour les longitudes; qu'il ne s'agissait uniquement que d'en rendre le calcul pratiquable au commun des marins'. . ' Boistel, IIpart IIIBoistel, 'L'astronomie nautique', part II, pp. 158-60; part III, pp. 386-412. Mémoire présenté par le Sieur Griffon. Quai ContiParisArchives de l'Académie des sciences. Pochette de séance'Mémoire présenté par le Sieur Griffon', 23 August 1748, Archives de l'Académie des sciences, Paris, Quai Conti, Pochette de séance. . ' Boistel, IIIBoistel, 'L'astronomie nautique', part III, pp. 479-500. Secretary to the Royal Society. Nevil Maskelyne ; Nevil Maskelyne, M A , F R S Thomas To Rev, D D Birch, Containing the Results of Observations of the Distance of the Moon from the Sun […] in Order to Determine the Longitude of the Ship from Time to Time', Philosophical Transactions, LII (1761Nevil Maskelyne, 'A Letter from the Rev. Nevil Maskelyne M.A., F.R.S., to Rev. Thomas Birch, D.D., Secretary to the Royal Society, Containing the Results of Observations of the Distance of the Moon from the Sun […] in Order to Determine the Longitude of the Ship from Time to Time', Philosophical Transactions, LII (1761), 558-77; L'astronomie nautique', part III and part IV for the computations of the horary motion of the Moon in the tables of Clairaut. H Donald, Sadler, Vistas in Astronomy. 20Mayer and d'AlembertLunar Distances and the Nautical AlmanacDonald H. Sadler, 'Lunar Distances and the Nautical Almanac', Vistas in Astronomy, 20 (1976), 113-21; see also Boistel, 'L'astronomie nautique', part III and part IV for the computations of the horary motion of the Moon in the tables of Clairaut, Mayer and d'Alembert. Lunar Distances. Sadler, Sadler, 'Lunar Distances'. Jérôme Lalande, Connaissance des temps pour l'année 1761. Paris: Imprimerie royale, 1759)Jérôme Lalande, Connaissance des temps pour l'année 1761 (Paris: Imprimerie royale, 1759), pp. 192-93. De quelle précision a-t-on réellement besoin en mer? Quelques aspects de la diffusion des méthodes de détermination astronomique et chronométrique des longitudes en mer en France, de Lacaille à Mouchez (1750-1880. Guy Boistel, Histoire & Mesure. 22XXIGuy Boistel, 'De quelle précision a-t-on réellement besoin en mer? Quelques aspects de la diffusion des méthodes de détermination astronomique et chronométrique des longitudes en mer en France, de Lacaille à Mouchez (1750-1880)', Histoire & Mesure, XXI (2006), 121-56. 22 Histoire générale des Mathématiques par M. Montucla, ed. by Jérôme Lalande, 4 vols. ParisHistoire générale des Mathématiques par M. Montucla, ed. by Jérôme Lalande, 4 vols (Paris: On a simplifié par des tables toutes les autres parties des calculs de la longitude […] Cependant cette partie allonge beaucoup la méthode des longitudes et empêche beaucoup de navigateurs de s'occuper de ces recherches ; s'ils négligent encore ces observations au risque de leurs fortunes et de leurs vies, c'est le devoir des astronomes de leur aplanir les difficultés et de les rappeler à de pressants intérêts. H Agasse, IV. ParisImprimerie de Prault581See also, Alexis Rochon, Mémoire sur l'astronomie nautiqueH. Agasse, 1799-1802), IV, 581: 'On a simplifié par des tables toutes les autres parties des calculs de la longitude […] Cependant cette partie allonge beaucoup la méthode des longitudes et empêche beaucoup de navigateurs de s'occuper de ces recherches ; s'ils négligent encore ces observations au risque de leurs fortunes et de leurs vies, c'est le devoir des astronomes de leur aplanir les difficultés et de les rappeler à de pressants intérêts.' See also, Alexis Rochon, Mémoire sur l'astronomie nautique (Paris: Imprimerie de Prault, 1798). Elles [les méthodes graphiques] ont l'inconvénient d'habituer à un travail en quelque sorte automatique, des hommes qui n'y sont déjà que trop disposés. REF : Borda & Lévêque, 1798 (an VII). 472ref note 30"Elles [les méthodes graphiques] ont l'inconvénient d'habituer à un travail en quelque sorte automatique, des hommes qui n'y sont déjà que trop disposés" . REF : Borda & Lévêque, 1798 (an VII), (ref note 30), p. 472. Rapport sur le mémoire et la carte trigonométrique présentés par le Citoyen Maingon, Lieutenant de Vaisseau', Procès-Verbaux de l'Académie des sciences de l'Institut de France, I, séance du 11 Vendémiaire an VII. Le Chevalier De Borda, Pierre Lévêque, Gauthier-Villars473ParisIl est temps que les marins cessent de regarder les sciences mathématiques et physiques comme inutiles à la pratique de la navigation et à ses progrès. Sans le secours des sciences. la Marine seroit encore dans l'enfance'Le Chevalier de Borda and Pierre Lévêque, 'Rapport sur le mémoire et la carte trigonométrique présentés par le Citoyen Maingon, Lieutenant de Vaisseau', Procès-Verbaux de l'Académie des sciences de l'Institut de France, I, séance du 11 Vendémiaire an VII (2 October 1798) (Paris: Gauthier-Villars, 1910), pp. 465-73 (p. 473): 'Il est temps que les marins cessent de regarder les sciences mathématiques et physiques comme inutiles à la pratique de la navigation et à ses progrès. Sans le secours des sciences, la Marine seroit encore dans l'enfance'. For more detailed studies, see Boistel, 'De quelle précision. For more detailed studies, see Boistel, 'De quelle précision'; Training Seafarers in Astronomy: Methods, Naval Schools and Naval Observatories in 18 th -and 19 th -century. Guy Boistel, Guy Boistel, 'Training Seafarers in Astronomy: Methods, Naval Schools and Naval Observatories in 18 th -and 19 th -century France, The Heavens on Earth: Observatories and Astronomy in Nineteenth-Century Science and Culture. David Aubin, Charlotte Bigg and H. Otto SibumDurham, NCDuke University PressFrance', in, The Heavens on Earth: Observatories and Astronomy in Nineteenth-Century Science and Culture, ed. by David Aubin, Charlotte Bigg and H. Otto Sibum (Durham, NC: Duke University Press, 2010), pp. 148-73. Archives nationales, fonds Marine, 2 JJ 69 (Delisle's papers). fol. 109Archives nationales, fonds Marine, 2 JJ 69 (Delisle's papers), fol. 109. . ' Boistel, IIBoistel, 'L'astronomie nautique', part II, especially pp. 160-2. O Planetario Lusitano de Eusébio da Veiga e a Astronomia em Portugal no século XVIII', Revista Brasileira de História da Ciência. Jefferson , Santos Alves, 6Jefferson dos Santos Alves, 'O Planetario Lusitano de Eusébio da Veiga e a Astronomia em Portugal no século XVIII', Revista Brasileira de História da Ciência, 6 (2013), 340. 23 The Travels and Trials of Mr Harrison's Timekeeper', in Instruments, Travel and Science. Itineraries of Precision from the Seventeenth to the Twentieth Century. Jim Bennett, Jim Bennett, 'The Travels and Trials of Mr Harrison's Timekeeper', in Instruments, Travel and Science. Itineraries of Precision from the Seventeenth to the Twentieth Century, ed. by . Marie-Noëlle Bourguet, Christian Licoppe, Heinz Otto Sibum, London: RoutledgeMarie-Noëlle Bourguet, Christian Licoppe and Heinz Otto Sibum (London: Routledge, 2002), pp. 75-95. . See, Jeaurat Letters Between Maskelyne, REG09/000037 <cudl.lib.cam.ac.uk/view/MS-REG-00009-00037/670> (1775) and <cudl.lib.cam.ac.uk/view/MS-REG-00009-00037/684 and 685>GreenwichNational Maritime Museumaccessed 10 FebruarySee, for example, letters between Maskelyne and Jeaurat, National Maritime Museum, Greenwich, REG09/000037 <cudl.lib.cam.ac.uk/view/MS-REG-00009-00037/670> (1775) and <cudl.lib.cam.ac.uk/view/MS-REG-00009-00037/684 and 685> (1789) [accessed 10 February Danielle See Also, Fauque, La correspondance Jérôme Lalande et Nevil Maskelyne: un exemple de collaboration internationale au XVIII e siècle', in Jérôme Lalande. Boistel, Lamy and Le Laysee also, Danielle Fauque, 'La correspondance Jérôme Lalande et Nevil Maskelyne: un exemple de collaboration internationale au XVIII e siècle', in Jérôme Lalande, ed. by Boistel, Lamy and Le Lay, pp. 109-28. . ' Boistel, ' L'astronomie Nautique, Boistel, 'L'astronomie nautique', part II. . Flaugergues Lalande To, Simone Dumont and Jean-Claude Pecker, Jérôme Lalande. Lalandiana I. Lettres à Madame du Pierry et au jugeViviersrepeated 27 JulyLalande to Flaugergues, Viviers, 14 May 1803 (repeated 27 July), quoted in Simone Dumont and Jean-Claude Pecker, Jérôme Lalande. Lalandiana I. Lettres à Madame du Pierry et au juge Nous nous sommes occupés ces jours-ci à recalculer par les observations les 34 étoiles de Maskelyne dont nous nous servions avec une pleine sécurité, et je trouve 5 à 6 secondes à ajouter aux ascensions droites. En sorte qu'il faudra corriger tous nos catalogues, toutes nos tables et toutes nos longitudes des planètes observées! Ce vieux barbouillon, ivrogne paresseux, avare avait usurpé notre confiance. Il est fort riche, il aurait dû se procurer un calculateur et vérifier plus d'une fois cet important résultat. Honoré Flaugergues (Paris: J. Vrin Dumont176Thanks to Simon Schaffer for help with the translationHonoré Flaugergues (Paris: J. Vrin Dumont, 2007), p. 176: 'Nous nous sommes occupés ces jours-ci à recalculer par les observations les 34 étoiles de Maskelyne dont nous nous servions avec une pleine sécurité, et je trouve 5 à 6 secondes à ajouter aux ascensions droites. En sorte qu'il faudra corriger tous nos catalogues, toutes nos tables et toutes nos longitudes des planètes observées! Ce vieux barbouillon, ivrogne paresseux, avare avait usurpé notre confiance. Il est fort riche, il aurait dû se procurer un calculateur et vérifier plus d'une fois cet important résultat'. Thanks to Simon Schaffer for help with the translation. Clairaut to Dr. Bevis, dated Paris. 208from the English Original, in his own Hand', Gentleman's Magazine, XXXV (1765'Copy of a Letter from M. Clairaut to Dr. Bevis, dated Paris, 11 April 1765, from the English Original, in his own Hand', Gentleman's Magazine, XXXV (1765), 208. L'astronomie nautique', part IV and bibliography. Boistel, Boistel, 'L'astronomie nautique', part IV and bibliography. Recherches sur quelques points d'astronomie physique. De la manière la plus simple de calculer analytiquement & astronomiquement les mouvements de la. Jean-Le-Rond D'alembert, Jean-Le-Rond d'Alembert, 'Recherches sur quelques points d'astronomie physique. De la manière la plus simple de calculer analytiquement & astronomiquement les mouvements de la On peut meme observer encore que dans les equations que M. Mayer emploie pour ses tables, les valeurs des coefficients ne sont pas exactement les mêmes que celles qu'il a tirées de la théorie ; d'où il paroît résulter que les tables de la M. Mayer ont été dressées en partie sur les observations, par une espèce de tâtonnement. ' Lune, Opuscules Mathématiques, David Vi (paris, ) , combiné avec les résutats principaux que la théorie fournitLune', Opuscules Mathématiques, VI (Paris, David, 1773), mémoire XLV, pp. 1-46 (pp. 43-4): 'On peut meme observer encore que dans les equations que M. Mayer emploie pour ses tables, les valeurs des coefficients ne sont pas exactement les mêmes que celles qu'il a tirées de la théorie ; d'où il paroît résulter que les tables de la M. Mayer ont été dressées en partie sur les observations, par une espèce de tâtonnement, combiné avec les résutats principaux que la théorie fournit.' La correspondance astronomique entre l'abbé Nicolas-Louis de Lacaille et Tobias Mayer', Revue d'histoire des sciences. ' Boistel, ' L'astronomie Nautique, Jacques Gapaillard, 49Boistel, 'L'astronomie nautique', part IV; see also, Jacques Gapaillard, 'La correspondance astronomique entre l'abbé Nicolas-Louis de Lacaille et Tobias Mayer', Revue d'histoire des sciences, 49 (1996), 483-541. L'astronomie nautique', part IV for a study of Clairaut's lunar tables. Boistel, Boistel, 'L'astronomie nautique', part IV for a study of Clairaut's lunar tables. Monteiro da Rocha wrote about lunar-distance methods and his trials on a voyage between Brazil and Portugal. His readings and inspiration were Lacaille, Lalande (longitudes and Connaissance des temps) and Maskelyne's British Mariner's Guide. See also Fernando Figueiredo. Fernando B For, Guy Figueiredo, Boistel, The Biographical Encyclopedia of Astronomers. Thomas Hockey et al.Lisbon, Colecção Pombalina, Ms; DordrechtSpringer511José Monteiro da Rocha (1734-1819) and the international debate in the 1760s on astronomical methods to find longitude at sea: its proposals and criticisms of the method of lunar distances of Lacaille' (forthcoming), concerning a manuscript of 1765-66. 2nd editionFor another example, see Fernando B. Figueiredo and Guy Boistel, 'José Monteiro da Rocha (1734-1819) and the international debate in the 1760s on astronomical methods to find longitude at sea: its proposals and criticisms of the method of lunar distances of Lacaille' (forthcoming), concerning a manuscript of 1765-66, Biblioteca Nacional de Portugal, Lisbon, Colecção Pombalina, Ms.511. Monteiro da Rocha wrote about lunar-distance methods and his trials on a voyage between Brazil and Portugal. His readings and inspiration were Lacaille, Lalande (longitudes and Connaissance des temps) and Maskelyne's British Mariner's Guide. See also Fernando Figueiredo, 'José Monteiro da Rocha (1734-1819)', in The Biographical Encyclopedia of Astronomers, ed. by Thomas Hockey et al. (2nd edition, Dordrecht: Springer, 2014), pp. 513- 15. Journal of Voyage to St Helena. Nevil Maskelyne, RGO 4/150 <cudl.lib.cam.ac.uk/view/MS-RGO-00004-00150/1>1761Cambridge University LibraryNevil Maskelyne, 'Journal of Voyage to St Helena', 1761, Cambridge University Library, RGO 4/150 <cudl.lib.cam.ac.uk/view/MS-RGO-00004-00150/1> [accessed 10 February 2015].	[]
[ "Shrinkage with shrunken shoulders: Gibbs sampling shrinkage model posteriors with guaranteed convergence rates", "Shrinkage with shrunken shoulders: Gibbs sampling shrinkage model posteriors with guaranteed convergence rates" ]	[ "Akihiko Nishimura ", "Marc A Suchard " ]	[]	[]	Use of continuous shrinkage priors -with a "spike" near zero and heavy-tails towards infinity -is an increasingly popular approach to induce sparsity in parameter estimates. When the parameters are only weakly identified by the likelihood, however, the posterior may end up with tails as heavy as the prior, jeopardizing robustness of inference. A natural solution is to "shrink the shoulders" of a shrinkage prior by lightening up its tails beyond a reasonable parameter range, yielding a regularized version of the prior. We develop a regularization approach which, unlike previous proposals, preserves computationally attractive structures of original shrinkage priors. We study theoretical properties of the Gibbs sampler on resulting posterior distributions, with emphasis on convergence rates of the Pólya-Gamma Gibbs sampler for sparse logistic regression. Our analysis shows that the proposed regularization leads to geometric ergodicity under a broad range of global-local shrinkage priors. Essentially, the only requirement is for the prior π local (·) on the local scale λ to satisfy π local (0) < ∞. If π local (·) further satisfies lim λ→0 π local (λ)/λ a < ∞ for a > 0, as in the case of Bayesian bridge priors, we show the sampler to be uniformly ergodic.MSC2020 subject classifications: Primary 60J20, 62F15; secondary 62J07.	10.1214/22-ba1308	[ "https://arxiv.org/pdf/1911.02160v5.pdf" ]	237,532,778	1911.02160	8c98e57f07ba3e8c8179ed9e703d42e7fcf93290	Shrinkage with shrunken shoulders: Gibbs sampling shrinkage model posteriors with guaranteed convergence rates Akihiko Nishimura Marc A Suchard Shrinkage with shrunken shoulders: Gibbs sampling shrinkage model posteriors with guaranteed convergence rates Bayesian inferencesparsitygeneralized linear modelMarkov chain Monte Carloergodicity Use of continuous shrinkage priors -with a "spike" near zero and heavy-tails towards infinity -is an increasingly popular approach to induce sparsity in parameter estimates. When the parameters are only weakly identified by the likelihood, however, the posterior may end up with tails as heavy as the prior, jeopardizing robustness of inference. A natural solution is to "shrink the shoulders" of a shrinkage prior by lightening up its tails beyond a reasonable parameter range, yielding a regularized version of the prior. We develop a regularization approach which, unlike previous proposals, preserves computationally attractive structures of original shrinkage priors. We study theoretical properties of the Gibbs sampler on resulting posterior distributions, with emphasis on convergence rates of the Pólya-Gamma Gibbs sampler for sparse logistic regression. Our analysis shows that the proposed regularization leads to geometric ergodicity under a broad range of global-local shrinkage priors. Essentially, the only requirement is for the prior π local (·) on the local scale λ to satisfy π local (0) < ∞. If π local (·) further satisfies lim λ→0 π local (λ)/λ a < ∞ for a > 0, as in the case of Bayesian bridge priors, we show the sampler to be uniformly ergodic.MSC2020 subject classifications: Primary 60J20, 62F15; secondary 62J07. Introduction Bayesian modelers are increasingly adopting continuous shrinkage priors to control the effective number of parameters and model complexity in a data-driven manner. These priors are designed to shrink most of the parameters towards zero while allowing for the likelihood to pull a small fraction of them away from zero. To achieve such effects, a shrinkage prior has a density with a "spike" near zero and heavy-tails towards infinity, encoding information that parameter values are likely close to zero but otherwise could be anywhere. Originally developed for the purpose of sparse regression (Carvalho et al., 2009), shrinkage priors have found applications in trend filtering of time series data (Kowal et al., 2019), (dynamic) factor models (Kastner, 2019), graphical models (Li et al., 2019), compression of deep neural networks (Louizos et al., 2017), among others. Shrinkage priors are often expressed as a scale mixture of Gaussians on the unknown parameter β = (β 1 , . . . , β p ) (Polson and Scott, 2010): π(β j \| τ, λ j ) ∼ N (0, τ 2 λ 2 j ), λ j ∼ π loc (·). (1.1) This global-local representation simplifies the posterior conditionals and lead to straightforward inference via Gibbs sampling. The global scale τ controls the average magnitude of β j 's and hence overall sparsity level. The local scale λ j is specific to individual β j and its density π loc (·) controls the size of the spike and tail behavior of the marginal β j \| τ . For instance, the popular horseshoe prior of Carvalho et al. (2010) uses π loc (λ) ∝ (1+λ 2 ) −1 , inducing a marginal π(β j \| τ ) with the spike proportional to − log(\|β j /τ \|) as \|β j /τ \| → 0 and the tail proportional to (β j /τ ) −2 as \|β j /τ \| → ∞. Another notable example is the Bayesian bridge prior of Polson et al. (2014), which generalizes the Bayesian lasso of Park and Casella (2008) with π(β j \| τ ) having a larger spike as \|β j /τ \| → 0 and heavier tails as \|β j /τ \| → ∞. Most importantly from the computational efficiency perspective, the bridge prior possesses a closed-form expression π(β j \| τ ) ∝ exp(−\|β j /τ \| a ) for a ∈ (0, 1) and thus allows for a collapsed Gibbs update from τ \| β with λ j 's marginalized out. For a simple purpose such as estimating the unknown means of independent Gaussian observations, a broad class of shrinkage priors achieve theoretically optimal performance (van der Pas et al., 2016;Ghosh and Chakrabarti, 2017). The lack of prior information in the tail of the distribution is problematic, however, in more complex models where parameters are only weakly identified. In such models, the posterior may have a tail as heavy as the prior, resulting in unreliable parameter estimates (Ghosh et al., 2018). To address the above shortcoming of shrinkage priors, we build on the work of Piironen and Vehtari (2017) and propose a computationally convenient way to regularize shrinkage priors. The basic idea is to modify the prior so that the marginal distribution of \|β j \| has light-tails beyond a reasonable range. Our formulation has computational advantages over that of Piironen and Vehtari (2017) due to a subtle yet important difference. By preserving the global-local structure (1.1), our regularized shrinkage priors can benefit from partial marginalization approaches that substantially improve mixing of Gibbs samplers (Polson et al. 2014;Johndrow et al. 2018;Appendix F). In addition, our regularization leaves the posterior conditionals of λ j 's unchanged, allowing their conditional updates via existing specialized samplers (Griffin and Brown 2010;Polson et al. 2014;Appendix G). 1 Our regularized shrinkage priors allow for posterior inference via Gibbs sampler whose convergence rates often are provably fast. As an illustrative example, we consider Bayesian sparse logistic regression models, whose need for regularization motivated the work of Piironen and Vehtari (2017). Gibbs sampling via the Pólya-Gamma data augmentation of Polson et al. (2013) is a state-of-the-art approach to posterior computation under logistic model. When combined with advanced numerical linear algebra techniques, this Gibbs sampler is highly scalable to large data sets (Nishimura and Suchard, 2018), but its theoretical convergence rate has not been investigated. Assuming that the prior density π loc (λ) is continuous and bounded except possibly at λ = 0, we establish that the Gibbs sampler is geometrically ergodic whenever π loc (0) < ∞. Stronger uniform convergence is achieved when λ −1 π loc (λ) dλ < ∞. The integrability condition holds in particular when π loc (λ) = O(λ a ) for a > 0 as λ → 0, which is the case for normal-gamma priors with shape parameter larger than 1/2 (Griffin and Brown, 2010) and for Bayesian bridge priors (Polson et al. 2014 and Appendix F). Previous studies of the convergence rates under shrinkage models have focused exclusively on linear regression with specific parametric families of shrinkage priors (Pal and Khare, 2014;Johndrow et al., 2018). In contrast, our analysis requires no parametric assumptions on the shrinkage prior, at the same time extending the convergence results to the logistic model and, in Appendix B, to the probit model. To summarize, this work provides two major contributions to the Bayesian shrinkage literature. First, we propose an effective and Gibbs-friendly approach to suitably modify shrinkage priors for use in weakly-identifiable models (Section 2). Second, we develop theoretical tools to study the behavior of shrinkage model Gibbs samplers near the spike β j = 0 without any parametric assumption on π loc (·), thereby unifying convergence analyses of the logistic regression Gibbs samplers under a range of shrinkage priors (Section 3). We conclude the article in Section 4 by demonstrating a practical use case of regularized shrinkage models via simulation study, which emulates increasingly common situations where the sample sizes are large yet the signals are difficult to detect. 2 Regularized shrinkage prior Piironen and Vehtari (2017) proposes to control the tail behavior of a global-local shrinkage prior by defining its regularized version with slab width ζ > 0 as β j \| τ, λ j , ζ ∼ N   0, 1 ζ 2 + 1 τ 2 λ 2 j −1   , (2.1) with the prior π loc (·) on the local scale λ j unmodified. This regularization ensures that the variance of β j \| τ, λ j , ζ is upper bounded by ζ 2 and hence β j \| ζ marginally has a density with Gaussian tails beyond \|β j \| > ζ. The slab width ζ can be either given a prior distribution or fixed at a reasonable value. 2 While beneficial in improving statistical properties (Piironen and Vehtari, 2017), regularization the form (2.1) compromises the posterior conditional structures of shrinkage models. Specifically, the conditional distribution of τ, λ is altered through their dependency on ζ. This structural change is at best an inconvenience and potentially a cause of computational inefficiency, prohibiting the use of common acceleration techniques. For instance, the global scale τ is known to mix slowly when updating from its full conditional, so the state-of-the-art Gibbs samplers for Bayesian sparse regression marginalize out a subset of parameters when updating τ (Johndrow et al., 2018;Nishimura and Suchard, 2018). The analytical tractabilities of the integrals, which these marginalization strategies rely on, is lost when using the regularization as in (2.1). We propose a more computationally convenient formulation, which induces regularization similar to that of (2.1) while keeping τ and λ conditionally independent of ζ given β. Intuitively, we achieve regularization indirectly through fictitious data that makes values \|β j \| ζ unlikely. The use of such fictitious data is technically unnecessary in defining our regularization strategy (Appendix A), but makes the mechanism and resulting posterior properties more transparent. We visually illustrate in Figure 2.1 the construction of our regularized prior as well as the corresponding posterior structure when data y and X inform β through the likelihood L(y \| X, β). Given a global-local prior β j \| τ, λ j ∼ N (0, τ 2 λ 2 j ), we introduce fictitious data z j whose realized value and underlying distribution are assumed to be z j = 0, z j \| β j , ζ ∼ N (β j , ζ 2 ) (2.2) for j = 1, . . . , p. We then define the regularized prior as the distribution of β j conditional on z j = 0. Under this model, the distribution of β j \| τ, λ j , ζ, z j = 0 coincides with that of (2.1). On the other hand, the scale parameters τ, λ are conditionally independent of the others given β, so that the posterior full conditional τ, λ \| β, ζ, z, y, X ( d = τ, λ \| β) has the same density as in the unregularized version. Our regularization thus allows the Gibbs sampler to update τ, λ with the exact same algorithm as the one designed for the original shrinkage prior. We summarize our discussion as Proposition 2.1 below. Proposition 2.1. Consider a global-local shrinkage prior β j \| τ, λ j ∼ N (0, τ 2 λ 2 j ), λ j ∼ π loc (·) and τ ∼ π glo (·). Introducing the fictitious data z = 0 as in (2.2) is equivalent to using the regularized prior (??) on (β j , λ j ), yielding β j \| τ, λ j , ζ, z j = 0 ∼ N   0, 1 ζ 2 + 1 τ 2 λ 2 j −1   . Or, with λ j marginalized out, we have π(β j \| τ, ζ, z j = 0) ∝ π(β j \| τ ) exp − β 2 j 2ζ 2 . When the likelihood depends only on β, the posterior full conditional of τ, λ has density π(τ, λ \| β) ∝ π glo (τ ) j 1 τ λ j exp − β 2 j 2τ 2 λ 2 j π loc (λ j ). (2.3) 3 Geometric and uniform ergodicity under regularized sparse logistic regression Shrinkage priors' popularity stems from, to a considerable extent, the ease of posterior computation via Gibbs sampling (Bhadra et al., 2017). As we have shown in Section 2, τ λ β X y ζ (a) Of the form (2.1) as previously proposed. The posterior conditional of (τ, λ) is affected by their dependency on ζ through β. shrinkage models can incorporate regularization without affecting its computational tractability. We now investigate how fast such Gibbs samplers converge. As a representative example where regularization is essential, we focus on Bayesian sparse logistic regression (Piironen and Vehtari, 2017;Nishimura and Suchard, 2018). To be explicit, we consider the model y i \| x i , β ∼ Bernoulli logit −1 (x i β) , z j = 0 \| β j ∼ N (0, ζ 2 ), β j \| τ, λ j ∼ N (0, τ 2 λ 2 j ), τ ∼ π glo (·), λ j ∼ π loc (·). (3.1) The Pólya-Gamma data-augmentation of Polson et al. (2013) is a widely-used approach to carry out the posterior computation under the logistic model. By introducing an auxiliary parameter ω = (ω 1 , . . . , ω n ) having a Pólya-Gamma distribution, the Gibbs sampler induces a transition kernel: (ω * , β * , λ * , τ * ) → (ω, β, λ, τ ) through the following cycle of conditional updates: 1. Draw τ \| β * , λ * from the density proportional to (2.3). When using Bayesian bridge priors, draw from the collapsed distribution τ \| β * (Appendix F). 2. Draw λ \| β * , τ from the density proportional to (2.3). 3. Draw ω i \| β * , X ∼ PolyaGamma(shape = 1, tilting = x i β * ) for i = 1, . . . , n. 4. Draw β \| ω, τ, λ, y, X, z = 0 from the multivariate-Gaussian β \| ω, τ, λ, y, X, z = 0 ∼ N Φ −1 X y − 1 2 , Φ −1 for Φ = X ΩX + ζ −2 I + τ −2 Λ −2 ,(3.2) where Ω = diag(ω) and Λ = diag(λ). Note that the transition kernel actually depends neither on ω * nor τ * (nor λ * in the Bayesian bridge case) because of conditional independence. We refer readers to Polson et al. (2013) for more details on this data augmentation scheme. In our analysis, we do not use any specific properties of the Pólya-Gamma distribution aside from a couple of results from Choi and Hobert (2013) and Wang and Roy (2018). The Pólya-Gamma Gibbs sampler for the logistic model has previously been analyzed under a Gaussian or flat prior on β (Choi and Hobert, 2013;Wang and Roy, 2018), but not under shrinkage priors. We establish geometric and uniform ergodicity -critical properties for any practical Markov chain Monte Carlo algorithms (Jones and Hobert, 2001). These properties imply the Markov chain central limit theorem and enables consistent estimation of Monte Carlo errors, ensuring that the Gibbs sampler reliably estimates quantities of interest (Flegal and Jones, 2011). To avoid cluttering notations and obscuring the main ideas, our analysis below assumes the slab width ζ to be fixed; however, the same conclusions hold if we only assume a prior constraint of the form ζ ≤ ζ max < ∞ (Remark 3.9). We verify that the Gibbs sampler satisfies the minorization and drift condition upon on which geometric and uniform ergodicity are immediately implied by the well-known theory of Markov chains (Meyn and Tweedie, 2009;Roberts and Rosenthal, 2004). In the statements to follow, we assume that a transition kernel P (θ * , dθ) has a corresponding density function which, with slight abuse of notation, we denote by P (θ \| θ * ); in other words, the two satisfy a relation P (θ * , A) = A P (θ \| θ * ) dθ. A chain on the space θ ∈ Θ with transition kernel P (θ * , dθ) is said to satisfy a minorization condition with a small set S if there are δ > 0 and a probability density π(·) such that P (θ \| θ * ) ≥ δ π(θ) for all θ * ∈ S. The chain is uniformly ergodic when S = Θ. Otherwise, the chain is geometrically ergodic if it additionally satisfies a drift condition i.e. there is a Lyapunov function V (θ) ≥ 0 such that, for γ < 1 and b < ∞, P V (θ * ) := V (θ)P (θ \| θ * ) dθ ≤ γV (θ * ) + b and S = {θ : V (θ) ≤ d} is a small set for some d > 2b/(1 − γ) (Rosenthal, 1995). For a two-block component-wise sampler on the space (θ, φ), alternately sampling θ ∼ P ( · \| φ) and φ ∼ P ( · \| θ), the geometric and uniform ergodicity of the joint chain follows from that of the marginal chain with the transition kernel P (θ \| θ * ) = P (θ \| φ)P (φ \| θ * ) dφ (Roberts and Rosenthal, 2001). In establishing the uniform ergodicity under Bayesian bridge (Theorem 3.1), we decompose the collapsed Gibbs sampler into components β and (ω, τ, λ) and study the marginal chain in β. In the subsequent analysis establishing the geometric ergodicity under a more general class of regularized shrinkage priors (Theorem 3.2), we decompose the Gibbs sampler into components (β, λ) and (ω, τ ) and study the marginal chain in (β, λ). Below are the main ergodicity results we will establish in this section, the uniform rate under Bayesian bridge and geometric rate under more general shrinkage priors: Theorem 3.1 (Uniform ergodicity in the Bayesian bridge case). If the prior π glo (·) is supported on [τ min , ∞) for τ min > 0, then the Pólya-Gamma Gibbs sampler for regularized Baysian bridge logistic regression is uniformly ergodic. Theorem 3.2 (Geometric ergodicity). Suppose that the local scale prior satisfies π loc ∞ < ∞ and that the global scale prior π glo (·) is supported on [τ min , τ max ] for 0 < τ min ≤ τ max < ∞. Then the Pólya-Gamma Gibbs sampler for regularized sparse logistic regression is geometrically ergodic. Remark. Uniform / geometric ergodicity is an essential requirement for, yet not a guarantee of, practically efficient Markov chains (Roberts and Rosenthal, 2004). In fact, the simulation results of Section 4 show that the benefit of regularization is greatest when ζ is chosen small enough to impose a reasonable prior constraint on the value of β j 's. Behavior of shrinkage model Gibbs samplers near β j = 0 In many models, establishing minorization and drift condition amounts to quantifying the chain's behavior in the tail of the target. In studying convergence rates under shrinkage models, however, we are faced with an additional and distinctive challenge: the need to establish that the chain does not get "stuck" near the spike at β j = 0 (Pal and Khare, 2014;Johndrow et al., 2018). Regularization effectively eliminates the possibility of the chain meandering to infinity, making it relatively routine to analyze its behavior as β j → ∞. On the other hands, the existing results provide no general insights into the behavior near β j = 0. In fact, a careful examination of the proofs by Pal and Khare (2014) and Johndrow et al. (2018) reveals that the analyses under various shrinkage priors could have been unified if we had a more general characterization of shrinkage model Gibbs samplers' behavior near β j = 0. To fill in this theoretical gap, we start our analysis by abstracting key model-agnostic results from our proofs of minorization and drift condition for the sparse logistic regression Gibbs sampler. Our Proposition 3.3 and 3.4 below characterize properties of the distribution of λ j \| β j , τ -this distribution, due to conditional independence, typically coincides with the full posterior conditional of λ j and critically informs behavior of the subsequent update of β j in a shrinkage model Gibbs sampler. Our proof techniques apply to a broad range of shrinkage priors, essentially requiring only that π loc ∞ := max λ π loc (λ) < ∞. 3 Proposition 3.3 below plays a critical role in our proof of minorization condition. The proposition tells us that a sample from λ j \| β * j , τ has a uniformly lower-bounded probability of λ j ≥ a as long as \|β * j /τ \| is bounded away from zero. In turn, the subsequent update of β j conditional on λ j should also have a guaranteed chance of landing away from zero. Intuitively, we can thus interpret the proposition as suggesting that a shrinkage model Gibbs sampler should not get "absorbed" to the spike at β j = 0. The difference in the limiting behavior as \|β * j /τ \| → 0, depending on whether λ −1 π loc (λ) dλ < ∞, is also significant and leads to the difference between geometric and uniform convergence under the sparse logistic regression example through Theorem 3.6. Proposition 3.3. For any a > 0, the tail probability P(λ j ≥ a \| β * j , τ ) is a decreasing function of \|β * j /τ \|. If λ −1 π loc (λ) dλ = ∞, then as \|β * j /τ \| → 0 the tail probability converges to 0, i.e. the conditional λ j \| β * j , τ converges in distribution to a delta measure at 0. If λ −1 π loc (λ) dλ < ∞, then the conditional λ j \| β * j , τ converges in distribution to π(λ j ) ∝ λ −1 j π loc (λ j ) as \|β * j /τ \| → 0. Another key property of λ j \| β j , τ , featured prominently in our proof of the drift condition (Theorem 3.8), is provided by Proposition 3.4 below. To briefly provide a context, a Lyapunov function of the form V (β) = j \|β j \| −α has proven effective in analyzing a shrinkage model Gibbs sampler (Pal andKhare 2014, Johndrow et al. 2018, Section 3.3). And bounding the conditional expectation of τ −α λ −α j as below often constitutes a critical step in establishing the drift condition. Proposition 3.4. Let R > 0 and α ∈ [0, 1). If π loc ∞ < ∞, then there is an increasing function γ(r) > 0 with lim r→0 γ(r) = 0, for which the expectation with respect to λ j \| β * j , τ satisfies E τ −α λ −α j \| τ, β * j ≤ γ(R/τ ) β * j −α + \|R\| −α . (3.3) Proposition 3.3 and 3.4 are substantial theoretical contributions on their own, but we defer their proofs to Appendix C so that we can without interruption proceed to establish ergodicity results in the regularized sparse logisitic regression case. Remark. The assumption π loc ∞ < ∞ is sufficient but not necessary one for the conclusion of Proposition 3.4 and later of Theorem 3.8. Following the analysis by Pal and Khare (2014), we can show that the conclusions also hold under normal-gamma priors with any shape parameter a > 0. These priors have the property π loc (λ) ∼ O(λ 2a−1 ) as λ → 0 and hence lim λ→0 π(λ) = ∞ for a < 1/2. We leave it as future work to characterize the behavior of general shrinkage priors with π loc ∞ = ∞. Remark. In Appendix B, we show that Proposition 3.3 and 3.4 can also be applied to establish uniform/geometric ergodicity of a Gibbs sampler for Bayesian sparse probit regression, demonstrating their relevance beyond the sparse logistic regression example. Minorization -with uniform ergodicity in special cases Having described the noteworthy model-agnostic results within our proofs, from now on we focus exclusively on the regularized sparse logistic regression case. We first consider the Gibbs sampler with fixed τ in Lemma 3.5 and Theorem 3.6. While fixing the global scale parameter is a common assumption in the ergodicity proofs for shrinkage models (Pal and Khare, 2014), we subsequently show that this assumption can be replaced with much weaker ones; we only require τ ∼ π glo (·) to be supported away from 0 in Theorem 3.1 and additionally away from +∞ in Theorem 3.7. Let P (β \| β * , τ, λ) denote the transition kernel corresponding to Step 3 and 4 of the Gibbs sampler as described in Page 4 and P (β \| β * , τ ) corresponding to Step 2 -4. In other words, we define P (β \| β * , τ, λ) = π(β \| ω, τ, λ, y, X, z = 0) π(ω \| β * , X) dω, P (β \| β * , τ ) = P (β \| β * , τ, λ) π(λ \| β * ) dλ. The following lemma builds on a result of Choi and Hobert (2013) and plays a prominent role, along with Proposition 3.3, in our proofs of minorization conditions. Lemma 3.5. Whenever min j τ λ j ≥ R > 0, there is δ > 0 -independent of τ and λ except through R -such that the following minorization condition holds: P (β \| β * , τ, λ) ≥ δ N (β; µ R , Φ −1 R ), where Φ R = 1 2 X X + ζ −2 I + R −2 I and µ R = Φ −1 R X (y − 1/2). We defer the proof to Appendix D. We now establish a minorization condition for the Gibbs sampler with fixed τ . Theorem 3.6 (Minorization). Let , R > 0. On a small set {β * : min j \|β * j /τ \| ≥ }, the marginal transition kernel satisfies a minorization condition P (β \| β * , τ ) ≥ δ(τ ) N (β; µ R , Φ −1 R ), where µ R and Φ R are defined as in Lemma 3.5, and δ(τ ) > 0 is increasing in τ and otherwise depends only on , R, and π loc . Moreover, the minorization holds uniformly on β * ∈ R p in case the prior satisfies ∞ 0 λ −1 π loc (λ) dλ < ∞. Proof. Using Lemma 3.5, we have P (β \| β * , τ ) = P (β \| β * , τ, λ)π(λ \| β * , τ ) dλ ≥ {minj τ λj ≥R} P (β \| β * , τ, λ)π(λ \| β * , τ ) dλ ≥ δ N (β; µ R , Φ −1 R ) j ∞ R/τ π(λ j \| β * j , τ ) dλ j , for δ > 0 depending only on R. Also, Proposition 3.3 implies that whenever \|β * j /τ \| ≥ ∞ R/τ π(λ j \| β * j , τ ) dλ j ≥ ∞ R/τ π λ \|β * /τ \| = dλ > 0. Hence, j ∞ R/τ π(λ j \| β * j , τ ) dλ j is lower bounded by a positive constant depending only on and R/τ . In case C = ∞ 0 λ −1 π loc (λ) dλ < ∞, we can forgo the assumption \|β * j /τ \| ≥ and obtain a uniform lower bound since ∞ R π(λ j \| β * j , τ ) dλ j ≥ 1 C ∞ R λ −1 π loc (λ) dλ > 0. We now relax the assumption of fixed τ . The results of van der Pas et al. (2017) suggest that a constraint of the form 0 < τ min ≤ τ ≤ τ max < ∞ can improve the statistical property of shrinkage priors. As it turns out, such a constraint also enables us to establish minorization conditions for the full Gibbs sampler under sparse logistic regression with τ update incorporated. We can in fact take τ max = ∞ in case of the Bayesian bridge prior, whose unique structure allows us to marginalize out λ j 's when updating τ (Polson et al. 2014;Appendix F). This collapsed update of τ from τ \| β makes it possible to deduce the uniform ergodicity result of Theorem 3.1 as an immediate consequence of Theorem 3.6 by studying the marginal transition β * → β with kernel P (β \| β * ) = ∞ τmin P (β \| β * , τ ) π(τ \| β * ) dτ. (3.4) Proof of Theorem 3.1. It suffices to establish uniform minorization for the marginal transition kernel (3.4). Under the Bayesian bridge prior, we have π loc (λ) ∝ O(λ 2a ) as λ → 0 (Appendix F) and hence λ −1 π loc (λ) < ∞. The minorization condition of Theorem 3.6 thus holds uniformly in β * , yielding ∞ τmin P (β \| β * , τ ) π(τ \| β * ) dτ ≥ N (β; µ R , Φ −1 R ) ∞ τmin δ(τ ) π(τ \| β * ) dτ, (3.5) for R > 0. Theorem 3.6 further tells us that δ(τ ) > 0 is increasing in τ , so we have ∞ τmin δ(τ ) π(τ \| β * ) dτ ≥ δ(τ min ) > 0. (3.6) The inequalities (3.5) and (3.6) together establish uniform minorization. For more general shrinkage priors, the global scale τ must be updated from the full conditional τ \| β, λ. This makes it necessary to study the marginal transition (β * , λ * ) → (β, λ), jointly in regression coefficients and local scales, with kernel P (β, λ \| β * , λ * ) = τmax τmin P (β \| β * , τ, λ) j π(λ j \| β * j , τ ) π(τ \| β * , λ * ) dτ. (3.7) We establish a minorization condition for this general case in Theorem 3.7. Theorem 3.7. If the prior π glo (·) is supported on [τ min , τ max ] for 0 < τ min ≤ τ max < ∞, then the marginal transition kernel P (β, λ \| β * , λ * ) of the Pólya-Gamma Gibbs sampler for regularized sparse logistic regression satisfies a minorization condition on a small set (β * , λ * ) : 0 < ≤ \|β * j \| ≤ E < ∞ for all j . Proof. By Lemma 3.5 and the fact τ λ j ≥ τ min λ j , we know that for R > 0 P (β \| β * , τ, λ) ≥ 1{min j τ min λ j ≥ R} δ N (β; µ R , Φ −1 R ). (3.8) To lower bound the term j π(λ j \| β * j , τ ) in (3.7), we first recall that π(λ j \| β * j , τ ) = λ −1 j exp −β * 2 j /2τ 2 λ 2 j π loc (λ j ) ∞ 0 λ −1 exp −β * 2 j /2τ 2 λ 2 π loc (λ) dλ . When τ min ≤ τ ≤ τ max and ≤ \|β * j \| ≤ E, we have exp −E 2 /2τ 2 min λ 2 ≤ exp −β 2 j /2τ 2 λ 2 ≤ exp − 2 /2τ 2 max λ 2 . It follows from the above inequalities that π(λ j \| β * j , τ ) ≥ λ −1 j exp −E 2 /2τ 2 min λ 2 j π loc (λ j ) ∞ 0 λ −1 exp(− 2 /2τ 2 max λ 2 ) π loc (λ) dλ := η π lower (λ j ) (3.9) for η > 0 and density π lower (·) independent of β * j and τ . Combining (3.8) and (3.9), we can lower bound the transition kernel (3.7) as P (β, λ \| β * , λ * ) ≥ δ η 1 min j λ j ≥ R τ min N (β; µ R , Φ −1 R ) j π lower (λ j ) τmax τmin π(τ \| β * , λ) dτ = δ η N (β; µ R , Φ −1 R ) j 1 λ j ≥ R τ min π lower (λ j ). Drift condition and geometric ergodicity Here we establish a drift condition for geometric ergodicity under sparse logistic regression. As discussed in Section 3.1, the regularization prevents the Markov chain from meandering to infinity, so the main question is whether the chain can get "stuck" for a long time near β * j = 0. The following result shows that this does not happen as long as the global scale τ is bounded away from zero. Theorem 3.8. Suppose that the local scale prior satisfies π loc ∞ < ∞ and that the global scale prior π glo (·) is supported on [τ min , ∞) for τ min > 0. Then the marginal transition kernel P (β, λ \| β * , λ * ) satisfies a drift condition with a Lyapunov function V (β) = j \|β j \| −α for any 0 ≤ α < 1. Proof. Note that P V (β * ) can be expressed as a series of iterated expectations with respect to (1) β \| ω, τ, λ, y, X, z = 0, (2) ω \| β * , (3) λ \| β * , τ , and (4) τ \| β * , λ * . We will bound the iterated expectations of \|β j \| −α one by one. Since β \| ω, τ, λ, y, X, z = 0 is distributed as Gaussian, denoting by µ j and σ 2 j the conditional mean and variance of β j , Proposition 3.10 below tells us that E \|β j \| −α \| ω, τ, λ, y, X, z = 0 ≤ C α (µ j /σ j ) σ −α j where sup t C α (t) ≤ Γ 1−α 2 2 α/2 √ π and C α (t) = O(\|t\| −α ) as \|t\| → ∞. For the purpose of this proof, we can simply set C α to be its global upper bound; however, a tighter bound may be obtained when the posterior concentrates away from zero and thereby resulting in \|µ j /σ j \| → ∞ and C α (µ j /σ j ) → 0 as the sample size increases. Combined with Proposition 3.11 below, the above inequality implies 1 C α E \|β j \| −α \| ω, τ, λ, y, X, z = 0 ≤ τ −α λ −α j + ζ −α + 1 − α 2 + α 2 n i=1 ω i x 2 ij . (3.10) In taking the expectation of (3.10) with respect to ω \| β * , we use the result E[ ω j \| β * ] ≤ 1/4 of Wang and Roy (2018) to obtain 1 C α E \|β j \| −α \| τ, λ ≤ τ −α λ −α j + ζ −α + 1 − α 2 + α 8 n i=1 x 2 ij . (3.11) Taking the expectation of (3.11) with respect to λ \| τ, β * , we have 1 C α E \|β j \| −α \| τ, β * ≤ E τ −α λ −α j \| τ, β * j + C (α, X) where C (α, X) = ζ −α + 1 − α 2 + α 8 n i=1 x 2 ij . ( 3.12) Now choose R > 0 small enough that γ(R/τ ) ≤ γ(R/τ min ) < C −1 α in Proposition 3.4. Then we have the following inequality for γ := C α γ(R/τ min ) < 1: C α E τ −α λ −α j \| τ, β * j ≤ γ \|β * j \| −α + \|R\| −α for all τ ≥ τ min . Incorporating the above inequality into (3.12), we obtain E \|β j \| −α \| τ, β * ≤ γ \|β * j \| −α + γ \|R\| −α + C α C (α, X). Since π(τ \| β * , λ * ) is supported on τ ≥ τ min by our assumption, taking the expectation with respect to τ \| β * , λ * yield E \|β j \| −α \| β * , λ * ≤ γ \|β * j \| −α + γ \|R\| −α + C α C (α, X). Theorem 3.7 and 3.8 together imply the geometric ergodicity result of Theorem 3.2: Proof of Theorem 3.2. We show that V (β) = j \|β j \| −α + β 2 is a Lyapunov function for the marginal transition kernel P (β, λ \| β * , λ * ). Note that E β 2 \| ω, τ, λ, y, X, z = 0 = E[β \| ω, τ, λ, y, X, z = 0] 2 + j var β 2 j \| ω, τ, λ, y, X, z = 0 = ΣX y − 1 2 2 + j e j Σe j for Σ = X ΩX + ζ −2 I + τ −2 Λ −2 −1 . Since Σ ≺ ζ 2 I, we have e j Σe j ≤ ζ 2 and ΣX (y − 1 2 ) 2 ≤ ζ 2 X (y − 1 2 ) 2 . Thus we have E β 2 \| ω, τ, λ, y, X, z = 0 ≤ ζ 2 ΣX y − 1 2 2 + nζ 2 . (3.13) Since the right-hand side does not depend on ω, τ, λ, the expectation with respect to P (β, λ \| β * , λ * ) satisfies the same bound: E β 2 \| β * , λ * ≤ ζ 2 ΣX y − 1 2 2 + nζ 2 . In addition to the above bound, we know that j \|β j \| −α is a Lypunov function by Theorem 3.8. Hence, V (β) = j \|β j \| −α + β 2 is again a Lyapunov function. Moreover, by Theorem 3.7, we know that the Gibbs sampler satisfies a minorization condition on the set β * : 0 < ≤ \|β * j \| ≤ E < ∞ for all j for > 0 and E < ∞. Thus the sampler is geometrically ergodic. Remark 3.9. As mentioned earlier, the geometric and uniform ergodicity as well as analogues of the intermediate results continue to hold when we relax the assumption of fixed ζ to a prior constraint of the form ζ ≤ ζ max < ∞. The proof goes as follows. Due to the conditional independence, the Gibbs sampler on the joint space draws alternately from ζ \| β, z = 0 and β, ω, τ, λ \| y, X, z = 0, ζ. By repeating all the previous arguments with ζ max in place of ζ, we obtain essentially the identical minorization and drift bounds that hold for all ζ ≤ ζ max . Since the bounds hold uniformly on the support ζ ≤ ζ max , the identical bounds again hold when taking the expectation over ζ \| β, z = 0. Auxiliary results for proof of geometric ergodicity Proposition 3.10 and 3.11 below are used in the proof of Theorem 3.8 and are proved in Appendix E. Proposition 3.10 is a refinement of Proposition A1 in Pal and Khare (2014) and of Equation (41) in Johndrow et al. (2018), neither of which have the D(µ/σ) term. Proposition 3.10. For α ∈ (0, 1) and β ∼ N (µ, σ 2 ), we have E\|β\| −α ≤ Γ 1−α 2 2 α/2 √ π σ −α min{1, D(µ/σ)} , where D(t) = O(\|t\| −α ) → 0 as \|t\| → ∞ and can be chosen as D(t) = 1 B α 2 , 1−α 2 2 5 2 −α 1 − α exp − t 2 4 + 2 1 2 +α Γ α 2 \|t\| −α . (3.14) Proposition 3.11. The diagonals σ j of Σ = X ΩX + ζ −2 I + τ −2 Λ −2 −1 satisfy the following inequality for 0 ≤ α < 1: σ −α j ≤ τ −α λ −α j + ζ −α + 1 − α 2 + α 2 n i=1 ω i x 2 ij . Simulation We run a simulation study to assess the computational and statistical properties of the regularized sparse logistic regression model. We use the Bayesian bridge prior π(β j \| τ ) ∝ τ −1 exp(−\|β j /τ \| a ) to take advantage of the efficient global scale parameter update scheme. This prior also allows us to experiment with a range of spike and tail behavior by varying the exponent a, inducing larger spikes and heavier tails as a → 0. For the global scale parameter, we chose the objective prior π glo (τ ) ∝ τ −1 (Berger et al., 2015, Appendix F) with the range restriction 10 −6 ≤ E[ \|β j \| \| τ ] ≤ 1 to ensure posterior propriety, though in practice we never observe a posterior draw of τ outside this range. For the posterior computations, we use the Pólya-Gamma Gibbs sampler provided by the bayesbridge package available from Python Package Index (pypi.org); the source code is available at the GitHub repository https://github.com/aki-nishimura/bayes-bridge. 4.1 Data generating process: "large n, but weak signal" problems Piironen and Vehtari (2017) demonstrate the benefits of regularizing shrinkage priors in the "p > n" case, when the number of predictors p exceeds the sample size n. To complement their study, we consider the case of rare outcomes and infrequently observed features, another common situation in which regularizing shrinkage priors becomes essential. For example in healthcare data, many outcomes of interests have low incidence rates and many treatments are prescribed to only a small fraction of patients (Tian et al., 2018). This results in binary outcomes y and features x j filled mostly with 0's, making the amount of information much less than otherwise expected (Greenland et al., 2016). To simulate under these "large n, but weak signal" settings, we generate synthetic data with n = 2,500 and p = 500 as follows. We construct binary features with a range of observed frequencies by first drawing 2w j ∼ Beta(1/2, 2) for j = 1, . . . , 500; this in particular means 0 ≤ w j ≤ 0.5 and E[w j ] = 0.1. For each j, we then generate x ij ∼ Bernoulli(w j ) for i = 1, . . . , n. We choose the true signal to be β j = 1 for j = 1, . . . , 10 and β j = 0 for j = 11, . . . , 500. To simulate an outcome with low incidence rate, we choose the intercept to be β 0 = 1.5 and draw y i ∼ Bernoulli(logit(−x i β)), resulting in y i = 1 for approximately 5% of its entries. Convergence and mixing: with and without regularization With the above data generating process, outcome y and design matrix X barely contain enough information to estimate all the coefficients β j 's. In particular, sparse logistic model without regularization can lead to a heavy-tailed posterior, for which uniform and geometric ergodicity of the Pólya-Gamma Gibbs sampler becomes questionable. These potential convergence and mixing issues are evidenced by the traceplot (Figure 4.1a) of the posterior samples based on bridge exponent a = 1/16. As we are particularly concerned with the Markov chain wandering off to the tail of the target, we examine the estimated credible intervals to identify the coefficients with potential convergence and mixing issues. Plotted in Figure 4.1 are the coefficients with the widest 95% credible intervals; these coefficients also have some of the smallest estimated effective sample sizes, though the accuracy of such estimates is not guaranteed without geometric ergodicity. When regularizing the shrinkage prior with a slab width ζ = 1, the posterior samples indicate no such convergence or mixing issues (Figure 4.1b) and yield more sensible posterior credible intervals (Figure 4.2). We emphasize that there is no fundamental change in the Gibbs sampler itself when incorporating the regularization, the only change being the addition of the ζ −2 I term in the conditional precision matrix (3.2) when updating β. It is the change in the posterior -more specifically the guaranteed light tails of the β marginal -that induces faster convergence and mixing. We also assess sensitivity of convergence and mixing rates on the slab width ζ. The regularized prior recovers the unregularized one as ζ → ∞. This means that, as seen from the problematic computational behavior of the unregularized model, ζ cannot be taken too large in this limited data setting. In other words, the choice of ζ has to reflect some degree of prior information on β j 's. We need not assume strong prior information, however; Figure 4.3 demonstrates that even small amount of regularization (e.g. ζ = 2 or 4) can noticeably improve the computational behavior over the unregularized case. Statistical properties of shrinkage model for weak signals To study the shrinkage model's ability to separate out the non-zero β j from the β j = 0, we simulate 10 replicate data sets and estimate the posterior for each of them. In total, we obtain 5,000 marginal posterior distributions -10 independent replication for each of the p = 500 regression coefficients -with 100 for the signal β j = 1 and 4,900 for the non-signal β j = 0. As all the predictors x j 's are simulated in an exchangeable manner, the 100 (and 4,900) posterior marginals for the signal (and non-signal) are also exchangeable. Figure 4.4 show the posterior credible intervals. Due to the low incidence rate and infrequent binary features, many of the signals are too weak to be detected. We also find that the credible intervals seemingly do not achieve their nominal frequentist coverage for signals below detection strength. This finding is consistent with the existing theoretical results on shrinkage priors and is unsurprising in light of the impossibility theorem by Li (1989) -confidence intervals cannot be optimally tight and have nominal coverage at the same time. Credible intervals produced by Bayesian shrinkage models tend to be optimally tight and thus require appropriate manual adjustments to achieve the nominal coverage (van der Pas et al., 2017). No statistical procedure is immune to this tightness-coverage trade-off; therefore, the apparent under-coverage should be seen not as a flaw but more as a feature of Bayesian shrinkage models. We benchmark the signal detection capability of the posterior against the frequentist lasso, arguably the most widely-used approach to feature selection. Obtaining the lasso point estimates requires a selection of the hyper-parameter commonly referred to as the penalty parameter. For its choice, we first follow the standard practice of minimizing the ten-fold cross-validation errors (Hastie et al., 2009). However, this approach yields inconsistent and poor overall performance, detecting only 13 out of the 100 signals (Figure 4.4). Cross-validation likely fails here because each fold does not capture the characteristics of the whole data when the signals are so weak. As a more stable alternative for calibrating the penalty parameter, we try an empirical Bayes procedure based on the Bayesian interpretation of the lasso (Park and Casella, 2008). We first estimate the posterior marginal mean of the penalty parameter from the Bayesian lasso Gibbs sampler. Conditionally on this value, we then find the posterior mode of β. This procedure seems to yield more consistent performance, detecting 39 out of the 100 signals albeit with the estimates more shrunk towards null than the Bayesian posterior means. The empirical Bayes procedure demonstrates more consistent behavior for the non-signals as well (Figure 4.5). We also assess how the spike size and (pre-regularization) tail behavior of the prior influence statistical properties of the resulting posterior. For this purpose, we fit the regularized bridge model with the exponent a −1 ∈ {2, 4, 8, 16} to the same data sets. Figure 4.6 summarizes the credible intervals under the a = 1/4 case. The credible intervals are centered around the values similar to the a = 1/16 case (Figure 4.4), but are much wider overall. We observe the same pattern throughout the range of the exponent values: similar median values, but tighter intervals for the smaller exponents. In particular, as can be seen in Figure 4.7, more "extreme" shrinkage priors with larger spikes and heavier-tails seem to yield tighter credible intervals for the same coverage. Discussion Shrinkage priors have been adopted in a variety of Bayesian models, but the potential issues arising from their heavy-tails are often overlooked. Our method provides a simple and convenient way to regularize shrinkage priors, making the posterior inference more robust. Both the theoretical and empirical results demonstrate the benefits of regularization in improving the statistical and computational properties when parameters are only weakly identified. Much of the systematic investigations into the shrinkage prior properties has so far focused on rather simple models and situations in which signals are reasonable strong. Our work adds to the emerging efforts to better understand the behavior of shrinkage models in more complex settings. Appendix A: Alternative definition of proposed regularization In Section 2, we described our regularization approach as effectively modifying the prior on β j through the likelihood of fictitious data z j . While many properties of the resulting posterior are most apparent from this formulation, we can forgo the use of fictitious data and achieve the identical effect via direct modification of a shrinkage prior as follows. We define the regularized prior π reg (·) by setting the distribution of β j , λ j \| τ, ζ as π reg (β j , λ j \| τ, ζ) ∝ exp − β 2 j 2ζ 2 1 τ λ j exp − β 2 j 2τ 2 λ 2 j π loc (λ j ) ∝ N   β j 0, 1 ζ 2 + 1 τ 2 λ 2 j −1   1 + τ 2 λ 2 j ζ 2 −1/2 π loc (λ j ) where N ( · \| 0, σ 2 ) denotes the centered Gaussian density with variance σ 2 . In other words, in addition to defining π(β j \| τ, λ j , ζ) as in (2.1), we alter the prior on λ j as π(λ j \| τ, ζ) ∝ π loc (λ j )/ 1 + τ 2 λ 2 j /ζ 2 . Incidentally, we see that our regularized prior is very similar to that of Piironen and Vehtari (2017), but has a slightly lighter tail due to the factor 1/ 1 + τ 2 λ 2 j /ζ 2 which, as λ j → ∞, behaves like ζ/τ λ j . Appendix B: Further results on behavior of shrinkage model Gibbs samplers: probit regression as example As we discussed in Section 3.1, Proposition 3.3 and 3.4 are quite general in scope and can provide insight into behavior of shrinkage model Gibbs samplers more broadly. Here we demonstrate the broader relevance of these results, as well as of a few additional results, by applying them to establish uniform/geometric ergodicity of a Gibbs sampler for regularized Bayesian sparse probit regression. More explicitly, we consider the model y i \| x i , β ∼ Bernoulli Φ(x i β) , z j = 0 \| β j ∼ N (0, ζ 2 ), β j \| τ, λ j ∼ N (0, τ 2 λ 2 j ), τ ∼ π glo (·), λ j ∼ π loc (·), where Φ(t) denotes the cumulative distribution function of the standard Gaussian. The corresponding Gibbs sampler induces a transition kernel (β * , λ * , τ * ) → (β, λ, τ ) through the following cycle of conditional updates: 1. Draw τ \| β * , λ * from the density proportional to (2.3). When using Bayesian bridge priors, draw from the collapsed distribution τ \| β * (Appendix F). 3. Draw β \| τ, λ, y, X, z = 0 from the density proportional to π (β \| τ, λ, y, X, z = 0) ∝ L probit (y \| X, β) L(z = 0 \| β) π(β \| τ, λ) ∝ L probit (y \| X, β) π(β \| τ, λ, z = 0) (B.1) where L probit (y \| X, β) = i Φ(x i β) yi 1 − Φ(x i β) 1−yi is the probit likelihood. The density (B.1) belongs to a unified skew-normal family, from which we can draw independent samples by the algorithm of Durante (2019). Borrowing a terminology from Durante (2019), we refer to the above Gibbs sampler as the conjugate Gibbs sampler for probit model to distinguish it from the more traditional one based on the data augmentation scheme of Albert and Chib (1993). Theorem B.1 and B.2 below provide uniform and geometric ergodicity results for the conjugate Gibbs sampler and are exact analogues of the corresponding results Theorem 3.1 and 3.2 for the logistic case. Theorem B.1 (Uniform ergodicity for probit model). If the prior π glo (·) is supported on [τ min , ∞) for τ min > 0, then the conjugate Gibbs sampler for regularized Baysian bridge probit regression is uniformly ergodic. Theorem B.2 (Geometric ergodicity for probit model). Suppose that the local scale prior satisfies π loc ∞ < ∞ and that the global scale prior π glo (·) is supported on [τ min , τ max ] for 0 < τ min ≤ τ max < ∞. Then the conjugate Gibbs sampler for regularized sparse probit regression is geometrically ergodic. B.1 Proofs of Theorem B.1 and B.2 The proof of Theorem B.1 (and B.2) above follows a path essentially identical to the proof of Theorem 3.1 (and 3.2) with most arguments carrying through verbatim or with trivial modifications; we only need to replace a few model-specific inequalities with the corresponding ones for the probit model. For establishing minorization conditions, Lemma B.3 below replaces Lemma 3.5. For establishing drift conditions, the bound on the conditional expectation of \|β j \| −α in Lemma B.4 replaces Eq. (3.11), and the bound on the conditional expectation of β 2 in Lemma B.5 replaces Eq. (3.13). Remarkably, Lemma B.4 and B.5 only requires a likelihood L(y \| X, β) to be a bounded function of β and thus may be applicable beyond the probit case. We sketch out the proofs of Theorem B.1 and B.2 below. Again, the omitted details are essentially identical to the logistic case or, in fact, simpler because the probit case does not involve the additional Pólya-Gamma parameter. Proof of Theorem B.1. A minorization result analogous to Theorem 3.6 follows from Proposition 3.3 and Lemma B.3. This minorization result straightforwardly implies a uniform minorization under Bayesian bridge priors as in Theorem 3.1. See the proofs of Theorem 3.6 and 3.1 for details. Proof of Theorem B.2. A minorization result analogous to Theorem 3.7 follows from Lemma B.3. Proposition 3.4, Lemma B.4, and Lemma B.5 together imply that V (β) = j \|β j \| −α + β 2 is a Lyapunov function as in the proofs of Theorem 3.8 and 3.2. The geometric ergodicity then follows from the minorization and drift condition. See the proofs of Theorem 3.7, 3.8, and 3.2 for details. B.2 Minorization lemma for probit model Lemma B.3. Whenever min j τ λ j ≥ R > 0, there areδ,δ > 0 -independent of τ and λ except through R -such that the following minorization condition holds: π(β \| τ, λ, y, X, z = 0) ≥δ L probit (y \| X, β) N β; 0, (ζ −2 + R −2 ) −1 I ≥δ N β; 0, X X + (ζ −2 + R −2 )I −1 . (B.2) Proof. The conditional distribution of β \| τ, λ, y, X is given by π(β \| τ, λ, y, X, z = 0) = L probit (y \| X, β) π(β \| τ, λ, z = 0) L probit (y \| X, β ) π(β \| τ, λ, z = 0) dβ . (B.3) Since Φ(t) = 1 − Φ(−t) ≤ 1 for all t, we have L probit ∞ ≤ 1 and L probit (y \| X, β ) π(β \| τ, λ, z = 0) dβ ≤ π(β \| τ, λ, z = 0) dβ = 1. (B.4) Also, we can easily verify that the following inequality holds whenever min j τ λ j ≥ R: π(β \| τ, λ, z = 0) = j 1 √ 2π ζ −2 + τ −2 λ −2 j 1/2 exp − 1 2 ζ −2 + τ −2 λ −2 j β 2 j ≥ j 1 √ 2πζ exp − 1 2 ζ −2 + R −2 β 2 j . (B.5) Combining (B.4) and (B.5), we can lower bound (B.3) withδ > 0 as π(β \| τ, λ, y, X, z = 0) ≥δ L probit (y \| X, β) N β; 0, (ζ −2 + R −2 ) −1 I , (B.6) establishing the first inequality in (B.2). To establish the second inequality in (B.2), we will show that min{Φ(t), 1 − Φ(t)} ≥ min 1 − Φ(1), 1 2 √ 2π exp −t 2 ; (B.7) this will imply L probit (y \| X, β) ≥ min 1 − Φ(1), (2 √ 2π) −1 exp(− Xβ 2 ) and complete the proof. Eq 7.1.13 of Abramowitz and Stegun (1965) tells us that 1 − Φ(t) ≥ 1 √ 2π t t 2 + 1 exp − t 2 2 . (B.8) We therefore have 1 − Φ(t) ≥ 1 2 √ 2π 1 t exp − t 2 2 ≥ 1 2 √ 2π exp −t 2 for t ≥ 1; (B.9) the latter inequality follows from the fact that t −1 ≥ exp(−t 2 /2) for t ≥ 1, which can be proven, for example, by noting that d dt t exp(−t 2 /2) ≤ 0 for t ≥ 1. For t ≤ 1, we have 1 − Φ(t) ≥ 1 − Φ(1) since Φ(t) is increasing in t. Combining the lower bounds for t ≥ 1 and t ≤ 1, we obtain 1 − Φ(t) ≥ min 1 − Φ(1), 1 2 √ 2π exp −t 2 ≥ min 1 − Φ(1), 1 2 √ 2π exp −t 2 . Since Φ(t) = 1 − Φ(−t), the same lower bound also holds for Φ(t), yielding (B.7). B.3 Drift condition lemmas for bounded likelihood models As we mentioned in Section B.1, Lemma B.4 and B.5 here apply not only to the probit case but also to any model whose likelihood is a bounded function of β. Lemma B.4 in particular holds with or without the fictitious likelihood L(z = 0 \| β) for regularization. While stated in terms of a generic bounded likelihood L(y \| X, β), Lemma B.4 can be applied to regularized models simply by replacing the likelihood β → L(y \| X, β) in its statement with the regularized one β → L(y \| X, β)L(z = 0 \| β). Lemma B.4. Let α ∈ [0, 1). Suppose the likelihood satisfies L ∞ := sup β L(y \| X, β) < ∞ and is strictly positive and continuous at β = 0. Then the following inequality holds for the conditional expectation under β \| τ, λ, y, X with constants C, C < ∞ depending only on α and functionals of the likelihood β → L(y \| X, β): E \|β j \| −α \| τ, λ, y, X ≤ C\|τ λ j \| −α + C . (B.10) Proof. The conditional distribution of β \| τ, λ, y, X is given by π(β \| τ, λ, y, X) = L(y \| X, β) π(β \| τ, λ) L(y \| X, β ) π(β \| τ, λ) dβ . (B.11) We consider the conditional expectation (B.10) under two separate cases: max j τ λ j ≤ and min j τ λ j ≥ , where > 0 is any value small enough to guarantee the likelihood to be positive on the set β ∞ = max j \|β j \| ≤ . When max j τ λ j ≤ , we have L(y \| X, β ) π(β \| τ, λ) dβ ≥ β ∞ ≤ L(y \| X, β ) π(β \| τ, λ) dβ ≥ min β ∞ ≤ L(y \| X, β ) j − π(β j \| τ, λ j ) dβ j ≥ min β ∞ ≤ L(y \| X, β ) Φ(1) − Φ(−1) p , where Φ(·) is the cumulative distribution function of the standard Gaussian. Using the above lower bound on the numerator, we can bound (B.11) as π(β \| τ, λ, y, X) ≤ C π(β \| τ, λ) (B.12) for C = L ∞ min β ∞ ≤ L(y \| X, β ) Φ(1) − Φ(−1) p . It now follows that E \|β j \| −α \| τ, λ, y, X ≤ C E \|β j \| −α \| τ, λ = C α C \|τ λ j \| −α , (B.13) where the latter equality with C α = Γ 1−α 2 2 α/2 √ π derives from the formula for negative moments of Gaussians (Winkelbauer, 2012). Turning to the case min j τ λ j ≥ , we have π(β \| τ, λ, y, X) = L(y \| X, β) j exp − β 2 j 2τ 2 λ 2 j L(y \| X, β ) j exp − β 2 j 2τ 2 λ 2 j dβ ≤ L ∞ L(y \| X, β ) j exp −β 2 j /2 2 dβ := C . (B.14) Using the above bound on the conditional density, we obtain E \|β j \| −α \| τ, λ, y, X ≤ 1 + E \|β j \| −α 1 \|β j \| ≤ 1 \| τ, λ, y, X ≤ 1 + C 1 −1 \|β j \| −α dβ j = 1 + 2C /(1 − α). (B.15) The bounds (B.13) and (B.15) together show that an inequality of the form (B.10) holds for any value of τ and λ, whether in {max j τ λ j ≤ } or {min j τ λ j ≥ }. Lemma B.5. Suppose the likelihood satisfies the assumptions as in Lemma B.4. Then the conditional expectation of β 2 j under β \| τ, λ, y, X, z = 0 is bounded by a constant which depends only on ζ and functionals of the likelihood β → L(y \| X, β). Proof. We will derive the following bound on the conditional density π(β \| τ, λ, y, X, z = 0) ≤ C N (β; 0, ζ 2 I) 1 + N β; 0, τ 2 Λ 2 = C N (β; 0, ζ 2 I) + C τ 2 λ 2 j + ζ 2 −1/2 N β; 0, τ −2 Λ −2 + ζ −2 I −1 , (B.16) which will imply the desired bound on the conditional expectation: E β 2 j \| τ, λ, y, X, z = 0 ≤ Cζ 2 + C τ 2 λ 2 j + ζ 2 −1/2 τ −2 λ −2 j + ζ −2 −1 = Cζ 2 + C ζ 2 τ 2 λ 2 j τ 2 λ 2 j + ζ 2 −3/2 ≤ Cζ 2 + C ζ 2 τ 2 λ 2 j + ζ 2 −1/2 ≤ Cζ 2 + Cζ. To complete the proof, therefore, it remains to establish (B.16). Our argument here closely follows those we use in deriving the bounds (B.12) and (B.14) in the proof of Lemma B.4. The conditional distribution of β \| τ, λ, y, X, z = 0 is given by π(β \| τ, λ, y, X, z = 0) = L(y \| X, β)L(z = 0 \| β) π(β \| τ, λ) L(y \| X, β )L(z = 0 \| β) π(β \| τ, λ) dβ . (B.17) As before, we choose > 0 to be any value small enough to guarantee the likelihood to be positive on the set β ∞ = max j \|β j \| ≤ . We can repeat an argument analogous to the derivation of the bound (B.12) to conclude that, when max j τ λ j ≤ , π(β \| τ, λ, y, X, z = 0) ≤ C L(z = 0 \| β) π(β \| τ, λ) (B.18) for C = L(y \| X, β) ∞ min β ∞ ≤ L(y \| X, β )L(z = 0 \| β) Φ(1) − Φ(−1) p with the · ∞ norm taken with respect to β. For the case min j τ λ j ≥ , we follow the derivation of the bound (B.14) to conclude that Appendix C: Proofs for Section 3.1 C.1 Proof of Proposition 3.3 π(β \| τ, λ, y, X) = C L(z = 0 \| β) where C = L(y \| X, β) ∞ L(y \| X, β )L(z = 0 \| β) j exp −β 2 j /2 2 dβ . (B. The key ingredient in our proof of Proposition 3.3 is the following general result on the stochastic ordering of tilted densities. The result allows us to study the behavior of π(λ \| β * , τ ) viewed as a product of f (λ) = λ −1 π loc (λ) and G(λ) = exp(−β * 2 /2τ 2 λ 2 ). Proposition C.1. Consider probability densities π G (λ) ∝ G(λ)f (λ) and π H (λ) ∝ H(λ)f (λ) on λ ∈ [0, ∞) for f, G, H ≥ 0. Suppose that f satisfies ∞ u f (λ)dλ < ∞ for u > 0. Suppose also that G and H are absolutely continuous and increasing, G ≤ H, and lim λ→∞ G(λ) = lim λ→∞ H(λ). Then π G is stochastically dominated by π H i.e. ∞ a π G (λ)dλ ≤ ∞ a π H (λ)dλ for any a ∈ R. (C.1) Proof. Multiplying G and H with an appropriate constant if necessary, without loss of generality we can assume lim λ→∞ G(λ) = lim λ→∞ H(λ) = 1 so that G and H can be interpreted as cumulative distribution functions. We first deal with the case G(0) = H(0) = 0; when f (λ)dλ = ∞, this assumption is in fact implied by the integrability of G(λ)f (λ) and H(λ)f (λ). In this case, we have G(λ) = λ 0 g(u)du and H(λ) = λ 0 h(u)du for density functions g, h ≥ 0. As can be verified using Fubini's theorem for positive functions, we can express π G and π H as π G (·) = f ( · \| u)g(u)du and π H (·) = f ( · \| u)h(u)du, where f ( · \| u) for u > 0 denote a probability density f ( · \| u) = f (λ)1{λ > u} ∞ u f (λ)dλ . Again by Fubini's theorem for positive functions, we have ∞ a π G (λ)dλ = F a (u)g(u)du and ∞ a π H (λ)dλ = F a (u)h(u)du (C.2) where F a (u) = ∞ a f (λ \| u)dλ = ∞ max{a,u} f (λ)dλ ∞ u f (λ)dλ . Note that the integrals in (C.2) can be represented as expectations with respect to distributions G and H: ∞ a π G (λ) dλ = E U ∼G [F a (U )] and ∞ a π H (λ) dλ = E U ∼H [F a (U )] . (C.3) Since F a is an increasing function and G is stochastically dominated by H by our assumption, the representation (C.3) implies the desired inequality (C.1). Earlier, we made a simplifying assumption G(0) = H(0) = 0. More generally, we have the relation G(λ) − G(0) = λ 0 g(u)du and H(λ) − H(0) = λ 0 h(u)du for integrable functions g, h ≥ 0. Essentially the identical arguments as before show that the identity (C.3) and hence the conclusion (C.1) still hold in this case. Proof of Proposition 3.3. Note that π(λ j \| β * j , τ ) ∝ exp −c 2 /λ 2 j λ −1 j π loc (λ j ) for c = c(β * j /τ ) = β * j √ 2τ . Applying Proposition C.1 with f (λ) = λ −1 π loc (λ), we see that P λ j > a \| β * j , τ ≤ P λ j > a \| β * j , τ whenever \|β * j /τ \| ≥ \|β * j /τ \|. Suppose now that λ −1 π loc (λ) dλ = ∞. For any β * j /τ , we have ∞ a exp − β * 2 j 2τ 2 λ 2 j λ −1 j π loc (λ j ) dλ j ≤ ∞ a λ −1 j π loc (λ j ) dλ j ≤ 1/a. (C.4) On the other hand, by Fatou's lemma, lim inf \|β * j /τ \|→0 exp − β * 2 j 2τ 2 λ 2 λ −1 π loc (λ) dλ ≥ λ −1 π loc (λ) dλ = ∞. (C.5) From (C.4) and (C.5), we conclude that for any a > 0 P(λ j > a \| β * j , τ ) = ∞ a exp − β * 2 j 2τ 2 λ 2 j λ −1 j π loc (λ j ) dλ j exp − β * 2 j 2τ 2 λ 2 λ −1 π loc (λ) dλ → 0 as \|β * j /τ \| → 0, i.e. π(λ j \| β * j , τ ) converges in distribution to a delta measure at 0. We now turn to quantifying the limiting behavior when λ −1 π loc (λ) dλ < ∞. For any a ∈ [0, ∞], the dominated convergence theorem yields lim \|β * j /τ \|→0 a 0 exp − β * 2 j 2τ 2 λ 2 j λ −1 j π loc (λ j ) dλ j = a 0 λ −1 π loc (λ) dλ. The above convergence result implies the point-wise convergence of the cumulative distribution function: lim \|β * j /τ \|→0 P(λ j ≤ a \| β * j , τ ) = a 0 λ −1 j π loc (λ j ) dλ j λ −1 π loc (λ) dλ . C.2 Proof of Proposition 3.4 Proof. In upper-bounding E λ −α j \| τ, β * , we can without loss of generality assume that π(0) > 0 by virtue of Proposition C.2 below. In terms of the constants and C (α, π loc ) as defined in Lemma C.3 below, let γ(r) = C (α, π loc ) log 1 + 4 2 r 2 . (C.6) By Lemma C.3 and the monotonicity of γ(r), we then have E τ −α λ −α j \| τ, β * j ≤ γ(R/τ ) β * j −α whenever \|β * j \| ≤ R. On the other hand, since the distribution λ j \| τ, β * j stochastically dominates λ j \| τ, β * j whenever β * j ≥ β * j (Proposition 3.3), we have E τ −α λ −α j \| τ, β * j ≤ E τ −α λ −α j \| τ, \|β * j \| = R whenever \|β * j \| ≥ R. (C.7) Combining (C.6) and (C.7) yields the inequality (3.3). Proposition C.2. Given a prior π loc (·) such that π loc (0) = 0 and π loc ∞ < ∞, there is a density π loc (·) such that π loc (λ) is continuous at λ = 0, π loc (0) > 0, π loc ∞ < ∞, and π loc (λ) ∝ G(λ)π loc (λ) for a bounded increasing function G ≥ 0. Consequently, a density π(·) stochastically dominates π (·) when π(λ) ∝ f (λ)π loc (λ) and π (λ) ∝ f (λ)π loc (λ) for f ≥ 0. By taking f (λ) = λ −1 exp(−β * 2 j /2τ 2 λ 2 j ) in particular, we have the following inequality between the expectations with respect to π(·) and π (·): E λ −α j \| τ, β * j ≤ E λ −α j \| τ, β * j for α ≥ 0. (C.8) Proof. Redefining π loc (λ) as π loc (λ − λ min ) for λ min = inf {λ : π loc (λ) > 0} if necessary, we can without loss of generality assume that π loc (λ) > 0 for all sufficiently small λ > 0. Define G(λ) = min π loc ∞ , λ 0 max 0, dπ loc dλ (u) du . (C.9) Then G is clearly increasing and bounded. The definition (C.9) further guarantees that lim λ→0 π loc (λ)/G(λ) = 1, π loc ≤ G, and lim λ→∞ G(λ) = π loc ∞ . Define π loc (·) via the relation π loc (λ) ∝ π loc (λ)/G(λ) for λ > 0 and π loc (0) := lim λ→0 π loc (λ). Then π loc (·) satisfy π loc ∞ = π loc (0) = π(λ)/G(λ) dλ −1 > 0, as well as all the other desired properties. When π(λ) ∝ f (λ)π loc (λ) and π (λ) ∝ f (λ)π loc (λ), the densities satisfies the relation π (λ) ∝ G(λ)π(λ). By applying Proposition C.1 with H = G ∞ , we conclude that π(·) stochastically dominates π (·). The inequality (C.8) is an immediate consequence of this stochastic ordering. Lemma C.3. Suppose that π loc (λ) is continuous at λ = 0 and π loc (0) > 0. For α ∈ [0, 1) and > 0 small enough that min λ∈[0, ] π loc (λ) ≥ π loc (0)/2, we have the following inequality: E τ −α λ −α j \| τ, β * ≤ C (α, π loc ) \|β * j \| −α log 1 + 4τ 2 2 \|β * j \| 2 , where C (α, π loc ) > 0 is a constant depending only on α and π loc (·) given by C (α, π loc ) = 2 2+α/2 π loc ∞ π loc (0) ∞ 0 1 λ 1+α exp − 1 λ 2 dλ. Proof. Observe that E λ −α j τ, β * = ∞ 0 1 λ 1+α exp − c 2 j λ 2 π loc (λ)dλ ∞ 0 1 λ exp − c 2 j λ 2 π loc (λ)dλ, (C.10) where c j = c(τ, β j ) = \|β j \|/ √ 2τ . With the change of variable λ → λ/c j , we can write the right-hand side of (C.10) as 1 c α j ∞ 0 1 λ 1+α exp − 1 λ 2 π loc (c j λ) dλ ∞ 0 1 λ exp − 1 λ 2 π loc (c j λ)dλ. (C.11) We can upper bound the numerator as 1 c α j ∞ 0 1 λ 1+α exp − 1 λ 2 π loc (c j λ) dλ ≤ 1 c α j π loc ∞ ∞ 0 1 λ 1+α exp − 1 λ 2 dλ. (C.12) To lower bound the denominator, we restrict the range of integration to [0, /c j ] for > 0 and apply the change of variable φ = λ −2 : ∞ 0 1 λ exp − 1 λ 2 π loc (c j λ)dλ ≥ min [0, ] π loc /cj 0 1 λ exp − 1 λ 2 dλ = min [0, ] π loc ∞ c 2 j / 2 φ −1 exp(−φ) dφ. The inequality of Gautschi (1959) tells us that ∞ a φ −1 exp(−φ)dφ ≥ log(1 + 2a −1 )/2, so we obtain ∞ 0 1 λ exp − 1 λ 2 π loc (c j λ)dλ ≥ min [0, ] π loc 1 2 log 1 + 2 2 c 2 j . (C.13) From the upper bound (C.12) of the numerator and lower bound (C.13) of the denominator, it follows that the ratio (C.11) is upper bounded by c −α j 2 π loc ∞ min [0, ] π loc log 1 + 2 2 c −2 j ∞ 0 1 λ 1+α exp − 1 λ 2 dλ. Substituting c j = \|β j \|/ √ 2τ into the above expression completes the proof. Appendix D: Proof of Lemma 3.5 Our proof of Lemma 3.5 builds on the known fact below. Proposition D.1 (Choi and Hobert, 2013). For fixed τ and λ, the marginal transition kernel satisfies the minorization condition P (β \| β * , τ, λ) ≥ δ τ λ N (β; µ τ λ , Φ −1 τ λ ) where Φ τ λ = 1 2 X X + ζ −2 I + τ −2 Λ −2 , µ τ λ = Φ −1 τ λ X (y − 1/2), and δ τ λ = C n \|ζ −2 I + τ −2 Λ −2 \| 1/2 \|Φ τ λ \| 1/2 exp 1 2 w Φ −1 τ λ − ζ −2 I + τ −2 Λ −2 −1 w (D.1) for w = X (y − 1/2) and C n > 0 depending only on n. Proposition D.2 and D.3 below are the main workhorses for our proof of Lemma 3.5 along with Proposition D.1. We first state the results and use them to prove Lemma 3.5, before proceeding to prove the results themselves. Proposition D.2. As a function of τ λ, the minorization constant (D.1) is uniformly bounded below by a positive constant on the set min j τ λ j ≥ R > 0. Proposition D.3. If two precision matrices Φ and Φ satisfy Φ ≺ Φ , then a minorization N (β; µ, Φ −1 ) ≥ δ N (β; µ , Φ −1 ) holds for δ > 0 given by δ = inf β N (β; µ, Φ −1 ) N (β; µ , Φ −1 ) = \|Φ\| 1/2 \|Φ \| 1/2 exp − 1 2 (µ − µ) Φ (Φ − Φ) −1 (Φ µ − Φµ) − µ . (D.2) When the means take the form µ = Φ −1 w and µ = Φ −1 w, (D.2) simplifies to δ = \|Φ\| 1/2 \|Φ \| 1/2 exp 1 2 w (Φ −1 − Φ −1 )w ≥ \|Φ\| 1/2 \|Φ \| 1/2 . Proof of Lemma 3.5. On the set {λ : min j τ λ j ≥ R}, Proposition D.1 implies that P (β \| β * , τ, λ) ≥ min τ λj ≥R δ τ λ N (β; µ τ λ , Φ −1 τ λ ), where min τ λj ≥R δ τ λ is guaranteed to be strictly positive by Proposition D.2. We complete the proof by showing that the following inequality holds whenever min j τ λ j ≥ R: N (β; µ τ λ , Φ −1 τ λ ) ≥ \|Φ ∞ \| 1/2 \|Φ R \| 1/2 N (β; µ R , Φ −1 R ). (D.3) When min j τ λ j > R, we have R −2 − τ −2 λ −2 j > 0 and hence Φ R − Φ τ λ = (R −2 I − τ −2 Λ −2 ) 0. By Proposition D.3, it follows that N (β; µ τ λ , Φ −1 τ λ ) ≥ \|Φ τ λ \| 1/2 \|Φ R \| 1/2 N (β; µ R , Φ −1 R ). (D.4) The above inequality in fact holds not only on the set {λ : τ λ j > R} but also on the closure {λ : min j τ λ j ≥ R} since all the quantities depend continuously on τ λ j . The inequality (D.3) follows from (D.4) by observing that Φ τ λ Φ ∞ and hence \|Φ τ λ \| ≥ \|Φ ∞ \|. Proof of Proposition D.2 and D.3 In the proofs to follow, we will make use of the following elementary linear algebra facts about positive definite matrices. We will denote the largest, ith largest, and smallest eigenvalue of a matrix A as ν max (A), ν i (A), and ν min (A). The determinant of A is denoted by \|A\| and the trace by tr(A). The notation A ≺ B means that B − A is positive definite or, equivalently, v Av < v Bv for any vector v = 0. Proposition D.4. Given positive definite matrices A and B, we have 1. (A + B) −1 ≺ A −1 . 2. (A + B) −1 A −1 − A −1 BA −1 3. ν i (A) + ν min (B) ≤ ν i (A + B) ≤ ν i (A) + ν max (B) for all i. 4. \|A\| < \|A + B\|. 5. \|A + B\| ≤ \|A\| exp ν max (B) tr(A −1 ) . When A ≺ C for another positive definite matrix C, we can apply above results with B = C − A 0 to obtain analogous inequalities. Proof. The eigenvalues of I + B are given by 1 + ν i (B) and those of (I + B) −1 by 1/(1 + ν i (B)) < 1, so we have (I + B) −1 ≺ I. This result holds when B is replaced by A −1/2 BA −1/2 and thus implies that v (A + B) −1 v = v A −1/2 I + A −1/2 BA −1/2 −1 A −1/2 v < v A −1/2 A −1/2 v for v = 0. Hence we have (A + B) −1 < A −1 . To prove Property 2, we first show (I + B) −1 I − B. By applying a change of basis if necessary, we can assume that B is diagonal. Since (1 + B ii ) −1 > 1 − B ii , we have v (I + B) −1 v = i (1 + B ii ) −1 v 2 i > i (1 − B ii )v 2 i = v (I − B)v. Since the result (I + B) −1 I − B holds when B is replaced by A −1/2 BA −1/2 , we obtain (A + B) −1 = A −1/2 I + A −1/2 BA −1/2 −1 A −1/2 A −1/2 I − A −1/2 BA −1/2 A −1/2 = A −1 − A −1 BA −1 . Property 3 is Theorem 8.1.5 of Golub and Van Loan (2012) and immediately implies Property 4. For Property 5, observe that \|A + B\| = i ν i (A + B) ≤ i {ν i (A) + ν max (B)} . Taking the logarithm and applying the inequality log(1 + x) ≤ x, we have log \|A + B\| − log \|A\| ≤ i log 1 + ν max (B) ν i (A) ≤ i ν max (B) ν i (A) = ν max (B) tr(A −1 ). Proof of Proposition D.2. Throughout the proof, we use the notation Φ ∞ = 1 2 X X + ζ −2 I so that Φ τ λ = Φ ∞ + τ −2 Λ −2 . By Proposition D.4, we have ζ −2 I + τ −2 Λ −2 ≥ \|ζ −2 I\| \|Φ ∞ + τ −2 Λ −2 \| ≤ \|Φ ∞ \| exp max j τ −2 λ −2 j tr Φ −1 ∞ . The above inequalities imply that \|ζ −2 I + τ −2 Λ −2 \| 1/2 \|Φ\| 1/2 ≥ \|ζ −2 I\| \|Φ ∞ \| exp − 1 min j τ 2 λ 2 j tr Φ −1 ∞ . (D.5) Also by Proposition D.4, we have ζ −2 I + τ −2 Λ −2 −1 ≺ ζ 2 I Φ ∞ + τ −2 Λ −2 −1 Φ −1 ∞ − Φ −1 ∞ τ −2 Λ −2 Φ −1 ∞ . We therefore have w Φ −1 τ λ − ζ −2 I + τ −2 Λ −2 −1 w ≥ w Φ −1 ∞ w − w Φ −1 ∞ τ −2 Λ −2 Φ −1 ∞ w − ζ −2 w 2 ≥ w Φ −1 ∞ w − 1 min j τ 2 λ 2 j Φ −1 ∞ w 2 − ζ −2 w 2 . (D.6) From (D.5) and (D.6), we see that for all min j τ λ j ≥ R δ τ λ ≥ C n \|ζ −2 I\| 1/2 \|Φ ∞ \| 1/2 exp w Φ −1 ∞ w − ζ −2 w 2 − tr Φ −1 ∞ + Φ −1 ∞ w 2 R 2 . Proof of Proposition D.3. Note that inf β N (β; µ, Φ −1 ) N (β; µ , Φ −1 ) = \|Φ\| 1/2 \|Φ \| 1/2 exp 1 2 inf β ∆(β) , where ∆(β) = (β − µ ) Φ (β − µ ) − (β − µ) Φ(β − µ). The quadratic function ∆(β) has a unique global minimum since the Hessian ∂ 2 β ∆ = Φ − Φ is positive definite by our assumption. Differentiating ∆(β), we see that the minimum occurs atβ such that Φ (β − µ ) − Φ(β − µ) = 0, or equivalentlyβ = (Φ − Φ) −1 (Φ µ − Φµ) . The minimum ∆ = ∆(β) can be expressed as ∆ = −(µ − µ) Φ(β − µ) = −(µ − µ) Φ (Φ − Φ) −1 (Φ µ − Φµ) − µ . In the special case µ = Φ −1 w and µ = Φ −1 w, we have ∆ = −(µ − µ) Φµ = − Φ −1 w − Φ −1 w w = w Φ −1 − Φ −1 w ≥ 0, where the last inequality follows from Φ −1 Φ −1 . Appendix E: Proof of Proposition 3.10 and 3.11 Proof of Proposition 3.10. Winkelbauer (2012) tells us that a negative moment of Gaussian is given by E\|β\| −α = Γ 1−α 2 2 α/2 √ π σ −α M α 2 , 1 2 , − µ 2 2σ 2 , where M (·, ·, ·) is Kummer's confluent hypergeometric function (see Proposition E.1). To complete the proof, therefore, it suffices to show that M α 2 , 1 2 , − µ 2 2σ 2 is bounded by the smaller of 1 and the function D(µ/σ) as given in (3.14). Since α/2 < 1/2, Proposition E.1 tells us that M α 2 , 1 2 , − µ 2 2σ 2 is bounded by 1 and admits the integral representation M α 2 , 1 2 , − µ 2 2σ 2 = 1 B α 2 , 1−α 2 1 0 (1 − u) 1−α 2 −1 u α 2 −1 exp − µ 2 2σ 2 u du. (E.1) To bound the integral, we break up the domain of integration into [0, 1/2] and [1/2, 1] and observe that 1 1/2 (1 − u) 1−α 2 −1 u α 2 −1 exp − µ 2 2σ 2 u du ≤ 2 1− α 2 exp − µ 2 4σ 2 1 1/2 (1 − u) 1−α 2 −1 du = 2 5 2 −α 1 − α exp − µ 2 4σ 2 , (E.2) and that 1/2 0 (1 − u) 1−α 2 −1 u α 2 −1 exp − µ 2 2σ 2 u du ≤ 2 1− 1−α 2 1/2 0 u α 2 −1 exp − µ 2 2σ 2 u du Proof of Proposition 3.11. A conditional precision (in expectation) is always larger than the marginal one, so we have σ −2 j ≤ Σ −1 jj = ζ −2 + τ −2 λ −2 j + n i=1 ω i x 2 ij . Exponentiating both sides of the inequality, we obtain σ −α j ≤ ζ −2 + τ −2 λ −2 j + n i=1 ω i x 2 ij α/2 ≤ ζ −α + τ −α λ −α j + n i=1 ω i x 2 ij α/2 (E.8) ≤ ζ −α + τ −α λ −α j + 1 + α 2 n i=1 ω i x 2 ij − 1 , (E.9) where (E.8) follows from the property of L α -norm (\|a\| + \|b\|) α ≤ \|a\| α + \|b\| α and (E.9) from the Taylor expansion of the concave function x → x α at x = 1. Appendix F: Properties of Bayesian bridge prior Bayesian bridge is characterized by the density of β j \| τ given as π(β \| τ ) ∝ τ −1 exp(−\|β/τ \| a ). (F.1) We obtain the global-local representation of (F.1) with the conditional β \| τ, λ ∼ N (0, τ 2 λ 2 ) when π loc (λ) ∝ λ −2 π st (λ −2 /2), where π st (·) denote the density of the one-sided stable distribution, characterized by location µ = 0, skewness β = 1, characteristic exponent a/2, and scale c = cos(aπ/4) 2/a (Hofert, 2011). This follows from the Laplace transform identity for the stable distribution: exp(−\|β/τ \| a ) = 1 2 ∞ 0 exp(−φβ 2 /2τ 2 ) π st (φ/2) dφ ∝ ∞ 0 N (β; 0, τ 2 φ −1 ) π(φ) dφ, for π(φ) ∝ φ −1/2 π st (φ/2), the density of φ = λ −2 . We can characterize the behavior of π loc (λ) at λ ≈ 0 from the following asymptotic behavior of the stable distribution as x → 0 (Nolan, 2018). Polson et al. (2014). For the two different values of b = β 2 j /2τ 2 , the autocorrelations at stationarity are computed from 10,000 iterations of the sampler to demonstrate how the mixing rate degrades as b → 0. The task of sampling from the local scale posterior, therefore, boils down to that of sampling from the family of univariate densities π(η) ∝ 1 1 + η exp(−bη) for b > 0. (G.1) To sample from (G.1), the online supplement of Polson et al. (2014) describes a slice sampling approach and Makalic and Schmidt (2015) a data augmentation method. However, we find that both approaches suffer from slow-mixing as b → 0 and the slowdecaying term (1 + η) −1 becomes significant (Figure G.1 and G.2). G.1 Rejection sampler algorithm Our rejection sampler acts on a transformed parameter ψ = log(1 + η) that maps back as η = e ψ − 1. The density of ψ is given by π(ψ) ∝ π(η)\|dη/dψ\| = 1 e ψ exp(−be ψ )e ψ = exp(−be ψ ) on ψ ≥ 0. We now define a function g b that upper bounds the unnormalized target density Makalic and Schmidt (2015). The auto-correlations at stationarity are computed from 10,000 iterations of the sampler. For b ≥ 1, we set g b (ψ) = exp{−b(1 + ψ)}, which coincides with an unnormalized density of the distribution Exp(rate = b). For b < 1, we set g b (ψ) = exp(−b) for ψ ≤ log(1/b) exp{−1 − (ψ − log(1/b))} for ψ ≥ log(1/b) , which coincides with an unnormalized density of a mixture of Uniform(0, log(1/b)) and Exp(1) shifted by log(1/b). To draw a random variable X from this mixture, we set X ∼ Uniform(0, log(1/b)) with probability log(1/b) / log(1/b) + e b−1 and X − log(1/b) ∼ Exp(1) otherwise. R and Python code of the rejection sampler are available at https: //github.com/aki-nishimura/horseshoe-scale-sampler. G.2 Analysis of acceptance probability The acceptance probability of a rejection sampler is given by the ratio of the integrals of the target to the bounding density (Ripley, 2009). In particular, the rejection sampler described in Section G.1 has the acceptance probability Proof. We can show that both the denominator and numerator of (G.2) depend continuously on b, and so does A(b), by a simple application of the dominated convergence theorem. The continuity of A(b) implies a uniform lower bound on b ∈ (0, ∞) as soon as we establish A(b) → 1 towards the boundary b → 0 and b → ∞. A(b) = ∞ 0 f b (η) dη We establish a lower bound on the acceptance probability (G.2) by explicitly computing the denominator and then lower bounding the numerator. We first consider the case b ≥ 1, when the denominator is given by (G.5) From (G.3) and (G.5), we obtain the following lower bound on the acceptance probability, which holds for any L > 0: A(b) ≥ exp(−bL 2 e L ) 1 − e −bL . Choosing L = log(κb)/b with κ > 1, for example, we obtain the lower bound A(b) ≥ exp − (log κb) 2 b κ 1/b b 1/b 1 − 1 κb . (G.6) It is straightforward to show that, for example by the derivative test, the function b → b 1/b has the global maximum exp(e −1 ) on b > 0. We can therefore simplify the lower bound (G.6) to A(b) ≥ exp − exp(e −1 )κ 1/b (log κb) 2 b 1 − 1 κb . (G.7) The lower bound in (G.7), and hence A(b), converges to 1 as b → ∞. We now turn to establishing a lower bound on the acceptance probability in the case b < 1. We have not affect the posterior conditional of (τ, λ) as the parameters remains decoupled from ζ. Figure 2 . 1 : 21Directed acyclic graphical model (a.k.a. Bayesian network) representation of regularized shrinkage priors under the two alternative formulations. regularization, the Markov chain takes multiple "excursions" -each lasting over hundreds of iterations -into the unreasonable value range of the coefficients. The deviation in β172 is particularly prominent around the 42,000th iteration. More severe deviations may occur if the chain is run longer. regularization, the Markov chain does not display any serious mixing issues. The noticeable auto-correlation is due to the multi-modality of the posterior, an unavoidable feature of shrinkage models. Note that the coefficients with widest credible intervals do not coincide with the unregularized setting. Figure 4 . 1 : 41Traceplot under the Bayesian bridge logistic regression with exponent 1/16. Shown are the three coefficients with most potentially problematic mixing behaviors; see the main text for the details on our criteria. Figure 4 . 2 : 42Ten widest 95% posterior credible intervals under the Bayesian bridge logistic regression with (right) and without (left) regularization. Without regularization, the intervals are unrealistically large compared to the signal size of β j = 1 for j = 1, . . . , 10. Figure 4 . 3 : 43Traceplots under different slab widths: ζ = 2 (bottom) and ζ = 4 (top). The settings are otherwise identical to those of Figure 4.1. As before, the three coefficients with most problematic mixing behaviors do not always coincide across different slab widths. Figure 4 . 4 : 44The 95% posterior credible intervals for the signals β j = 1 (top) and nonsignals β j = 0 (bottom) under the Bayesian bridge logistic regression with the bridge exponent 1/16. The intervals are sorted by the posterior means. To avoid clutter, the top plot shows only the non-zero values of the lasso estimates. The lasso estimates for the non-signals are summarized in Figure 4.5 and are not shown in the bottom plot. Figure 4 . 5 : 45Comparison of the 4,900 Bayesian bridge posterior means and lasso estimates for the non-signals β j = 0. Lasso with cross-validation produces a larger number of false positives. Lasso with the empirical Bayes calibration yields the estimates more in line with the bridge posterior. Figure 4 . 6 : 46The 95% posterior credible intervals under the Bayesian bridge logistic regression with the bridge exponent 1/4. Compared with the 1/16 exponent case(Figure 4.4), the posterior distributions have similar means but much wider credible intervals. Figure 4 . 7 : 47Average width v.s. coverage of the credible intervals. The plots are produced by computing the equal-tailed credible intervals at a range of credible levels. The x-axis is in the log 10 scale for the non-signals. Figure G.1: Trace and auto-correlation plots when slice sampling η from (G.1) as proposed in Figure f b (ψ) := exp(−be ψ ). G.2: Trace and auto-correlation plots when sampling η from (G.1) with the data-augmentation scheme of Figure G. 3 : 3Acceptance probability of the proposed rejection sampler as a function of b = β 2 j /2τ 2 . The probability is uniformly lower-bounded and increases to 1 as b → 0 and b → ∞ (see Theorem G.1). The minimum probability is ≈ 0.6975. Figure G.2 plots the acceptance probability A(b), evaluated to high accuracy via numerical integration of the integrals in (G.2), and supports the theoretical results below.Theorem G.1. The acceptance probability A(b) is uniformly lower bounded over b > 0 by a positive constant. Moreover, A(b) converges to 1 as b → 0 and b → ∞. ψ)} dψ = b −1 e −b . (G.3)Then, using Taylor's theorem and the fact d 2 dψ 2 e ψ = e ψ , we have0 ≤ e ψ − (1 + ψ) ≤ ψ 2 max ψ ∈[0,ψ] e ψ = ψ 2 e ψ .The above inequality in particular implies thatf b (ψ) = exp(−be ψ ) ≥ exp{−b(1 + ψ)} exp(−bψ 2 e ψ ). (G.4)We now apply (G.4) to lower bound the numerator of (G.2); for any L > 0, −1 e −b exp(−bL 2 e L ) 1 − e −bL . bound f b (ψ) dψ, we first observe that, by the change of variable ψ = ψ/ log(1/b), interval ψ ∈ [0, 1), the integrand converges to 1 as b → 0 and hence the dominated convergence theorem implies C(b) → 1 as b → 0. On the interval ψ ∈ [log(1/b), ∞), last inequality follows from (G.5) with b = 1 and L = log(κ) for κ > 1. It follows from (G.8), (G.9), and (G.10) that for b < 1A(b) ≥ log(1/b) C(b) + e −1 C (κ) e −b log(1/b) + e −1 ,(G.11) Appendix G describes a simple and provably efficient rejection-sampler for the conditional distributions of local scale parameter λ j 's under the horseshoe prior. Despite the horseshoe's popularity, we find that no existing algorithm for the conditional update comes with theoretically guaranteed efficiency. While an appropriate choice of ζ is application specific, by way of illustration, we suggest ζ = 2 as a weakly informative and sensible starting point in biomedical applications with standardized predictors.Schuemie et al. (2018) surveys 59,196 published effect estimates in the observational study literature and finds only a small portion of them exceeds 2. The results presented in this article, specifically those that depend on Proposition C.2 and Lemma C.3, implicitly assume that π loc (λ) is absolutely continuous at λ min = inf {λ : π loc (λ) > 0}. This is a purely technical assumption as any shrinkage prior in practice should satisfy π loc (λ) > 0 for λ > 0 and be a differentiable function of λ. . Draw λ \| β * , τ from the density proportional to (2.3). AcknowledgmentsWe are indebted to Andrew Holbrook for the alliteration in the article title. This work was partially supported through National Institutes of Health grants R01 AI107034 and U19 AI135995 and through Food and Drug Administration grant HHS 75F40120D00039.Proof. Kummer's function can be represented as the following infinite series (Gradshteyn and Ryzhik 2014, Section 9.210):M (a, b, z) = 1 + a b z 1! + a(a + 1) b(b + 1) z 2 2! + a(a + 1)(a + 2) b(b + 1)(b + 2) z 3 3! + . . . .Since b > a > 0, the series representation immediately impliesby the identity (9.212.1) inGradshteyn and Ryzhik (2014). Since b > b − a > 0 and −z ≥ 0, we can apply our previous bound (E.6) to conclude that M (b − a, a, −z) ≤ exp(−z). Combined with (E.7), this yields M (a, b, z) ≤ 1 for z ≤ 0.The integral representation (E.4) is given in Section 9.211 ofGradshteyn and Ryzhik (2014). To obtain (E.5), we apply the change of variable v = (1 + u)/2:where ≈ 3.14159 is Archimedes' constant. In particular, we haveThe availability of the marginal π(β j \| τ ) = N (β j ; 0, τ 2 λ 2 j ) π loc (λ j ) dλ j allows for a Gibbs update of τ from the posterior with the local scale parameters λ j 's marginalized out. More precisely, instead of drawing from τ \| β, λ, the Bayesian bridge Gibbs sampler can directly target the conditionalSince β \| τ belongs to the location-scale family, the reference prior is π glo (τ ) ∝ τ −1(Berger et al., 2015), which also happens to be a conjugate prior. More generally, in terms of the parametrization φ = τ −α , a prior φ ∼ Gamma(shape = s, rate = r) belongs to a conjugate family, yielding the posterior conditional π(φ \| β) ∼ Gamma shape = s + p, rate = r + p j=1 \|β j \| .In the limit s, r → 0, the gamma prior on φ recovers the reference prior π glo (τ ) ∝ τ −1 which is invariant under reparametrization,Appendix G: Sampler for local scale posterior under horseshoe priorOur theoretical results on convergence rate assume the ability to sample independently from the conditionals λ j \| β j , τ for j = 1, . . . , p. While not necessarily trivial, this task is typically quite manageable given the wide range of algorithms available to deal with univariate distributions(Devroye, 2006;Ripley, 2009).As an illustration, we present a simple rejection sampler for the conditional λ j \| β j , τ under the prior π loc (λ j ) ∝ 1/(1 + λ 2 j ) -corresponding to the horseshoe prior, arguably the most popular of the existing shrinkage priors(Bhadra et al., 2017). The rejection sampler, as we will show, has uniformly high acceptance probability for all β j and τ with the minimum acceptance probability ≈ 0.6975 (Figure G.3). On the precision scale η j = λ −2 j , the prior is given by π loc (η j ) = π loc (λ j )\|dλ/dη j \| ∝ 1 1 + η −1 j η −3/2 j = 1 η 1/2 j (1 + η j ).The full conditional η j \| β j , τ has the density π(η j \| β j , τ ) ∝ π loc (η j ) π(β j \| τ, η j ) ∝ 1 1 + η j exp −η j β 2 j 2τ 2 . Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 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Electronic Journal of Statistics, 12(2): 3295-3311. 5, 11 Moments and absolute moments of the normal distribution. A Winkelbauer, arXiv:1209.43402736Winkelbauer, A. (2012). "Moments and absolute moments of the normal distribution." arXiv:1209.4340 . 27, 36 where lim b→0 C(b) = 1 and C (κ) ≈ 0.264 for κ = 1.57. The lower bound in (G.11), and hence A(b). converges to 1 as b → 0where lim b→0 C(b) = 1 and C (κ) ≈ 0.264 for κ = 1.57. The lower bound in (G.11), and hence A(b), converges to 1 as b → 0.	[ "https://github.com/aki-nishimura/bayes-bridge." ]
[ "The unramified Brauer group of homogeneous spaces with finite stabilizer", "The unramified Brauer group of homogeneous spaces with finite stabilizer" ]	[ "Giancarlo Lucchini Arteche [email protected] \nDepartamento de Matemáticas\nFacultad de Ciencias\nUniversidad de Chile Las Palmeras 3425\nÑuñoa, SantiagoChile\n" ]	[ "Departamento de Matemáticas\nFacultad de Ciencias\nUniversidad de Chile Las Palmeras 3425\nÑuñoa, SantiagoChile" ]	[]	We give formulas for calculating the unramified Brauer group of a homogeneous space X of a semisimple simply connected group G with finite geometric stabilizer F over a wide family of fields of characteristic 0. When k is a number field, we use these formulas in order to study the Brauer-Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on X(k v ) for every v outside from an explicit finite set of non archimedean places of k.	10.1090/tran/7796	[ "https://arxiv.org/pdf/1709.01170v3.pdf" ]	119,578,141	1709.01170	8966361b52fb1ba7b15758987ab507702af9f7ba	The unramified Brauer group of homogeneous spaces with finite stabilizer Sep 2017 Giancarlo Lucchini Arteche [email protected] Departamento de Matemáticas Facultad de Ciencias Universidad de Chile Las Palmeras 3425 Ñuñoa, SantiagoChile The unramified Brauer group of homogeneous spaces with finite stabilizer Sep 2017Brauer grouphomogeneous spacesfinite groupsrational points MSC classes: 14F2214M1714G2014G25 We give formulas for calculating the unramified Brauer group of a homogeneous space X of a semisimple simply connected group G with finite geometric stabilizer F over a wide family of fields of characteristic 0. When k is a number field, we use these formulas in order to study the Brauer-Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on X(k v ) for every v outside from an explicit finite set of non archimedean places of k. Introduction The Brauer group Br X of a (smooth, geometrically integral) variety X over a field k is defined as the second étale cohomology group H 2 (X, G m ). A variant of this group is the unramified Brauer group Br un X ⊂ Br X. The latter has the advantage of being a stably birational invariant (and as such, it admits a definition that only depends on the fields k(X) and k, cf. [CT95]). Moreover, it corresponds to the Brauer group of a smooth compactification of X and as such it can be used for the definition of the Brauer-Manin obstruction to the Hasse principle and weak approximation. Since these arithmetic properties and the corresponding obstruction involves both global and local fields, it is important to develop formulas for Br un X over arbitrary fields. In this paper we are interested in the unramified Brauer group of homogeneous spaces of (connected) linear groups. In this context, when X is a homogeneous space of a linear group G with connected or abelian stabilizer (in general, of "ssumult" type), Borovoi, Demarche and Harari have given formulas for the "algebraic part" of Br un X in [BDH13]. The same part in the case of arbitrary stabilizers was covered by the author in [LA15], but only for a semisimple simply connected group G. It is worth noting that, for connected and abelian stabilizers, the algebraic part corresponds to the whole group Br un X (i.e. there is no "transcendental part"). This is not true anymore for general non connected stabilizers, in particular for finite stabilizers, as it was proven over k = C by Bogomolov in [Bog87]. Passing then to the whole group Br un X is a completely different task and it is only recently that Colliot-Thélène managed to do this in [CT14] for homogeneous spaces of G = SL n with finite constant stabilizer (in particular X(k) = ∅) by generalizing Bogomolov's method. In this paper, we use the same methods in order to generalize this result in three directions: • we replace SL n with a semisimple simply connected group (although this is a simple application of a remark by Colliot-Thélène himself, cf. the appendix of [LA15]); • we do not assume the finite stabilizer to be constant; • we do not even assume that there are k-points in X. The price we have to pay for such a generalization is a small restriction on the base field: namely we have to avoid a certain family of fields which, for lack of a better term, we have named essentially real fields. These include all real-closed fields, but not much more, cf. section 2.1. In particular, all global fields and non-archimedean local fields, for which we want to find arithmetic applications, are considered in our formulas. Moreover, the general intermediate results in this paper are enough to get the expected application for the field of real numbers. The structure of the article is as follows. In section 2 we fix notations, give the definition of essentially real fields and we study some subgroups of a finite gerb which are necessary to state the main formulas. In section 3 we interpret the group Br X in terms of group cohomology using a natural gerb that is associated to any homogeneous space X (which is nothing but the fundamental group of X). We also reinterpret the Brauer-Manin evaluation pairing in terms of restrictions in group cohomology. In section 4 we study the particular case of finite abelian stabilizers, for which we prove the triviality of Br un X under some extra hypotheses. This is the heart of Bogomolov and Colliot-Thélène's method. We finally get to the main formulas in section 5 and give an arithmetic application in section 6, which consists in giving an explicit set of bad places away from which the Brauer-Manin obstruction does not detect anything. Notations and preliminaries All throughout the text, k denotes a field of characteristic 0. We denotek an algebraic closure of k and Γ k the absolute Galois group Gal(k/k). For L/k any other Galois extension, we denote the corresponding Galois group by Γ L/k . For X a k-variety, we denote byX thek-variety X × kk . In particular, we will be denoting everyk-variety by a letter with a bar. We will denote by G a semisimple simply connected group and by F (resp.F ) a finite k-group (resp.k-group). For such a finite group, we will make the following abuse of notation: we will always identify the finite algebraic group with the abstract group of itsk-points (eventually with an action of Γ k ), thus avoiding everywhere the notation F (k) orF (k). We think that the context will be clear enough for the reader to see whether we mean the group or the scheme. By a procyclic group, we mean a commutative profinite group which is topologically generated by a single element. If Γ is procyclic, then one can always canonically write Γ ∼ = p Γ p , where p ranges through the set of all prime numbers and Γ p is a procyclic pro-p-group, hence a quotient of (Z p , +). By a strictly procyclic group, we mean a procyclic group Γ such that each one of its p-Sylow subgroups Γ p is either trivial or infinite (and hence isomorphic to Z p ). Essentially real fields Recall that, by the Artin-Schreier Theorem, an absolute Galois group is finite if and only if it is isomorphic to Z/2Z. In particular, a finite closed subgroup of an arbitrary Galois group must be isomorphic to Z/2Z. Definition 2.1. We will say that a field k is essentially real if the 2-Sylow subgroups of Γ k are finite and non trivial (so a fortiori of order 2). Examples of essentially real fields are real closed fields (including R), but also fields like k = n∈N R((t 1 2 n )). This last field has L = n∈N C((t 1 2 n )) as a Galois extension of order 2, and sincek = n∈N C((t 1 n )), it is easy to see that Γ L ⊂ Γ k is isomorphic to the product p =2 Z p , so in particular it has trivial 2-Sylow subgroups, hence Γ k has 2-Sylow subgroups of order 2. These fields happen to be the only ones for which our techniques for calculating the unramified Brauer group of a homogeneous space do not work. Note however that every such field has a unique quadratic extension for which our formulas work, so a simple restriction-corestriction argument tells us that we can calculate the prime-to-2 part of the unramified Brauer group for these fields as well. Moreover, aside from the real numbers, there seem to be no other interesting fields if one has arithmetic applications in mind. And we can use some of the results here below that are independent of the field to get the expected applications in the real case, cf. section 6. Subgroups of finite gerbs Let k be a field of characteristic 0,F be a finitek-group. A (F , k)-gerb (or simply a finite gerb) in this context is just a group extension 1 →F ι − → E π − → Γ k → 1, such that π is continuous. Note that in particular E is a profinite group. Following Springer, one can naturally associate a finite gerb to every homogeneous space with finite stabilizer (cf. [Spr66]). We will do the same thing here below by considering the étale fundamental group. Then, in order to calculate the unramified Bruaer group of the homogeneous space, we will consider certain families of subgroups of a given gerb E, defined as follows. Recall that a strictly procyclic group is a procyclic group such that each one of its p-Sylow subgroups is either trivial or infinite. Definition 2.2. Let E be a (F , k)-gerb as above and let A be a (discrete) E-module. For x ∈ {ab, bic, cyc} and y ∈ {scyc, 0}, we denote by (E) x y the set of closed subgroups D of E such that • ι −1 (D) = D ∩F is abelian (x = ab), resp. bicyclic (x = bic), resp. cyclic (x = cyc); • π(D) is strictly procyclic (y = scyc), resp. trivial (y = 0). We also set, for i ≥ 1: X i x,y (E, A) := Ker   H i (E, A) → D∈x(E)y H i (D, A)   . For example, X 2 bic,scyc (E, A) is the subgroup of H 2 (E, A) of elements that are trivialized by restriction to every closed subgroup D of E such that π(D) is strictly procyclic and D ∩F is bicyclic; while X 2 ab,0 (E, A) is the subgroup of elements that are trivialized by restriction to every closed abelian subgroup D ofF (seen as a subgroup of E). Brauer group and Brauer pairing in group cohomology Let X be a homogeneous space of a semisimple simply connected group G with finite geometric stabilizerF . We claim that there is a canonical (F , k)-gerbF X associated to X, i.e. a group extension 1 →F →F X → Γ k → 1,(1) In fact, such a group can be obtained by taking the étale fundamental group of X. Consider the Galois coverX → X and fix a geometric pointx ofX. Then we get an isomorphismX ∼ =F \Ḡ, whereF is thek-group stabilizingx. In particular,Ḡ →X is anF -torsor and hence a Galois cover. SinceḠ is simply connected, we know by the theory of étale covers (see [SGA1, Exp. V]) that: • the group π 1 (X,x) corresponds to the automorphism group of the coverḠ → X; • the groupF can be seen as a subgroup of π 1 (X,x), corresponding to those automorphisms fixingX; • sinceX → X is Galois, the subgroupF is normal in π 1 (X,x); • the quotient of π 1 (X,x) byF is isomorphic to Γ k , the group of automorphisms of X → X. For such a homogeneous space, we define thenF X to be the group π 1 (X,x), which fits in the group extension (1) and hence corresponds to a finite gerb. It is well-known that the isomorphism class of such a group does not depend on the choice ofx and it is hence canonically associated to X. Note that if X(k) = ∅, then we may consider the étale cover G → X defined by sending g ∈ G to xg ∈ X for some x ∈ X(k). This is in fact an F -torsor, where F is the stabilizer of x and hence a k-group. Once again, by the theory of étale covers, G → X corresponds to a subgroup of F X which is easily seen to correspond to a splitting of sequence (1). As such, it defines an action of Γ k on F by conjugation in F X which turns out to be the natural action of Γ on the geometric points of F as a k-group. In other words, if X(k) = ∅, the natural extension (1) associated to X is the semidirect product of Γ k and F for the natural action of Γ k on the stabilizer F of a k-point. Remarks. 1. The class of the extension we have just presented is the Springer class associated to the homogeneous space X in the nonabelian 2-cohomology set H 2 (k/k,F rel G), as defined by Springer in [Spr66]. Elements in this set are gerbs (which justifies our use of the word), as introduced by Giraud in his theory of nonabelian cohomology, cf. [Gir71]. 2. Note that the use of either the étale fundamental group or the Springer class can be avoided in the case where X(k) = ∅, where one can just put F X := F ⋊ Γ k by definition. The need for these nonabelian tools appears in the case where we do not have a rational point, case in which (to our knowledge) no formula for calculating the unramified Brauer group of a homogeneous space with finite stabilizer had been given until now, with the exception of [Dem10] for the algebraic part Br un,al X in the particular case where k is a number field and G = SL n . The proposition below is also a simple consequence of the theory of étale covers as explained in [SGA1, Exp. V]. Proposition 3.1. Let k, G and X be as above. Let E be a closed subgroup ofF X and denoteH = E ∩F . Denote by L the subfield ofk fixed by the image of E in Γ k . Then the evidentk-morphismH\Ḡ →X =F \Ḡ descends to a G L -equivariant L-morphism Y → X L , where Y is a right homogeneous space of G L with geometric stabilizerH. Moreover, one has E ∼ =H Y .⌣ We will use the gerbF X to study the unramified Brauer group Br un X as follows. Proposition 3.2. Let k, X and G be as above and letF X be the canonical gerb associated to X. Then Br X = H 2 (F X ,k * ), whereF X acts onk * via its quotient Γ k in the obvious way. Moreover, for any algebraic extension L/k and any homogeneous space Y of G L with finite stabilizerH equipped with a G L -equivariant map Y → X L , one has a commutative diagram Br un X / / Res H 2 (F X ,k * ) Res Br un Y / / H 2 (H Y ,k * ). Proof. By construction, we know that the arrowḠ → X is anF X -cover. We have the right then to consider the Hochschild-Serre spectral sequence in étale cohomology: H p (F X , H q (Ḡ, G m )) ⇒ H p+q (X, G m ). Since Pic (Ḡ) = 0 andk[X] * =k * , we deduce from this spectral sequence the following exact sequence 0 → H 2 (F X ,k * ) → Br X → BrḠ, where we note thatF X acts onk * via its quotient Γ k in the obvious way. Now, by [Gil09,§0], we know that BrḠ is trivial for semisimple simply connected groups. This gives the equality Br X = H 2 (F X ,k * ). Finally, the unramified Brauer group is functorial for dominant morphisms (cf. [CTS07, Lem. 5.5]) and the spectral sequence is compatible with restrictions. This, along with Proposition 3.2, proves the commutativity of the diagram.⌣ Let us study now how the classic Brauer-Manin pairing Br X × X(k) → Br k, behaves with respect to this new interpretation of Br X. Consider a point x ∈ X(k). Then, as we remarked before, we get a homomorphic section s x : Γ k →F X , so thatF X becomes a semi-direct productF ⋊ Γ k for the action of Γ k on the stabilizer of x, which is a k-form ofF that we note F x . Such a point allows us moreover to see X as a quotient G/F x , hence we can consider the coboundary morphism δ x : X(k) → H 1 (k, F x ). Now, it is well-known that the set H 1 (k, F x ) classifies, up to conjugation, the continuous sections Γ k →F X . Then any point y ∈ X(k) gives us as the same time a new homomorphic section s y ofF X and a class δ x (y) ∈ H 1 (k, F x ). The following result is then an easy exercise: Lemma 3.3. Let y ∈ X(k). Then the class in H 1 (k, F x ) corresponding to the homomorphic section s y is δ x (y).⌣ Thus for y ∈ X(k), we obtain the following restriction morphism H 2 (F X ,k * ) s * y − → H 2 (Γ k ,k * ) = Br k, which only depends on the class δ x (y) ∈ H 1 (k, F x ): indeed, any other point having the same image by δ x corresponds to a section that is conjugate to s y and conjugation induces the identity on H 2 (F X ,k * ). We have thus found compatible pairings: H 2 (F X ,k * ) × X(k) / / δx Br k H 2 (F X ,k * ) × H 1 (k, F x ) / / Br k,(2) which are easily seen to be functorial in k. Proof. We may assume that X(k) = ∅ (otherwise there is nothing to prove). In particular, we may assume that X = G/F for some finite k-group F (once again, this amounts to fixing a point in X(k)) and henceF X = F X is a semi-direct product as above. Consider now, for x ∈ X(k), the cartesian commutative diagram P / / Ḡ P / / G Spec k x / / X, whereP is the trivial F -torsor overk, i.e. copies of Speck permutated by F . The diagram induces a morphism of Hochschild-Serre spectral sequences H p (F X , H q (Ḡ, G m )) + 3 H p+q (X, G m ) x * H p (F X , H q (P , G m )) + 3 H p+q (k, G m ), which gives us in particular the following commutative square H 2 (F X , H 0 (Ḡ, G m )) / / H 2 (X, G m ) x * H 2 (F X , H 0 (P , G m )) / / H 2 (k, G m ). The Brauer-Manin pairing for the group H 2 (F X ,k * ) with respect to the point x corresponds to passing through the upper part of the diagram, whereas the pairing in (2) corresponds to passing through the lower part. Indeed, this part of the diagram can be restated as H 2 (F X ,k * ) → H 2 (F X , (k * ) ⊕n ) → Br k, where n denotes the order of F and where the left arrow is defined by the diagonal inclusion. A direct computation shows that the F X -module in the middle term is the induced module Ind sx(Γ k ) F X (k * ) and hence the middle term is just H 2 (s x (Γ k ),k * ) by Shapiro's Lemma. This proves that the composition of both arrows corresponds to the restriction with respect to the subgroup s x (Γ k ) of F X .⌣ The case of abelian stabilizers Our formula for the unramified Brauer group of a homogeneous space with finite stabilizer is inspired by Colliot-Thélène's formula in the constant case (cf. [CT14,thm. 5.5]), where he emulates Bogomolov's work (which was over C) over a non algebraically closed field. In particular, the formula uses the triviality of the unramified Brauer group of homogeneous spaces with constant abelian stabilizer. In order to generalize the formula to the non-constant case, we also need to generalize the result on triviality to non-constant abelian stabilizers. Let k be a field of characteristic 0, G be a semisimple simply connected k-group and X a homogeneous space of G with finite abelian stabilizerĀ. We get then as before the finite gerb 1 →Ā →Ā X → Γ k → 1, which gives us a natural action of Γ k onĀ by conjugation inĀ X (which is well defined sinceĀ is abelian). We get then a natural k-form A ofĀ associated to X. Proposition 4.1. Let k be a field of characteristic 0, G be a semisimple simply connected k-group and X a homogeneous space of G with finite abelian stabilizer. Let A denote the natural k-form of the stabilizer, L/k be an extension splitting A and n be the exponent of A. Assume that for every p r dividing n with p a prime number, the extension L(µ p r )/k is cyclic. Then Br un X = Br 0 X. Remark. Note that if L = k the last hypothesis is automatically satisfied for odd p, so that it is only a condition on the ciclicity of the extension k(µ 2 r )/k. As such, this is a generalization of Colliot-Thélène's condition Cyc(G, k) for a constant k-group G, which is precisely the particular case of L = k, see [CT14]. Proof. Since Br un X is a torsion group, it will suffice to prove the proposition for Br un X{p} for a given prime number p. Consider then the maximal subextension k ⊂ L p ⊂ L such that [L p : k] is prime to p (note that L/k is necessarily cyclic). By the classic restrictioncorestriction argument, we may assume that k = L p and hence that Γ L/k is a finite cyclic p-group. Consider now the p-Sylow subgroup S of A. Since the group H 2 (k, A) from which A X comes from splits into the product of its Sylow subgroups, we know that there exists a commutative diagram of extensions 1 / / S / / E / / Γ k / / 1 1 / / A / /Ā X / / Γ k / / 1. In particular, we see that E corresponds, on one side, to a closed subgroup ofĀ X of index prime to p, and on the other side, by Proposition 3.1, to the groupS Y for a homogeneous space Y of G with geometric stabilizerS lying above X. Proposition 3.2 tells us then that the restriction of Br un X{p} to H 2 (S Y ,k * ) is contained in Br un Y {p} and once again the classic restriction-corestriction argument tells us that we may reduce to the case where X = Y , i.e. we may assume that A is an abelian p-group split by a cyclic p-primary extension. Since A is abelian, we get that Br unX = 0 by [LA15, Prop. 26] and [CT14, Prop. 3.2], hence any element α ∈ Br un X is algebraic. By [BDH13, Thm. 8.1], we know then that the imageᾱ of α in Br un X/Br 0 X is in X 1 cyc (k,Â) ⊂ H 1 (k,Â), whereÂ = Hom(A, µ p r ), p r is the exponent of A and X 1 cyc denotes the elements that are trivialized by restriction to every procyclic subgroup of Γ k . Now, the inflation-restriction sequence for L(µ p r )/k reads 1 → H 1 (L(µ p r )/k,Â) → H 1 (k,Â) → H 1 (L(µ p r ),Â), andᾱ ∈ H 1 (k,Â) is clearly sent to X 1 cyc (L(µ p r ),Â). SinceÂ is constant over L(µ p r ), the term on the right corresponds to homomorphisms Γ L(µ p r ) →Â, hence X 1 cyc (L(µ p r ),Â) is trivial andᾱ comes from H 1 (L(µ p r )/k,Â). But since L(µ p r )/k is cyclic by condition 2, there exists a procyclic subgroup C of Γ k surjecting onto Γ L(µ p r )/k , so that we have the following commutative diagram H 1 (L(µ p r )/k,Â) 1 Inf / / # w Inf ) ) ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ H 1 (k,Â) Res H 1 (C,Â). Since the restriction ofᾱ to C is trivial, we deduce thatᾱ is trivial, hence α ∈ Br 0 X, which concludes the proof.⌣ Remark. Using the same kind of restriction-corestriction argument one can prove a similar result for homogeneous spaces with meta-abelian stabilizer (i.e. whose Sylow subgroups are abelian), at least under the hypothesis of the existence of a rational point. In particular, the formulas appearing in next section could also be stated with meta-abelian subgroups of the stabilizer. General formulas for Br un X Let k be a field of characteristic 0, G a semisimple simply connected k-group and X a homogeneous space of G with finite geometric stabilizerF . LetF X be the finite gerb canonically associated to X, cf. section 3. Recall that Proposition 3.2 gives us an isomorphism Br X ∼ − → H 2 (F X ,k * ). Moreover, if we denote by µ the subgroup ofk * consisting of all the roots of unity, then we have an exact sequence 1 → µ →k * →k * /µ → 1, in which the groupk * /µ is a uniquely divisibleF X -module, hence cohomologically trivial. This means that H 2 (F X , µ) = H 2 (F X ,k * ) and hence we can replacek * by µ on the statement here below. Theorem 5.1. Let k be a field that is not essentially real (i.e. the 2-Sylows of Γ k are either infinite or trivial). Let G, X,F be as above. Then the following subgroups of H 2 (F X ,k * ) coincide and correspond to Br un X: • X 2 ab,scyc (F X ,k * ); • X 2 bic,scyc (F X ,k * ); • X 2 cyc,scyc (F X ,k * ) ∩ X 2 ab,0 (F X ,k * ); • X 2 cyc,scyc (F X ,k * ) ∩ X 2 bic,0 (F X ,k * ). Remark. In [CT14, Thm. 5.5], Colliot-Thélène proves that in the case where the stabilizer of a k-point is a constant k-group F (i.e. whenF X =F × Γ k ) the normalized unramified Brauer group of X is isomorphic to some unspecified subgroup of X 2 ab (F , k * ). Using the inflation arrow H 2 (F , k) → H 2 (F X ,k * ), which is easily seen to be injective in this particular case, we see that X 2 ab (F , k * ) falls into X 2 ab,0 (F X ,k * ). Then Theorem 5.1 gives a first description of this subgroup: it is the intersection with X 2 cyc,scyc (F X ,k * ). Proof of Theorem 5.1. Note that the following inclusions are evident X 2 ab,scyc (F X ,k * ) 1 / / X 2 bic,scyc (F X ,k * ) X 2 cyc,scyc (F X ,k * ) ∩ X 2 ab,0 (F X ,k * ) 1 / / X 2 cyc,scyc (F X ,k * ) ∩ X 2 bic,0 (F X ,k * ). Let us then prove first the inclusion Br un X ⊂ X 2 ab,scyc (F X ,k * ). Let α ∈ Br un X. Consider a closed subgroup E ofF X in (F X ) ab scyc . By Proposition 3.1, we know that E =H Y for some homogeneous space Y of G L whose geometric stabilizer H = E ∩F is abelian, where L is an extension of k such that Γ L is strictly procyclic. We have then that the restriction α E of α to H 2 (E,k * ) falls into Br un Y by Proposition 3.2. Now, Proposition 4.1 tells us that Br un Y = Br 0 Y . But since Γ L is strictly procyclic, the field L is of cohomological dimension ≤ 1 and hence Br L = 0. 1 Thus α E is trivial for all such E and α ∈ X 2 ab,scyc (F X ,k * ) as claimed. Let us prove now the inclusion X 2 cyc,scyc (F X ,k * ) ∩ X 2 bic,0 (F X ,k * ) ⊂ Br un X. Let α ∈ H 2 (F X ,k * ) ⊂ Br X and assume that α ∈ Br un X. We need to show that there exists a subgroup E ofF X in either (F X ) bic 0 or (F X ) cyc scyc such that the restriction α E ∈ H 2 (E,k * ) is non trivial. Recall that the Brauer group of X is embedded in Br k(X), where k(X) denotes the function field of X. Let then A ⊂ k(X) be a discrete valuation ring of rank one with fraction field k(X), residue field κ A ⊃ k and such that ∂ A (α) = 0, where ∂ A : Br k(X) → H 1 (κ A , Q/Z), is the residue map (cf. [CTS07, Def. 5.2]). We will follow Bogomolov's argument as it is presented in [CTS07, Thm. 6.1]. In particular, the facts that follow come from [Ser79, I.7]. 2 LetÃ be the integral closure of A ink(Ḡ). It is a semi-local Dedekind ring. We can choose then a prime ideal and consider the localization B ofÃ in it, which is also a discrete valuation ring with residue field κ B ⊃k. Let D ⊂F X be the associated decomposition subgroup and I ⊂ D the corresponding inertia group, that is, the kernel of the surjective morphism D → Gal(κ B /κ A ). Since k ⊂ κ A andk ⊂ κ B , it is easy to see that I is contained inF . The group I is thus finite and, by [Ser79, IV.2], we deduce that it is cyclic and central in D ∩F since the fieldk(Ḡ)F =k(X) contains all the roots of unity. On the other hand, one easily sees that D is of finite index inF X , hence its 2-Sylow subgroups are either infinite or contained inF by our hypothesis on Γ k . Denote by α I the restriction of α to H 2 (I,k * ) and similarly for D. If α I = 0, then we are done since I is cyclic and hence clearly belongs to (F X ) bic 0 . Assume then that α I = 0. Consider the tower of discrete valuation rings A ⊂ B D ⊂ B I ⊂ B with respective fraction fields k(X) ⊂k(Ḡ) D ⊂k(Ḡ) I ⊂k(Ḡ) and respective residue fields κ A = κ A ⊂ κ B = κ B . By [CT95, Prop. 3.3.1], we have the following commutative diagram Br k(X) / / ∂ A Brk(Ḡ) D / / ∂ B D Brk(Ḡ) I ∂ B I H 1 (κ A , Q/Z) H 1 (κ A , Q/Z) Res / / H 1 (κ B , Q/Z), with ∂ A (α) = 0 and ∂ B I (α I ) = 0. Now, recall that H 1 (κ, Q/Z) = Hom(Γ κ , Q/Z), hence there exists an elementf ∈ Gal(κ B /κ A ) ∼ = D/I such that ∂ A (α)(f ) = 0. Take a preimage f ∈ D and consider the closed procyclic subgroup C := f ⊂ D ⊂F X . We claim that we may choose f so that either C ⊂F or C ∩F = {1}. Assuming the claim, put E := I, C ⊂ D and insert Brk(G) E in the diagram above. We verify then that ∂ B E (α E ) = 0, which implies that α E = 0. Now, since I is normal in D and central in D ∩F , one easily deduces that either E ∈ (F X ) bic 0 or E ∈ (F X ) cyc scyc depending on the choice above. Given the trivial inclusions stated at the beginning, this proves the theorem. We give now a proof of the claim in order to finish. The group C splits into a direct product of pro-p-groups C p such that: • if C p is infinite, then C p ∩F = {1}; • if C p is finite, then C p ⊂F , except maybe if p = 2. Indeed, if C p is infinite, then C p ∼ = Z p and the intersection C p ∩F corresponds to a finite subgroup of Z p , hence it is trivial. On the other hand, if C p is finite, then its image in the quotient Γ k is finite and hence trivial if p = 2 by the Artin-Schreier theorem. Since ∂ A (α) is non trivial on C, it must be non trivial on one of the C p , so that up to changing f by a generator of such a C p we are done, unless the only such prime is p = 2 and C 2 is finite. In that case, either the 2-Sylow of D is contained inF and hence so is C 2 and we are done, or else the 2-Sylow of D is infinite. In this last case, we can modify f by a 2-primary element of infinite order in the kernel of ∂ A (α), so that we get an infinite C 2 , which allows us to conclude.⌣ We now give a lemma that will be useful for the applications in section 6. Recall that, given x ∈ X(k), we may normalize the Brauer group Br X by considering the subgroup of elements that are trivialized by evaluation at x, that is, we may define Br x X := Ker[Br X x * − → Br k], and define the normalized unramified Brauer group Br x un X as the intersection of Br x X with Br un X. Lemma 5.2. Let k be a field of characteristic 0 that is not essentially real, G a semisimple simply connected k-group, F ⊂ G a finite k-subgroup of order n and X the khomogeneous space G/F . Let x ∈ X(k) be the point corresponding to the subgroup F , so that the natural section Γ k → F X = F ⋊ Γ k is s x . Finally, let µ{n} denote the group generated by the p-primary roots of unity for every p dividing n and let L/k be a Galois extension splitting F and containing µ{n}. Then the subgroup s x (Γ L ) is normal in F X and every α in the normalized unramified Brauer group Br x un X ⊂ H 2 (F X ,k * ) can be lifted to the group H 2 (F ⋊ Γ L/k , µ{n}) via the composite map H 2 (F ⋊ Γ L/k , µ{n}) Inf − → H 2 (F X , µ{n}) → H 2 (F X ,k * ), where F ⋊ Γ L/k is the quotient of F X by s x (Γ L ). Proof. The fact that s x (Γ L ) is normal comes from the fact that Γ L is normal in Γ k and acts trivially on F by definition. Then the subgroup F ⋊ Γ L of F X is actually a direct product and the normality of s x (Γ L ) follows. It is easy to see then that the quotient is F ⋊ Γ L/k . Concerning the second assertion of the lemma, note that, by definition, α ∈ Br x un X ⊂ H 2 (F X ,k * ) is trivial when restricted to the subgroup s x (Γ k ) of F X . Since this subgroup is of index n, the classic restriction-corestriction argument tells us that α is an n-torsion element. Consider then the short exact sequence 1 → µ{n} →k * →k * /µ{n} → 1, and note that the quotient is a uniquely p-divisible group for every p dividing n. This means that multiplication by p is an automorphism of H i (F X ,k * /µ{n}) for every i ≥ 1. In particular, these groups have no p-primary torsion, which tells us that α can be uniquely lifted to an n-torsion element of H 2 (F X , µ{n}) and so it goes for any restriction of α to any closed subgroup of F X . We will thus make an abuse of notation and still call α the element in H 2 (F X , µ{n}). Consider now the Hochschild-Serre spectral sequence E p,q 2 = H p (F ⋊ Γ L/k , H q (L, µ{n})) ⇒ H p+q (F X , µ{n}) = E p+q , associated to the short exact sequence 1 → Γ L sx − → F X → F ⋊ Γ L/k → 1. Since L contains µ{n}, Γ L acts trivially on µ{n} and hence the map E 2,0 2 → E 2 corresponds to the inflation map of the statement of the theorem. In order to prove the lemma, we have to prove then that α ∈ E 2 maps to 0 in E 0,2 2 = H 2 (L, µ{n}) F ⋊Γ L/k and then to 0 in E 1,1 2 = H 1 (F ⋊ Γ L/k , H 1 (L, µ{n})). Now, note that the map E 2 → E 0,2 2 is nothing but the restriction map H 2 (F X , µ{n}) s * x − → H 2 (Γ L , µ{n}), associated to the inclusion s x : Γ L → F X . Now, it is by hypothesis that α is trivialized by restriction to s x (Γ k ) ⊃ s x (Γ L ), so α is indeed trivial in E 0,2 2 . For the second arrow, note that the section s x is well defined on the quotient F ⋊ Γ L/k and the Hochschild-Serre spectral sequence is compatible with restrictions, so that we have a commutative diagram Ker(H 2 (F X , µ{n}) → H 2 (L, µ{n})) s * x / / H 1 (F ⋊ Γ L/k , H 1 (L, µ{n})) s * x Ker(H 2 (k, µ{n}) → H 2 (L, µ{n})) / / H 1 (L/k, H 1 (L, µ{n})), which tells us that the image of α in H 1 (L/k, H 1 (L, µ{n})) is trivial because the restriction of α to H 2 (k, µ{n}) is. Now, the split exact sequence 1 / / F / / F ⋊ Γ L/k / / Γ L/k / / sx v v 1, induces an inflation-restriction exact sequence with a retraction H 1 (L, µ{n})), which tells us that H 1 (L/k, H 1 (L, µ{n})) is a direct factor of H 1 (F ⋊Γ L/k , H 1 (L, µ{n})) and since the restriction of α to this factor is trivial, all we are left to prove is that its restriction to H 1 (F, H 1 (L, µ{n})) is trivial. Consider then the sum of restriction maps 0 / / H 1 (L/k, H 1 (L, µ{n})) / / H 1 (F ⋊ Γ L/k , H 1 (L, µ{n})) / / s * x t t H 1 (F,H 1 (F, H 1 (L, µ{n})) → A,K H 1 (A, H 1 (K, µ{n})), where A ranges over all abelian subgroups of F and K ranges over all the extensions of L that correspond to a (strictly) procyclic subgroup of Γ L . Since all actions are trivial, all H 1 groups correspond to Hom groups and hence it is an easy exercise to see that this map is injective. Finally, once again, since the spectral sequence is compatible with restrictions, for each such pair (A, K) we have a commutative diagram and since α ∈ Br x un X, we know by Proposition 3.2 and Theorem 5.1 that the restriction of α to H 2 (A × Γ K , µ{n}) is trivial (recall that pre-images in H 2 ( * , µ{n}) are unique for the n-torsion). This proves the triviality of the image of α in the direct sum A,K H 1 (A, H 1 (K, µ{n})) and hence its triviality in H 1 (F, H 1 (L, µ{n})), which concludes the proof of the lemma.⌣ Definition 6.1. Let k, G, X be as above. Let F be the stabilizer of a k-point in X and denote by L/k the Galois extension obtained via the kernel of the natural morphism Γ k → Aut (F ) given by the action of Γ k on the points of F . Then v ∈ Ω k is said to be a bad place if it is a non archimedean place that either ramifies in L/k or divides the order of F . Otherwise, we say that v is a good place. Note that by this definition all archimedean places are assumed to be good places. This is due to the fact that the Brauer-Manin obstruction doesn't interact with archimedean places. In fact, we prove that this is true for all good places via the following two theorems: Theorem 6.2. Let G a semisimple simply connected R-group, F ⊂ G a finite Rsubgroup of order n and X the R-homogeneous space G/F . Then the Brauer pairing Br un X × X(R) → Br R, is constant on X(R). Theorem 6.3. Let k be a non archimedean local field of residue characteristic p, G a semisimple simply connected k-group, F ⊂ G a finite k-subgroup of order n and X the k-homogeneous space G/F . Assume that p is prime to the order of F and that F is split by an unramified extension of k. Then the Brauer pairing Br un X × X(k) → Br k, is constant on X(k). With these results, we can relate Colliot-Thélène's conjecture on the Brauer-Manin obstruction (cf. [CT03,Introduction]) with a conjecture given by Demarche, Neftin and the author in [DLAN17, §1.2] on "tame approximation", as it was suggested in [DLAN17,§2.5]. Indeed, Theorems 6.2 and 6.3 tell us that the Brauer set ( Ω k X(k v )) Brun surjects onto the product S X(k v ) for every finite set S of good places. Assuming the first conjecture, we get then the density of X(k) in S X(k v ). Recalling [DLAN17,Prop. 2.4], we have thus proved that: Corollary 6.4. Assume that the Brauer-Manin obstruction is the only obstruction to weak approximation for homogeneous spaces with finite stabilizers. Then the Tame approximation problem has a positive solution for every finite k-group F , i.e. the natural restriction map H 1 (k, F ) → v∈S H 1 (k v , F ), is surjective for every finite set S of good places.⌣ Proof of Theorem 6.2. Let x ∈ X(R) be the point corresponding to the subgroup F . Then we may consider the normalized subgroup Br x un X ⊂ Br un X of the elements that are trivial when evaluated in x. It is clear then that Br un X ≃ Br 0 X × Br x un X and hence it will suffice to prove that the pairing is trivial on the subgroup Br x un X. Consider the group F X = F ⋊ Γ R . This group comes with a splitting s x : Γ R → F X associated to x. Consider then an element α ∈ Br x un X ⊂ H 2 (F X , C * ) and take a point y ∈ X(R). Let s y : Γ R → F X denote the corresponding splitting. We know then by Proposition 3.4 that the image of (α, y) by the Brauer-Manin pairing is the restriction of α to H 2 (Γ R , C * ) = Br R via s y . Let σ be the non trivial element in Γ R . Then, since it is well-known that sections Γ R → F X are classified by the cocycle set Z 1 (R, F ), we know that s y (σ) is of the form f s x (σ) with f ∈ F such that f σ = f −1 . This tells us that there is a cyclic σ-stable subgroup C = f of F such that the section s y factors through C ⋊ Γ R . In other words, we have the folllowing commutative diagram of finite groups F X C ⋊ Γ R @ Ø 5 5 ❦ ❦ ❦ ❦ ❦ ❦ Γ R , T sy i i ❙ ❙ ❙ ❙ ❙ ❙ c 1 sy O O which induces the following commutative diagram of cohomology groups H 2 (F X , C * ) s * y Res s s ❣ ❣ ❣ ❣ ❣ ❣ Br X evy H 2 (C ⋊ Γ R , C * ) s * y + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ H 2 (Γ R , C * ) Br R. On the other hand, since C is σ-stable, it corresponds to an R-subgroup of F . We can consider then the morphism Z = G/C → G/F = X and the point z ∈ Z above x corresponding to the subgroup C. Since α ∈ Br x un X, we know by Proposition 3.2 that its restriction α Z to Br Z = H 2 (C ⋊ Γ R , C * ) actually falls into Br z un Z. Now, since C and Γ R are cyclic, Proposition 4.1 tells us that α Z is in Br 0 Z ∩ Br z un Z and hence is trivial. We conclude then that its restiction to s y (Γ R ) is trivial and hence so is its evaluation at y for every y ∈ X(R). This proves the theorem.⌣ Proof of Theorem 6.3. Let x ∈ X(k) be the point corresponding to the subgroup F . As before, we may consider the normalized subgroup Br x un X ⊂ Br un X and prove that the pairing is trivial on Br x un X. Consider the group F X = F ⋊ Γ k . This group comes with a splitting s x : Γ k → F X associated to x and Proposition 3.4 tells us then that the elements in Br x un X are elements that are trivialized when restricted to s x (Γ k ). Since the index of this subgroup is n, the classic restriction-corestriction argument tells us then that Br x un X is an n-torsion group. Let µ{n} ⊂ k * be the group generated by the q-primary roots of unity for every q dividing n. Then it is easy to see that H 2 ( * , µ{n}) corresponds to the subgroup of H 2 ( * ,k * ) generated by the elements of q-primary torsion. In particular, H 2 (k, µ{n}) conatins the n-torsion of Br k and H 2 (F X , µ{n}) contains the n-torsion of Br X. Consider then an element α ∈ Br x un X ⊂ H 2 (F X , µ{n}) and take a point y ∈ X(k). Let s y : Γ k → F X denote the corresponding splitting. We know then by Proposition 3.4 that the image of (α, y) by the Brauer-Manin pairing is the restriction of α to H 2 (Γ k , µ{n}) ⊂ Br k via s y . Moreover, if we denote by W ⊂ Γ k the subgroup of wild ramification, then (s y )\| W = (s x )\| W . Indeed, W is a pro-p-group which acts trivially on F (and hence sections of F × W are classified by group morphisms W → F ) and the order of F is prime to p. Let now L/k be the maximal unramified extension of k. This extension splits F and contains µ{n}, so that Lemma 5.2 tells us that s x (Γ L ) is normal in F X and we have a projection π L x : F X → F ⋊ Γ L/k . Let us look then at the image of the morphism π L x • s y : Γ k → F ⋊ Γ L/k . The image of W ⊂ Γ k is trivial since s y (W ) = s x (W ) ⊂ s x (Γ L ) = Ker(π L x ). And since it is well-known that there is an isomorphism of profinite groups Γ k /W ≃ σ, τ \| στ σ −1 = τ q , where q is the order of the residue field of k, we see that we only have to care about the images of σ and τ , which represent respectively a lift of the Frobenius (which generates Γ L/k ) and a generator of the tame inertia subgroup (cf. [NSW08,7.5.3]). Since L/k is unramified and s y is a section, the image of τ by π L x • s y must be in F and generate a s y (σ)-stable cyclic subgroup T of F . Consider then a lift of σ to Γ k and the procyclic subgroup C ⊂ Γ k generated by it. Then C is an extension of σ ≃Ẑ and some pro-p procyclic group P . We have the following commutative diagram of profinite groups F ⋊ Γ L/k F X π L x o o o o T s y (C) E ; ; ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ π L x x x x x q q q q q q q q q q π L x (s y (Γ k )) c 1 O O Γ k , π L x •sy o o o o c 1 sy O O where T s y (C) is a subgroup of F X whose image in the quotient Γ k is C and whose intersection with F is T since T is stable by the action of s y (C). This diagram induces the following diagram of 2-cohomology groups H 2 (F ⋊ Γ L/k , µ{n}) H 2 (π L x (s y (Γ k )), µ{n}) s * y •(π L x ) * / / (π L x ) * 4 4 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ H 2 (Γ k , µ{n}) 1 / / Br k, where the (π L x ) * arrows are inflation arrows since π L x is surjective. Using Lemma 5.2, we can lift α ∈ H 2 (F X , µ{n}) to an element in H 2 (F ⋊ Γ L/k , µ{n}) and we know that its image in H 2 (T s y (C), µ{n}) is trivial by Theorem 5.1. Assume that the map (π L x ) * : H 2 (π L x (s y (Γ k )), µ{n}) → H 2 (T s y (C), µ{n}), is injective. Then we get that the image of the lift of α in H 2 (π L x (s y (Γ k )), µ{n}) is trivial, which tells us that its image in H 2 (Γ k , µ{n}) is trivial and hence its evaluation at y is trivial, which concludes the proof of the theorem. Let us then prove the injectivity of (π L x ) * in order to conclude. This is an inflation map coming from the surjective morphism T s y (C) → π L x (s y (Γ k )), whose kernel is the pro-p group P . An application of the Hochschild-Serre spectral sequence H p (π L x (s y (Γ k )), H q (P, µ{n})) ⇒ H p+q (T s y (C), µ{n}), tells us then that injectivity follows from the triviality of H 1 (P, µ{n}), which is easily seen since P is a pro-p-group and µ{n} is a torsion group whose elements are all of order prime to p.⌣ Remark. All the results in this paper are most probably true for fields of characteristic p > 0 as long as one assumes that the finite groupF is of order prime to p. Note that, accordingly with Theorem 6.3, this should imply in particular that, for the function field k of a curve over F q , there is no Brauer-Manin obstruction to weak approximation for homogeneous spaces X of G with X(k) = ∅ and finite stabilizer F such that the natural map Γ k → Aut F factors through Γ Fq (in particular for constant F ). Proving (or disproving) that such homogeneous spaces have weak approximation should then be a very interesting result. Proposition 3. 4 . 4The pairing on the upper line of (2) is the Brauer-Manin pairing. Ker(H 2 2(F X , µ{n}) → H 2 (L, µ{n}))Res / / H 1 (F ⋊ Γ L/k , H 1 (L, µ{n}))ResKer(H 2 (A × Γ K , µ{n}) → H 2 (K, µ{n})) / / H 1 (A, H 1 (K, µ{n})), (T s y (C), µ{n}) This is a point where our proof fails for essentially real fields, since in general one should rather consider general procyclic subgroups, for which we may then get Br L ≃ Z/2Z. For non essentially real fields, one can avoid this by only restricting to strictly procyclic subgroups, cf. the rest of the proof.2 We should remark that, unlike Serre in[Ser79], we are dealing here with an infinite Galois extension k(Ḡ)/k(X) with Galois groupF X , but since the infinite part falls in the residue field extension, the reader will easily see through this "problem". An arithmetic application: the set of bad places for the Brauer-Manin obstruction Let k be a number field, Ω k the set of its places, G a semisimple simply connected k-group and X a homogeneous space of G with finite geometric stabilizer. When X has a k-point, we gave in [DLAN17, §2.2] a definition of "bad places" for X with respect to the Brauer-Manin obstruction to weak approximation. The Brauer group of quotient spaces of linear representations. F A Bogomolov, Izv. Akad. Nauk SSSR Ser. Mat. 513F. A. Bogomolov. The Brauer group of quotient spaces of linear represen- tations. Izv. Akad. Nauk SSSR Ser. Mat. 51(3), 485-516, 688, 1987. Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes. M Borovoi, C Demarche, D Harari, Ann. Sci. Ec. Norm. Sup. 46M. Borovoi, C. Demarche, D. Harari. Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes. Ann. Sci. Ec. Norm. Sup. 46, 651-692, 2013. 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[ "Measuring Unified Dark Matter with 3D Cosmic Shear", "Measuring Unified Dark Matter with 3D Cosmic Shear" ]	[ "Stefano Camera \nDipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nTorinoItaly\n\nDipartimento di Fisica Teorica\nUniversità degli Studi di Torino\nTorinoItaly\n\nINFN †\nSezione di Torino\nTorinoItaly\n", "Thomas D Kitching \nInstitute for Astronomy\nSUPA ‡\nUniversity of Edinburgh\nRoyal Observatory\nEdinburghUK\n", "4⋆Alan F Heavens \nInstitute for Astronomy\nSUPA ‡\nUniversity of Edinburgh\nRoyal Observatory\nEdinburghUK\n", "Daniele Bertacca \nDipartimento di Fisica \"Galileo Galilei\"\nUniversità di Padova\nPadovaItaly\n\nINFN †\nSezione di Padova\nPadovaItaly\n\nInstitute of Cosmology and Gravitation\nUniversity of Portsmouth\nPortsmouthUK\n", "Antonaldo Diaferio \nDipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nTorinoItaly\n\nINFN †\nSezione di Torino\nTorinoItaly\n\nHarvard-Smithsonian Center for Astrophysics\nCambridgeMAUSA\n" ]	[ "Dipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nTorinoItaly", "Dipartimento di Fisica Teorica\nUniversità degli Studi di Torino\nTorinoItaly", "INFN †\nSezione di Torino\nTorinoItaly", "Institute for Astronomy\nSUPA ‡\nUniversity of Edinburgh\nRoyal Observatory\nEdinburghUK", "Institute for Astronomy\nSUPA ‡\nUniversity of Edinburgh\nRoyal Observatory\nEdinburghUK", "Dipartimento di Fisica \"Galileo Galilei\"\nUniversità di Padova\nPadovaItaly", "INFN †\nSezione di Padova\nPadovaItaly", "Institute of Cosmology and Gravitation\nUniversity of Portsmouth\nPortsmouthUK", "Dipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nTorinoItaly", "INFN †\nSezione di Torino\nTorinoItaly", "Harvard-Smithsonian Center for Astrophysics\nCambridgeMAUSA" ]	[ "Mon. Not. R. Astron. Soc" ]	We present parameter estimation forecasts for future 3D cosmic shear surveys for a class of Unified Dark Matter (UDM) models, where a single scalar field mimics both Dark Matter (DM) and Dark Energy (DE). These models have the advantage that they can describe the dynamics of the Universe with a single matter component providing an explanation for structure formation and cosmic acceleration. A crucial feature of the class of UDM models we use in this work is characterised by a parameter, c ∞ (in units of the speed of light c = 1), that is the value of the sound speed at late times, and on which structure formation depends. We demonstrate that the properties of the DM-like behaviour of the scalar field can be estimated with very high precision with large-scale, fully 3D weak lensing surveys. We found that 3D weak lensing significantly constrains c ∞ , and we find minimal errors ∆c ∞ = 3.0 · 10 −5 , for the fiducial value c ∞ = 1.0 · 10 −3 , and ∆c ∞ = 2.6 · 10 −5 , for c ∞ = 1.2 · 10 −2 . Moreover, we compute the Bayesian evidence for UDM models over the ΛCDM model as a function of c ∞ . For this purpose, we can consider the ΛCDM model as a UDM model with c ∞ = 0. We find that the expected evidence clearly shows that the survey data would unquestionably favour UDM models over the ΛCDM model, for the values c ∞ 10 −3 .	10.1111/j.1365-2966.2011.18712.x	[ "https://arxiv.org/pdf/1002.4740v2.pdf" ]	118,697,822	1002.4740	80586bac94b0d96ccbc8143de085217d6ba164c2	Measuring Unified Dark Matter with 3D Cosmic Shear 2010 Stefano Camera Dipartimento di Fisica Generale "Amedeo Avogadro" Università degli Studi di Torino TorinoItaly Dipartimento di Fisica Teorica Università degli Studi di Torino TorinoItaly INFN † Sezione di Torino TorinoItaly Thomas D Kitching Institute for Astronomy SUPA ‡ University of Edinburgh Royal Observatory EdinburghUK 4⋆Alan F Heavens Institute for Astronomy SUPA ‡ University of Edinburgh Royal Observatory EdinburghUK Daniele Bertacca Dipartimento di Fisica "Galileo Galilei" Università di Padova PadovaItaly INFN † Sezione di Padova PadovaItaly Institute of Cosmology and Gravitation University of Portsmouth PortsmouthUK Antonaldo Diaferio Dipartimento di Fisica Generale "Amedeo Avogadro" Università degli Studi di Torino TorinoItaly INFN † Sezione di Torino TorinoItaly Harvard-Smithsonian Center for Astrophysics CambridgeMAUSA Measuring Unified Dark Matter with 3D Cosmic Shear Mon. Not. R. Astron. Soc 0002010Accepted 00 -0000. Received 00 -0000; in original form 28 March 2011arXiv:1002.4740v2 [astro-ph.CO] Printed 28 March 2011 (MN L A T E X style file v2.2)gravitationgravitational lensingcosmology: theory -observations - dark matter -dark energy -large-scale structure of Universe We present parameter estimation forecasts for future 3D cosmic shear surveys for a class of Unified Dark Matter (UDM) models, where a single scalar field mimics both Dark Matter (DM) and Dark Energy (DE). These models have the advantage that they can describe the dynamics of the Universe with a single matter component providing an explanation for structure formation and cosmic acceleration. A crucial feature of the class of UDM models we use in this work is characterised by a parameter, c ∞ (in units of the speed of light c = 1), that is the value of the sound speed at late times, and on which structure formation depends. We demonstrate that the properties of the DM-like behaviour of the scalar field can be estimated with very high precision with large-scale, fully 3D weak lensing surveys. We found that 3D weak lensing significantly constrains c ∞ , and we find minimal errors ∆c ∞ = 3.0 · 10 −5 , for the fiducial value c ∞ = 1.0 · 10 −3 , and ∆c ∞ = 2.6 · 10 −5 , for c ∞ = 1.2 · 10 −2 . Moreover, we compute the Bayesian evidence for UDM models over the ΛCDM model as a function of c ∞ . For this purpose, we can consider the ΛCDM model as a UDM model with c ∞ = 0. We find that the expected evidence clearly shows that the survey data would unquestionably favour UDM models over the ΛCDM model, for the values c ∞ 10 −3 . the mass distribution, gravitational lensing is an excellent tool for cosmological parameter estimation, complementing Cosmic Microwave Background (CMB) studies. One of the most useful manifestations of gravitational lensing by intervening matter is the alignment of nearby images on the sky. Detection of DM on large scales through such cosmic shear measurements -the small, coherent distortion of distant galaxy images due to the large-scale distribution of matter in the cosmos -has recently been shown to be feasible. At a statistical level, it has been shown (Hu & Tegmark 1999;Hu 1999) that there is some extra information on cosmological parameters which can be gained by dividing the sample into several redshift bins; this technique is known as weak lensing tomography. However, a more comprehensive representations of the shear field can be called 3D weak lensing (Heavens 2003;Castro et al. 2005;Heavens et al. 2006;Kitching et al. 2007), in which, by using the formalism of spin-weighted spherical harmonics and spherical Bessel functions, one can relate the two-point statistics of the harmonic expansion coefficients of the weak lensing shear and convergence to the power spectrum of the matter density perturbations. Such a tool is relevant in view of the present and next generations of large-scale weak lensing surveys, which will provide distance information of the sources through photometric redshifts. Recently, rather than considering DM and DE as two distinct components, it has been suggested the alternative hypothesis that DM and DE are two states of the same fluid. This has been variously referred to as "Unified Dark Matter" or "quartessence" models. Compared with the standard DM plus DE models (e.g. ΛCDM), these models have the advantage that we can describe the dynamics of the Universe with a single scalar field which triggers both the accelerated expansion at late times and the LSS formation at earlier times. Specifically, for these models, we can use Lagrangians with a non-canonical kinetic term, namely a term which is an arbitrary function of the square of the time derivative of the scalar field, in the homogeneous and isotropic background. Originally this method was proposed to have inflation driven by kinetic energy, called k-inflation (Armendariz-Picon et al. 1999;Garriga & Mukhanov 1999), to explain early Universe's inflation at high energies. Then this scenario was applied to DE (Chiba et al. 2000;de Putter & Linder 2007;Linder & Scherrer 2009). In particular, the analysis was extended to a more general Lagrangian (Armendariz-Picon et al. 2000, 2001 and this scenario was called k-essence (see also Chiba et al. 2000;Rendall 2006;Li et al. 2006;Calcagni & Liddle 2006;Babichev 2006;Fang et al. 2007;Bazeia et al. 2007;Kang et al. 2007;Babichev et al. 2008;Babichev 2008;Ahn et al. 2009). For zmodels, several adiabatic or, equivalently, purely kinetic models have been investigated in the literature: the generalised Chaplygin gas (Kamenshchik et al. 2001;Bilic et al. 2002;Bento et al. 2002;Carturan & Finelli 2003;Sandvik et al. 2004), the single dark perfect fluid with a simple two-parameter barotropic equation of state Quercellini et al. 2007;Pietrobon et al. 2008) and the purely kinetic models studied by Scherrer (2004), , Chimento et al. (2009). Alternative approaches have been proposed in models with canonical Lagrangians with a complex scalar field (Arbey 2006). One of the main issues of these UDM models is to see whether the single dark fluid is able to cluster and produce the cosmic structures we observe in the Universe today. In fact, a general feature of UDM models is the possible appearance of an effective sound speed, which may become significantly different from zero during the evolution of the Universe. In general, this corresponds to the appearance of a Jeans length (or sound horizon) below which the dark fluid does not cluster. Thus, the viability of UDM models strictly depends on the value of this effective sound speed (Hu 1998;Garriga & Mukhanov 1999;Mukhanov 2005), which has to be small enough to allow structure formation (Sandvik et al. 2004;Giannakis & Hu 2005;Bertacca & Bartolo 2007) and to reproduce the observed pattern of the CMB temperature anisotropies (Carturan & Finelli 2003;Bertacca & Bartolo 2007). In general, in order for UDM models to have a very small speed of sound and a background evolution that fits the observations, a severe fine tuning of their parameters is necessary. In order to avoid this fine tuning, alternative models with similar goals have been analysed in the literature: Piattella et al. (2010) studied in detail the functional form of Jeans scale in adiabatic UDM perturbations and introduced a class of models with a fast transition between an early Einstein-de Sitter cold DM-like era and a later ΛCDMlike phase. If the transition is fast enough, these models may exhibit satisfactory structure formation and CMB fluctuations, thus presenting a small Jeans length even in the case of a non-negligible sound speed; Gao et al. (2009) explore unification of DM and DE in a theory containing a scalar field of non-Lagrangian type, obtained by direct insertion of a kinetic term into the energy-momentum tensor. Here, we choose to investigate the class of UDM models studied in Bertacca et al. (2008), who designed a reconstruction technique of the Lagrangian, which allows one to find models where the effective speed of sound is small enough, and the k-essence scalar field can cluster (see also Camera et al. 2009, Camera 2010. In particular, the authors require that the Lagrangian of the scalar field is constant along classical trajectories on cosmological scales, in order to obtain a background identical to the background of the ΛCDM model. Here, we wish to investigate whether this class of UDM models can be scrutinised in realistic scenarios. Specifically, we compute the weak lensing signals expected in these models as they would be measured by a Euclid-like survey. The structure of this paper is as follows. In Section 2 we describe the UDM model we use in this work. In Section 3 we detail the theory of weak gravitational lensing on the celestial sphere, with a particular interest in the cosmic shear observable (Section 3.1). In Section 4 we outline the Fisher matrix formalism we use to calculate the expected statistical errors on cosmological parameters, and with the same formalism we compute the expected Bayesian evidence for UDM models over the standard ΛCDM model as a function of the sound speed parameter c∞ (Section 5). In Section 6 we present our results, such as the matter power spectrum obtained in these UDM models (Section 6.1) and the corresponding 3D shear signal (Section 6.2); the parameter estimations for a Euclid-like survey are presented in Section 6.3, while in Section 6.4 we use the Bayesian approach to ask the data whether our UDM model is favoured over the ΛCDM model or not. Finally, in Section 7, conclusions are drawn. UNIFIED DARK MATTER MODELS We consider a UDM model where the Universe is filled with a perfect fluid of radiation, baryons and a scalar field ϕ(t), the latter mimicking both DM and DE in form of a cosmological constant. In particular, Bertacca et al. (2008), by using scalar-field Lagrangians L (X, ϕ) with a non-canonical kinetic term, where 3 X = − 1 2 ∇ µ ϕ∇µϕ,(1) have outlined a technique to reconstruct UDM models such that the effective speed of sound is small enough to allow the clustering of the scalar field. Specifically, once the initial value of the scalar field is fixed, the scalar field Lagrangian is constant along the classical trajectories, namely Lϕ = −Λ/(8πG), and the background is identical to the background of ΛCDM. In other words, the energy density of the UDM scalar field presents two terms ρUDM(t) = ρDM(t) + ρΛ,(2) where ρDM behaves like a DM component (ρDM ∝ a −3 ) and ρΛ like a cosmological constant component (ρΛ = const.). Consequently, ΩDM = ρDM(a = 1)/ρc and ΩΛ = ρΛ/ρc are the density parameters of DM and DE today, where ρc is the present day critical density; hence, the Hubble parameter in these UDM models is the same as in ΛCDM, H(z) = H0 Ωm(1 + z) 3 + ΩΛ,(3) with H0 = 100 h km s −1 Mpc −1 and Ωm = ΩDM + Ω b , where Ω b = ρ b /ρc is the baryon density in units of the critical density. Now we introduce small inhomogeneities of the scalar field δϕ(t, x), and in the linear theory of cosmological perturbations and in the Newtonian gauge, the line element is ds 2 = −(1 + 2Φ)dt 2 + a 2 (t)(1 + 2Ψ)dx 2 ,(4) in the case of a spatially flat Universe, as supported by CMB measurements (e.g. Spergel et al. 2007). This scalar field presents no anisotropic stress, thus Φ = −Ψ. With this metric, when the energy density of radiation becomes negligible, and disregarding also the small baryonic component, the evolution of the Fourier modes of the Newtonian potential Φ k (a) are described by (Garriga & Mukhanov 1999;Mukhanov 2005) v k ′′ + cs 2 k 2 v k − θ ′′ θ v k = 0,(5) where a prime denotes a derivative with respect to the con- formal time dτ = dt/a, k = \|k\| and v ≡ Φ √ ρUDM + pUDM ,(6)θ ≡ 1 a 1 + p UDM ρ UDM ;(7) here, cs 2 (a) = pUDM ,X ρUDM ,X(8) is the effective speed of sound, where ,X denotes a derivative w.r.t. X. By following the technique outlined by Bertacca et al. (2008), it is possible to construct a UDM model in which the sound speed is small enough to allow the formation of the LSS we see today and is capable of reproducing the observed pattern of the temperature anisotropies in the CMB radiation. We choose a Lagrangian of the form Lϕ ≡ pUDM(ϕ, X) = f (ϕ)g(X) − V (ϕ),(9) with a Born-Infeld type kinetic term g(X) (Born & Infeld 1934), where M is a suitable mass scale. Such a kinetic term can be thought as a field theory generalisation of the Lagrangian of a relativistic particle (Padmanabhan & Choudhury 2002;Abramo & Finelli 2003;Abramo et al. 2004). It was also proposed in connection with string theory, since it seems to represent a low-energy effective theory of D-branes and open strings, and has been conjectured to play a role in cosmology (Sen 2002a,b,c;Padmanabhan & Choudhury 2002). By using the equation of motion of the scalar field ϕ(t, x) and by imposing that the scalar field Lagrangian is constant along the classical trajectories, i.e. pUDM = −ρΛ, we obtain the following expressions for the potentials = − √ 1 − 2XM −4f (ϕ) = Λc∞ 1 − c∞ 2 cosh(ξϕ) sinh(ξϕ) 1 + (1 − c∞ 2 ) sinh 2 (ξϕ) ,(10)V (ϕ) = Λ 1 − c∞ 2 1 − c∞ 2 2 sinh 2 (ξϕ) + 2(1 − c∞ 2 ) − 1 1 + (1 − c∞ 2 ) sinh 2 (ξϕ) ,(11) with ξ = 3Λ/[4(1 − c∞ 2 )M 4 ]. Hence, the sound speed takes the parametric form cs(a) = ΩΛc∞ 2 ΩΛ + (1 − c∞ 2 )ΩDMa −3 ,(12) and it is easy to see that the parameter c∞ represents the value of the speed of sound when a → ∞. Moreover, when a → 0, cs → 0. In UDM models the fluid which triggers the accelerated expansion at late times is also the one which has to cluster in order to produce the structures we see today. Thus, from recombination to the present epoch, the energy density of the Universe is dominated by a single dark fluid, and therefore the gravitational potential evolution is determined by the background and perturbation evolution of this fluid alone. As a result, the general trend is that the possible appearance of a sound speed significantly different from zero at late times corresponds to the appearance of a Jeans length (Bertacca & Bartolo 2007) λJ (a) = θ θ ′′ cs(a) below which the dark fluid does not cluster any more, causing a strong evolution in time of the gravitational potential. In Fig. 1 we show λJ (a), the sound horizon, for different values of c∞. WEAK LENSING ON THE CELESTIAL SPHERE In the linear régime, corresponding to the Born approximation, where the lensing effects are evaluated on the nullgeodesic of the unperturbed (unlensed) photon (Hu 2000;Bartelmann & Schneider 2001), it is possible to relate the weak lensing potential φ for a given source at a 3D position in comoving space x = (χ,n) to the Newtonian potential Φ(x) via φ(x) = χ 0 dχ ′ W (χ ′ ) χ ′ Φ(χ ′ ,n)(14) where W (χ ′ ) = −2 ∞ χ ′ dχ χ − χ ′ χ n(χ)(15) is the weight function of weak lensing, with n [χ(z)] representing the redshift distribution of sources, for which dχ n(χ) = 1 holds, and χ(z) being the radial comoving distance, such that 1 H(z) = dχ(z) dz .(16) Spin-weighted spherical harmonics and spherical Bessel functions are a very natural expansion for weak lensing observables, such as the potential φ(x) (Heavens 2003;Castro et al. 2005). Since cosmic shear depends on the Newtonian potential, the use of this basis allows one to relate the expansion of the shear field to the expansion of the mass density field. The properties of the latter depend in a calculable way on cosmological parameters, so this opens up the possibility of using 3D weak shear to estimate these quantities. In the flat-sky approximation, the weak lensing potential (14) reads φ(k, ℓ) = 2 π d 3 x φ(x)kj ℓ (kχ) e −iℓ·n ,(17) where ℓ = \|ℓ\| is a 2D angular wavenumber, k a radial wavenumber and j ℓ (kχ) a spherical Bessel function of order ℓ. The covariances of these coefficients define the power spectrum of the weak lensing potential via φ(k, ℓ)φ * (k ′ , ℓ ′ ) = (2π) 2 δD(ℓ − ℓ ′ )C φφ (k, k ′ ; ℓ),(18) where δD is the Dirac delta. The 3D shear field In this paper we are interested in the information brought by the cosmic shear. We now introduce a distortion tensor (Kaiser 1998;Bartelmann & Schneider 2001) φ,ij(x) = χ 0 dχ ′ χ ′ W (χ ′ )Φ,ij (χ ′ ,n),(19) where commas denote derivatives w.r.t. directions perpendicular to the line of sight. The trace of the distortion tensor represents the convergence κ(x) = 1 2 (φ,11(x) + φ,22(x))(20) and, defining γ1(x) = 1 2 (φ,11(x) − φ,22(x)) and γ2(x) = φ,12(x), the linear combination γ(x) = γ1(x) + iγ2(x)(21) is the differential stretching, or shear. Castro et al. (2005) have shown that the complex shear is the second "edth" derivative of the weak lensing potential γ(x) = 1 2 ððφ(x),(22) where, in Cartesian coordinates {x, y}, ð = ∂x + i∂y. We can now express the power spectrum of the 3D cosmic shear as a function of the gravitational potential via C γγ (k1, k2; ℓ) = ℓ 4 π 2 dk k 2 I Φ ℓ (k1, k)I Φ ℓ (k2, k)P Φ (k, 0),(23) where P Φ (k, z) is the Newtonian potential power spectrum and, for a generic field X, we have defined I X ℓ (ki, k) = dχ X k (χ) X k (0) W (χ)j ℓ (kiχ) .(24) FISHER MATRIX ANALYSIS Cosmological parameters influence the shear in a number of ways: the matter power spectrum P δ (k, z) is dependent on Ωm, h and the linear amplitude σ8. The linear power spectrum is dependent on the growth rate, which also has some sensitivity to the parameter of the Λ-like component equation of state wΛ = pΛ/ρΛ. It is well know that the speed of sound (Eq. 8) is strictly related to wΛ(z), and it also affects the χ(z) relation and hence the angular diameter distance sinK [χ(z)]. These parameters {ϑα} may be estimated from the data using likelihood methods. Assuming uniform priors for the parameters, the maximum a posteriori probability for the parameters is given by the maximum likelihood solution. We use a Gaussian likelihood where C = (d − d th )(d − d th ) T is the covariance matrix and D = (d − d th )(d − d th ) T is the data matrix, with d the data vector and d th the theoretical mean vector. The expected errors on the parameters can be estimated with the Fisher information matrix (Fisher 1935;Jungman et al. 1996;Tegmark et al. 1997). This has the great advantage that different observational strategies can be analysed and this can be very valuable for experimental design. The Fisher matrix gives the best errors to expect, and should be accurate if the likelihood surface near the peak is adequately approximated by a multivariate Gaussian. The Fisher matrix is the expectation value of the second derivative of the ln L w.r.t. the parameters {ϑα}: F αβ = − ∂ 2 ln L ∂ϑα∂ϑ β(26) and the marginal error on parameter ϑα is F −1 αα 1 2 . If the means of the data are fixed, the Fisher matrix can be calculated from the covariance matrix and its derivatives (Tegmark et al. 1997) by F αβ = 1 2 Tr C −1 C,αC −1 C ,β .(27) For a square patch of sky, the Fourier transform leads to uncorrelated modes, provided the modes are separated by 2π/Θ rad where Θ rad is the side of the square in radians, and the Fisher matrix is simply the sum of the Fisher matrices of each ℓ mode: F αβ = 1 2 ℓ (2ℓ + 1)Tr C ℓ −1 C ℓ ,α C ℓ −1 C ℓ ,β ,(28) where C ℓ is the covariance matrix for a given ℓ mode. BAYESIAN EVIDENCE In this paper we compute parameter forecasts from 3D cosmic shear for UDM models. It is important to notice that we are dealing with an alternative model with respect to the standard ΛCDM model; hence, besides determining the best-fit value (and the errors) on a set of parameters within a model, we can also ask if this particular alternative model is preferable to the standard. Model selection is in a sense a higher-level question than parameter estimation. While in estimating parameters one assumes a theoretical model within which one interprets the data, in model selection, one wants to know which theoretical framework is preferred given the data. Clearly if our alternative model has more parameters than the standard one, chi-square analysis will not be of any use, because it will always reduce if we add more degrees of freedom. From a Bayesian point of view, this involves computation of the Bayesian evidence and of the Bayes factor B. We refer to the two models under examination with MUDM and MΛCDM. We know that, in this context, MΛCDM is simpler than MUDM because it has one fewer parameter, i.e. c∞; in the same way, it is also contained in MUDM, because, if ϑ ΛCDM α and ϑ UDM α ′ are the parameters of the two models (with α = 1, . . . , n and α ′ = 1, . . . , n + 1), respectively, then {ϑ ΛCDM α , c∞} = {ϑ UDM α ′ }(29) holds; here, c∞ ≡ ϑ UDM n+1 . The posterior probability for each model M is given by Bayes' theorem p(M \|d) = p(d\|M )p(M ) p(d) .(30) The Bayesian evidence is defined as the marginalisation over the parameters p(d\|M ) = d m ϑ p(d\|ϑ, M )p(ϑ\|M ),(31) where ϑ is the parameter vector, whose length m is n for the ΛCDM model and n + 1 for UDM models. The posterior relative probabilities of our two models given the data d and with flat priors in their parameters p(M ) = const., is then Heavens 2009) p(MΛCDM\|d) p(MUDM\|d) = p(MΛCDM) p(MUDM) × d n ϑ ΛCDM p(d\|ϑ ΛCDM , MΛCDM)p(ϑ ΛCDM \|MΛCDM) d n+1 ϑ UDM p(d\|ϑ UDM , MUDM)p(ϑ UDM \|MUDM) .(32) If we choose non-committal priors p(MUDM) = p(MΛCDM), the posterior evidence probability reduces to the ratio of the evidences, which takes the name of the Bayes factor and in the present case reads B ≡ d n ϑ ΛCDM p(d\|ϑ ΛCDM , MΛCDM)p(ϑ ΛCDM \|MΛCDM) d n+1 ϑ UDM p(d\|ϑ UDM , MUDM)p(ϑ UDM \|MUDM) .(33) Now, let us focus on the priors p(ϑ\|M ). If we assume flat priors in each parameter, over the range ∆ϑ, then p(ϑ ΛCDM \|MΛCDM) = α ∆ϑ ΛCDM α −1 and B = d n ϑ ΛCDM p(d\|ϑ ΛCDM , MΛCDM) d n+1 ϑ UDM p(d\|ϑ UDM , MUDM) ∆c∞.(34) The Bayes factor B still depends on the specific dataset d. For future experiments, we do not yet have the data, so we compute the expectation value of the Bayes factor, given the statistical properties of d. The expectation is computed over the distribution of d for the correct model (assumed here to be MUDM). To do this, we make two further approximations: first we note that B is a ratio, and we approximate B by the ratio of the expected values, rather than the expectation value of the ratio. This should be a good approximation if the likelihoods are sharply peaked. We also make the Laplace approximation, that the expected likelihoods are given by multivariate Gaussians, i.e., p(d\|ϑ, M ) = L0e − 1 2 (ϑ−ϑ 0 ) α F αβ (ϑ−ϑ 0 ) β ,(35) where F αβ is the Fisher matrix, given in Eq. (26). Heavens et al. (2007) have shown that, if we assume that the posterior probability densities are small at the boundaries of the prior volume, then we can extend the integration to infinity, and the integration over the multivariate Gaussians can be easily performed. In the present case, this gives B = √ det F UDM √ det F ΛCDM L ΛCDM 0 L UDM 0 ∆c∞ √ 2π .(36) One more subtlety has to be taken into account to compute the ratio L ΛCDM 0 /L UDM 0 : if the correct underlying model is MUDM, in the incorrect model MΛCDM the maximum of the expected likelihood will not, in general, be at the correct parameter values (see Heavens et al. 2007, Fig. 1). The n parameters of the ΛCDM model shift their values to compensate the fact that c∞ is being kept fixed at the incorrect fiducial value c∞ = 0. With these offsets in the maximum likelihood parameters in the ΛCDM model, the Bayes factor takes the form B = √ det F UDM √ det F ΛCDM ∆c∞ √ 2π e − 1 2 δϑαF UDM αβ δϑ β ,(37) where the shifts δϑα can be computed under the assumption of a multivariate Gaussian distribution (Taylor et al. 2007), and read δϑα = − F ΛCDM −1 αβ G UDM β,n+1 δc∞,(38) with G UDM β,n+1 a subset of the UDM Fisher matrix (a vector in the present case). It is usual to consider the logarithm of the Bayes factor, for which the so-called "Jeffreys' scale" gives empirically calibrated levels of significance for the strength of evidence (Jeffreys 1961), 1 < \| ln B\| < 2.5 is described as "substantial" evidence in favour of a model, 2.5 < \| ln B\| < 5 is "strong," and \| ln B\| > 5 is "decisive." These descriptions seem too aggressive: \| ln B\| = 1 corresponds to a posterior probability for the less-favoured model which is 0.37 of the favoured model (Kass & Raftery 1995). Other authors have introduced different terminology (e.g. Trotta 2007). RESULTS AND DISCUSSION We use a fiducial cosmology with the following parameters: Hubble constant (in units of 100 km s −1 Mpc −1 ) h = 0.71, present-day total matter density (in units of critical density) Ωm ≡ ΩDM + Ω b = 0.3, baryon contribution Ω b = 0.045, cosmological constant contribution ΩΛ = 0.7, spectral index ns = 1, linear amplitude (within a sphere of radius 8 h −1 Mpc) σ8 = 0.8. In Section 6.1 we compute the predicted matter power spectrum for UDM models, with a comparison to ΛCDM. In Section 6.2 the 3D shear matrix C γγ (k1, k2; ℓ) is shown. In Section 6.3 we present the parameter forecasts, and in Section 6.4 we show the expected Bayesian evidence for UDM models over the ΛCDM model. The matter power spectrum Our class of UDM models allows the value w = −1 for a → ∞. In other words they admit an effective cosmological constant energy density at late times. Therefore, in order to compare the predictions of our UDM model with observational data, we follow the same prescription used by Piattella et al. (2010), where the density contrast of the clustering fluid is δ ≡ δρm ρm = ρDMδUDM + ρ b δ b ρm ,(39) where δ b and δUDM are the baryon and the scalar field density contrasts, respectively, and we emphasise that ρDM = ρUDM − ρΛ is the only component of the scalar field density which clusters. Linear régime The today matter power spectrum P (k) ≡ P δ (k, z = 0) is the present value of the Fourier transform of the density perturbation correlation function. To construct P (k) in the ΛCDM model, we need the growth factor D(z) = δ(x, z)/δ(x, z = 0) on linear scales (i.e. in absence of free-streaming) and the transfer function T (k), that describes the evolution of perturbations through the epochs of horizon crossing and radiaton-matter transition. Here, we use the transfer function suggested by , which, with an accurate, general fitting formula, calculates the power spectrum as a function of the cosmological parameters quite efficiently. show that baryons are effective at suppressing power on small scales compared to DM-only models. Moreover, the small-scale limit of this transfer function can be calculated analytically as a function of the cosmological parameters (Hu & Eisenstein 1998). Hence, we can write the matter power spectrum as P (k) = 2π 2 δH 2 k H0 3 ns T 2 (k) D(z) D(z = 0) 2 ;(40) here, δH is a normalisation. To obtain P (k) in UDM models, it is useful to remember that the class of UDM models we use here is constructed to have the same properties of the ΛCDM model in the early Universe; in Eq. (5), which describes the time evolution of Fourier modes of the Newtonian potential Φ k (a), we thus set the same initial conditions for both the UDM and the ΛCDM potentials. Gravity is GR, so we can use the Poisson equation Φ k (a) = − 3 2 ΩmH0 2 δ k (a) k 2 a ,(41) which relates Φ k (a) to the matter power spectrum via δ k (a)δ k ′ * (a) = (2π) 3 δD k − k ′ P δ (k, a).(42) Clearly, if we solve Eq. (5) with cs = 0, we obtain the standard ΛCDM matter power spectrum. Fig. 2 shows the matter power spectrum P (k) for ΛCDM and UDM models, for a number of values of c∞. By increasing the sound speed, the potential starts to decay earlier in time, oscillating around zero afterwards (Camera et al. 2009); at large scales, if c∞ is small enough, these UDM models reproduce the ΛCDM model. This feature reflects the dependence of the gravitational potential on the effective Jeans length λJ (a). It is easy to see that if c∞ 10 −3 the perturbations of the UDM reproduce the behaviour of the concordance model within the linear régime (the UDM curve for c∞ = 10 −3 is virtually on top of the ΛCDM one). Instead, a larger sound speed inhibits structure formation earlier in time, thus we observe less power on small scales; in this case, the consequence of the oscillatory feature of the gravitational potential, due to the non-negligible speed of sound, can be clearly seen. In principle, the large-scale distribution of galaxies could constrain the value of c∞. However, the shape of the power spectrum also depends on the normalisation σ8 and the spectral index ns: therefore, for a given c∞ as large as 10 −2 , an appropriate choice of σ8 and ns can provide a power spectrum in agreement with observations, at least on scales where non-linear effects are not dominant. In addition, in UDM models it is still unclear how the galaxy distribution is biased against the gravitational potential of the scalar field on small scales. Therefore the large-scale distribution of galaxies does not appear to be the best tool to constrain this family of UDM models. On the contrary, a weak lensing analysis can constrain the matter power spectrum without a fine-tuning of either σ8 or the galaxy bias. Non-linear régime For wavenumbers k > k nl ≃ 0.2 h Mpc −1 , non-linear contributions to the evolution of the Newtonian potential (i.e. to matter overdensities) become important. In the ΛCDM model, the gravitational potential satisfies Eq. (5), but in this case cs is the sound speed of the hydrodynamical fluid, and therefore can be set equal to zero in the matterdominated epoch. For cs = 0, Eq. (5) has an analytic solution Hu 2002;Mukhanov 2005 ; Bertacca & Bartolo 2007) Φ k (a) = A k 1 − H(a) a a 0 da ′ H(a ′ ) ,(43) where the constant of integration is A k = Φ k (0)T (k), with T (k) the matter transfer function and Φ k (0) the large-scale potential during the radiation-dominated era. To perform further calculations on a wider range of scales than that allowed by linear theory, we will use the Smith et al. (2003) non-linear fitting formulae for P (k) in the ΛCDM model. However, currently there is no linearto-non-linear mapping in UDM models. Nevertheless, as we have seen, differences between the ΛCDM and UDM models arise at scales smaller than the sound horizon. With a crossover wavenumber k ≃ 1/λJ , if the sound speed is small enough to guarantee that λJ is well within the non-linear regime we can assume that the non-linear evolution of the UDM power spectrum will be similar to the ΛCDM one. A deeper knowledge on this aspect will be the next step of the development of UDM models and has to be explored in future work. The 3D shear signal For a 20, 000 deg 2 Euclid-like survey (Cimatti et al. 2009;Refregier et al. 2010), we assume that the source distribution over redshifts has the form (Smail et al. 1994) n(z) ∝ z 2 e − z z 0 1.5 ,(44) where z0 = zm/1.4, and zm = 0.8 is the median redshift of the survey. The source number density with photometric redshift and shape estimates is 35 per square arcminute. We also assume that the photometric redshift errors are Gaussian, with a dispersion given by σ(z) = 0.05(1 + z). In order to avoid the high-wavenumber régime where the fitting formulae of Smith et al. (2003) may be unreliable, or where baryonic effects might alter the power spectrum (k > 10 h Mpc −1 ; White 2004; Zhan & Knox 2004), we do not analyse modes with k > 1.5 Mpc −1 . Note that the non-local nature of gravitational lensing does mix modes to some degree, but these modes are sufficiently far from the uncertain highly non-linear régime that this is not a concern (Castro et al. 2005). We include angular modes as small as each survey will allow, and analyse up to ℓmax = 5000 (but note the wavenumber cut). In Fig. 3 we present the 3D shear matrix C γγ (k1, k2; ℓ). The first three rows show log 10 C γγ (k1, k2; ℓ) for the ΛCDM model and for a UDM model with c∞ = 5 · 10 −4 and c∞ = 5·10 −3 (respectively) in the (k1, k2)-plane in blue(gray)-scale for a number of values of ℓ. In the fourth row we present the diagonal elements k 2 C γγ (k, k; ℓ) of the 3D shear matrix, where the upper (green) curve refers to the smaller speed of sound and the lower (green) curve to the greater c∞; the ΛCDM (red) curve is virtually on top of the small-c∞ UDM curve. Finally, in the bottom row we show k 2 times the ratio of the diagonal elements C γγ (k, k; ℓ) of UDM models over the ΛCDM model. The oscillatory features of the UDM gravitational potential (Camera et al. 2009), whose power spectrum enters the shear via Eq. (23), can be clearly seen in the shear signal of the UDM model with c∞ = 4 · 10 −3 . The bumps in the diagonal signal can be easily understood by looking at the log 10 C γγ (k1, k2; ℓ) plot, where it is interesting to notice how the oscillations take place along any direction, with the obvious symmetry along the k1-and k2-axes. Instead, as we have noticed in Fig. 2, when the sound speed is small enough we do not see any oscillations and the matter power spectrum of UDM models is in agreement with ΛCDM. This agreement holds even at non-linear scales k 0.2 h Mpc −1 . Beyond the oscillations, these signals, expected for two different values of c∞, show us the effect of the effective Jeans length of the gravitational potential. In fact, The Newtonian potential in UDM models behaves like ΛCDM at scales much larger than λJ (a) (Eq. 13), while at smaller scales it starts to decay and oscillate. Hence, at high values of ℓ and k, which correspond to small angular and physical scales, respectively, the signal of weak lensing observables, Figure 3. The 3D shear matrix log 10 C γγ (k 1 , k 2 ; ℓ) for five values of ℓ in (blue)gray-scale. In the first row we show the ΛCDM signal, while in the second and third rows we present the UDM signal for c∞ = 1.0 · 10 −3 and c∞ = 5.4 · 10 −3 , respectively. The fourth row shows the diagonal elements k 2 C γγ (k, k; ℓ), and each curve, from top to bottom, refers to the corresponding matrix above. The ΛCDM curve is virtually on top of the small-c∞ UDM curve. The fifth row shows the fractional error. like cosmic shear, shows the decay of the gravitational potential. Although the UDM signal for c∞ = 5 · 10 −4 appears to be in agreement with the ΛCDM signal (fourth row of Fig. 3), their fractional difference shown in the fifth row is still of order unit at k 1 h Mpc −1 and is not negligible. In fact, we will see below in Section 6.4, that this low value of c∞ still yields a Bayesian evidence which indicates a statistically very large difference between this UDM model and ΛCDM. Finally, in Fig. 3, we can also notice that, the higher the value of ℓ, the smaller the physical scales are those which contribute to the shear signal. This effect is due to the approximate Bessel function inequality, ℓ kχ, in Eq. (24). As the ℓ value increases the diagonal terms of the covariance matrix do not become significant until kχmax ∼ ℓ, where χmax ≡ χ(zmax) is the upper limit imposed on the integration over the radial comoving distance. Estimation of cosmological parameters Once having introduced the method (Section 4) and the survey design formalism (Section 6.2), now we show cosmological parameter forecasts for such a survey and we explore the variation in the marginal errors with changes in the sound speed parameter c∞. By using the Fisher matrix analysis outlined in Taylor et al. (2007), we calculate predicted Fisher matrices and parameter constraints for a 20, 000 squaredegree Euclid-like survey. In all Fisher matrix calculations we use a seven-parameter cosmological set {Ωm = ΩDM + Ω b , Ω b , h, ΩΛ, σ8, ns, c∞} with fiducial values {0.3, 0.045, 0.71, 0.7, 0.8, 1.0} for the first six. The Fisher matrix is sensitive to c∞, so we compute the evidences at twenty c∞ fiducial values from 5 · 10 −4 to 5 · 10 −2 . We find that the Fisher matrices are unstable for c∞ 10 −3 . This is because, when the sound speed is small, the UDM 3D shear signal is virtually indistinguishable from that of ΛCDM, and the numerical derivatives w.r.t. c∞ thus become unreliable. Fig. 4 shows the Fisher matrix elements marginalised over all other parameters. In dark blue(gray) we present the results for a UDM model with c∞ = 1.0 · 10 −3 and in light blue(gray) for c∞ = 5.4 · 10 −3 . Notice that results are shown for universes which are not necessarily flat. In non-flat geometries, the spherical Bessel functions j ℓ (kχ) should be replaced by ultraspherical Bessel functions Φ ℓ β (y) (Heavens et al. 2006). For the case considered here ℓ ≫ 1 and k ≫ (curvature scale) −1 , then Φ ℓ β (y) → j ℓ (kχ) (Abbott & Schaefer 1986;Zaldarriaga & Seljak 2000). The expansion used is not ideal for curved universes, but it should however be an adequate approximation given current constraints on flatness (e.g. Larson et al. 2010). The Fisher constraints for lensing are large enough that for some parameters (σ8, Ω b ) the 1σ confidence region has an unphysical lower bound. We note that this is a symptom of the Fisher matrices Gaussian approximation. Taylor & Kitching (2010) address this concern by suggesting a semi-analytic approach that only assumes Gaussianity in particular parameter directions; we leave an implementation of this type of parameter error prediction, or a more sophisticated likelihood parameter search for future investigation. Before starting the interpretation of such results, it is important to underline that what deeply affects the matter power spectrum in UDM models, and thus the lensing signal, is the presence of an effective Jeans length for the Newtonian potential. Let us focus on Eq. (5): we can consider the asymptotic solutions, i.e. long wavelength and short wavelength perturbations, depending on whether k ≪ 1/λJ or k ≫ 1/λJ , respectively. In the former case, the term in Eq. (5) involving the speed of sound of the scalar field is negligible, therefore the solution is formally the same that in the ΛCDM model (Eq. 43), and the Fourier modes Φ k (a) read (Bertacca & Bartolo 2007) Φ k≪1/λ J (a) ∝ 1 − H(a) a a 0 da ′ H(a ′ ) (k ≪ 1/λJ ); (45) instead, in the opposite régime we have Φ k≫1/λ J (a) ∝ 1 cs(a) cos k a 0 da ′ cs(a ′ ) a ′2 H(a ′ ) (k ≫ 1/λJ ).(46) This means that what enters in the oscillatory dynamics is not only c∞, which however plays an important role, but also ΩDM and ΩΛ, as described in Eq. (12). Therefore, the links which connect the expected marginal errors in Fig. 4 with the corresponding fiducial c∞ are not quite straightforward. Moreover, we find that the Fisher matrix is rather sensitive to c∞. The errors we find on the sound speed parameter are almost constant, and go from ∆c∞ = 3.0 · 10 −5 , for the fiducial value c∞ = 1.0 · 10 −3 , to ∆c∞ = 2.6 · 10 −5 , when c∞ = 1.2 · 10 −2 . It is already well known that weak lensing can tightly constrain the (Ωm, σ8)-plane, using standard cosmic shear techniques (see Brown et al. 2003;Semboloni et al. 2006), and 3D weak lensing constrains σ8 in the same way by measuring the overall normalisation of the matter power spectrum. The expected marginal errors on Ωm and σ8 are in fact very promising, particularly in the perspective of combining the cosmic shear data with other cosmological observables, i.e. CMB or SNeIa (Heavens et al. 2006). However, the presence of a sound speed can be mimicked in the power spectrum, at least in the non-linear régime, by an accurate tuning of some parameter values, on top of all σ8 and ns (Camera et al. 2009). This is why the ellipses of those parameters get worse for larger values of c∞. In UDM models, there is another aspect which is particularly interesting to notice: we are able to lift the degeneracy between Ωm and Ω b without using early-Universe data. That is because ΩDM and Ω b enter in the growth of structures in two different ways. The expansion history of the Universe takes into account only their joint effect, through Ωm, whereas the speed of sound is determined by ΩDM alone. In fact we have to keep in mind that in UDM models there is a scalar field which mimics both DM and Λ, but it still has proper dynamics different from that of its respective in the ΛCDM model. Model selection In Section 5 we showed how the Bayes factor can be used to determine which model is favoured by the data. By using the Fisher matrix formalism for a Euclid-like survey, we compute the Bayes factor B for UDM models over the standard ΛCDM cosmology. We fix flat prior ∆c∞ = 1. The large values of − ln B derive from the large deviations δϑα in Eq. (38) which yield an extremely small exponential. On turn, the deviations δϑα are large because, as shown in the right-most column of Fig. 4, (i) the ellipsoidal confidence regions are narrow, and (ii) they are almost vertical; in other words, the ΛCDM parameters that one would derive if living in a universe with a non-null c∞ would be largely biased. We conclude that, if UDM is the correct model, there would be large evidence for UDM models over ΛCDM for values of c∞ 10 −3 . However, if c∞ is so small that the UDM peculiar features in the matter power spectrum only appear at k ≫ 1 h Mpc −1 , namely on galactic or smaller scales, in principle, we might be unable to distinguish UDM from ΛCDM, unless the non-linear dynamics and/or the effects of the baryonic physics on the DM-like dynamics of the scalar field are largely different from what we expect in ΛCDM. CONCLUSIONS In this work, we calculate the expected error forecasts for a 20, 000 square degree survey with median redshift zm = 0.8 such as Euclid (Cimatti et al. 2009;Refregier et al. 2010) in the framework of unified models of DM and DE (UDM models). We focus on those UDM models which are able to reproduce the same Hubble parameter as in the ΛCDM model (Bertacca & Bartolo 2007;Bertacca et al. 2008). In these UDM models, beyond standard matter and radiation, there is only one exotic component, a classical scalar field with a non-canonical kinetic term in its Lagrangian, that during the structure formation behaves like DM, while at the present time contributes to the total energy density of the Universe like a cosmological constant Λ. In order to avoid the strong integrated Sachs-Wolfe effect which typically plagues UDM models, we follow the technique outlined by Bertacca et al. (2008), that allows one to construct a UDM model in which the sound speed is small enough to let the cosmological structures grow and reproduce the LSS we see today. This can be achieved by parameterising the sound speed with its value at late times, c∞. An effect of the presence of a non-negligible speed of sound of the UDM scalar field is the emerging of an effective time-dependent Jeans length λJ (a) of the gravitational potential. It causes a strong suppression, followed by oscilla-tions, of the Fourier modes Φ k (a) with k ≡ \|k\| > 1/λJ . This reflects on the predicted lensing signal, because the latter is an integrated effect of the potential wells of the LSS over the path that the photons travel from the sources to the observer. We calculate the 3D shear matrix C γγ (k1, k2; ℓ) in the flat-sky approximation for a large number of values of c∞. In agreement with Camera et al. (2009), we see that, whilst the agreement with the ΛCDM model is good for small values of c∞, when one increases the sound speed parameter, the lensing signal appears more suppressed at small scales, and moreover the 3D shear matrix does show bumps related to the oscillations of the gravitational potential. We also compute the Fisher matrix for a Euclid-like survey. It has been shown that 3D lensing is a powerful tool in constraining cosmological parameters (e.g. Castro et al. 2005), and Heavens et al. (2006) have demonstrated that it is particularly useful in unveiling the properties of the dark components of the Universe. By using a seven-parameter cosmological set {Ωm = ΩDM + Ω b , Ω b , h, ΩΛ, σ8, ns, c∞}, with one fiducial value for each parameter, except for c∞, for which we use twenty values in the range 5·10 −4 . . . 5·10 −2 , we obtain the expected marginal errors. However, the c∞ Fisher matrix elements are unstable in the parameter range c∞ 10 −3 , because the UDM signal is degenerate with respect to ΛCDM. Therefore, we restrict our analysis by considering only sound speed fiducial values larger than ∼ 10 −3 . We get minimal errors that go from ∆c∞ = 3.0 · 10 −5 , for the fiducial value c∞ = 1.0 · 10 −3 , to ∆c∞ = 2.6 · 10 −5 , when c∞ = 1.2 · 10 −2 . In the case of UDM models, 3D lensing is revealed to be even more useful for estimating cosmological parameters, because since it encodes information from both the geometry and the dynamics of the Universe, it can lift the usual degeneracy between the DM and the baryon fractions, ΩDM and Ω b . This is because in the Hubble parameter, which determines the background evolution of the geometry of the Universe, both ΩDM and Ω b enter in the usual way, through the total matter fraction Ωm. On the other side, the speed of sound, which affects the structure formation, and thus the dynamics of the Universe, is sensitive only on the DM-like behaviour of the scalar field, since for baryons cs = 0 holds. Finally, we compute the Bayesian expected evidence (e.g. Trotta 2007) for UDM models over the ΛCDM model as a function of the sound speed parameter c∞. The expected evidence clearly shows that the survey data would unquestionably favour UDM models over the standard ΛCDM model, if its sound speed parameter exceed ∼ 10 −3 . Figure 1 . 1Sound horizon λ J (a) for c∞ = 10 −4 , 10 −3 , 10 −2 , 10 −1 from bottom to top. Figure 2 . 2Matter power spectra P (k) ≡ P δ (k, 0) for ΛCDM (solid) and UDM (dot-dashed), with c∞ = 10 −3 , 10 −2 , 10 −1 from top to bottom. Figure 5 . 5Bayes factor − ln B for UDM models over the standard ΛCDM model as a function of the sound speed parameter c∞. Figure 4 . 4Expected marginal errors on UDM model cosmological parameters from a 20, 000 deg 2 Euclid-like survey with a median redshift zm = 0.8. Ellipses show the 1σ errors for two parameters (68% confidence regions), marginalised over all the other parameters. Dark(light) ellipses refer to a UDM model with c∞ = 1.0 · 10 −3 (c∞ = 5.4 · 10 −3 ). c 2010 RAS, MNRAS 000, 1-12 We use units such that c = 1 and signature {−, +, +, +}, where Greek indices run over spacetime dimensions, whereas Latin indeces label spatial coordinates. ln L = −Tr ln C − C −1 D ,(25)c 2010 RAS, MNRAS 000, 1-12 ACKNOWLEDGMENTSWe thank the referee for her/his useful comments which contributed to remove some ambiguities in the presentation of our results. SC and AD gratefully acknowledge partial support from the INFN grant PD51. 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[ "Pu nuclear magnetic resonance in the candidate topological insulator PuB 4", "Pu nuclear magnetic resonance in the candidate topological insulator PuB 4" ]	[ "A P Dioguardi \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n\nInstitute for Solid State Research\nP.O. Box 270116D-01171Dresden, DresdenGermany\n", "H Yasuoka \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n\nMax Planck Institute for Chemical Physics of Solids\n01187DresdenGermany\n", "S M Thomas \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "H Sakai \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n\nAdvanced Science Research Center\nJapan Atomic Energy Agency\n319-1195Tokai, NakaIbarakiJapan\n", "S K Cary \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "S A Kozimor \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "T E Albrecht-Schmitt \nDepartment of Chemistry and Biochemistry\nFlorida State University\n95 Chieftan Way32306TallahasseeFlorida\n", "H C Choi \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "J.-X Zhu \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "J D Thompson \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "E D Bauer \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "F Ronning \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n" ]	[ "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Institute for Solid State Research\nP.O. Box 270116D-01171Dresden, DresdenGermany", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Max Planck Institute for Chemical Physics of Solids\n01187DresdenGermany", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Advanced Science Research Center\nJapan Atomic Energy Agency\n319-1195Tokai, NakaIbarakiJapan", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Department of Chemistry and Biochemistry\nFlorida State University\n95 Chieftan Way32306TallahasseeFlorida", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA" ]	[]	We present a detailed nuclear magnetic resonance (NMR) study of 239 Pu in bulk and powdered single-crystal plutonium tetraboride (PuB4), which has recently been investigated as a potential correlated topological insulator. This study constitutes the second-ever observation of the 239 Pu NMR signal, and provides unique on-site sensitivity to the rich f -electron physics and insight into the bulk gap-like behavior in PuB4. The 239 Pu NMR spectra are consistent with axial symmetry of the shift tensor showing for the first time that 239 Pu NMR can be observed in an anisotropic environment and up to room temperature. The temperature dependence of the 239 Pu shift, combined with a relatively long spin-lattice relaxation time (T1), indicate that PuB4 adopts a non-magnetic state with gap-like behavior consistent with our density functional theory (DFT) calculations. The temperature dependencies of the NMR Knight shift and T −1 1 -microscopic quantities sensitive only to bulk states-imply bulk gap-like behavior confirming that PuB4 is a good candidate topological insulator. The large contrast between the 239 Pu orbital shifts in the ionic insulator PuO2 (∼ +24.7 %) and PuB4 (∼ −0.5 %) provides a new tool to investigate the nature of chemical bonding in plutonium materials. arXiv:1812.09202v1 [cond-mat.str-el]	10.1103/physrevb.99.035104	[ "https://arxiv.org/pdf/1812.09202v1.pdf" ]	119,189,588	1812.09202	9a538bf00be6a1ae9128d95ce9ae71b3900bf210	Pu nuclear magnetic resonance in the candidate topological insulator PuB 4 A P Dioguardi Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Institute for Solid State Research P.O. Box 270116D-01171Dresden, DresdenGermany H Yasuoka Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Max Planck Institute for Chemical Physics of Solids 01187DresdenGermany S M Thomas Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA H Sakai Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Advanced Science Research Center Japan Atomic Energy Agency 319-1195Tokai, NakaIbarakiJapan S K Cary Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA S A Kozimor Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA T E Albrecht-Schmitt Department of Chemistry and Biochemistry Florida State University 95 Chieftan Way32306TallahasseeFlorida H C Choi Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA J.-X Zhu Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA J D Thompson Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA E D Bauer Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA F Ronning Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Pu nuclear magnetic resonance in the candidate topological insulator PuB 4 (Dated: December 24, 2018)239 We present a detailed nuclear magnetic resonance (NMR) study of 239 Pu in bulk and powdered single-crystal plutonium tetraboride (PuB4), which has recently been investigated as a potential correlated topological insulator. This study constitutes the second-ever observation of the 239 Pu NMR signal, and provides unique on-site sensitivity to the rich f -electron physics and insight into the bulk gap-like behavior in PuB4. The 239 Pu NMR spectra are consistent with axial symmetry of the shift tensor showing for the first time that 239 Pu NMR can be observed in an anisotropic environment and up to room temperature. The temperature dependence of the 239 Pu shift, combined with a relatively long spin-lattice relaxation time (T1), indicate that PuB4 adopts a non-magnetic state with gap-like behavior consistent with our density functional theory (DFT) calculations. The temperature dependencies of the NMR Knight shift and T −1 1 -microscopic quantities sensitive only to bulk states-imply bulk gap-like behavior confirming that PuB4 is a good candidate topological insulator. The large contrast between the 239 Pu orbital shifts in the ionic insulator PuO2 (∼ +24.7 %) and PuB4 (∼ −0.5 %) provides a new tool to investigate the nature of chemical bonding in plutonium materials. arXiv:1812.09202v1 [cond-mat.str-el] Topological insulators have received much attention recently due to the experimental verification of the theoretical prediction of topologically nontrivial symmetryprotected surface states [1,2]. Kondo insulators are felectron systems with strong correlations in which hybridization of the f -electrons with conduction electrons forms a gap at the Fermi level [3]. Strong spin-orbit coupling can result in a topological Kondo insulator in which band inversion drives the emergence of nontrivial topologically protected gapless surface states [4,5]. Samarium hexaboride (SmB 6 ) is the primary candidate example of a topological Kondo insulator [6][7][8][9]. As compared with rare-earth 4f -electron systems, the actinide 5felectron systems have more spatially-extended f -electron wave functions, which generally results in an enhancement of the energy scales involved [10][11][12]. Plutonium (Pu) materials display particularly complex physical properties due to the 5f -electrons lying on the brink between bonding and non-bonding configurations [13,14]. For example, elemental Pu forms in six allotropes at ambient pressure that vary in density by up to 25% [15,16]. Pu compounds display a wide variety of electronic ground states including heavy-fermion behavior, magnetism, superconductivity [17], and most recently the prediction of topologically non-trivial states [12,18]. Very recently, plutonium tetraboride (PuB 4 ) has been theoretically predicted to be a strong topological insulator in which electronic correlations play an important role [19]. The density functional theory (DFT) calculations predict a band gap ∆ ∼ 254 meV and dynamical mean-field theory (DMFT) calculations find that electronic correlations significantly reduce the magnitude of the predicted energy gap. Experimental measurements from the same work find an increase of the resistivity with decreasing temperature and saturation at low temperature reminiscent of the behavior of SmB 6 [20]. Fits to the temperature dependent resistivity yield an energy gap ∆ = 35 meV, which is taken as evidence for correlation-induced suppression of the expected gap value. PuB 4 forms in the tetragonal ThB 4 -type crystal structure with space group P4/mbm (# 127) as shown in Fig. 1(a) and was first reported nearly 60 years ago [21][22][23]. Magnetic measurements of PuB 4 indicated that the Pu magnetic moment is very small, on the order of 7.2 × 10 −4 emu/mol and shows little temperature dependence [24]. This small magnetic susceptibility and insulating-like electrical transport make PuB 4 an ideal material in which to search for 239 Pu nuclear magnetic resonance (NMR). NMR is a powerful tool for the investigation of the physics and chemistry of condensed matter in general [27][28][29][30]. The 239 Pu nucleus has nuclear spin I = 1 2 and is of great interest as an on-site probe of the rich f -electron physics of Pu. The first attempt to observe 239 Pu NMR was performed on α-Pu more than 50 years ago [31], however to date there is only a single report of 239 Pu NMR [26] in the ionic insulator PuO 2 . The main difficulty involved in observing 239 Pu NMR-and other f -electron nuclei, in general-can be traced to the very strong hyperfine fields at the nucleus produced by onsite hyperfine coupling to the f -electrons. Consequently, the resulting spectral width can be very large, and the spin-lattice (T 1 ) and spin-spin (T 2 ) relaxation times can be extremely short, which makes detection of the signal difficult. These effects can be minimized in systems with a gap in the electronic and spin excitation spectrum, as evident in the case of PuO 2 , UO 2 , and YbB 12 [26,32,33]. Here we report the observation of, and the microscopic properties extracted from 239 Pu NMR in powdered and single crystalline PuB 4 . Crystals were grown by an Alflux method and sample preparation details are provided in the Supplementary Material. We deduce the resonant condition of 239 Pu in PuB 4 239 γ(1 + K PuB4 )/2π = 2.288 ± 0.001 MHz/T from the powder spectra, and find axial symmetry of the hyperfine interaction on the Pu site. Both the powder and the single crystal Knight shift K(T ) of 239 Pu show temperature dependence consistent with gap-like behavior with a static energy gap (extracted from the single crystalline K c (T ) data) ∆ K ≈ 156 meV. The relaxation time is quite long-on the order of milliseconds to seconds-even at the 239 Pu site, indicating that the f -electron configuration is non-magnetic. The dominant temperature dependence of the spin-lattice relaxation rate T −1 1 (T ) also shows gap-like behavior with a dominant dynamic gap ∆ T1 ≈ 251 meV. We compare our experimental NMR results with the density of states, calculated within density functional theory including spinorbit coupling, which finds a gap of similar order of mag-nitude. A weak low-temperature peak in T −1 1 (T ) indicates the presence of bulk in-gap magnetic states with a gap δ ≈ 2 meV. Our DFT calculations including spin-orbit coupling reveal a gap in the density of states (DOS) at the Fermi energy E F of roughly 254 meV as shown in Fig. 1(b). To account for the presence of correlations we also performed DFT + DMFT calculations. Using a U of 4.5 eV and high-order Slater integrals amounting to an effective J = 0.512 eV [34,35] and attempting to stabilize a magnetic solution, we find that the self-consistent solution recovers a non-magnetic state with a band gap at the Fermi level of order 10.3 meV (see Supplemental Material for further calculation details). The appreciable calculated gap in the DOS combined with an expected non-magnetic ground state indicate the probable absence of strong spin-and charge-relaxation channels, and therefore, we expect the spin-lattice relaxation rate in PuB 4 to be long enough to observe the 239 Pu signal. The 239 Pu nucleus has I = 1 2 and the bare gyromagnetic ratio was determined based on the initial observation in PuO 2 to be 239 γ/2π = 2.29±0.001 MHz/T [26]. Consequently, we would expect to find an NMR signal in the field range of roughly 7 to 9 T with an rf excitation frequency f 0 ∼ 20 MHz. Indeed, for f 0 = 20.222 MHz we discovered an asymmetric powder spectrum between 8.80 and 8.92 T as shown in Fig. 2(a-b). To establish that the observed signal is indeed due to 239 Pu from PuB 4 field-swept spectra were collected at several frequencies. These spectra are shown in Fig. 2(a) and they confirm the intrinsic nature of the NMR signal. The crystal structure of PuB 4 has a single Pu site with oriented site symmetry m.2m (see Fig. 1(a)). For each crystallite in the powdered sample the resonance condition can be expressed as 2πf 0 = γB 0 (1 + K i ) where K i are the elements of the shift tensor for a given field orientation and B 0 is the magnetic field at which the resonance occurs for frequency f 0 . Although the local symmetry is orthorhombic in principle, the non-axial components of the shift tensor are found to be extremely close to zero from the spectral pattern in Fig. 2(b), i.e., it can be practically regarded to be tetragonal. Assuming tetragonal symmetry for the hyperfine interaction on Pu, the isotropic and axial shifts (K iso and K ax , respectively) are extracted from the observed K c and K ab using K iso = (K c + 2K ab )/3 and K ax = (K c − K ab )/3, where the angular dependence of the shift is given by K(θ) = K iso + K ax (3 cos 2 θ − 1). The isotropic shift of 239 Pu in PuB 4 is K iso (T = 4 K) = −0.09 ± 0.04 % is obtained from the slope in the frequency vs. field plot in Fig. 2(a). This value is notably different from the shift K(T = 4 K) = 24.72 ± 0.04 % of 239 Pu in PuO 2 [26]. To calculate these shifts we have assumed the bare 239 γ/2π = 2.29 MHz/T as determined from the study of PuO 2 [26]. K ax (T = 4 K) = −0.48 ± 0.01 % is also significantly different from the shift found in PuO 2 [26] at the same temperature. It is worth noting that the relatively small absolute value of K ax was crucial to find the 239 Pu signal in an anisotropic environment. The temperature dependence of the field-swept spectra at f 0 = 20.222 MHz and the corresponding least-squares fits are shown in Fig. 2(b). An axially symmetric shift tensor remains a good approximation for all temperatures measured. Fig. 2(c) illustrates that K iso has a small negative value with a positive temperature dependence, and K ax has a larger negative value with a smaller temperature dependence relative to K iso . In general, K iso originates from the spin-polarized Fermi contact interaction and couples to the uniform spin susceptibility via the hyperfine interaction. K ax may be dominantly attributed to the temperature independent orbital hyperfine interaction with a small temperature dependence resulting from a reduction of the anisotropy of the spin susceptibility with increasing temperature. The facts that the spin-lattice relaxation time in PuB 4 is sufficiently long to enable the observation of 239 Pu NMR, and that Knight shifts are weakly temperature dependent imply that the electronic state of Pu in PuB 4 is nearly nonmagnetic. Assuming a local picture this implies either that Pu has a 5f 6 configuration or PuB 4 adopts a Kondo insulating state. Finally, we performed measurements on a single crystal of PuB 4 for the external field applied along theĉ-axis. We measured both theĉ-axis 239 Pu shift K c and T −1 1 as a function of temperature up to 300 K as shown in Fig. 3. We fit the 239 Pu inversion recovery curves to the form M N (t) = M N (∞) 1 − αe −(t/T1) β ,(1) where M N (∞) is the equilibrium nuclear magnetization, α is the inversion fraction, T 1 is the spin-lattice relaxation time, and β is a stretching exponent that modifies the expected single exponential behavior (β = 1). We find that β avg = 0.813, which is a measure of the width of the probability distribution of T 1 [36], is independent of temperature and may indicate sensitivity to self-irradiation induced disorder [37]. Both K c and T −1 1 are consistent with gap-like behavior, and T −1 1 exhibits a low temperature maximum consistent with the presence of in-gap states which are suppressed with applied magnetic field as shown in the inset of Fig. 3. From a chemistry perspective, the 239 Pu orbital shift is very different between PuO 2 (∼ +24.7 % [26]) and PuB 4 (∼ −0.5 %). The origin of the difference in magnitude of the orbital shift is clear from the fact that in the case of PuO 2 the Pu ion has a completely ionic Pu 4+ (5f 4 ) state and experiences strong cubic crystalline electronic field giving rise to a non-magnetic ground state with a Van Vleck orbital magnetism, which is the main source of the hyperfine interaction to the Pu nuclear moment. In contrast, DFT + DMFT calculations point to PuB 4 being a strongly correlated insulator with possible strong topological character, similar to the case of SmB 6 . In SmB 6 the gap arises from hybridization between 4f and ligand electrons that give rise to a pronounced non-integral vs. temperature and fit (solid line) to gap-like behavior as discussed in the text which yields energy gaps ∆T 1 = 251.3 ± 49.4 meV and δ = 1.8 ± 2.4 meV. The external field was adjusted (from 8.5900 T at 5 K to 8.5455 T at 300 K) such that the observed frequency was f0 = 19.465 MHz for all temperatures. The inset shows the field dependence of T −1 1 at T = 5 K indicating the suppression of in-gap states with applied external field. value of the 4f configuration. Our results suggest that this is also the case in PuB 4 . The large difference in orbital shift between PuO 2 and PuB 4 clearly indicates that 239 Pu NMR is highly sensitive to the degree of bond mixing and the f -electron configuration. Furthermore, the relaxation time is roughly two orders of magnitude shorter than in PuO 2 [26], which likely reflects the difference in chemical environments between PuB 4 and PuO 2 . The capability to measure 239 Pu was key to observing gap-like behavior in the static and dynamic spinsusceptibilities as evidenced by the temperature dependencies of K c and T −1 1 shown in Fig. 3. Our 11 B measurements of the temperature dependence of the Knight shift (see Supplemental Material) do not show any evidence of gap-like behavior, likely due to the much smaller value of the hyperfine coupling of the 11 B nuclei to the electrons as compared to the 239 Pu hyperfine coupling, which is expected to be on the order of 150 T/µ B . Therefore, our 239 Pu NMR results are sensitive to otherwise enigmatic physics in PuB 4 . There exist a number of previous NMR studies that find gap-like behavior of f -electron systems, e.g. SmB 6 [38,39], YbB 12 [33], Ce 3 Bi 4 Pt 3 [40]. Here we follow the analysis scheme of SmB 6 [39] by fitting the temperature dependence of the 239 Pu Knight shift and spinlattice relaxation rate by assuming a simple model for the density of states near the Fermi energy. The Knight shift is given by, K(T ) ∝ f (E, T )[1 − f (E, T )]ρ(E)dE,(2) where f (E, T ) is the Fermi function and ρ(E) is the density of states. The spin-lattice relaxation rate is given by, T −1 1 (T ) ∝ f (E, T )[1 − f (E, T )]ρ(E) 2 dE.(3) We assume a simplified model of the density of states (equivalent to that of Caldwell et al. [39]) given by, ρ(E) = ρ i (T ) for δ < \|E\| < W i = ρ for ∆ < \|E\| < W,(4) and zero otherwise as shown in the inset of Fig. 3(a). We perform least-squares fits using a Levenberg-Marquardt minimization algorithm which iteratively recalculates the model function via numerical integration of Eqns. 2 and 3 (see Supplemental Material for a full description of the curve fitting). The energy gap extracted from the Knight shift ∆ K = 155.6 ± 11.0. For the Knight shift we find no indication of the presence of in-gap states, that is ρ i (T ) = 0, similar to the static susceptibility of SmB 6 [39]. The dominant energy gap extracted from the spin-lattice relaxation ∆ T1 = 251.3±49.4 meV. A smaller in-gap density of states ρ i (T ) = ρ i0 e −T /T0 with an energy gap δ = 1.8 ± 2.4 meV was also found to be consistent with the small low temperature enhancement of T −1 1 (T ). The discrepancy between the static gap ∆ K and the dynamic gap ∆ T1 has been observed in numerous spin-gap systems [41] and is related to differences in the processes that contribute to the Knight shift and the spin-lattice relaxation. In the majority of these spin-gap systems the dynamic gap ∆ T1 > ∆ K and on average ∆ T1 /∆ K = 1.73. In the case of PuB 4 we find ∆ T1 /∆ K = 1.6 ± 0.3. While NMR is not sensitive to the surface states in bulk powders or single crystals [42], it is a powerful microscopic probe of the bulk properties of topological materials. Our 239 Pu NMR results are consistent with a bulk gap which is only slightly suppressed from the DFT+SOC calculated value of 254 meV. In addition to the dominant gap-like behavior evidenced by K c (T ) and T −1 1 (T ), we also find a small peak at low temperature that is reminiscent of the 11 B T −1 1 (T ) in SmB 6 [38] and YbB 12 [33]. In SmB 6 the peak is thought to be due to bulk magnetic in-gap states, and while the nature of these states is still controversial, it has been suggested that these states are identical to the topologically protected surface states [43]. In PuB 4 we find that T −1 1 (T = 5 K) is strongly field dependent as shown in the inset of Fig. 3(b), which is similar to previous field dependent measurements of SmB 6 [39]. These results motivate further investigation of the field and Pu-substitution dependence of T −1 1 over a wide temperature range. Previous transport measurements find a much smaller gap ∆ = 35 meV [19]. This is also the case in YbB 12 , where NMR finds a larger gap than resistivity, and may be related to the presence of in-gap states which account for the low temperature enhancement in T −1 1 . This discrepancy motivates Hall coefficient measurements in PuB 4 (which in YbB 12 agree with the NMR-measured gap), as well as surface-sensitive tunneling or spin-polarized ARPES measurements. Finally, we note that measurements comparing 11 B and 10 B T −1 1 in YbB 12 and Yb 0.99 Lu 0.01 B 12 provide evidence for another interpretation of the low temperature relaxation enhancement, namely that it may be driven by fluctuations of defect-induced magnetic centers and spindiffusion-assisted relaxation [44]. These YbB 12 results motivate further measurements and comparison of 11 B and 10 B T −1 1 in PuB 4 . To conclude, we have performed 239 Pu NMR measurements for the second time ever in powdered and single crystalline PuB 4 . We extracted the isotropic and anisotropic shifts from the uniaxially symmetric powder pattern and demonstrate that one can observe the 239 Pu NMR signal in anisotropic environments and up to room temperature. The large contrast of the orbital shift between the purely ionic insulator PuO 2 (∼ +24.7 %) and band insulator PuB 4 (∼ −0.5 %) provide us with new tool to investigate the nature of the chemical bond based on the value of the 239 Pu shift. Single crystal 239 Pu NMR measurements of K c (T ) and T −1 1 (T ) provide unique access to bulk gap-like behavior with an energy gap that is only slightly suppressed with respect to DFT+SOC calculations, and T −1 1 (T ) also evidences the existence of bulk in-gap states. Our confirmation of a bulk gap motivate future surface sensitive measurements to confirm the theoretical prediction that PuB 4 is a topological insulator. Los Alamos National Laboratory was performed with the support of the Los Alamos LDRD program. TEA-S was supported as part of the Center for Actinide Science and Technology (CAST), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award Number de-sc0016568. HS was also partly supported by JSPS KAK-ENHI Grant Number JP16KK0106. APD acknowledges a Director's Postdoctoral Fellowship supported through the Los Alamos LDRD program. EXPERIMENTAL AND THEORETICAL METHODS Single crystals of PuB 4 were grown from aluminum flux. Elemental α-Pu (isotopic mixture: 0.013 wt. % 238 Pu, 93.96% 239 Pu, 5.908% 240 Pu, 0.098% 241 Pu, and 0.025% 242 Pu) was placed in a 10 ml alumina crucible along with boron pieces (99.9999%) and aluminum shot (99.999%) in the molar ratio Pu:B:Al=1:6:600. The constituents were heated in flowing purified argon to 1450 • C at a rate of 50 • C/hour, held at 1450 • C for 10 hours, then cooled slowly at 5 • C/hour to 1000 • C at which point the furnace was turned off. After the furnace reached room temperature the crucibles were removed from the furnace. The aluminum was dissolved from the crucible by etching with a NaOH (500 mL, 8 M) solution. The etching was preformed inside a 1 L Pyrex beaker, which was capped with Teflon foil, all of which sat in an 1170 mL Pyrex crystallization dish. After two days all of the aluminum dissolved leaving behind well-faceted PuB 4 crystals with typical dimensions 1×1×0.3 mm 3 . No trace of the cubic PuB 6 structure was found in the sample batch. The single crystal X-ray diffraction results indicate full occupancy of the B sites with no interstitial or free B. The PuB 4 single crystals were ground to a powder using an agate mortar and pestle. The powder was then sifted through a 125-µm sieve, then through a 30-µm sieve to remove the finest particles. 125 mg of powder between 30 and 125 µm was loaded into an NMR coil embedded in a Stycast 1266 epoxy cube with dimensions 20×20×20 mm 3 to avoid radioactive contamination. The cube was hollow along the coil's axis with both of its ends sealed by titanium frits with 2-µm diameter pores, allowing for thermal contact via He gas heat transfer. All NMR spectral data were collected by performing an optimized spin-echo π/2 − τ − π pulse sequence. T −1 1 was measured by integration of the phase-corrected real part of the spin echo following an inversion or saturation recovery π−t wait −π/2−τ −π pulse sequence. Home-built NMR probes with variable cryogenic capacitors were used in conjunction with superconducting NMR magnets that have a field homogeneity better than 10 ppm over a 1-cm diameter spherical volume. DFT calculations were performed using the WIEN2k code [1]. For the exchange-correlation potential we used the parameterization of Perdew-Burke-Ernzerhof based on the generalized gradient approximation [2], and spinorbit interactions were included through a second variational method. The electric field gradient is subsequently determined directly as a second derivative of the electrostatic potential, which is obtained by solving Poisson's equation using the self-consistently determined charge density from the all-electron DFT calculation [3]. The effects of correlations were included using a DFT+DMFT approach [4,5]. Allowing a spin-polarized solution for the DFT calculation incorrectly leads to a metallic solution with a magnetic moment of 4.4 µ B per formula unit. Fully self-consistency calculations in both charge density and impurity Green's functions were carried out at a temperature of T = 116 K. The Coulomb interaction U = 4.5 eV was used together with remaining Slater integrals F 2 = 6.1 eV, F 4 = 4.1 eV and F 6 = 3.0 eV reduced by 30%. The quantum impurity model was solved by using the strong coupling version of the continuous time quantum Monte Carlo method [6,7]. The calculations involved at least 10 iterations of charge self-consistency (each containing 1 DMFT iteration and 10 LDA iterations). CONTROL EXPERIMENT As 239 Pu nuclear magnetic resonance (NMR) has only been observed in one previous compound, it was prudent to perform a control experiment to confirm that the observed signal was indeed intrinsic to the PuB 4 . The same experiment was conducted after replacing the sample and encapsulation cube with an identical but empty cube. The rest of the experimental apparatus was unchanged and operated under identical conditions of temperature, magnetic field, and observation frequency. No NMR signal was observed in this control experiment. Furthermore, we performed a two-dimensional rf pulse power optimization experiment on the empty control-cube at the resonance conditions stated in the main text. No signal was observed for any excitation conditions. PU HYPERFINE COUPLING ANALYSIS We attempt a naive analysis of the 239 Pu hyperfine coupling A c in a single crystal sample of PuB 4 for external field aligned with the crystalline c-axis by plotting K c vs χ c with temperature as an implicit parameter. We observe a positive slope of the K-χ plot and use the slope of a linear fit to estimate the hyperfine coupling A c . The K-χ plot is not linear, and therefore we performed fits to two separate regions. The high temperature region was taken to be 200-300 K and the intermediate temperature region was chosen to be 100-200 K. The hyperfine couplings quoted in Fig. S1 should be taken with caution as it is known that the relationship between the hyperfine coupling and the slope of the K-χ plot is no longer trivial in the case of strong spin-orbit coupling [8]. In spite of this caveat we find a relatively large hyperfine coupling at high temperatures on the order of the free-ion calculation of 258.3 T/µ B for Pu 4+ [9]. Below 100 K the Knight shift saturates and remains constant, whereas the susceptibility increases down to the lowest temperatures. This low temperature tail in the bulk susceptibility likely results from paramagnetic impurities. The Knight shift is unaffected by the small concentration of impurities and is able to probe the small of density of states within the gap. At high temperatures the susceptibility increases only slightly, likely due to the nearly nonmagnetic state of the Pu f -electrons in PuB 4 . PU GAP FITTING PROCEDURE In the main text we briefly describe the fitting procedure for extracting the gap values of the static gap ∆ k and the dynamic gap ∆ T1 . Here we detail the exact method and all optimized fit parameters for the benefit of the reader (with some repetition for clarity) .The general form of the Knight shift is given by, K(T ) ∝ f (E, T )[1 − f (E, T )]ρ(E)dE,(S1) where f (E, T ) is the Fermi function and ρ(E) is the density of states (DOS). The spin-lattice relaxation rate is given by, T −1 1 (T ) ∝ f (E, T )[1 − f (E, T )]ρ(E) 2 dE.(S2) In the above equations the Fermi function given by, f (E, T ) = 1 exp E−E F 2k B T + 1 ,(S3) where E F is the Fermi energy and k B is the Boltzmann constant. We model the DOS as, ρ(E) = ρ i (T ) for δ < \|E\| < W i = ρ for ∆ < \|E\| < W,(S4) and zero otherwise. Within this model the in-gap DOS must have some temperature dependence due to the fact that the relaxation rate is peak-like and not simply activated. Following the same analysis scheme used by Caldwell et al. for SmB 6 [10], we express the in-gap DOS as ρ i (T ) = ρ i0 e −T /T0 ,(S5) where ρ i0 is the maximum value of the in-gap DOS and T 0 is a characteristic temperature scale. For the actual fits we must include a temperature independent offset K 0 for the case of K(T ) and (T −1 1 ) 0 is a temperatureindependent residual relaxation rate. The fitting procedure was written in the Python programming language and makes use of the numpy (numerical python) and lmfit (Levenberg-Marquardt fit) packages. We write objective functions for K(T )(Eqn. S1) and T −1 1 (T ) (Eqn. S2) which are integrated over energy for each temperature at which experimental data exists. We then input the respective objective function into a second minimization function that uses the lmfit package's Minimizer class. We hold the Fermi energy to be at the gap center as the gross features of the experimental data are not well reproduced for E F ≈ 0. Initially we explore the parameter space for physically reasonable parameter guesses for the free parameters (rho i0 , T 0 , δ, W i , ρ, ∆, W , and either K 0 or (T −1 1 ) 0 ). An initial minimization was performed using the Nelder-Mead method which is fast and better at finding the global minimum of χ 2 , but is unable to calculate the standard errors. Then we use the minimized output fit coefficients of the Nelder-Mead method as an input for the Levenberg-Marquardt χ 2 minimizer, yielding the gap values and standard errors quoted in the main text of the manuscript and all coefficients shown in Table S1. The fitting procedure for K(T ) is straightforward. Due to the fact that we do not observe any low temperature enhancement in the Knight shift, we hold the in-gap states to be zero by holding rho i0 = 0 and. We also verified that allowing ρ i (T ) to vary results in rho i0 ≈ 0. The bandwidth W was found to evolve to values larger than 1 eV, and to have little effect on the functional form of the objective function for W 1000 meV and therefore we hold W = 1500 meV (this was also true for the T −1 1 (T )). So the only remaining parameters that were allowed to vary are ρ, ∆, and K 0 . The optimized values obtained for the parameters for K(T ) are given in the first row of Table S1. The numerical integral is carried out over the range E = −3 to 3 eV with an energy resolution of 0.06 meV. ∆ (meV) δ (meV) K0 (%) (T −1 1 )0 (s −1 ) ρ (arb. units) ρi0 (arb. units) T0 (K) W (meV) Wi (meV) K(T ) 155.6 ± 11. For T −1 1 (T ) the fitting procedure was more complicated due to the appearance of a low temperature peak similar to SmB 6 and YbB 1 2 for example. We hold the Fermi energy E F = 0 and W = 1500 meV as mentioned above, and used the Nelder-Mead method to search the parameter space for reasonable initial guesses. We found that the bandwidth of the in-gap states had a similar lack of effect on the model function for W i 50 meV, and so we chose to hold W i = 100 meV. The best fit coefficients can be found in the second row of Table S1. We found good stability of the optimized parameters associated with the dominant gap-like behavior (ρ and ∆) independent of the initial guesses, indicating that we are indeed finding the global minimum of these coefficients in the parameter space. Even for the case where we ignore the in-gap states and hold ρ i0 = 0 we find ∆ ∼ 250 meV. On the other hand, the best fit coefficients associated with the in-gap states have larger standard errors (e.g. the small gap δ). This is partially due to the small effect of the in-gap states on T −1 1 (T ), but also due to correlations between the parameters. As such, we do not draw conclusions in the main text from the exact values of these quantities. PU SPIN-LATTICE RELAXATION RATE ANISOTROPY For an initial study of nuclear spin-lattice relaxation behavior in powdered single crystals of PuB 4 , we measured the 239 Pu nuclear magnetization recovery curves for both the B 0 c shoulder of the spectrum (high-field side) and the B 0 ⊥ c peak in a powder of PuB 4 (shown in Fig. S2). We fit the 239 Pu inversion recovery curves to the form M N (t) = M N (∞) 1 − αe −(t/T1) β ,(S6) where M N (∞) is the equilibrium nuclear magnetization, α is the inversion fraction, T 1 is the spin-lattice relaxation time, and β is a stretching exponent that modifies the expected single exponential behavior (β = 1). is the median of the probability distribution of relaxation rates and β is a measure of the logarithmic full-width at half maximum, with the distribution approaching a Dirac delta function as β → 1 [11]. The appearance of stretched exponential behavior is not completely clear, but is partially caused by a sensitivity to crystalline imperfections caused by grinding to produce the powdered sample. In our single crystal measurements discussed in the main manuscript we find that β is much closer to unity and nearly temperature independent (β avg = 0.813); this may indicate sensitivity to self-irradiation induced disorder [12]. Even though we cannot extract precise values of the spin-lattice relaxation rates, we still glean useful information from the relaxation measurements. The median of the probability distribution for the 239 Pu relaxation times is in the range of hundreds of milliseconds to seconds and agrees with the single crystal data. B POWDER NMR RESULTS Data collected with 11 B NMR in PuB 4 support our observations from 239 Pu NMR. 11 B is an excellent NMR nucleus with I = 3 2 and 11 γ/2π = 13.6629814 MHz/T [13]. This is advantageous for the discovery of the 239 Pu signal, as one can first optimize the pulse conditions using 11 B NMR before searching for 239 Pu NMR. Furthermore, it is useful to have a well-studied nucleus with which to compare the shift and therefore gain insight into the hyperfine coupling to the Pu f -electrons that dominate the magnetic properties of the material. A comparison can therefore be made between the on-site hyperfine coupling to the 239 Pu and the transferred hyperfine coupling to the 11 B. Spectral Measurements There are three crystallographic B sites with local symmetries Tetragonal (4 ), orthorhombic (m.2m), and monoclinic (m). As such, all B sites are allowed by symmetry to have a nonzero electric field gradient (EFG). This, coupled with the fact that 11 B has I = 3 2 , results in a more complex powder spectrum than that of the 239 Pu. The field-swept spectrum at T = 4 K is shown in Fig. S3(a) and (b). The spectrum consists of three overlapping but well-distinguished powder patterns with differing shifts and EFG parameters; the quantities used to generate the simulation in Fig. S3 are listed in Table S2. We also compare the experimentally determined EFGs to those computed by DFT in Table S2. We note that there exists some ambiguity in the assignment of B(1) vs B(2) as the B(2) site has a very small experimental value of the EFG asymmetry parameter η. To extract the temperature dependence of the 11 B shift, we collected fast Fourier transforms of the sharp central transitions over a wide temperature range and employed a multipeak fitting scheme as shown in Fig. S3(c). We performed double-Gaussian fits on the two spectrally dominant central transitions of 11 B (see Fig. S3). In this process we neglect the effect of secondorder quadrupole perturbation, which will slightly shift and broaden the powder pattern by between 0% (for H 0 V zz , which is the principle axis of the EFG tensor) and 0.003% (for H 0 ⊥ V zz ). The maximum second order perturbation of the central transition is roughly an order of magnitude less than the shifts that we find in Fig. S3(d). The extracted shifts for the two spectrally dominant peaks are shown as a function of temperature in Fig. S3(d). We expect a transferred hyperfine coupling to the Pu f -electrons mediated by B p-orbital hybridization, which can be quite small in comparison to the on-site hyperfine coupling of the 239 Pu. We find little temperature dependence of the 11 B Knight shift, which is in agreement with the bulk susceptibility [14]. The temperature independence of the 11 B Knight shift highlights the importance of the large on-site hyperfine coupling of 239 Pu which allows for the observation of gap-like behavior in PuB 4 . Spin-Lattice Relaxation The spin-lattice relaxation rate of 11 (S7) Because the 11 B sites all have roughly the same shift and similar EFG, we find that the central transitions overlap as shown in Fig. S3(b). This results in a situation where deviations from β = 1 in Eq. S7 could be caused by different relaxation at the three 11 B sites and/or anisotropy of the relaxation rate. The fact that the stretching exponent 11 β ∼ 0.35 (fit shown in Fig. S4) is smaller than 239 β ∼ 0.6 indicates that this spectral overlap likely contributes to deviation from the expected relation function. Single crystal studies where one can measure T 1 at a wellseparated satellite transition of a single site are required to extract the precise value of T 1 for 11 B. As the focus of this work is the 239 Pu, we chose to study a powder which enables extraction of both 239 K c and 239 K a from a single powder pattern at each temperature. In spite of this overlap, we find that the relaxation rate distribution of the 239 Pu is peaked roughly an order of magnitude faster than that of the 11 B relaxation rate distribution, indicative of the difference in strength of the hyperfine fields at the nuclei. The magnitude of the relaxation rates reflect that the Pu 5f -electrons are in a non-magnetic configuration. If the 5f -electrons were magnetic, then fluctuations of the transferred hyperfine fields at the 11 B site would drive fast much faster relaxation of the 11 B, and fluctuations of the on-site hyperfine field would drive extremely fast relaxation at the 239 Pu site such that the NMR signal would relax too quickly to observe. FIG. 1 . 1(a) Unit cell of PuB4 illustrating the single plutonium site and three inequivalent boron sites[25]. (b) Density of states (DOS) and partial DOS as a function of energy calculated within density functional theory including spin-orbit coupling. The inset shows an expanded region near the Fermi energy where there is an energy gap of ∆ ∼ 254 meV. FIG. 2 . 2(a) 239 Pu nuclear magnetic resonance (NMR) field-swept spectra of powdered single crystals of PuB4 at several frequencies at T = 4 K. Spectra are normalized to the maximum value and offset vertically so as to correspond to the observed frequency f0 on the left axis. Red circles and line indicate the resonant condition of 239 Pu in PuB4 239 γ(1 + KPuB 4 )/2π = 2.288 ± 0.001 MHz/T (at T = 4 K) and green dashed line shows 239 γ(1 + KPuO 2 )/2π = 2.856 ± 0.001 MHz/T as determine previously [26]. (b) 239 Pu NMR field-swept spectra at f0 = 20.222 MHz offset vertically for several temperatures. Solid red curves are best fits as described in the text. Vertical dashed line indicates zero shift K = 0 using 239 γ/2π = 2.29 MHz/T. (c) 239 Kiso and 239 Kax vs temperature extracted from fits in (b). FIG. 3 . 3Single crystal 239 Pu NMR data for external field aligned along the crystallineĉ direction. (a) Shift Kc vs. temperature and fit (solid line) to gap-like behavior as discussed in the text which yields an energy gap ∆K = 155.6 ± 11.0 meV, but is not sensitive to the in-gap states. The inset shows the model density of states ρ(E) vs. energy E employed in the fits. (b) Spin-lattice relaxation rate T −1 1 . S1. Knight shift Kc for external field aligned along the crystalline c-axis vs the bulk magnetic susceptibility χc for the same crystal orientation with temperature as an implicit parameter. FIG. S2. Inversion recovery curve in powdered PuB4 showing the real part of the integrated 239 Pu spin-echo signal as a function of delay time t. The recovery curves are well fit by a stretched exponential function for an I = 1 2 nucleus (Eqn. S6) and show moderate anisotropy as discussed in the text. The inset shows the spectral positions where the recovery curves were acquired. For the crystallites with B 0 ⊥ c we find T −1 1 = 14.81 ± 0.93 s −1 and β = 0.55 ± 0.03, and for crystallites with B 0 c we find T −1 1 = 7.16 ± 92 s −1 and β = 0.56 ± 0.05. In this case, T −1 1 B was measured at the central transition of the powder pattern. The corresponding normal modes relaxation equation for inversion/saturation of the central transition of 11 B is given by, M N (t) = M N (∞ FIG. S3. (a) and (b) 11 B field-swept spectrum at T = 4 K with overlaid computer simulation shown as red lines. Experimental simulation parameters are shown in Table S2. (c) Multipeak fitting scheme employed for extraction of shift as a function of temperature from the two dominant central lines 11 B. (d) Temperature dependence of the 11 B shift 11 K of the two spectrally dominant overlapping central transitions determined from the two peaks shown in (c). FIG . S4. Nuclear magnetization saturation recovery curve for 11 B superposed central transition at T = 5 K with stretched exponential fit to Eqn. S7. Experimental results Theoretical calculations Site Wyckoff Position Occupancy Shift (%) \|Vzz\| (MHz) η (unitless) \|Vzz\| (MHz) η (unitless)TABLE S2. 11 B experimental spectral parameters and DFT calculated EFGs.B(1) 4e 1 -0.0295 0.6 0 0.400 0 B(2) 4h 1 -0.0295 0.42 < 0.03 0.579 0.052 B(3) 8j 2 -0.0525 0.353 0.664 0.343 0.642 * [email protected] Pu nuclear magnetic resonance in the candidate topological insulator PuB 4 -Supplemental Material A. P. Dioguardi, 1, 2, * H. Yasuoka, 1, 3 S. M. Thomas, 1 H. Sakai, 4, 1 S. K. Cary, 1 S. A. Kozimor, 1 T. E. Albrecht-Schmitt, 5 H. C. Choi, 1 J.-X. Zhu, 1 J. D. Thompson, 1 E. D. Bauer, 1 and F. Ronning 1 1 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2 IFW Dresden, Institute for Solid State Research, P.O. 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[ "Yield ratio of hypertriton to light nuclei in heavy-ion collisions from √ s NN = 4.9 GeV to 2.76 TeV", "Yield ratio of hypertriton to light nuclei in heavy-ion collisions from √ s NN = 4.9 GeV to 2.76 TeV" ]	[ "Tianhao Shao *[email protected]†[email protected]‡[email protected]§[email protected]¶[email protected] \nInstitute of Modern Physics\nKey Laboratory of Nuclear Physics and Ion-beam Application (MOE)\nFudan University\n200433ShanghaiChina\n\nShanghai Institute of Applied Physics\nChinese Academy of Science\n201800ShanghaiChina\n\nUniversity of Chinese Academy of Science\n100049BeijingChina\n", "Jinhui Chen \nInstitute of Modern Physics\nKey Laboratory of Nuclear Physics and Ion-beam Application (MOE)\nFudan University\n200433ShanghaiChina\n", "Che Ming Ko \nDepartment of Physics and Astronomy\nCyclotron Institute\nTexas A&M University\n77843College StationTexasUSA\n", "Kai-Jia Sun \nDepartment of Physics and Astronomy\nCyclotron Institute\nTexas A&M University\n77843College StationTexasUSA\n", "Zhangbu Xu \nBrookhaven National Laboratory\n11973UptonNew YorkUSA\n" ]	[ "Institute of Modern Physics\nKey Laboratory of Nuclear Physics and Ion-beam Application (MOE)\nFudan University\n200433ShanghaiChina", "Shanghai Institute of Applied Physics\nChinese Academy of Science\n201800ShanghaiChina", "University of Chinese Academy of Science\n100049BeijingChina", "Institute of Modern Physics\nKey Laboratory of Nuclear Physics and Ion-beam Application (MOE)\nFudan University\n200433ShanghaiChina", "Department of Physics and Astronomy\nCyclotron Institute\nTexas A&M University\n77843College StationTexasUSA", "Department of Physics and Astronomy\nCyclotron Institute\nTexas A&M University\n77843College StationTexasUSA", "Brookhaven National Laboratory\n11973UptonNew YorkUSA" ]	[]	We resolve the difference in the yield ratio S 3 = N 3 Λ H /N Λ N 3 He /Np measured in Au+Au collisions at √ s NN = 200 GeV and in Pb-Pb collisions at √ s NN = 2.76 TeV by adopting a different treatment of the weak decay contribution to the proton yield in Au+Au collisions at √ s NN = 200 GeV. We then use the coalescence model to extract information on the Λ and nucleon density fluctuations at the kinetic freeze-out of heavy ion collisions. We also show from available experimental data that the yield ratio S 2 = N 3 Λ H N Λ N d is a more promising observable than S 3 for probing the local baryon-strangeness correlation in the produced medium. PACS numbers: 25.75.-q, 25.75.Dw	10.1088/1674-1137/abadf0	[ "https://arxiv.org/pdf/2004.02385v1.pdf" ]	214,802,493	2004.02385	e8d545684759ceaaa3c5a63f1e1ffc3f071ee6a4	Yield ratio of hypertriton to light nuclei in heavy-ion collisions from √ s NN = 4.9 GeV to 2.76 TeV 6 Apr 2020 Tianhao Shao *[email protected]†[email protected]‡[email protected]§[email protected]¶[email protected] Institute of Modern Physics Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) Fudan University 200433ShanghaiChina Shanghai Institute of Applied Physics Chinese Academy of Science 201800ShanghaiChina University of Chinese Academy of Science 100049BeijingChina Jinhui Chen Institute of Modern Physics Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) Fudan University 200433ShanghaiChina Che Ming Ko Department of Physics and Astronomy Cyclotron Institute Texas A&M University 77843College StationTexasUSA Kai-Jia Sun Department of Physics and Astronomy Cyclotron Institute Texas A&M University 77843College StationTexasUSA Zhangbu Xu Brookhaven National Laboratory 11973UptonNew YorkUSA Yield ratio of hypertriton to light nuclei in heavy-ion collisions from √ s NN = 4.9 GeV to 2.76 TeV 6 Apr 2020(Dated: April 7, 2020)1 We resolve the difference in the yield ratio S 3 = N 3 Λ H /N Λ N 3 He /Np measured in Au+Au collisions at √ s NN = 200 GeV and in Pb-Pb collisions at √ s NN = 2.76 TeV by adopting a different treatment of the weak decay contribution to the proton yield in Au+Au collisions at √ s NN = 200 GeV. We then use the coalescence model to extract information on the Λ and nucleon density fluctuations at the kinetic freeze-out of heavy ion collisions. We also show from available experimental data that the yield ratio S 2 = N 3 Λ H N Λ N d is a more promising observable than S 3 for probing the local baryon-strangeness correlation in the produced medium. PACS numbers: 25.75.-q, 25.75.Dw I. INTRODUCTION The correlation coefficient C BS = −3 BS − B S S 2 − S 2 between the baryon number B and the strangeness S in a strongly interacting matter was first proposed in Refs. [1][2][3] as a probe of the properties of the matter produced in relativistic heavy ion collisions. A later study suggested, however, that the strangeness population factor S 3 = N 3 Λ H /N Λ N 3 He /Np measured in these collisions could serve as a better probe of the baryon number and strangeness correlation in the produced matter because of its different behaviors in QGP and the hadronic matter [4,5]. Experimentally, results on S 3 show an increase of its value from heavy ion collisions at AGS [6] to RHIC energy [7] and then a decrease to a small value in collisions at the LHC energy [8]. Compared to the predictions from the statistical model [5,9,10], the values of S 3 extracted from the RHIC data, which has large statistical uncertainties, are larger, and this has led to the questioning of the data and its interpretation. As to the different values of S 3 measured at RHIC and LHC, a possible explanation was provided in Ref. [11] by assuming an early freeze-out of the Λ than nucleons from the hadronic matter and a longer freeze-out time difference at RHIC than at LHC. This idea was further extended in Ref. [12] to study light nuclei production in relativistic heavy ion collisions by taking into consideration of their finite sizes compared to the size of produced hadronic matter at kinetic freeze out. Since the first theoretical estimate on the abundance of hypernuclei that could be produced in heavy-ion collisions in 1970s [13], there have been many improved studies on this very interesting problem [6][7][8][14][15][16][17], see more details on recent topical review articles [18][19][20]. For the lightest hypernucleus 3 Λ H, the separation energy of its Λ is very small in early measurements with a typical value of 130 ± 50 KeV [21] but changes to a larger value of 410 ± 120 ± 110 KeV in recent measurements using more precise method [22]. Since this value is significantly smaller than that of normal nuclei with a similar mass number [23], the 3 Λ H can be considered as a loosely bound d − Λ 2-body system, and the ratio S 2 = N 3 Λ H N Λ N d can also be used as an observable for probing the correlation between baryon and strangeness in relativistic heavy ion collisions. In this paper, we study the ratios S 3 = N Λ N d in the framework of the coalescence model. We first revisit the study of S 3 and find that the discrepancy between the ALICE and STAR measurements may be partially due to the difference in the primordial proton yield used in the two analyses. We then show that the ratio S 2 , particularly its ratio S 2 /B 2 with respect to B 2 , which is the coalescence parameter for the production of a deuteron from a proton and neutron pair, is a cleaner probe of the baryon-strangeness correlation in the produced hadronic matter from relativistic heavy ion collisions. II. THE S 3 = N 3 Λ H /N Λ N 3 He /Np RATIO IN RELATIVISTIC HEAVY ION COLLISIONS In this Section, we first review the experimental data on S 3 from relativistic heavy ion collisions and then use the coalescence model to extract from these data the correlation between Λ and nucleon density fluctuations in the produced matter. A. Experimental results on S 3 For the value of S 3 , it was measured to be 0.36 ± 0.26 in central 11.5A GeV/c Au + Pt collisions [6] and increases to 1.08 ± 0.22 ± 0.16 for Au+Au collisions at √ s NN = 200 GeV with a mixed event sample of central trigger and minimum bias trigger [7]. The measured value of S 3 decreases, however, to 0.60 ± 0.13 ± 0.21 in central Pb-Pb collisions at √ s NN = 2.76 TeV [8]. Preliminary data with an improved precision from Au+Au collisions at √ s NN = 200 GeV results in a 20% reduction of the value of S 3 [24], which makes the results from STAR and ALICE comparable within their experimental uncertainties. Replacing the proton yield in the S 3 ratio in the STAR analysis, which is based on the subtraction of protons from measured hyperon data, with the proton yield in the PHENIX data, which is obtained from a theoretical model for the same collision system and energy [25], also reduces the value of S 3 at √ s NN = 200 GeV to approximately same as the value measured by ALICE [7,8,24]. The difference between the results from STAR and ALICE is thus partially due to the different treatments in the subtraction of the weak decay contribution to the primordial proton yield. [26] indicate that more precise measurements of 3 Λ H in high energy heavy-ion collisions are needed. To see the physics that can be extracted from the ratio S 3 , we adopt the coalescence model for the present study. According to the COAL-SH formula in Ref. [27], the yield of certain nucleus consisting of N i number of the constituent species i (proton, neutron and Λ) of mass m i from the kinetically freezed-out hadronic matter of local temperature T and volume V K in a heavy ion collision can be written as N A = g rel g size g A A i m i 3/2 A i=1 N i m 3/2 i × A−1 i=1 (4π/ω) 3/2 V K x(1 + x 2 ) x 2 1 + x 2 l i G(l i , x).(1) In the above, g A = (2S + 1)/( A i=1 (2s i + 1)) is the coalescence factor for A nucleons and/or Λ of spin s i = 1/2 to form a nucleus of spin S, g rel is the relativistic correction to the effective volume in momentum space and is set to 1 in the present study due to the much larger nucleon mass than the effective temperature of the hadronic matter at kinetic freeze out, g size is the correction due to the finite size of produced nucleus and is also taken to be 1 in our study because of the much larger size of the hadronic matter than the sizes of produced light nuclei. The symbols l i and ω denote, respectively, the orbital angular momentum of the nucleon or Λ in the nucleus and the oscillator constant used in its wave function. For the value of x = (2T/ω) 1/2 , it is significantly larger than one because of the much larger size of the nucleus than the thermal wavelength of its constituents in the hadronic matter. Since the light nuclei considered in the present study all involve only the l = 0 s-wave, the suppression factor G(l i , x) due to the orbital angular momentum in the above equation is simply one. For the production of 3 Λ H and 3 He, their yields according to Eq.(1) after taking into account the approximations mentioned above are given by N3 Λ H = g3 Λ H (m Λ + m p + m n ) 3/2 m 3/2 Λ m 3/2 p m 3/2 n 2π T 3 N Λ N p N n V 2 K , N3 He = g3 He (2m p + m n ) 3/2 m 3 p m 3/2 n 2π T 3 N 2 p N n V 2 K .(2) Allowing possible nucleon and Λ density fluctuations in heavy ion collisions at lower energies due to the spinodal instability during the QGP to hadronic matter phase transition [27][28][29][30], we rewrite the density distributions of nucleons and Λ as n( r) = 1 V K n( r)d r + δn( r) = n + δn( r).(3) According to Ref. [31], this modifies Eq.(2) to N3 Λ H = g3 Λ H (m Λ + m p + m n ) 3/2 m 3/2 Λ m 3/2 p m 3/2 n 2π T 3 N Λ N p N n V 2 K (1 + α Λp + α Λn + α np ), N3 He = g3 He (2m p + m n ) 3/2 m 3 p m 3/2 n 2π T 3 N 2 p N n V 2 K (1 + ∆p + 2α np ),(4) if one neglects higher-order correlation coefficients of density fluctuations. In the above, ∆p = (δp) 2 / p 2 is the proton relative density fluctuation, and α Λp , α Λn , and α np are, respectively, the Λ-proton, Λ-neutron, and proton-neutron density fluctuation correlation coefficients α n 1 n 2 = δn 1 δn 2 /( n 1 n 2 ) with n 1 and n 2 denoting Λ or nucleon. Taking the same mass for proton and neutron, i.e., m p = m n = m, the yield ratio S 3 = N 3 Λ H /N Λ N 3 He /Np is then S 3 = g 1 + α Λp + α Λn + α np 1 + ∆p + 2α np ,(5)with g = m Λ +2m 3m Λ 3/2 ≈ 0.845. From the above equation, the sum of the correlation coefficients α Λp +α Λn between the Λ density and the proton or neutron density fluctuations can be determined from S 3 , ∆p, and α np according to α Λp + α Λn = S 3 g × (1 + ∆p + 2α np ) − α np − 1.(6) For the value of α np , we follow the method in Ref. [27] by using the deuteron yield after including in the coalescence formula COAL-SH the proton and neutron density fluctuations, i.e., N d = 2 3/2 g d 2π mT 3/2 N p N n V K (1 + α np ).(7) In terms of g d−p = 1 2 3/2 g d (2π) 3 = 2 1/2 3(2π) 3 ≈ 0.0019, O d−p = N d /N 2 p , R np = N p /N n , and V ph = (2πmT) 3/2 V K , the value of α np can then be calculated from α np = g d−p R np V ph O d−p − 1.(8) For the proton density fluctuation ∆p, we consider the ratio N3 He N 3 p × N p N n = 3 3/2 g3 He 2π mT 3 1 V 2 K (1 + ∆p + 2α np ),(9) and determine it from the relation ∆p = g3 He−p V 2 ph R np O3 He−p − 2α np − 1,(10) where g3 He−p = 1 3 3/2 g 3 He (2π) 6 = 4 3 3/2 (2π) 6 ≈ 1.25 × 10 −5 and O3 He−p = N3 He /N 3 p , and the factors V ph , R np and α np are the same as above. For the effective phase-space volume V ph occupied by nucleons in the hadronic matter at kinetic freeze-out, they can be evaluated from its value at chemical freeze-out by using the relation T 3/2 V K = λT 3/2 ch V ch , where T ch and V ch are, respectively, the temperature and volume of the system at chemical freeze-out, and λ is a parameter. For collisions at RHIC energies, we take the value of T ch from the grand canonical ensemble fits to the particle yields in Ref. [32] and that of V ch to be V ch = 4πR 3 /3 as in Ref. [32] for collisions at various centralities except for collisions at 0 − 80% centrality, where it is taken to be proportional to the charged particle multiplicity obtained from Ref. [33]. Using the results from Ref. [32] based on the strangeness suppressed canonical ensemble fits gives almost the same results. The values of T ch and V ch used in the present study for collision at AGS energy are taken from Ref. [34], and for collision at the LHC energy, they are taken from the COAL-SH model used in Ref. [35]. In our calculations, all hadrons are taken as point particles. Including an exclusive volume for each hadron would give a larger value for V ch [34]. For the value of λ, it is determined by assuming that the entropy per baryon is the same at the chemical and the kinetic freeze out after including in the hadronic matter all the resonances in PDG. This approach thus differs from the naive approach of a hadronic matter of constant number of nucleons expanding isentropically after chemical freeze-out, which would give a λ of value of unity, and also that in Ref. [27] based on a multiphase transport model. As to the value of R np = N p /N n , it can be determined from measured ratio of charged pions according to the relation N p /N n = (N π + /N π − ) 1/2 from the statistical model. In Table II, we summarize the values of above parameters used in the present study for central heavy ion collisions. For reference, we also provide in Table III their values for collisions at the centralities of 0-80% or 0-60%. Tables II and III, . These results will be shown in the next Section to compare with the correlation coefficient α Λd of the Λ and deuteron density fluctuations that is extracted from As shown in measured S 2 = N 3 Λ H N Λ N d ratio. III. THE S 2 = N 3 Λ H N Λ N d RATIO In this Section, we consider the S 2 = N 3 Λ H N Λ N d ratio in the framework of the coalescence model and its ratio S 2 /B 2 with respect to the coalescence parameter B 2 for the production of deuteron from the coalescence of proton and neutron as well as the extracted correlation coefficient α Λd between the Λ and deuteron density fluctuations using the experimental data from RHIC and the LHC. N3 Λ H = g3 Λ H (m Λ + m d ) 3/2 m 3/2 Λ m 3/2 d 2π T 3/2 N Λ N d V K (1 + α Λd ),(11) with α Λd = δn Λ δn d /( n Λ n d ) being the correlation coefficient between deuteron and Λ density fluctuations. The S 2 ratio is then S 2 = N3 Λ H N Λ N d = g3 Λ H (m Λ + m d ) 3/2 m 3/2 Λ m 3/2 d 2π T 3/2 1 V K (1 + α Λd ),(12)with g S 2 = 1 3 (m Λ +m d ) 3/2 m 3/2 Λ m 3/2 d (2π) 3/2 −1 ≈ 0.12, and from which we can express the density fluctuation correlation coefficient α Λd in terms of S 2 as In the left window of Fig. 1, we show by solid symbols the extracted ratio S 2 = N 3 Λ H N Λ N d using experimental data from AGS [6,38,39], RHIC [7,40,41] and LHC [8,36,42]. Open symbol represents the result for the 3 Λ H yield in collision at 0-10% centrality where the RHIC data is obtained from multiplying the measured 3 Λ H data at 0-80% centrality by a factor of 3. We have checked using data available from the RHIC BES program that the yield ratio of deuteron to Λ is a factor 3 larger in collisions at 0-10% centrality than at 0-80% centrality, independent of the collision energy [40,43]. Solid and dashed lines in the left window of Fig. 1 are results from statistical model calculations [27] using the parameters in Tables II and III. One sees that although there are missing data points in the large collision energy range, the ratio S 2 in collisions at 0-10% centrality seems independent of the collision energy √ s NN considering the large uncertainty of the AGS data. Also, the model calculation describes reasonably well the data at the LHC energy but over-predicts the results for collisions at AGS and RHIC energies. α Λd = g S 2 S 2 T 3/2 V K − 1.(13)3 10 × d) × Λ H/( Right window of Fig. 1 TeV. The negative α Λd and the negative α np shown in Tables II and III could be due to an underestimate of the value of the λ parameter or the kinetic freeze-out volume used in our study. To fully understand these results requires detailed studies based on microscopic models for light cluster production in high energy heavy-ion collisions [44][45][46]. Compared with the correlation coefficient α Λp + α Λn extracted from S 3 shown by solid and open squares for collisions at centralities of 0-10% and 0-80%, respectively, which seems to vary very little over a broad range of collision energies, the deviation of its value from zero is larger than that of α Λp + α Λn , as it shows a more visible √ s NN dependence. This may suggest that α Λd is a cleaner observable than α Λp + α Λn for studying the √ s NN dependence of baryon density fluctuations and their correlations as seen from the comparison of Eq. (12) to Eq. (5). Future experimental measurements in a broad collision energy range from AGS to RHIC will be very useful for shedding lights on the underlying physics. B. The S 2 /B 2 ratio In the coalescence model, the yield ratio S 2 = N3 Λ H /(N Λ N d ) is the coalescence parameter for the production of 3 Λ H if it is considered as a bound system of Λ and deuteron. Because of the strangeness carried by Λ, the S 2 may be different from the coalescence parameter B 2 for the production of deuteron from the coalescence of proton and neutron [19,[47][48][49]. From Eq.(12) for S 2 and a similar equation for B 2 , given by B 2 = N d N p N n = g d 1 m 3/2 p 2π T 3/2 1 V K (1 + α np ),(14) taking N n = N p then leads to S 2 B 2 = N3 Λ H N Λ N d N d N p N n = g 1 + α Λd 1 + α np ,(15) where g = Similarly, we can introduce the coalescence parameter B 3 = N 3 He NpNpNn for the production of 3 He from the three-body coalescence of two protons and a neutron, and the coalescence parameter B s3 = N 3 Λ H N Λ NpNn for the production of 3 Λ H from the three-body coalescence of Λ, proton and neutron. Their ratio is exactly the S 3 discussed in Section II. Since the ratio S 2 /B 2 does not involve the proton density fluctuation ∆p and other mixed density fluctuation correlations, it seems a more sensitive observable than S 3 for studying the Λ density fluctuation. predicted by the statistical model (solid horizontal bars) for collision at 0 − 10% centrality [36,40]. Right window: Values of α Λd extracted from experimental results according to Eqs. (16) and (13). Left window of Fig. 2 shows results for the yield ratio S 2 /B 2 = N3 Λ H /(N Λ N d )/(N d /N 2 p ) from experimental data (solid triangles) and predicted by the statistical model (solid horizontal bars) using the proton, deuteron and 3 Λ H yields from the full p T range. It is seen that the measured yield ratio increases slightly with increasing collision energy as in predictions from the statistical model. We note that the value of B 2 has also been determined in experiments from the proton, neutron and deuteron momentum spectra in a small p T window [40]. The S 2 /B 2 ratio obtained from collisions at the LHC energy for momentum per constituent p T /A = 1.4 GeV/c [8,36,42] is 0.899±0.171. Within their uncertainties, this value is similar to that obtained using yields from the full p T range. However, the S 2 /B 2 ratio measured for particular p T bins are unavailable from experiments in the energy range between those available at AGS and RHIC, and this is due to the lack of 3 Λ H p T spectra at these energies. Future measurements of 3 Λ H spectra over a broad energy range are needed for extracting the Λ and deuteron density correlation coefficient α Λd discussed below. The Λ and deuteron correlation coefficient α Λd can also be extracted from the yield ratio S 2 /B 2 given in Eq. (15), that is α Λd = N3 Λ H N Λ N d (N d /N 2 p ) g × (1 + α np ) − 1,(16) IV. CONCLUSION In summary, we have argued that both the ratio S 2 and the ratio S 2 /B 2 , where S 2 and B 2 are, respectively, the coalescence parameter for the production of hypertriton from Λ and deuteron and of deuteron from proton and neutron, are more sensitive observables than the previously proposed ratio S 3 = N 3 Λ H /N Λ N 3 He /Np for studying the local baryon-strangeness correlation in the matter produced in relativistic heavy ion collisions. We have substantiated this argument in the framework of baryon coalescence by demonstrating that the correla-tion coefficient α Λd between Λ and deuteron density fluctuations extracted from measured S 2 /B 2 shows a stronger dependence on the energy of heavy ion collisions than the correlation coefficients α Λp + α Λn between Λ and nucleon density fluctuations extracted from the measured S 3 . Experimental measurement of the ratio S 2 /B 2 is expected to provide a promising way to study the strangeness and baryon correlation in the matter produced from heavy ion collisions as the collision energy or the baryon chemical potential of produced matter is varied, which in turn can shed lights on the properties of the QGP to hadronic matter phase transition during the collisions. He /Np and S 2 = B. S 3 in the coalescence model and the nucleon and Λ density fluctuations the extracted values for the neutron and proton density fluctuation correlation α np are negative with appreciable magnitude, indicating that neutrons and protons are anti-correlated in the matter produced in relativistic heavy ion collisions, similar to that found in Ref.[27]. For the proton density fluctuation ∆p, the extracted values are 0.656 ± 0.049, 0.536 ± 0.083, and 0.727 ± 0.085 for collisions at √ s NN = 4.9, 200, and 2760 GeV, respectively[6,7,36]. It shows a non-monotonic behavior as a function of collision energy from √ s NN = 7.7 GeV to 200 GeV using the preliminary data of 3 He yield from Ref.[24], similar to that of the neutron density fluctuation extracted from the yield N Λ N d ratio in the coalescence modelApproximating 3 Λ H as a bound system of Λ and deuteron, the yield of 3 Λ H in heavy ion collisions can be calculated from the coalescence of Λ and deuteron using Eq.(1). Including the effect of deuteron and Λ density fluctuations, it is given by N Λ N d ratio (left window) and the density fluctuation correlation coefficients α Λp + α Λn and α Λd (right window) extracted from the experimental data (symbols) and calculated from the statistical model (horizontal bars) using parameters inTables II and III. See texts for details. shows the value of the correlation coefficients α Λd as a function of the collision energy for collisions at 0 − 10% centrality (solid symbols) and at 0 − 80% centrality (open symbols). The large uncertainty at √ s NN = 2.76 TeV is due to the standard error propagation from the large volume V ch used in collisions at the LHC energy. Within current experimental uncertainties, the value of α Λd , which is negative and thus indicates an anti-correlation between the Λ and deuteron density fluctuations, becomes slightly less negative as the collision energy increases and approaches towards zero at √ s NN = 2.76 .23. The ratio S 2 /B 2 thus carries information on the difference between α np and α Λd and thus the difference between the baryon-baryon correlation and the baryon-strangeness correlation. FIG. 2 . 2Left window: The S 2 /B 2 ratio extracted from experimental data (solid triangles) and N by taking advantage of the empirical fact that the neutron and proton density fluctuation correlation α np is less affected by T and V K . Shown in the right window ofFig. 2by trianglesare the values of α Λd extracted from the experimental results using Eq.(16). They are seen to have similar values as those obtained from Eq.(13) using S Λ N d , which are shown by filled circles and also inFig. 1where it is compared with the density fluctuation correlation coefficient α Λp + α Λn extracted from S of T.S. and J.C. was supported in part by the National Natural Science Foundation of China under Contract Nos. 11890710, 11775288, 11421505 and 11520101004, while that of C.M.K. and K.J.S. was supported by the US Department of Energy under Contract No. de-sc0015266 and the Welch Foundation under Grant No. A-1358. Table I Isummarizes the published S 3 results, together with the value using the proton yield from the PHENIX data in Au+Au collisions at √ s NN = 200 GeV. It is seen that the values of S 3 from STAR and ALICE are now comparable within their large uncertainties. Experimentally, there also exists a puzzle related to the 3 Λ H/ 3 He ratio [10, 11]. Its value is 0.82±0.16±0.12 in 0−80% centrality of Au+Au collisions at √ s NN = 200 GeV at RHIC [7], which is considerably larger than the value of 0.47 ± 0.10 ± 0.13 in Pb-Pb collisions at √ s NN TABLE I . IValues of S 3 from AGS, STAR and ALICE, where PH means that the proton yield istaken from PHENIX [25]. See texts for details. Experiment S 3 AGS 0.36 ± 0.26 STAR 1.08 ± 0.22 ± 0.16 STAR + PH 0.90 ± 0.22 ± 0.15 ALICE 0.60 ± 0.13 ± 0.21 = 2.76 TeV and 0-10% centrality at the LHC [8]. Although the preliminary measurement with improved precision by STAR in Au+Au collisions at √ s NN = 200 GeV results in a reduction of the 3 Λ H/ 3 He ratio to a value comparable with that from ALICE, its large uncertainties TABLE II . IIValues of parameters used for 0-10% central collisions.√ s NN (GeV) T ch (GeV) V ch (fm 3 ) R np α np λ 4.9 0.132 640 0.925 −0.781 ± 0.026 2.23 7.7 0.144 806 0.966 −0.744 ± 0.024 2.60 11.5 0.151 875 0.977 −0.763 ± 0.019 2.76 19.6 0.158 843 0.987 −0.830 ± 0.014 2.92 27 0.160 846 0.988 −0.848 ± 0.012 2.97 39 0.160 951 0.990 −0.834 ± 0.013 3.00 62.4 0.164 1215 0.992 −0.792 ± 0.037 3.16 200 0.168 1334 0.992 −0.726 ± 0.038 3.30 2760 0.156 4320 1.00 −0.717 ± 0.023 2.94 TABLE III . IIISame as Table IIfor values of parameters used in calculations for 0-80% or 0-60% centralities in collisions at RHIC energies and the LHC energy, respectively. 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[ "QUANTIZING BRAIDS AND OTHER MATHEMATICAL OBJECTS: THE GENERAL QUANTIZATION PROCEDURE", "QUANTIZING BRAIDS AND OTHER MATHEMATICAL OBJECTS: THE GENERAL QUANTIZATION PROCEDURE" ]	[ "Samuel J Lomonaco ", "Louis H Kauffman " ]	[]	[]	Extending the methods from our previous work on quantum knots and quantum graphs, we describe a general procedure for quantizing a large class of mathematical structures which includes, for example, knots, graphs, groups, algebraic varieties, categories, topological spaces, geometric spaces, and more. This procedure is different from that normally found in quantum topology. We then demonstrate the power of this method by using it to quantize braids.This general method produces a blueprint of a quantum system which is physically implementable in the same sense that Shor's quantum factoring algorithm is physically implementable. Mathematical invariants become objects that are physically observable.	10.1117/12.883681	[ "https://arxiv.org/pdf/1105.0371v1.pdf" ]	118,386,231	1105.0371	7ea72cacd6671ec7cf17b8a1a855a3a407d7ef4e	QUANTIZING BRAIDS AND OTHER MATHEMATICAL OBJECTS: THE GENERAL QUANTIZATION PROCEDURE Samuel J Lomonaco Louis H Kauffman QUANTIZING BRAIDS AND OTHER MATHEMATICAL OBJECTS: THE GENERAL QUANTIZATION PROCEDURE Extending the methods from our previous work on quantum knots and quantum graphs, we describe a general procedure for quantizing a large class of mathematical structures which includes, for example, knots, graphs, groups, algebraic varieties, categories, topological spaces, geometric spaces, and more. This procedure is different from that normally found in quantum topology. We then demonstrate the power of this method by using it to quantize braids.This general method produces a blueprint of a quantum system which is physically implementable in the same sense that Shor's quantum factoring algorithm is physically implementable. Mathematical invariants become objects that are physically observable. Introduction Extending the methods found in previous work [12,13] on quantum knots and quantum graphs, we describe a general procedure for quantizing a large class of mathematical structures which includes, for example, knots, graphs, groups, algebraic varieties, categories, topological spaces, geometric spaces, and more. This procedure is different from that normally found in quantum topology. We then demonstrate the power of this method by using it to quantize braids. We should also mention that this general method produces a blueprint of a quantum system which is physically implementable in the same sense that Shor's quantum factoring algorithm is physically implementable. Moreover, mathematical invariants become objects that are physically observable. The above mentioned general quantization procedure consists of two steps: Step 1. Mathematical construction of a motif system S , and Step 2. Mathematical construction of a quantum motif system Q from the system S . Caveat. The term "motif" used in this paper should not be confused with the use of the term "motive" (a.k.a., "motif") found in algebraic geometry. Part I. A General Procedure for Quantizing Mathematical Structures We now outline a general procedure for quantizing mathematical structures. One useful advantage to this quantization procedure is that the resulting system is a multipartite quantum system, a property that is of central importance in quantum computation, particularly in regard to the design of quantum algorithms. In a later section of this paper, we illustrate this quantization procedure by using it to quantize braids. Examples of the application of this quantization procedure to knots, graphs, and algebraic structures can be found in [12,13,8]. 2.1. Stage 1. Construction of a motif system S n . Let T = {t 0 , t 1 , . . . , t −1 } be a finite set of symbols, with a distinguished element t 0 , called the trivial symbol, and with a linear ordering denoted by '<'. Let T ×N be the the N -fold cartesian product of T with an induced LEX ordering also denoted by '<', and let S (n) be the group of all permutations of T ×N . For positive integers N and N (N < N ), let ι : T ×N −→ T ×N be the injection defined by t j(0) , t j(1) , t j(2) , . . . , t j(N −1) −→   t j(0) , t j(1) , t j(2) , . . . , t j(N −1) , N −N t 0 , t 0 , t 0 , . . . , t 0   Next, let N 0 < N 1 < N 2 < . . . be a monotone strictly increasing infinite sequence of positive integers. For each positive integer n ≥ 0, let M (n) be a subset of T ×Nn such that ι M (n) lies in M (n+1) , i.e., ι M (n) ⊂ M (n+1) . Moreover, for each non-negative integer n, let A(n) be a subgroup of the permutation group S (N n ) having M (n) as an invariant subset, and such that the injection ι : T ×Nn −→ T ×Nn+n induces a monomorphism ι : A (N n ) −→ A (N n+1 ), also denoted by ι. We define a motif system S n = S M (n) , A(n) of order n as the pair M (n) , A(n) , where M (n) is called the set of motifs, and where A(n) is called the ambient group. Finally, we define a nested motif system S * = S * M ( * ) , A( * ) as the following sequence of sets, groups, injections, and monomorphisms: S 1 M (1) , A(1) ι −→ S 2 M (2) , A(2) ι −→ · · · ι −→ S n M (n) , A(n) ι −→ · · · Remark 1. There is also one more symbolic motif system that is often of use, the direct limit motif system defined by S ∞ M (∞) , A(∞) = lim −→ S * M ( * ) , A( * ) , where lim −→ denotes the direct limit. Stage 2. Motif equivalence and motif invariants. Let S n = S M (n) , A(n) be a motif system of order n. Two motifs m 1 and m 2 of the set M (n) are said to be of the same n-motif type, written m 1 ∼ n m 2 , if here exists an element g of the ambient group A(n) which takes m 1 to m 2 , i.e., such that gm 1 = m 2 . The motifs m 1 and m 2 are said to be of the same motif type, written m 1 ∼ m 2 , if there exists a non-negative integer k such that ι k m 1 ∼ n+k ι k m 2 . We now wish to answer the question: Question. What is meant by a motif invariant? Definition 1. Let S n = S M (n) , A(n) be a motif system, and let D be some yet to be chosen mathematical domain. By an n-motif invariant I (n) , we mean a map I (n) : M (n) −→ D such that, when two motifs m 1 and m 2 are of the same n-type, i.e., m 1 ∼ n m 2 , then their respective invariants must be equal, i.e., I (n) (m 1 ) = I (n) (m 2 ) . In other words, I (n) : M (n) −→ D is a map that is invariant under the action of the ambient group A(n), i.e., I (n) (m) = I (n) (gm) for all elements of g in A(n). Stage 3. Construction of the corresponding quantum motif systems Q n . We now use the nested motif system S * to construct a nested sequence of quantum motif systems Q * . For each non-negative n, the corresponding n-th order quantum motif system Q n = Q M (n) , A(n) consists of a Hilbert space M (n) , called the quantum motif space, and a group A(n), also called the ambient group. The quantum motif space M (n) and the ambient group A(n) are defined as follows: • The quantum motif space M (n) is the Hilbert space with orthonormal basis \|m : m ∈ M (n) . The elements of M (n) are called quantum motifs. • The ambient group A(n) is the unitary group acting on the Hilbert space M (n) consisting of all linear transformations of the form g : M (n) −→ M (n) : g ∈ A(n) , where g is the linear transformation defined by g : M (n) −→ M (n) \|m −→ \|gm Since each element g in A(n) is a permutation, each g permutes the orthonormal basis \|m : m ∈ M (n) of M (n) . Hence, g is automatically a unitary transformation. It follows that A(n) and A(n) are isomorphic as groups. We will often abuse notation by denoting g by g, and A(n) by A(n). Next, for each non-negative integer n, let ι : M (n) −→ M (n+1) and ι : A(n) −→ A(n + 1) respectively denote the Hilbert space monomorphism and the group monomorphism induced by the injection ι : M (n) −→ M (n+1) and the group monomorphism ι : A(n) −→ A(n + 1) . Finally, we define the nested quantum motif system Q * = Q * M ( * ) , A( * ) as the following sequence of Hilbert spaces, groups, Hilbert space monomorphisms, and group monomorphisms: Q 1 M (1) , A(1) ι −→ Q 2 M (2) , A(2) ι −→ · · · ι −→ Q n M (n) , A(n) ι −→ · · · Remark 2. We should also mention one other quantum motif system that can be useful, namely, the quantum direct limit motif system defined by Q ∞ = Q ∞ M (∞) , A(∞) = lim −→ Q * M ( * ) , A( * ) , where lim −→ denotes the direct limit. This quantum system is often also physically implementable. 2.4. Stage 4. Quantum motif equivalence. Let Q n = Q M (n) , A(n) be a quantum motif system of order n. Two quantum motifs \|ψ 1 and \|ψ 2 of the Hilbert space M (n) are said to be of the same n-motif type, written \|ψ 1 ∼ n \|ψ 2 , if there exists an element g of the ambient group A(n) which takes \|ψ 1 to \|ψ 2 , i.e., such that g \|ψ 1 = \|ψ 2 . The quantum motifs \|ψ 1 and \|ψ 2 are said to be of the same motif type, written \|ψ 1 ∼ \|ψ 2 , if there exists a non-negative integer m such that ι m \|ψ 1 ∼ n+m ι m \|ψ 2 . Stage 5. Motif invariants as quantum observables. We consider the following question: Question: What do we mean by a physically observable quantum motif invariant? We answer this question with a definition. Definition 2. Let Q n = Q M (n) , A(n) be a quantum motif system of order n, and let Ω be an observable, i.e., a Hermitian operator on the Hilbert space M (n) of quantum motifs. Then Ω is a quantum motif n-invariant provided Ω is left invariant under the big adjoint action of the ambient group A(n), i.e., provided U ΩU −1 = Ω for all U in A(n). Proposition 1. If I (n) : M (n) −→ R is a real valued n-motif invariant, then Ω = m∈M (n) I (n) (m) \|m m\| is a quantum motif observable which is a quantum motif n-invariant. Much more can be said about this topic. For a more in-depth discussion of this issue, we refer the reader to [12,13]. Part II. Quantizing Braids We now illustrate the quantization procedure defined above by using it to quantize braids. For each integer n ≥ 2, let T (n) denote the following set of the 2n − 1 symbols , · · · , , , , , , · · · , called n-stranded braid tiles, or n-tiles, or simply tiles. We also denote these tiles respectively by the symbols b −(n−1) , . . . , b −2 , b −1 , b 0 = 1, b 1 , b 2 , . . . , b n−1 , as indicated in the table given below: · · · · · · b −(n−1) · · · b −2 b −1 b 0 = 1 b 1 b 2 · · · b n−1 n-stranded braid tiles Definition 3. An (n, )-braid mosaic β is defined as a sequence of n-stranded braid tiles β = b j(1) b j(2) . . . b j( ) of length . We let B (n, ) denote the set of all (n, )-braid mosaics. An example of a (3, 8)-braid mosaic is given below 1 b −1 b 1 b 2 1 1 b −1 b 2 The (3, 8)-braid mosaic β = 1b −1 b 1 b 2 11b −1 b 2 Remark 3. Please note that the set of all (n, )-braid mosaics B (n, ) is a finite set of cardinality (2n − 1) . Stage 1 (Cont.) Braid mosaic moves. Definition 4. Let and be positive integers such that ≤ . An (n, )-braid mosaic γ is is said to be an (n, )-braid submosaic of an (n, )-braid mosaic β provided γ is a subsequence of consecutive tiles of β. The (n, )-braid submosaic γ is said to be at position p in β if the first (leftmost) tile of γ is the p-th tile of β from the left. We denote the (n, )-braid submosaic γ of β at location p by γ = β p: . γ 3 ↔ γ = 3 ←→ Then γ 3 ↔ γ =   Braid Submosics Switched   γ 3 ↔ γ =   Braid Submosics Switched   γ 3 ↔ γ =   Braid Mosaic Unchanged   The following proposition is an almost immediate consequence of the definition of a braid move. Proposition 2. Each braid move is a permutation on the set B (n, ) of (n, )-braid mosaics. In fact, it is a permutation which is a product of disjoint transpositions. Stage 1. (Cont.) Planar isotopy moves. Our next objective is to translate all the standard topological moves on braids into braid mosaic moves. To accomplish this, we must first note that there are two types of standard topological moves, i.e., those which do not change the topological type of the braid projection, called planar isotopy moves, and those which do change the typological type of the braid projection but not of the braid itself, called Reidemeister moves. We begin with the planar isotopy moves. Definition 6. For braid mosaics, there are two types planar isotopy moves, i.e., types P 1 and P 2 , which are defined below as: 1b i λ ←→ P1 b i 1 for 0 < \|i\| < n Definition of a type P 1 planar isotopy move and b i b j λ ←→ P2 b i b j f or0 < \|i\| , \|j\| < n and \|\|i\| − \|j\|\| > 1 Definition of a type P 2 planar isotopy move Example 1. Examples of P 1 and P 2 moves are respectively given below: λ ←→ An example of a P 1 move. and λ ←→ An example of a P 2 move: Remark 5. The number of P 1 and P 2 moves are respectively 2 (n − 1) ( − 1) and (n − 1) (2n − 6) ( − 1) . Stage 1. (Cont.) Reidemeister moves. There are two types of topological moves, i.e., R 2 and R 3 . Definition 7. The Reidemeister R 2 moves are defined as b i b −i λ ←→ 1 2 where 0 < \|i\| < n Example 2. An example of a Reidemeister 2 move is given below λ ←→ Remark 6. The number of R 2 moves is 2(n − 1) ( − 1) Definition 8. The Reidemeister R 3 moves are defined for 0 < \|i\| < n , and given below: b i b i+1 b i b −(i+1) b −i b −(i+1) λ ←→ 1 6 b i b i+1 b i b −(i+1) b −i λ ←→ b i+1 1 4 b i b i+1 b i b −(i+1) λ ←→ b i+1 b i 1 2 b i b i+1 b i λ ←→ b i+1 b i b i+1 b i b i+1 1 2 λ ←→ b i+1 b i b i+1 b −i b i 1 4 λ ←→ b i+1 b i b i+1 b −i b −(i+1) 1 6 λ ←→ b i+1 b i b i+1 b −i b −(i+1) b −i Example 3. Two examples of Reidemeister R 3 are given below: λ ←→ λ ←→ Remark 7. The number of Reidemeister 3 moves R 3 is given by # R 3 Moves =                            n (n − 2) (6 − 21) if ≥ 6 n (n − 2) (5 − 16) if = 5 n (n − 2) (3 − 8) if = 4 n (n − 2) ( − 2) if = 3 0 if < 3 3.5. Stage 1. (Cont.) The ambient group A (n, ) and the braid mosaic system B n, * . At this point, we can define what is meant by the ambient group and the resulting braid mosaic system. We begin reminding the reader of a fact noted earlier in this paper, namely the fact that each braid move is a permutation on the set B (n, ) of (n, )-braid mosaics. Thus, since planar isotopy and Reidemeister moves are permutations, we can make the following definition: Definition 9. We define the ( (n, )-braid mosaic) ambient group A(n, ) as the group of all permutations on the set B (n, ) of (n, )-braid mosaics generated by (n, )-braid planar isotopy and Reidemeister moves. We need one more definition, before we can move to the objective of this section. from the (n, )-braid ambient group A(n, ) to the (n, + 1)-braid ambient group A(n, + 1). This monomorphism is called the braid mosaic monomorphism. Definition 11. We define an braid system B n, = B B (n, ) , A(n, ) of order (n, ) as the pair B (n, ) , A(n, ) , where B (n, ) is called the set of (n, )-braid mosaics, and where A(n, ) is called the ambient group. Finally, we define a nested motif system B n, * = B B (n, * ) , A(n, * ) as the following sequence of sets, groups, injections, and monomorphisms: Our next objective is to define what it means for two braid mosaics to represent the same topological braid. Two braid mosaics β 1 and β 2 of the set B (n, ) are said to be of the same n-braid mosaic type, written β 1 ∼ n β 2 , if there exists an element g of the ambient group A(n, ) which takes β 1 to β 2 , i.e., such that gβ 1 = β 2 . The braid mosaics β 1 and β 2 are said to be of the same braid mosaic type, written β 1 ∼ β 2 , if there exists a non-negative integer k such that ι k β 1 ∼ n+k ι k β 2 . We now wish to answer the question: Question. What is meant by a braid mosaic invariant? Definition 12. Let B n, = B B (n, ) , A(n) be a braid system, and let D be some yet to be chosen mathematical domain. By an n-braid mosaic invariant I (n) , we mean a map I (n) : B (n, ) −→ D such that, when two braid mosaics β 1 and β 2 are of the same n-type, i.e., when β 1 ∼ n β 2 , then their respective invariants must be equal, i.e., I (n) (β 1 ) = I (n) (β 2 ) . In other words, I (n) : B (n, ) −→ D is a map that is invariant under the action of the ambient group A(n), i.e., I (n) (β) = I (n) (gβ) for all elements of g in A(n). Stage 3. Construction of the corresponding quantum braid system. We now use the nested braid mosaic system B n, * to construct a nested sequence of quantum braid mosaic systems Q n, * . For pair of non-negative integers n and the corresponding (n, )-th order quantum braid system Q n, = Q B (n, ) , A(n, ) consists of a Hilbert space B (n, ) , called the quantum mosaic space, and a group A(n, ), also called the ambient group. The quantum motif space B (n, ) and the ambient group A(n, ) are defined as follows: • The quantum motif space B (n, ) is the Hilbert space with orthonormal basis \|β : β ∈ B (n, ) . The elements of B (n, ) are called quantum braids. Since each element g in A(n, ) is a permutation, each g permutes the orthonormal basis \|β : β ∈ B (n, ) of B (n, ) . Hence, g is automatically a unitary transformation. It follows that A(n, ) and A(n, ) are isomorphic as groups. We will often abuse notation by denoting g by g, and A(n, ) by A(n, ). Next, for each pair of non-negative integers n and , let Finally, we define the nested quantum braid system Q n, * = Q n, * B (n, * ) , A(n, * ) as the following sequence of Hilbert spaces, groups, Hilbert space monomorphisms, and group monomorphisms: Q 1, B (1, ) , A(1, ) ι −→ Q 2, B (2, ) , A(2, ) ι −→ · · · ι −→ Q n, B (n, ) , A(n, ) ι −→ · · · 3.8. Stage 4. Quantum braid equivalence. Let Q n, = Q B (n, ) , A(n, ) be a quantum motif system of order (n, ). Two quantum braids \|ψ 1 and \|ψ 2 of the Hilbert space B (n, ) are said to be of the same (n, )-braid type, written \|ψ 1 ∼ n \|ψ 2 , if there exists an element g of the ambient group A(n, ) which takes \|ψ 1 to \|ψ 2 , i.e., such that g \|ψ 1 = \|ψ 2 . The quantum motifs \|ψ 1 and \|ψ 2 are said to be of the same braid type, written \|ψ 1 ∼ \|ψ 2 , if there exists a non-negative integer m such that ι m \|ψ 1 ∼ n+m ι m \|ψ 2 . 3.9. Stage 5. Quantum braid invariants as quantum observables. We consider the following question: Question: What do we mean by a physically observable quantum braid invariant? We answer this question with a definition. Definition 13. Let Q n, = Q B (n, ) , A(n, ) be a quantum braid system of order (n, ), and let Ω be an observable, i.e., a Hermitian operator on the Hilbert space B (n, ) of quantum braids. Then Ω is a quantum braid (n, )-invariant provided Ω is left invariant under the big adjoint action of the ambient group A(n, ), i.e., provided U ΩU −1 = Ω for all U in A(n, ). I (n, ) (β) \|β β\| is a quantum motif observable which is a quantum motif (n, )-invariant. Conclusion Much more can be said about this topic. For more examples of the application of the quantization procedure discussed in this paper, we refer the reader to [12,13,8,2]. For knot theory and the braid group, we refer the reader to [3,16,4,11,1,10]; for topological quantum computation, [5,6,7,9,17,19]; and for quantum computation and information, [18,14,15]. 3. 1 . 1Stage 1. The set of braid mosaics B (n, ) . Remark 4 . 4The number of (n, )-braid submosaics of an (n, )-braid mosaic β is − + 1. Two examples of braid submosaics of the (3, 8)-braid mosaic β = 1b −1 b 1 b 2 11b −1 b 2 are given above are: b −1 b 1 b 2 The (3, 3)-braid submosaic β 2:3 of β at position 2 1 1 b −1 b 2 The (3, 4)-braid submosaic β 5:4 of β at position 5 Definition 5. Let and be positive integers such that ≤ . For any two (n, )-braid mosaics γ and γ , we define the -braid mosaic move at location p on the set of all (n, )-braid mosaics B (n, ) β p: replaced by γ if β p: = γ β with β p: replaced by γ if β p: Definition 10 . 10We define the braid mosaic injection ι : B (n, ) −→ B (n, +1) as the map β −→ β1 for each (n, )-braid mosaic in B (n, ) . It immediately follows that the braid mosaic injection induces a monomorphism ι : A(n, ) −→ A(n, + 1) . Stage 2. Braid mosaic type and braid mosaic invariants. • The ambient group A(n, ) is the unitary group acting on the Hilbert space B (n, ) consisting of all linear transformations of the form g : B (n, ) −→ B (n, ) : g ∈ A(n, ) , where g is the linear transformation defined by g : B (n, ) −→ B (n, ) \|β −→ \|gβ ι : B (n, ) −→ B (n+1, ) and ι : A(n, ) −→ A(n + 1, ) respectively denote the Hilbert space monomorphism and the group monomorhism induced by the injection ι : B (n, ) −→ B (n+1, ) and the group monomorphism ι : A(n, ) −→ A(n + 1, ) . Proposition 3 . 3If I (n) : B (n, ) −→ R is a real valued (n, )-braid invariant, then Ω = β∈B (n, ) Braids, Links, and Mapping Class Groups. Joan S Birman, Princeton University PressBirman, Joan S., "Braids, Links, and Mapping Class Groups," Princeton University Press, (1974). Quantum money from knots. Edward Farhi, David Gosset, Avinatan Hassidim, Andrew Lutomirski, Peter Shor, ArKiv. quant-phFarhi, Edward, David Gosset, Avinatan Hassidim, Andrew Lutomirski, and Peter Shor, Quantum money from knots,ArKiv[quant-ph], (2010). http://arXiv.org/abs/1004.5127 Introduction to Knot Theory. Richard H Crowell, Ralph H Fox, DoverCrowell, Richard H., and Ralph H. Fox, "Introduction to Knot Theory," Dover, (2008). Knots and Physics. Louis H Kauffman, World ScientificSecond Edition. Third EditionKauffman, Louis H., "Knots and Physics,", World Scientific, (1991), Second Edition (1993), Third Edition (2002). Louis H Kauffman, Samuel J Lomonaco, Quantum Knots, Quantum Information and Computation II -SPIE Proc. Kauffman, Louis H., and Samuel J. Lomonaco, Quantum Knots, Quantum Information and Computation II -SPIE Proc., 12-14, (2004). q-deformed networks, knot polynomials, anyonic topological quantum computation. Louis H Kauffman, Samuel J Lomonaco, J. 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Chicago, ILUniversity of Maryland Baltimore County (UMBC)60607-7045 USA E-mail address: kauffman@uicUniversity of Maryland Baltimore County (UMBC), Baltimore, MD 21250 USA E-mail address: [email protected] URL: http://www.csee.umbc.edu/~lomonaco University of Illinois at Chicago, Chicago, IL 60607-7045 USA E-mail address: [email protected] URL: http://www.math.uic.edu/~kauffman	[]
[ "arXiv:physics/0302068v1 [physics.chem-ph] Oscillator strength distribution in C 3 H 6 isomers studied with the time-dependent density functional method in the continuum", "arXiv:physics/0302068v1 [physics.chem-ph] Oscillator strength distribution in C 3 H 6 isomers studied with the time-dependent density functional method in the continuum" ]	[ "Takashi Nakatsukasa \nPhysics Department\nTohoku University\n980-8578SendaiJapan\n", "Kazuhiro Yabana \nInstitute of Physics\nUniversity of Tsukuba\n305-8571TsukubaJapan\n" ]	[ "Physics Department\nTohoku University\n980-8578SendaiJapan", "Institute of Physics\nUniversity of Tsukuba\n305-8571TsukubaJapan" ]	[]	We present photoabsorption oscillator strengths for C3H6 molecules with emphasis on the difference between isomers, cyclopropane and propylene. We use an iterative numerical method based on the time-dependent local density approximation with continuum, which we have recently developed. The oscillator strengths for the two isomers differ at photon energies above their ionization thresholds. The magnitude and the shape of the oscillator strength distribution are in good agreement with recent experiments. The differences between the isomers arise from difference in symmetry of electronic states and different behaviors of continuum excitations. *	10.1016/s0009-2614(03)00778-4	[ "https://arxiv.org/pdf/physics/0302068v1.pdf" ]	95,935,253	physics/0302068	fdd4048d69a585345ac4ce8f0ba69cce879f110f	arXiv:physics/0302068v1 [physics.chem-ph] Oscillator strength distribution in C 3 H 6 isomers studied with the time-dependent density functional method in the continuum 20 Feb 2003 Takashi Nakatsukasa Physics Department Tohoku University 980-8578SendaiJapan Kazuhiro Yabana Institute of Physics University of Tsukuba 305-8571TsukubaJapan arXiv:physics/0302068v1 [physics.chem-ph] Oscillator strength distribution in C 3 H 6 isomers studied with the time-dependent density functional method in the continuum 20 Feb 20031 We present photoabsorption oscillator strengths for C3H6 molecules with emphasis on the difference between isomers, cyclopropane and propylene. We use an iterative numerical method based on the time-dependent local density approximation with continuum, which we have recently developed. The oscillator strengths for the two isomers differ at photon energies above their ionization thresholds. The magnitude and the shape of the oscillator strength distribution are in good agreement with recent experiments. The differences between the isomers arise from difference in symmetry of electronic states and different behaviors of continuum excitations. * Introduction The photoabsorption and photoionization cross sections of molecules are of significant interest in many fields of both fundamental and applied sciences. The oscillator strength distribution characterizing the optical response is the most important quantity in understanding the interaction of photon with electrons in atoms, molecules, and matters. The oscillator strength distribution in the whole spectral region has been extensively studied with the advanced synchrotron radiation and the high resolution electron energy loss spectroscopy [1,2]. In order to see how the oscillator strength changes with varying molecular structures, it is useful to study isomer molecules. Since the isomers consist of the same kind and the same number of atoms, we expect similarity of the oscillator strengths at high photon energies. This is because the molecular structure has little influence on the excitation of inner core electrons. However, the valence photoabsorption may differ according to the difference of electronic states between the isomers. In fact, Koizumi et al. have observed a prominent distinction for the cross sections of simple hydrocarbon isomers, C 3 H 6 (cyclopropane and propylene), at photon energies of 10 − 20 eV [3]. The photoabsorption and photoionization data of cyclopropane in this energy region were later improved by a measurement with metallic thin film windows [4]. These works clearly show that the oscillator strengths have different peaks and shoulders depending on the isomers in a continuous spectral region above the ionization potentials (IPs). Theoretical investigation for the isomer effect of C 3 H 6 has been demanded for a long time, however, none has been reported so far. This is due to difficulties in treatment of the electronic continuum in a non-spherical multicenter potential. There are several methods which are able to take into account correlations among valence electrons in the continuum [5,6,7,8,9]. Nevertheless, some of them are not suitable for calculating detailed structures of the spectra, and some are difficult to be applied to large molecules. We have recently developed an alternative theoretical method for this purpose [10]. The method is based on the time-dependent local-density approximation (TDLDA) in a grid representation of the three-dimensional Cartesian coordinate, and utilizes the Green's function to take account of the continuum boundary condition. We use iterative methods to solve linear algebraic equations for construction of the dynamical screening field above the first IP. The theoretical back-ground of our method is similar to the one of Ref. [5] in which the authors used a single-center expansion technique. However, the application was limited to small axially symmetric molecules, because of difficulties in the single-center expansion. Our method is based on direct calculation of the self-consistent screening potential in the three-dimensional grid representation, which does not rely on the expansion and requires no spatial symmetry. In the present Letter, we report the valence photoabsorption of the C 3 H 6 isomers studied with the continuum TDLDA method in Ref. [10], and would like to show the power of the method. Then, we give an interpretation of the continuous spectra and elucidate origins of the isomer effects. Theory and computational method Optical response of molecules is characterized by the oscillator strength, denoted as df /dω in the followings, which is given by df dω = − 2mω 3π Im ν=x,y,z d 3 rr ν δn ν (r, ω),(1) where the transition density δn ν is related to the Fourier component of a time-dependent external dipole perturbation in ν-direction, V ν (ω) = r ν , through a complex susceptibility (δn ν (ω) = χ(ω)V ν (ω)). The TDLDA describes a spin-independent Nelectron system in terms of the time-dependent Kohn-Sham equations. Correlations among electrons are taken into account through deformations of the self-consistent Kohn-Sham potential. Linearizing the Kohn-Sham potential with respect to the transition density, we obtain δn ν (r, ω) = d 3 r ′ χ 0 (r, r ′ ; ω) r ′ ν + d 3 r ′′ δV KS [n(r ′ )] δn(r ′′ ) δn ν (r ′′ ) .(2) The χ 0 (r, r ′ ; ω) is a complex susceptibility for a system without the correlations and is given by χ 0 (r, r ′ ; ω) = 2 occ i φ i (r) (G(r, r ′ ; ǫ i − ω * )) * + G(r, r ′ ; ǫ i + ω)} φ i (r ′ ).(3) Here, φ i 's are the ground-state Kohn-Sham orbitals and G is the Green's function for an electron in the static Kohn-Sham potential. In order to properly treat the electronic continuum, the outgoing boundary condition must be imposed on G. Construction of the Green's function is easily done for a rotationally invariant potential V 0 (r), using the partial wave expansion [11]. Thus, we split the Kohn-Sham potential into two parts, a long-range spherical part, V 0 (r), and a short- range deformed part,Ṽ (r) = V KS (r) − V 0 (r). First, we construct the Green's function, G 0 , for the spherical potential V 0 , then, G can be obtained from an identity G = G 0 + G 0Ṽ G.(4) We solve Eqs. (2), (3), and (4) simultaneously in the uniform grid representation of the threedimensional real space. Equations (2) and (4) are linear algebraic equations with respect to δn and G, respectively, for which an iterative method provides an efficient algorithm for the numerical procedure. We adopt the generalized conjugate residual method for these non-hermitian problems. We would like to refer the reader to Ref. [10] for detailed discussion of the methodology and the theoretical background. The exchange-correlation potential is a sum of the local density part given by Ref. [13], µ (PZ) [ρ], and the gradient correction of Ref. [14], µ (LB) [ρ, ∇ρ], which will be abbreviated to LB potential. This gradient correction is constructed so as to reproduce the correct Coulomb asymptotic behavior of the potential (−e 2 /r) and to describe the Rydberg states. It was also pointed out that the LB potential is necessary to reproduce the excitation energies of high-lying bound states in the TDLDA [15]. In our previous work [10], we have also found for simple molecules that the TDLDA with the LB potential reasonably accounts for resonances embedded in the continuum. Table 1 Calculated eigenvalues of occupied valence orbitals in units of eV. Values in brackets indicate eigenvalues calculated using a different value of the parameter in µ (LB) (β = 0.05). See text for details. Line styles in Figs. 1 (b), 2 (b), and 3 (b) are indicated in the third and sixth columns. We use following abbreviations: "S" for the solid, "Do" for the dotted, "Da" for the dashed, "LD" for the long-dashed, "DD" for the dot-dashed, and "T-" for the thick lines. Cyclopropane Propylene Orbital Calc. Line Orbital Calc. Line (3e ′ ) 4 −10.6 (−11.9) T-S (2a ′′ ) 2 −9.9 T-S (1e ′′ ) 4 −11.9 (−13.2) Do (10a ′ ) 2 −11.4 T-Do (3a ′ 1 ) 2 −14.8 (−16.1) S (9a ′ ) 2 −12.0 T-Da (1a ′′ 2 ) 2 −15.4 (−16.8) Da (1a ′′ ) 2 −13.4 T-LD (2e ′ ) 4 −17.5 (−19.0) DD (8a ′ ) 2 −13.6 DD (2a ′ 1 ) 2 −23.9 (−25.3) LD (7a ′ ) 2 −14.6 S (6a ′ ) 2 −16.6 Do (5a ′ ) 2 −19.6 Da (4a ′ ) 2 −22.3 LD Results and discussion Ground-state properties We fix the geometry of nuclei optimized for the ground state. This is based on a semiempirical method known as PM3 [12]. We only treat valence electrons in the TDLDA calculation. Thus, we use the norm-conserving pseudopotential [16] with a separable approximation [17] for the electron-ion potentials. The coordinate space is discretized in a square mesh of 0.3Å and we adopt all the grid points inside a sphere of 6Å radius. This results in a model space of 33,401 grid points. First, we calculate the ground state of cyclopropane and propylene by solving the Kohn-Sham equations with the exchange-correlation potential of µ (PZ) + µ (LB) . The LB potential µ (LB) contains a parameter β [14], and we adjust this value to make eigenvalues of the highest occupied molecular orbitals (HOMO) coincide with the empirical vertical IPs (10.54 eV for cyclopropane [18] and 9.91 eV for propylene [19]). The occupied valence orbitals in the ground state calculated with β = 0.015 are listed in Table 1. The HOMO eigenvalues are well reproduced for both isomers. Propylene has a geometry of the C s point group while cyclopropane has the D 3h group. Although these isomers possess equal number of valence electrons (eighteen valence electrons), the electron configuration of cyclopropane is more degenerate in energy because of the higher symmetry. Photoabsorption oscillator strength Now we calculate the photoresponse of the isomers. We use complex frequencies, ω + iΓ/2 with Γ = 0.5 eV. The Γ plays a role of a smoothing parameter to make the energy resolution finite. This also helps a convergence of the numerical iteration procedure [10]. In Figs. 1 and 2 respectively, we show the calculated photoabsorption oscillator strength, df /dω, for cyclopropane and propylene in a frequency (photon energy) range of 8−50 eV. The calculations have been done with a frequency mesh of ∆ω = 0.25 eV for a region of 8 ≤ ω ≤ 20 eV, and with ∆ω = 0.5 eV for the rest of frequencies. The oscillator strength distributions of the isomers are nearly identical at ω 22 eV. This energy roughly corresponds to the ionization energy of the lowest-lying σ orbital. The df /dω monotonically decreases as the frequency increases but has a large tail at high frequency. This behavior of the high-frequency tail in df /dω is one of the characteristics of the electronic excitations in the continuum, which was also found in our previous studies of simple molecules [10]. The molecular structure has a little influence on the electronic continuum in this energy region (ω 22 eV). In contrast, in the frequency region below 22 eV, different structures are observed among the isomers. The df /dω of propylene shows a single broad peak at ω = 13 ∼ 18 eV with small wiggles. On the other hand, distinctive three peaks at ω = 11.8, 13.5, and 15.7 eV, are found in cyclopropane. This difference exactly matches the experimental findings of the isomer effect (the thin solid line in Fig. 1 (a)). The energy positions of calculated peaks are lower than the experimental ones by about 1.5 eV. This is also true for propylene in Fig. 2 (a), in which the broad peak is shifted to lower energy by 1.5 eV compared to the experiment. We would like to mention that, if we treat the electrons as responding independently to the external dipole field, we cannot reproduce the main feature of the oscillator strength distributions. We call this approximation "independent-particle approximation (IPA)", which corresponds to neglecting the induced screening potential, the second term in the bracket in Eq. (2). In Figs. 1 (a) and 2 (a), the IPA calculations are shown by dashed lines. The Thomas-Kuhn-Reiche (TRK) sum rule tells us the integrated oscillator strength f (∞) = 18 in our calculations, since we only treat valence electrons in C 3 H 6 . In Table 2, partial sum values of the oscillator strengths are listed and compared to the experiment. Again, the IPA calculation cannot account for the data, while the TDLDA in the continuum well agrees with the experiment. Calculated total sum values for 8 < ω < 60 eV are 16.2 for cyclopropane and 15.8 for propylene. These values correspond to about 90 % of the TRK sum rule for valence electrons. Origin of the difference in photoab- sorption between the isomers We would like to discuss details of resonance peaks and an origin of the different behaviors between the isomers. First, let us compare the IPA results for the two isomers (See dashed lines in Figs. 1 (a) and 2 (a)). In the energy region of 10 − 20 eV, although the bulk structure is similar, cyclopropane shows a sharper main peak at 11.5 [4]. See text for details. (b) An energy region of 10 < ω < 25 eV is magnified and the total oscillator strength is decomposed into those associated with different occupied valence electrons. See Table 1 for correspondence between a line style and an occupied orbital. Figure 2. The same as Fig. 1 but for propylene. The experimental data are taken from Ref. [3]. eV and an additional peak at 13 eV. This may be due to higher-fold degeneracies in electronic eigenstates in cyclopropane. These peak structures in cyclopropane remain after inclusion of the dynamical screening effects, while, for propylene, the strong peak at 12 eV seen in the IPA is diminished. Next, we shall examine this difference in the dynamical screening effects. For this purpose, it is useful to calculate a partial oscillator strength [11,10] which corresponds to a contribution of each occupied orbital to the total oscillator strength. We display the partial df /dω in the energy range of 10 − 25 eV in Figs. 1 (b) and 2 (b). One can see that the major contributions to df /dω come from bound-to-continuum excitations of electrons near the Fermi level; the HOMO (3e ′ ) 4 and the second HOMO (1e ′′ ) 4 in case of cyclopropane, and the second (10a ′ ) 2 , the third (9a ′ ) 2 and the fourth HOMO (1a ′′ ) 2 for propylene. The sharp peaks at 11.8 eV and 13.5 eV in cyclopropane originate from bound-to-bound transitions of (3a ′ 1 ) 2 and (1a ′′ 2 ) 2 electrons. These resonances are also seen in the IPA calculation. Then, the electron-electron correlation brings out coherent contributions of bound-to-continuum excitations of (3e ′ ) 4 and (1e ′′ ) 4 electrons. The width of the resonances becomes slightly larger than that of the IPA, because of the autoionization process. The peak at 15.7 eV is also produced by coherent excitations of (2e ′ ) 4 (bound-to-bound) and (3e ′ ) 4 (bound-to-continuum) electrons. We only see a shoulder around 15.5 eV in the IPA calculation, however, the dynamical effect enhances the peak. In the case of propylene, we find several small peaks of bound-to-bound excitations in the partial df /dω in Fig. 2 (b). However, the bound-to-continuum excitations, which mostly contribute to the broad resonance in 13 − 18 eV, behave rather independently, and do not produce coherent enhancement of those peaks. As a result, the small peaks in the bound-to-bound transitions are mostly smeared out in the total oscillator strength. We think that this difference in the continuum response could be attributed to the difference in strength of bound-to-bound transitions. The df /dω of propylene shows a typical behavior of the dynamical screening effects. Namely, the oscillator strengths (and the peak at 12 eV) in the IPA calculation are significantly weakened by the induced screening field in Eq. (2). Conversely, those at energies above 16 eV are enhanced. This is because the real part of the dynamical polarizability changes its sign at ω ≈ 16 eV, then the screening field changes into the "anti-screening field" at higher energies. In the case of cyclopropane, the situation is slightly more complicated. Generally speaking, the real part of the dynamical polarizability changes its sign from positive to negative at bound resonances. Since the degeneracies of electronic orbitals are higher in cyclopropane, the bound-to-bound transitions have large oscillator strengths. Then, there appears a reminiscence of bound resonance in the continuum region. The screening field suddenly drops down at energies corresponding to boundto-bound transitions. This provides effective antiscreening effects to cause the peak structures in the bound-to-continuum transitions. Finally, we would like to comment on dependence of our results upon the parameter β in the LB potential µ (LB) . A choice of this parameter is rather arbitrary, since the value of β does not change the Coulomb asymptotic behavior. In fact, if we adopt β = 0.05, the value proposed by the original paper [14], we obtain better agreement to the photoabsorption spectra for both the isomers. In the case of using β = 0.05, the Kohn-Sham eigenvalues for occupied orbitals in cyclopropane are indicated as values in brackets in Table 1. A binding energy of each orbital becomes deeper by 1.3 − 1.5 eV, though spacings between the orbitals almost stay constant. The calculated oscillator strength distribution is shown in Fig. 3 for cyclopropane. The disagreement on the peak positions are removed in the calculation. A bound peak at 10 eV is also well reproduced in the calculation. This peak consists of the excitations of (3e ′ ) 4 and (1e ′′ ) 4 electrons. These excitations have an almost identical energy when we use β = 0.05, while the excitation of (1e ′′ ) 4 electrons is shifted to lower energy by 1 eV when using β = 0.015. Except for the bound peak at 10 eV, the characteristic features of the oscillator strength distribution are not changed, and we maintain the interpretation given above. Conclusion The oscillator strength distributions of C 3 H 6 isomer molecules are studied with the TDLDA in the continuum utilizing the three-dimensional Cartesian coordinate representation. The calculation shows good agreement with experiments. The oscillator strength in the energy region above 22 eV is almost identical among the isomers, however, different peaks appear below that. This isomer effect is analysed by calculating the partial oscillator strength of each occupied orbital. In addition to the difference in properties of bound electronic orbitals, it turns out that bound-tocontinuum excitations of electrons near the Fermi level behave differently between the isomers. The bound-to-bound transitions in cyclopropane possess large strengths, and the bound-to-continuum transitions exhibit coherent peak structures because of the anti-screening effects. On the other hand, in propylene, the bound-to-bound transitions are too weak to produce the anti-screening peaks for the continuum excitations. Although the molecular structure directly has minor influence on the continuum, the difference in boundto-bound transitions leads to variation in the dynamical screening effects to change the continuum excitations. Figure 1 . 1(a) Calculated (thick solid line) and experimental (thin solid) photoabsorption oscillator strength distribution for a cyclopropane molecule as a function of photon energy. The dashed line indicates the IPA calculation without dynamical screening effects. The experimental data are taken from Ref. Figure 3 . 3The same asFig. 1but for using a parameter β = 0.05 in the calculation. The calculation has been performed for an energy range of 8 − 43.5 eV with a mesh of ∆ω = 0.5 eV. Table 2 2Partially summed oscillator strengths for C 3 H 6 isomers. The experimental values are estimated from data in Refs. [3,4]. Cyclopropane Propylene Energy range TDLDA (IPA) Expt. TDLDA (IPA) Expt. 10 eV < ω < 25 eV 10.0 (11.1) 9.9 9.5 (10.9) 9.4 25 eV < ω < 35 eV 2.9 (1.2) 3.3 2.9 (1.8) 3.1 35 eV < ω < 60 eV 2.9 (2.3) 2.7 (1.8) AcknowledgementsThis work is supported in part by Grantsin-Aid for Scientific Research (No.1470146 and 14540369) from the Japan Society for the Promotion of Science. 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A. 492421R. van Leeuwen, E. J. Baerends, Phys. Rev. A 49 (1994) 2421. . M E Casida, C Jamorski, K C Caside, D R Salahub, J. Chem. Phys. 1084439M. E. Casida, C. Jamorski, K. C. Caside, D. R. Salahub, J. Chem. Phys. 108 (1998) 4439. . N Troullier, J L Martins, Phys. Rev. B. 43N. Troullier, J. L. Martins, Phys. Rev. B 43 (1991) 1993. . L Kreinman, D Bylander, Phys. Rev. Lett. 481425L. Kreinman, D. Bylander, Phys. Rev. Lett. 48 (1982) 1425. . V V Plemenkov, Y Y Villem, N V Villem, I G Bolesov, L S Surmina, N I Yakushkina, A A Formanovskii, Zh. Obshch. Khim. 512076V. V. Plemenkov, Y. Y. Villem, N. V. Villem, I. G. Bolesov, L. S. Surmina, N. I. Yakushk- ina, A. A. Formanovskii, Zh. Obshch. Khim. 51 (1981) 2076. . D A Krause, J W Taylor, R F Fenske, J. Am. Chem. Soc. 100718D. A. Krause, J. W. Taylor, R. F. Fenske, J. Am. Chem. Soc. 100 (1978) 718.	[]
[ "Chandra Observation of PWN G16.73+0.08 in SNR G16.7+0.1", "Chandra Observation of PWN G16.73+0.08 in SNR G16.7+0.1" ]	[ "H.-K Chang \nInstitute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan\n\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n", "S.-F Chung \nInstitute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan\n", "C.-Y Yang \nInstitute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan\n", "W W Tian \nKey Laboratory of Optical Astronomy\nNational Astronomical Observatories\nChinese Academy of Sciences\n100049BeijingChina\n\nDepartment of Physics and Astronomy\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada\n" ]	[ "Institute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan", "Department of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan", "Institute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan", "Institute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan", "Key Laboratory of Optical Astronomy\nNational Astronomical Observatories\nChinese Academy of Sciences\n100049BeijingChina", "Department of Physics and Astronomy\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada" ]	[ "Mon. Not. R. Astron. Soc" ]	We present X-ray observations of PWN G16.73+0.08/SNR G16.7+0.1 using archival data of Chandra ACIS. The X-ray emission peak location of this pulsar wind nebula is found to be offset by 24 arcsec from the centre of the 1.4-GHz emission of this nebula. The X-ray nebula is elongated in the direction from the X-ray peak to the 1.4-GHz emission centre. This offset suggests that G16.73+0.08 is an evolved pulsar wind nebula interacting with the supernova remnant reverse shock. We identify a point source, CXO J182058.16-142001.5, near the location of the X-ray peak. The spectrum of the X-ray nebula can be described by an absorbed power law of photon index 0.98 +0.79 −0.71 and hydrogen column density N H = 4.99 +2.75 −2.28 × 10 22 cm −2 . CXO J182058.16-142001.5 is likely a pulsar. We estimate its spin-down power to be about 2.6 × 10 36 erg s −1 . Assuming its age at 3000 and 10,000 years, its dipole magnetic field strength at the polar surface is estimated to be about 4.2 × 10 13 G and 1.3 × 10 13 G, respectively.	10.1093/mnras/stx2941	[ "https://arxiv.org/pdf/1711.04953v1.pdf" ]	54,667,849	1711.04953	7f95a70dd57fbb0ea6257986b0de754713e0073b	Chandra Observation of PWN G16.73+0.08 in SNR G16.7+0.1 2017 H.-K Chang Institute of Astronomy National Tsing Hua University 30013HsinchuTaiwan Department of Physics National Tsing Hua University 30013HsinchuTaiwan S.-F Chung Institute of Astronomy National Tsing Hua University 30013HsinchuTaiwan C.-Y Yang Institute of Astronomy National Tsing Hua University 30013HsinchuTaiwan W W Tian Key Laboratory of Optical Astronomy National Astronomical Observatories Chinese Academy of Sciences 100049BeijingChina Department of Physics and Astronomy University of Calgary T2N 1N4CalgaryAlbertaCanada Chandra Observation of PWN G16.73+0.08 in SNR G16.7+0.1 Mon. Not. R. Astron. Soc 0002017Accepted 2017 November 13 . Received 2017 November 13 ; in original form 2017 June 25Printed 11 (MN L A T E X style file v2.2)ISM: individual objects: G167+01 -X-rays: individual: G1673+008 - stars: neutron -pulsars: general -ISM: supernova remnants -radiation mechanisms: non-thermal We present X-ray observations of PWN G16.73+0.08/SNR G16.7+0.1 using archival data of Chandra ACIS. The X-ray emission peak location of this pulsar wind nebula is found to be offset by 24 arcsec from the centre of the 1.4-GHz emission of this nebula. The X-ray nebula is elongated in the direction from the X-ray peak to the 1.4-GHz emission centre. This offset suggests that G16.73+0.08 is an evolved pulsar wind nebula interacting with the supernova remnant reverse shock. We identify a point source, CXO J182058.16-142001.5, near the location of the X-ray peak. The spectrum of the X-ray nebula can be described by an absorbed power law of photon index 0.98 +0.79 −0.71 and hydrogen column density N H = 4.99 +2.75 −2.28 × 10 22 cm −2 . CXO J182058.16-142001.5 is likely a pulsar. We estimate its spin-down power to be about 2.6 × 10 36 erg s −1 . Assuming its age at 3000 and 10,000 years, its dipole magnetic field strength at the polar surface is estimated to be about 4.2 × 10 13 G and 1.3 × 10 13 G, respectively. INTRODUCTION The term 'Pulsar Wind Nebulae' (PWNe) was first proposed by Hester & Kulkarni (1988a), motivated by theoretical and observational works in the 1980's on the interpretation of the observed radio and X-ray nebulae or cores in supernova remnants (SNRs). It was realized that these objects are not part of remnants of supernova explosion events, but objects driven by the relativistic winds injected from the central pulsar and interacting with ISM or with supernova ejecta (e.g., Kennel & Coroniti (1984); Michel (1985); Chevalier (1987); Kulkarni & Hester (1988); Hester & Kulkarni (1988b)). The study of PWNe has provided information about physical properties of ISM and supernova ejecta, about the composition, energetics and evolution of the winds, and about the wind injection rate of the pulsar. In addition, pulsar winds are likely related to the so-called positron excess in cosmic rays, which otherwise may have a dark-matter origin (e.g., Erlykin & Wolfendale (2013); Feng & Zhang (2016)). PWNe are also found to be major TeV sources in the sky. To date, about 60 PWNe are found, although of nearly half of which E-mail: [email protected] the central pulsars are yet to be identified. For a more comprehensive review, readers are referred to, e.g., Kargaltsev et al. (2015), Porth et al. (2017), and Reynolds et al. (2017). G16.73+0.08 is believed to be a PWN in the SNR G16.7+0.1. Its radio emission from 90 cm to 2 cm was reported in Helfand et al. (1989). ASCA observation yielded a detection of an unresolved source (Sugizaki et al. 2001). Later in XMM observation an extended source was revealed, which is located near the geometric centre of SNR G16.7+0.1 (Helfand, Agueros & Gotthelf 2003). This gives a hint of possible location offset between the XMM-observed X-ray nebula and the radio nebula of G16.73+0.08, because the radio nebula is about 28 arcsec away from the geometric centre of G16.7+0.1. We noticed that Chandra had also observed this source in 2003 and therefore started the analysis of this Chandra data. Our results are reported in this paper. With Chandra's fine angular resolution, we confirm the location offset between the X-ray and radio nebulae of G16.73+0.08. This morphology indicates that G16.73+0.08 is likely an evolved system interacting with the supernovaremnant reverse shock ( Kolb et al. 2017). We also identify a point X-ray source near Figure 1. Chandra image of G16.73+0.08 overlaid with a JVLA 1.4-GHz contour map of G16.7+0.1/G16.73+0.08. The 1.4-GHz contours are at the levels of 10, 20, 35, and 45 mJy per beam, respectively. The abscissa is the right ascension and the ordinate is the declination (Epoch 2000). A zoom-in view of the central region is shown in Fig. 2. The two green rectangular boxes indicate the regions used for background estimate in our spectral analysis described in Section 3. the peak of the X-ray nebula. The spectrum of the X-ray nebula can be described by an absorbed power law, consistent with earlier XMM observation. We describe the Chandra data and image morphology in Section 2, spectral analysis in Section 3, and discuss the location offset between the X-ray and radio nebulae of G16.73+0.08 and other physical properties of the possible pulsar at the centre of the X-ray nebula in Section 4. DATA AND IMAGE MORPHOLOGY The Chandra X-ray Observatory (Weisskopf, O'Dell & van Speybroeck 1996) observed G16.7+0.1 on 2003 October 3 with ACIS-S without grating in a timed-exposure (TE) mode (ObsId: 3845; PI: D. Helfand). The usable data volume, after GTI filtering, is about 25 ksec. We processed this data using Chandra Interactive Analysis of Observations (CIAO) software version 4.9 to produce level 2 data for further analysis. We selected only those data in the energy range from 1 keV to 8 keV to have better energy calibration information, lower particle background and lower ACIS quantum efficiency contamination for this study. (2003)). The X-ray nebula is elongated in the direction from the X-ray peak to the 1.4-GHz nebula centre, with The X-ray emission peak, located at 24 arcsec away from the 1.4-GHz nebula centre to the southeast, is highly concentrated. To study the morphology in more details, we used a model composed of a delta function, a twodimensional Gaussian and a constant to fit the image data in a 20-arcsec × 20-arcsec region centred at the X-ray emission peak location, the black cross in Fig. 2. The model was convolved with a Chandra point spread function, which we obtained with the Chandra Ray Tracer (ChanRT) simulator (Carter et al. 2003) at photon energy 3.5 keV and with the MARX package version 5.3.2 (Davis et al. 2012). Results of this model fitting are shown in the 2nd column in Fig. 3. In the top panel is the convolved model. The middle panel is the residual image and the bottom panel is the smoothed residual image. Although this fitting is well acceptable with χ 2 ν = 0.19, mainly due to the small number of photons, 397, in this 40-pixel × 40-pixel region, a bright feature remains at the location of the X-ray emission peak. We therefore added one more two-dimensional Gaussian to account for that feature. Results of this fitting are shown in the 3rd column in Fig. 3. This fitting is similarly acceptable, but that bright feature is gone. This second Gaussian component, whose FWHM is about 2.6 arcsec, might be manifestation of a possible torus around the central neutron star represented here by the delta-function component. Values of its fitting parameters are shown in Table 1. We note, however, that a model without the delta-function component, that is, consisting of a constant and two two-dimensional Gaussians, also gives an equally well fitting and there is no bright featute at the centre either. Results of this fitting are shown in the 4th column in Fig. 3. It casts some doubt on the existence of a point source at the centre, as judged from the data. Nonetheless, the model with a delta function and two Gaussians besides a constant, seems more physical, since it presents a picture of a neutron star, a possible torus, and a pulsar wind nebula. We identify the central point source, represented here by the delta function component, as CXO J182058.16-142001.5. SPECTRAL ANALYSIS To study the X-ray spectrum of this elongated nebula, we defined the target region for spectral analysis as the rectangular region shown in Fig. 2. The background region was chosen to be the two rectangular regions outside the SNR, as shown in Fig. 1 (NE ∝ E −Γ ) and a neutral hydrogen column density NH = 5.16 +2.28 −1.92 ×10 22 cm −2 . The interstellar absorption model employed in our analysis is an photoelectric absorption model ('phabs' in XSPEC) with the assumed abundance taken from Anders & Grevesse (1989). These results are consistent with the earlier XMM observation (Helfand, Agueros & Gotthelf 2003). On the other hand, we actually expect this region contains not only the pulsar wind nebula, but also some contribituion from the SNR. Adding one more thermal spectral componet ('bremss' in XSPEC) to the model, however, yields a best fit with a larger χ 2 , 0.85. Although it is still an acceptable fit, the thermal component becomes the dominating one in such a fit, and the uncertainty ranges of its fitting parameters are so large that XSPEC (version 12.0.9n) cannot give a valid error estimate. We do not think this is an adequate fit. We conclude that the contribution from the SNR is negligible based on the current data. Although the above power-law fitting describes the spectrum very well, the central neutron star, together with the tiny extended source as revealed in our imaging analysis, can have a spectral behavior very different from the PWN. We therefore also performed another spectral analysis with the central region, defined as a circular region of radius 2 arcsec centred at the X-ray peak location, removed from the PWN. We fit the spectrum of the PWN target region without the central circle with a power-law model. Fitting results are shown in mal component to the spectral model does not improve the fitting. The fitting and residuals are shown in Fig. 4. The central circle contains only 39 photons in 1 − 8 keV, out of the 512 photons in the whole target region. It is difficult to have a meaningful spectral fitting. Its photon count is 11, 11, and 17 in the energy bands 1 -3.4, 3.4 -4.7, and 4.7 -8.0 keV, respectively. The corresponding number for the whole target region is 164, 180, and 168. In the Chandra X-ray image (Fig. 1), the X-ray emission in the SNR shell region is not obvious. Taking an annular region centred at the geometric centre of the radio SNR (the blue cross in Fig. 2) with a 45-arcsec inner ra- dius and a 135-arcsec outer radius as the SNR shell region, the same as that defined in Helfand, Agueros & Gotthelf (2003), and another annular region from 135 arcsec to 175 arcsec as the background region, we found that there are 13014 photons in the SNR shell region and 9391 counts in the background region. The number of net counts in the shell region is 748.6, after subtracting estimated background counts. This number is 6.5 times the square root of 13014. We therefore conclude that the X-ray emission in the SNR shell region is detected at a level of about 6.5σ. This region is so background-dominated that a meaningful spectral fitting cannot be obtained. DISCUSSION The offset between the 1.4-GHz and X-ray nebulae in G16.73+0.08 was revealed to some extent in the XMM data (Helfand, Agueros & Gotthelf 2003), in which, with a coarser spatial resolution, the X-ray nebula was found to be close to the geometric centre of the radio SNR, while its radio nebula was known to be off centre (Helfand et al. 1989). With Chandra's better resolution, we report the offset of 24 arcsec between the 1.4-GHz and X-ray nebulae in G16.73+0.08 in Fig. 2. We note that the nebula observed at 6-cm and 20-cm wavelengths has a weak extension to the location of the Xray nebula and that the 2-cm emission in fact has a high intensity region surrounding the location of CXO J182058.16-142001.5, in addition to the 2-cm nebula coincident with that at 6-cm, 20-cm, and 1.4-GHz (Figure 2 in Helfand et al. (1989)). Such an offset between radio and X-ray nebulae has been observed in several pulsar wind nebulae, for example, G327.1-1.1 (Temim et al. 2015(Temim et al. , 2009 Table 2. 11-62 (Slane et al. 2012), and Vela X (LaMassa, Slane & de Jager 2008). The morphology of these PWNe in X-ray and radio bands can be interpreted as PWNe being interacting with (or, crushed by) the supernova remnant reverse shock (Blondin, Chevalier & Frierson 2001;van der Swaluw, Downes & Keegan 2004;Temim et al. 2015;Kolb et al. 2017). The asymmetry of the SNR reverse-shock sweeping direction may come from inhomogeneity of ISM, anisotropic SN explosion, and/or the motion of pulsars. Our study indicates that G16.73+0.08 is a member of this reverse-shock-crushed PWN family. The age of SNR G16.7+0.1 could therefore be larger than previuosly thought, because in such a case the SNR has passed the free expansion phase and is likely dynamically evolved. Though, its absolute age could still be on the order of a few thousand years if the ambient density is high. It depends strongly on the environment property; see, e.g., Temim et al. (2015) and Kolb et al. (2017). From the spectral analysis of the X-ray nebula, we obtained a photon index of 0.99 +0.68 −0.62 and column density NH = 5.16 +2.28 −1.92 × 10 22 cm −2 , which are both in agreement with the photon index 1.17±0.29 and NH = 4.74±0.98×10 22 cm −2 obtained from XMM observation (Helfand, Agueros & Gotthelf 2003). The unabsorbed flux in 0.5 -10 keV from our result is 1.3×10 −12 erg s −1 cm −2 , comparable to the flux 1.9 × 10 −12 erg s −1 cm −2 obtained in Helfand, Agueros & Gotthelf (2003). Removing the contribution from the central source in a 2-arcsec-radius circular region does not change the fitting results much; see Table 2. Based on an empirical relationship between the photon index of the X-ray nebula and the central pulsar's spindown power (Gotthelf 2003), the photon index 0.99 gives a spin-down power ofĖ = 2.2 × 10 36 erg s −1 for the possible pulsar in G16.73+0.08. Using another empirical relationship between the X-ray luminosity LX (0.2 -4 keV) of the pulsar-powered nenula (including the pulsar) and the pulsar spin-down powerĖ, log LX = 1.39 logĖ − 16.6 (Seward & Wang 1988), we may also estimate the spin-down power. The derived unabsorbed flux from our fitting model in 0.2 -4 keV is 5.1 × 10 −13 erg s −1 cm −2 . Adopting a distance of 14 kpc (Zhang et al. 2017), the corresponding luminosity is 1.2 × 10 34 erg s −1 , which yields a spin-down power ofĖ = 2.9 × 10 36 erg s −1 . We will use the average of the above two estimates, that is,Ė = 2.6 × 10 36 erg s −1 , for the following discussion. In Helfand, Agueros & Gotthelf (2003), the age of the possible pulsar is estimated to be about 2100 years, based on the assumption that the SNR is in the free-expansion phase with an expansion speed of 3000 km/s and the SNR is at a distance of 10 kpc. Using a new distance measurement based on HI absorption and OH maser observations (Zhang et al. 2017), the distance to G16.7+0.1 is inferred to be 14 kpc. The age of the pulsar is then about 3000 years. However, considering the scenario that SNR reverse shock has already crushed the PWN, its age could be larger. Given the spindown powerĖ and characteristic age τ , one may infer the pulsar's period P and its time derivativeṖ . For τ = 3000 years, we have P = 0.29 s andṖ = 1.5 × 10 −12 s s −1 . For τ = 10000 years, they are P = 0.16 s andṖ = 2.5 × 10 −13 s s −1 . The estimated dipole magnetic field strength at the polar surface is 4.2 × 10 13 G and 1.3 × 10 13 G for the above two cases, respectively. In summary, the PWN G16.73+0.08 has its X-ray emission region separated from the radio one. It is likely that this PWN is in the stage of interacting with the SNR reverse shock. A point-like X-ray source, which we designate as CXO J182058.16-142001.5, is found near the brightness centre of the X-ray nebula, together with a small extended source, which may be a torus surrounding the neutron star. Future longer observations with adequate time resolution is very much desired to explore the properties of this possible pulsar. Observations with adequate spatial resolution is also needed to study spatially resolved spectra of the nebula. These are helpful to our understanding of how pulsars emit electron-positron pairs and how these pair plasmas, as the pulsar winds, propagate and evolve. Flight Center. This work was supported by the Ministry of Science and Technology of the Republic of China under grants MOST 105-2112-M-007-002 and MOST 106-2811-M-007-008. Fig. 1 1shows the Chandra image of G16.73+0.08 overlaid with JVLA 1.4-GHz continuum contours (see alsoZhang et al. (2017)). It is clear in this figure that the X-ray nebula does not coincident well with the 1.4-GHz nebula near the centre of SNR G16.7+0.1. Fig. 2 is a zoom-in view of this central region. The three crosses indicate the positions of the X-ray emission peak (black, α = 18 h 20 m 58 s .13, δ = −14 • 20 01 .5), the centre of the 1.4-GHz nebula (white, α = 18 h 20 m 57 s .09, δ = −14 • 19 42 .8), and the geometric centre of the radio SNR G16.7+0.1 (blue, α = 18 h 20 m 57 s .8, δ = −14 • 20 09 .6, as quoted in Helfand, Agueros & Gotthelf Figure 2 . 2A zoom-in view of the central region ofFig. 1. The Chandra image has been smoothed with a Gaussian filter of σ = 1.5 pixels. The smoothed intensity is displayed with a logarithmic scale and is described quantitatively with black contours. These X-ray contours are at the level of 0.25, 0.5, 0.75, and 1 count per pixel in the smoothed image. The 1.4-GHz contours are the same as that inFig. 1. The three crosses indicate the location of the Xray emission peak (black), the centre of the 1.4-GHz core (white), and the geometric centre of G16.7+0.1 at 6 cm (blue). The green rectangle encloses the target region for spectral analysis.The two point sources, one at 16 arcsec to the west of black cross and the other at the top right corner of this image, both have counterparts in the 2MASS point-source catalog. a size of about 30 arcsec × 20 arcsec. Two point-like sources in Fig. 2, located at 16 arcsec to the west of the X-ray peak and at the top-right in the figure, are spatially coincident with two point sources in the 2MASS catalog, 2MASS 18205707-1420034 (α = 18 h 20 m 57 s .071, δ = −14 • 20 03 .49) and 2MASS 18205669-1419295 (α = 18 h 20 m 56 s .690, δ = −14 • 19 29 .58), respectively. . The spectrum of this target region, in 1 − 8 keV, can be well fitted (χ 2 ν = 0.78 for 22 d.o.f) by an absorbed power law of a photon index Γ = 0.99 +0.68 −0.62 Figure 3 . 2 2 322Image morphology fitting in the 20-srcsec × 20-arcsec square area centred at the X-ray emission peak location (black cross inFig. 2). The first column (counted from the left) shows the original Chandra image in that area. The bottom panels in all columns are the corresponding middle-panel images smoothed with a Gaussian filter of σ = 1.5 pixels. In the 2nd, 3rd and 4th columns, the top panels are images of different fitting models convolved with the Chandra point spread function, and the middle panels are residual images for that fitting model. See the main text for details of components employed in different models.The whole target region Column density, N H 5.16 +2.28 −1.92 × 10 22 cm −Photon flux (0.5 -10 keV) 1.3 × 10 −12 erg cm −2 s −1Target region without the central circle Column density, N H 4.99 +2.75 −2.28 × 10 22 cm −Photon 10 −12 erg cm −2 s −1 , G292.0+1.8 (Bhalerao et al. 2015), MSH 15-56 (Temim et al. 2013), MSH Figure 4 . 4Spectral fitting with Chandra ACIS-S data of the G16.73+0.08 X-ray nebula. The upper panel shows the spectrum of the whole target region, while the lower panel shows that with the central circular region removed. The model used is an absorbed power law. The best fit parameters are shown in Blondin, Chevalier & Frierson 2001; van der Swaluw, Downes & Keegan 2004; Temim et al. 2015; Table 2 . 2do not change much. Similar to the spectrum of the whole target region with the central circle included, adding a ther-Model A (the 3rd column inFig. 3):The neutral hydrogen column density, 4.99 +2.75 −2.28 × 10 22 cm −2 and the photon index, Γ = 0.98 +0.79 −0.71 , Table 2 . 2Spectral fitting results. c 2017 RAS, MNRAS 000, 1-6 ACKNOWLEDGMENTSWe are very much appreciative of valuable suggestions from the anonymous referee to improve this paper significantly. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research centre Online Service, provided by the NASA/Goddard Space E Anders, N Grevesse, Geochimica et Cosmochimica Acta. 53197Anders, E., Grevesse, N., 1989, Geochimica et Cosmochim- ica Acta, 53, 197 . J Bhalerao, S Park, D Dewey, J P Hughes, K Mori, J.-J Lee, ApJ. 80065Bhalerao, J., Park, S., Dewey, D., Hughes, J. P., Mori, K., Lee, J.-J., 2015, ApJ, 800, 65 . J M Blondin, R A Chevalier, D M Frierson, ApJ. 563806Blondin, J. M., Chevalier, R. 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[]	[]	[]	[]	Possi bl e evi dence f rom l aboratory m easurem ents f ora l ati tude and l ongi tude dependence ofG J. P.M bel ek and M .Lachi eze-R ey Servi ce d' A strophysi que,C . E.Sacl ay F-91191 G i f-sur-Y vette C edex,France February 1,2022A bstractStabi l i ty argum ents suggest that the K al uza-K l ei n (K K ) i nternal scal ar el d, ,shoul d becoupl ed to som eexternal el ds.A n externalbul k realscal ar el d, , m i ni m al l y coupl ed to gravi ty i s proved to be sati sfactory. A t l ow tem perature,the coupl i ng of to the el ectrom agneti c (EM ) el d al l ow s to bem uch strongercoupl ed to the EM el d than i n the genui ne ve di m ensi onal K K theory. It i s show n that the coupl i ng of to the geom agneti c el d m ay expl ai n the observed di spersi on i n l aboratory m easurem ents ofthe (e ecti ve) gravi tati onalconstant. T he anal ysi s takes i nto account the spati alvari ati ons of the geom agneti c el d. Except the hi gh PT B val ue, the predi cti ons are found i n good agreem ent w i th al lofthe experi m entaldata.	null	[ "https://export.arxiv.org/pdf/gr-qc/0204064v1.pdf" ]	14,160,936	gr-qc/0204064	3fb2e1e0179e432b2f4b1d1f9fa98e5ae8bfe852	arXiv:gr-qc/0204064v1 19 Apr 2002 arXiv:gr-qc/0204064v1 19 Apr 2002PA C S num bers: 04. 50. + h,95. 30. Sf,06. 20. Jr,04. 80. C c Possi bl e evi dence f rom l aboratory m easurem ents f ora l ati tude and l ongi tude dependence ofG J. P.M bel ek and M .Lachi eze-R ey Servi ce d' A strophysi que,C . E.Sacl ay F-91191 G i f-sur-Y vette C edex,France February 1,2022A bstractStabi l i ty argum ents suggest that the K al uza-K l ei n (K K ) i nternal scal ar el d, ,shoul d becoupl ed to som eexternal el ds.A n externalbul k realscal ar el d, , m i ni m al l y coupl ed to gravi ty i s proved to be sati sfactory. A t l ow tem perature,the coupl i ng of to the el ectrom agneti c (EM ) el d al l ow s to bem uch strongercoupl ed to the EM el d than i n the genui ne ve di m ensi onal K K theory. It i s show n that the coupl i ng of to the geom agneti c el d m ay expl ai n the observed di spersi on i n l aboratory m easurem ents ofthe (e ecti ve) gravi tati onalconstant. T he anal ysi s takes i nto account the spati alvari ati ons of the geom agneti c el d. Except the hi gh PT B val ue, the predi cti ons are found i n good agreem ent w i th al lofthe experi m entaldata. Introduction T husthe currentstatusofthe G terrestri alm easurem ents (see [ 3] )i m pl i esei theran unknow n source oferrors (not taken i nto account i n the publ i shed uncertai nti es), or som e new physi cs [ 4] . In the l atter spi ri t,m any theori es have been proposed as candi dates for the uni cati on ofphysi cs. A s such,they i nvol ve a coupl i ng between gravi tati on and el ectrom agneti sm (hereafter G E coupl i ng), as wel l as w i th other el ds present. T he K al uza-K l ei n theori es [ 5,6]have been am ong the rst attem pts to uni fy el ectrom agneti sm and gravi tati on. A l though they are di squal i ed i n thei r ori gi nal form ,they consti tute the prototype for m any present theori es (i n parti cul ar,those i nvol vi ng extra-di m ensi ons). In the l aboratory condi ti onsforG -m easurem ents,fundam entaltheori es as wel las K K theori es are wel ldescri bed as an e ective theory, under the form of the Ei nstei n and M axwel l equati ons conveni entl y m odi ed, as descri bed bel ow . T hi s paper expl ores the possi bi l i ty that the di screpancy between the resul ts of the G -m easurem ents i s the e ect of the G E coupl i ng,by com pari ng the avai l abl e data to the predi cti ons of such an e ecti ve theory. W e do not pretend that the e ecti ve theory adopted here (K K ,see bel ow ) i s the onl y possi bi l i ty. R ather,we consi der that i t provi des the si m pl est opportuni ty to confront the i dea of a G E coupl i ng w i th realdata. T he success ofthe t suggests that thi s m ay be the real expl anati on. T he si m pl est theori es accounti ng for a G E coupl i ng are those ofK al uza-K l ei n [ 5,6] ,orsl i ghtm odi cati onsofthem .T hei ruse correspondsto the m osteconom i cal way (them i ni m um ofhypotheses)to testthehypothesi sofG E coupl i ng.A si ti swel l know n,such theori esare e ecti vel y wel ldescri bed by Ei nstei n equati ons(w i th thei r correctN ew toni an l i m i t),w here the new toni an gravi tati onalconstantG i srepl aced by G eff ,gi ven bel ow : thi s e ecti ve gravi tati onalconstant depends ofthe fteenth degree offreedom ,ĝ 44 ,ofthe (5-di m ensi onal ) bul k m etri c,w hi ch pl ays the rol e ofa (four di m ensi onal ) scal ar el d (ĝ 44 = 2 i n the Jordan-Fi erz fram e). T he genui ne ve di m ensi onalK al uza-K l ei n theori esbei ng subjectto i nstabi l i ti es ([ 7,8,9] ),vari ous authors ( [ 10,11,12] ) have suggested a m ore acceptabl e versi on w hi ch i ncl udes an addi ti onalstabi l i zi ng externalbul k el d: here we adoptthe m i ni m al hypothesi s of a scal ar el d m i ni m al l y coupl ed to gravi ty. In thi s theory (hereafter K K ),G eff vari es w i th the el ectrom gneti c el d,and thus i n spaceti m e. H owever,onl y vari ati ons w i th respect to the cosm i c di stance or ti m e have been i nvesti gated i n the l i terature hi therto. But,si nce the geom agneti c el d vari es,w i th l ati tude and l ongi tude,and thus at the di erent si tes ofthe G m easurem ents,T he G E coupl i ng i m pl i esthattheexperi m entsi n factm easure thedi sti nctcorrespondi ng val ues ofG eff ,rather than an uni que val ue ofG . N ote al so that thi s theory predi cts a vari ati on of the e ecti ve ne structure constant w i th the gravi tati onal el d, and thus w i th the cosm ol ogi cal ti m e. In a com pani on paper ( [ 13] ),we com pare (w i th success) the predi cted evol uti on w i th astrophysi caldata concerni ng the di stant quasars ( [ 14,15] ). In secti on 2 we recal lthe de ni ti on ofthe e ecti ve coupl i ng constants,and the A ppl yi ng the l east acti on pri nci pl e to the acti on (1) yi el ds : T heoreticalbackground S = Z p g [ c 4 16 G R + 1 4 " 0 3 F F + c 4 4 G @ @ ]d 4 x + Z p g [ 1 2 @ @ U J ]d 4 x;(1) the general i zed Ei nstei n equati ons R 1 2 R g = 8 G eff c 4 [T (E M ) + T ( ) + T ( ) ] ;(4) w here T (E M ) = " 0eff ( F F + 1 4 F F g ); (5) T ( ) = c 4 8 G (r r g r r )(6) and T ( ) : = [@ @ ( 1 2 @ @ U J )g ](7) the general i zed M axwel lequati ons r F + r F + r F = 0(8) and r (" 0eff F )= 0; (9) and the scal ar el ds equati ons r r = J @J @ @U @ (10) and r r = 4 G c 4 " 0 F F 3 + U + J + @J @ 2 1 2 (@ @ ) ;(11) w here the sym bolr standsforthe R i em anni an covari antderi vati ve. C l earl y, 1 r 2 [ @ @r (r 2 @ @r ) + 1 si n @ @ (si n @ @ ) + 1 si n 2 @ 2 @' 2 ]= 2 @f E M @ (v;1)v B 2 0(12) and (g 0 1 ) 2 + (g 1 1 ) 2 + (h 1 1 ) 2 denotes i ts m agneti c m om ent (see [ 18,19] ). Setti ng cos 1 r 2 [ @ @r (r 2 @ @r )+ 1 si n @ @ (si n @ @ )+ 1 si n 2 @ 2 @' 2 ]= 2 @f E M @ (v;1)v B 2 0 ;(13' 1 = g 1 1 = q (g 1 1 ) 2 + (h 1 1 ) 2 , si n ' 1 = h 1 1 = q (g 1 1 ) 2 + (h 1 1 ) 2 and tan = g 0 1 = q (g 1 1 ) 2 + (h 1 1 ) 2 , w here we have set k(r)= 1 0 @f E M @ (v;1)v( 0 M 4 r 2 ) 2(15) and x( ;')= cos 2 + ( ;') = cos 2 + cot 2 si n 2 cos 2 (' + ' 1 ) cot si n 2 cos(' + ' 1 ) Tabl e 1 : R esul ts of the m ost preci se l aboratory m easurem ents of G publ i shed duri ng the l ast si xty years and l ocati on ofthe l aboratori es. Sam pl e H 0 H 1 S1 17 poi nts 2 = 3. 607 (best t) 2 = 21. 523 Tabl e 2 : R educed 2 for the two di erent hypothesi s H 0 (H ypothesi s of a constant G ) and H 1 ((H ypothesi s of an e ecti ve G ), and di erent sam pl es S1 and w hol e (except the hi gh PT B val ue [ 41] ,see text). N ow ,peri odi c vari ati ons ofthe gravi tati onal"constant" w i th the l unar or di urnal peri od have yet been poi nted out i n the l i terature (see [ 48,26,29,43] ). A l though i t i s presentl y bel i eved that they are rel ated to ti des,the expl anati on coul d be thi s tem poralvari ati on. W e noti ce that the G m easurem ents of [ 38]are consi stent w i th an annualvari ati on. D iscussion and conclusion Besi des, a recent study [ 49]show s that hel ioseism ol ogy seem s to favor a l ow x(L, l) Fi gure 1: Laboratory m easurem ents w i th rel ati ve uncertai nty G lab G lab < 10 3 and m easuri ng ti m e t < 200 s (sam pl e S1,17 poi nts [ 23] , [ 25]- [ 28] , [ 31] , [ 35] , [ 38] - [ 40] , [ 42]- [ 45] ). T he l i ne i ndi cates the best t G lab versus the m i xed vari abl e x ( 2 = 1: 327).A ssum i ng a constantG woul d yi el d a bad tto the data ( 2 = 3: 607), m ostl y because ofthe H U ST val ue. T hus we concl ude that,apart from system ati c errors that need to be corrected (e.g.,by appl yi ng som e prescri pti ons l i ke that poi nted out by K uroda [ 55]for the the SEE project [ 56] ) and further anal yses taki ng i nto account hi gher harm oni cs and the vari ouski nd ofchanges ofthe geom agneti c el d shoul d bri ng m ore support to our cl ai m . T he general i zed Ei nstei n equati ons (4) rew ri te (i ncl udi ng the contri buti on of the ordi nary m atter,T (m ) g r = G 0 (r)M r 2 f1 3( a r ) 2 J 0 2 ( 3 2 cos 2 1 2 )+ k(r) ( ;')[1 3( a r ) 2 J 2 ( 3 2 cos 2 1 2 )] 9 2 k(r)( a r ) 2 J 2 cos 4 g ! 2 r(1 cos 2 ) (25) and anal ogousl y for g ,w here J 0 2 = J 2 (1 13 3 k(r)) 2 9 k(r)( r a ) 2 1 + k(r) 3 [13 + 3 2 ( a r ) 2 J 2 ] [ 2] Y .T .C hen and A .C ook,G ravitationalExperim entin the Laboratory(C am bri dge U ni versi ty Press,C am bri dge,G reat Bri tai n,1993) [ 3] G .T .G i l l i es,R ep.Prog. Phys.60,151 (1997),and references therei n. A l though the m ethods and techni ques have been greatl y i m proved si nce the l ate ni neteenth century,the preci si on on the m easurem ent ofthe gravi tati onalconstant, G , i s sti l l the l ess accurate i n com pari son w i th the other fundam ental constants ofnature [ 1] . M oreover,gi ven the rel ati ve uncertai nti es ofm ost ofthe i ndi vi dual experi m ents (reachi ng about 10 4 for the m ost preci se m easurem ents), they show an i ncom pati bi l i ty w hi ch l eadsto an overal lpreci si on ofonl y about1 parti n 10 3 [ 2] . e ecti ve M axwel l -Ei nstei n equati ons,i ssued from the ve di m ensi onalcom pacti ed K K theory stabi l i zed by a m i ni m al l y coupl ed bul k scal ar el d. In secti on 3, we cal cul ate the vacuum sol uti onson Earth,i n the weak el d l i m i t,taki ng i nto account the geom agneti c el d. In secti on 4,we confront the predi cted val ues ofG eff w i th the l aboratory m easurem ents. In secti on 5,we di scussthe consi stency ofourresul ts w i th respect to the orbi talm oti on ofthe LA G EO S satel l i te,the M oon and pl anets ofthe sol ar system ,as wel las the bi nary pul sar PSR 1913 + 16. n argum ent i ni ti al l y from Landau and Li fshi tz[ 16]m ay be appl i ed to the pure K al uza-K l ei n (K K ) acti on ([ 12] ): the negati ve si gn ofthe ki neti c term ofthe K K i nternalscal ar el d l eads to i nescapabl e i nstabi l i ty. Stabi l i zati on m ay however be obtai ned i fan external el d i s present ([ 7] ,[ 8] ,[ 12] ),and we assum e here a versi on K K oftheK K theory w hi ch i ncl udesan externalbul k scal ar el d m i ni m al l y coupl ed to gravi ty. A fter di m ensi onalreducti on ( = 0;1;2;3),thi s bul k el d reduces to a four di m ensi onalscal ar el d = (x ) and,i n the Jordan-Fi erz fram e,the l ow energy e ecti ve acti on takes the form (up to a totaldi vergence) w here A i sthe potenti al4-vectorofthe el ectrom agneti c el d,F = @ A @ A the el ectrom agneti c el d strength tensor,U the sel f-i nteracti on potenti alof and J i ts source term . Fol l ow i ng Li chnerow i cz [ 17] ,we i nterpret the quanti ty G eff = G (2) ofthe Ei nstei n-H i l bert term ,and the factor " 0eff = " axwel lterm respecti vel y as the e ecti ve gravi tati onal"constant" and the e ecti vevacuum di el ectri cperm i tti vi ty.T hee ecti vevacuum m agneti cperm eabi l i ty reads 0eff = 0 = 3 ,so thatthe vel oci ty ofl i ghti n vacuum rem ai nsa true uni versal constant. Both term s depend sol el y on the l ocal(for l ocalphysi cs) or gl obal(at cosm ol ogi calscal e) val ue ofthe K K scal ar el d ,assum ed to be posi ti ve de ned. T he source term of the -el d, J, i ncl udes the contri buti ons of the ordi nary m atter(otherthan the scal ar el ds and ),ofthe el ectrom agneti c el d and ofthe i nternalscal ar el d . Foreach,the coupl i ng i s de ned by a functi on (tem perature dependent,asforthepotenti alU )f X = f X ( ; ),w here thesubscri ptX standsfor "m atter","EM " and " ".In orderto recoverthe Ei nstei n-M axwel lequati onsi n the weak el ds l i m i t,these three functi ons are subject to the condi ti ons: f E M (v;1) = f m atter (v;1)= f (v;1)= 0,w here v denotes the vacuum expectati on val ue (V EV ) ofthe -el d. T he contri buti ons of m atter and are proporti onalto the traces of thei r respecti ve energy-m om entum tensors. Si nce the energy-m om entum tensorofthe el ectrom agneti c el d i s tracel ess, a contri buti on of the form " 0 f E M F F accounts for the coupl i ng w i th i t. T he t ofour m odelto the data (see bel ow ) show s that @f E M @ (v;1)v 4 G =c 4 ,asi tcan be expected nearthe vacuum atl ow tem perature. T hus,we w i l lnot take the l atter term i nto account. H owever,we m ay suspect that @f E M @ (v;1)v 4 G =c 4 athi gh tem perature,w hi ch m ay have consequences i n som e astrophysi calcondi ti ons (see bel ow ). T (E M ) and T ( ) de ne respecti vel y an e ecti ve energy-m om entum tensor for the el ectrom agneti c el d i n the presence ofthe K K scal ar el d and an e ecti ve energy-m om entum tensorforthe l atteri tsel f. R el ati ons(4-9)are form al l y the sam e as the Ei nstei n-M axwel l ones, but w i th the addi ti onalcontri buti on of the K K scal ar as a m atter source and the repl acem ent ofG and " 0 by thei r respecti ve e ecti ve val ues. 3 V acuum solutions in the presence of a dipolar m agnetic eld Si nce we are i n weak el d condi ti ons (we l ook forsm al ldevi ati ons from N ew toni an physi cs),we onl y keep rst order term s. T hus,we negl ect the exci tati ons of and w i th respect to thei r respecti ve V EV ' s 1 and v. A l so,the energy densi ty ofthe -el d m ust be l ower than that ofthe m agneti c el d. Let us study the spati alvari ati on of out ofthe el ds'source,but i n presence of a stati c di pol ar m agneti c el d,B =B (r;'; ). W e denote r,' and respecti vel y the radi us from the centre, the azi m uth angl e and the col ati tude. T hus, w ri ti ng = (r;'; )and = (r;'; ),and taki ng i nto accountthat @U @ (v)= 0 (de ni ti on ofthe V EV ),equati ons (10) and (11) si m pl i fy respecti vel y as ) w here we have dropped the pure gravi tati onal constant 4 G =c 4 w i th respect to v@f E M =@ ,as i ndi cated above. W e consi der a di pol ar m agneti c el d:B =r V . Forour purpose,i t i s su ci ent to l i m i tthe expansi on ofthe scal arpotenti al ,V ,to the term softhe Legendre functi on ofdegree one (n = 1) and order one (m = 1). H ence V = (a 3 =r 2 )the rel evant G auss coeci ents, a i s the Earth' s radi us and M = 4 0 a 3 q and si m i l arl y for by m aki ng the substi tuti on @f E , one deri ves the expressi on of G eff (r; ;') by i nserti ng the sol uti on(14) above i n rel ati on (2). T hus,si nce @f E M @ (v;1)v > 0 and the vari abl e x turns out to beposi ti veatany posi ti on i n space,i tfol l ow sthatthee ecti vegravi tati onalconstant G eff w i l lal ways be greater than the true gravi tati onalconstant, G . W hence the predi cti on ofan upward bi as i n the l aboratory m easurem ents ofG .Becauseofvari ousuncontrol l ed system ati cerrors,thedata publ i shed by thedi erent l aboratori eshave di erent preci si ons. In the fol l ow i ng,we test two hypotheses w i threspect to these resul ts: H 0 = H ypothesi s ofa constant G ( = n 1) and H 1 = H ypothesi s ofan e ecti ve G ( = n 2). H ere denotes the num ber ofdegrees of freedom ,n i s the num ber ofdata poi nts,and we have 1 or 2 param eters i n the t. T here are presentl y al m ost 45 resul ts ofm easurem ents G publ i shed si nce 1942 (see e. g. ,[ 3] ,Tabl e2,pp.168 and 169).W eexcl ude from thepresentstudy them i ne m easurem entsbecause ofthe too num erousuncontrol l ed system ati c bi asesi nvol ved. T he " accepted " val ues are presentl y G = 6: 67259 0: 00085 10 11 (C O D ATA 86, [ 20] )and G = 6: 670 0: 010 10 11 (C O D ATA 2000,[ 21] )i n M K S uni t.A tti ng ofthe 45 data w i th theseval uesgi verespecti vel y 2 = 145: 17 and 2 = 213: 25 ( 2 = 2 = , w here denotes the degrees offreedom ). Ifwe forget the accepted val ue and try a best t,assum i ng an arbi trary constant val ue ofG ,we obtai n G = 6: 6741 10 11 SI w i th 2 = 127: 04. T he m ore di scordant l aboratory m easurem ent (hi gh PT B val ue [ 41] ) i s controversi al . If we di scard i t, the previ ous ts l ead to 2 = 11: 128 (C O D ATA 86), 2 = 62: 498 (C O D ATA 2000) and 2 = 2: 341 (free val ue). Si nce i t seem s now certai n that thi s hi gh PT B val ue [ 41]su ers from som e system ati c error (see [ 36] , for m ore detai l s), we rem ove i t for our anal ysi s (i f we keep i t, our m odel i s sti l l m ore favored). N ote thatthe strongest contri buti ons to the 2 then com e from the BIPM 2001 [ 36]and the H U ST [ 25]m easurem ents. T hus, unl ess there are som e experi m entalsystem ati c errors presentl y not understood,these experi m ents do not m easure the sam e quanti ty. In the fram ework of the theory proposed here, they m easure G eff .Because ofthe G E coupl i ng,G eff shoul d depend on the geom agneti c el d at the l aboratory posi ti on. Fi rst, we sel ect a subset of resul ts w i th a good preci si on and w hi ch does not su er any si gni cant system ati c error: we i ncl ude onl y those poi nts w i th rel ati ve uncertai nty G lab =G lab 10 3 . A l so, we dem and a short m easuri ng ti m e ( t < 200 s),to m i ni m i ze the possi bl e bi asesdue to the ti m e vari ati onsofthe geom agneti c el d. T hi s gi ves the sam pl e S1, w i th 17 poi nts, show n i n gure 1. W e t these data usi ng the IG R F 2000 G auss coe ci ents 1 11 m 3 kg 1 s 2 ; (18) that we retai n as a true gravi tati onalconstant. T he rel ati ve uncertai nty i s onl y 1 parti n 10 4 :the m ajorpartofthe di erences between the l aboratory m easurem ents was generated by the predi cted vari ati on ofG eff w i th the m agneti c el d. Further, adjusti ng to the sam e set the m ean val ue M = 7: 87 10 22 A m 2 (i n the ti m e i nterval spanni ng from 1942 to 2001),i t fol l ow s @f E M @ (v;1)v = (5: 44 0: 66)10 6 fm T ev 1 ; (19) that we retai n too. W e observe al so that the H U ST val ue (the l owest m ost preci se m easured val ue ofG ) i s perfectl y tted: i t di ers from other val ues because ofthe proxi m i ty ofthi s l aboratory to the equator.T hen,we t the w hol e sam pl e (excl udi ng the PT B val ue,as stated above). ts. It gi ves 2 = 1: 669,to be com pared to 2 = 2: 255 for the best t assum i ng a constantG . T hese resul ts are sum m ari zed i n Tabl e 2. In orderto check the rel evance ofourresul t(w hi ch i nvol ves two free param etersratherthan one),we appl y the F test (Fi sher l aw ). T hi s yi el ds F = 2 2 W e have not taken i nto account the tem poralvari ati on of the geom agneti c el d.A l lthe data consi dered i n thi s paper are averaged on ti m e. N everthel ess,for those w hi ch woul d not be averaged on ti m e, one m ay expect sm al l ti m e vari ati ons of G eff both w i th the Sq and w i th the L el d di sturbances ofthe geom agneti c el d. Fi gure 2 : 2G lab versus x (w hol e sam pl e pl us the PT B 95 val ue,45 poi nts[ 24]-[ 47] ). val ue ofG ,cl ose to the H U ST val ue. T hi si si n accordance w i th ourpredi cti ons(see secti on 3 and rel ati ons (17) and(18)),si nce one expects the e ecti ve coupl i ng of the -el d to the EM el d to decrease towards i ts l owest val ue as the tem perature i ncreases. H ence,one gets a good agreem ent between our predi cti on(18) and the hel i osei sm i c data,on account ofthe hi gh tem perature m et i n the core ofthe Sun.To testourproposalofa spati aldependence ofG eff ,we cal lfora com pl ete coveri ng ofthe Earth (i n parti cul ari n the south hem i sphere) and atdi erentl ati tudes.O n the basi sthe currentavai l abl e data,the present study al l ow sto m ake som e predi cti ons. Fi rst, accordi ng to the authors [ 49] ,the Sudbury N eutri no O bservatory (SN O :L = 46 29'N and l= 80 59'O ) [ 50]shoul d consti tute a prom i si ng m eans ofdeterm i ni ng G w i th a qui te good accuracy. H ence,si nce the m easured quanti ty i s actual l y the e ecti ve gravi tati onalconstant,the val ue predi cted at SN O by the rel ati on (17) i s G S N O = (6: 6742 0: 0009) 10 11 m 3 kg 1 s 2 ,that i s practi cal l y the sam e as at the M SL [ 23] . In addi ti on,i t woul d be ofi nterest to test al so our predi cted val ue atthe surface ofthe pol es,thati s:G poles = (6: 6763 0: 0010)10 11 i n M K S uni ts. A t the equator, one shoul d get a dependence w i th respect to the l ongi tude (si nce the m agneti c and geographi c pol es do not coi nci de). A t the orbi t of the M oon, the rel ati ve devi ati on, (G eff G )=G , to the true gravi tati onalconstant i s predi cted to be as sm al las 1: 7 10 13 w hi ch i s consi stent w i th l unar l aser rangi ng (see, [ 51] ). In the Sol ar System , the rel evant m agneti c el d i s the di pol ar el d of the qui et Sun. Si nce the coupl i ng constant of theel d to the EM el d i s m uch weaker w i thi n the Sun than on Earth, one nds at the orbi tal radi us of M ercury (G eff G )=G < 10 6 and m uch sm al l er beyond, decreasi ng as 1=r 4 . T hus,at the orbi talradi us ofN eptune (G eff G )=G drops to l essthan 10 12 . Taki ng i nto accountthe overal lpl anetary constrai ntson G M [ 52] , the accord between the proposed m odeland observati on i s sti l lacceptabl e. Forarti ci alsatel l i tes i n quasi -ci rcul arorbi ts,the appropri ate quanti ty forcom -pari son w i th observati onaldata i s(G eff G 0 )=G 0 w hen the anal ysi si sbased onl y on thesatel l i tedata (seeappendi cesA and B),w herewehavesetG 0 = G [1+ k(2 J 2 )]and J 2 i s the true Earth quadrupol e m om ent coe ci ent. R eferri ng to theorbi talm oti on oftheLA G EO S satel l i te,w i th sem i m ajoraxi sequalto 12 270 km , i ncl i nati on i = 109: 94 and eccentri ci ty e = 0: 004,one nds the m axi m um devi ati ons (G eff G 0 )=G 0 ' 1: 9 10 8 (w hereas (G eff G )=G ' (G 0 G )=G ' 2 10 5 ), consi stentw i th the constrai ntj j< 10 5 5 10 8 on the Yukawa coupl i ng constant (see [ 53] , gure 3. 2a,p. 99 and secti on 6. 7). W e al so l ooked for a possi bl e rel ati ve devi ati on to the true gravi tati onalconstant i nduced by the strong m agneti c el ds ofpul sars. W e found thi s e ect qui te negl igi bl e,ofthe order 10 7 i n the case ofthe bi nary pul sar PSR 1913 + 16,for w hi ch observati on yi el ds G eff = G N (1: 00 + 0:14 0:11 ) [ 54] . T he devi ati on i s ti ny because ofthe sm al l radi us of the pul sar and of i ts com pani on as com pared to thei r respecti ve orbi talradi us around the center ofm ass. sw i ngi ng pendul um m ethod),the onl y possi bi l i ty to reconci l e the publ i shed val ues ofG i sto consi dera dependence on the l ati tude and l ongi tude,ofthe type proposed here. In parti cul ar,i fal lpresent system ati c errors coul d be rem oved i n the future, we predi ct G lab = (6: 6742 0: 0009) 10 11 m 3 kg 1 at the PT B l aboratory,that i s the sam e val ue as that predi cted at SN O and the current M SL. U p to now ,a l ot ofattenti on had been pai d to the dependence ofG on cosm i c ti m e orradi aldi stance onl y. Butthe dependence on l ati tude and l ongi tude,thatwe exam i ne here,has not been taken i nto account. M ore preci se m easurem ents (e. g. , appearsasthe e ecti ve quadrupol e m om entcoe ci entofthe centralbody atradi us r. T he quadrati c cosi ne term provi des an addi ti onal term to the e ecti ve zonal harm oni c coe ci ent of order 4. C l earl y, by i nterpreti ng the data from a si ngl e satel l i te i n ci rcul ar orbi t,J 0 2 w i l lappears as a constant param eter. R eferences [ 1] J.Luo and Z.K .H u,C l ass.Q uantum G rav.17,2351 (2000) [ 4 ] 4V .N .M el ni kov,1999,SISSA prepri nt gr-qc/9903110,(1999) [ 5] T h.K al uza,Si tzungsberi chte derK .Preussi schen A kadem i e derW i ssenschaften zu Berl i n,966 (1921) ; Engl i sh transl ati on i s avai l abl e i n M odern K al uza-K l ein T heories,T .A ppel qui st,A .C hodos and P.G .O .Freund eds. ,pp.61-68.[ 6] O .K l ei n,Zei tschri ftf ourPhysi k 37,895 (1926);Engl i sh transl ati on i savai l abl e i n M odern K al uza-K l ein T heories,T .A ppel qui st,A .C hodosand P.G .O .Freund eds. ,pp.76-86. T he IG R F coe ci ents are gi ven for ti m e i nterval s of ve years. H ence, for a m ore preci se tti ng,one shoul d use the m ost sui tabl e IG R F coe ci ents for a gi ven l aboratory val ue accordi ng to the years the m easurem ents were carri ed out. O fcourse,the necessi ty ofdoi ng so depends on the preci si on reached i n the l aboratory G m easurem ents. 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[ "Rheology of Granular Materials: Dynamics in a Stress Landscape", "Rheology of Granular Materials: Dynamics in a Stress Landscape" ]	[ "B Y Dapeng Bi \nDepartment of Physics\nBrandeis University\n02454WalthamMAUSA\n", "Bulbul Chakraborty \nDepartment of Physics\nBrandeis University\n02454WalthamMAUSA\n" ]	[ "Department of Physics\nBrandeis University\n02454WalthamMAUSA", "Department of Physics\nBrandeis University\n02454WalthamMAUSA" ]	[]	We present a framework for analyzing the rheology of dense driven granular materials, based on a recent proposal of a stress-based ensemble. In this ensemble fluctuations in a granular system near jamming are controlled by a temperature-like parameter, the angoricity, which is conjugate to the stress of the system. In this paper, we develop a model for slowly driven granular materials based on the stress ensemble and the idea of a landscape in stress space. The idea of an activated process driven by the angoricity has been shown byBehringer et al (2008)to describe the logarithmic strengthening of granular materials . Just as in the Soft Glassy Rheology (SGR) picture, our model represents the evolution of a small patch of granular material (a mesoscopic region) in a stress-based trap landscape. The angoricity plays the role of the fluctuation temperature in SGR. We determine (a) the constitutive equation, (b) the yield stress, and (c) the distribution of stress dissipated during granular shearing experiments, and compare these predictions to experiments ofHartley & Behringer (2003).	10.1098/rsta.2009.0193	[ "https://arxiv.org/pdf/0908.0970v1.pdf" ]	16,142,109	0908.0970	b7ca4f50533bc48f6cf7820dd1d3c2af66dc7064	Rheology of Granular Materials: Dynamics in a Stress Landscape B Y Dapeng Bi Department of Physics Brandeis University 02454WalthamMAUSA Bulbul Chakraborty Department of Physics Brandeis University 02454WalthamMAUSA Rheology of Granular Materials: Dynamics in a Stress Landscape granularrheologystress We present a framework for analyzing the rheology of dense driven granular materials, based on a recent proposal of a stress-based ensemble. In this ensemble fluctuations in a granular system near jamming are controlled by a temperature-like parameter, the angoricity, which is conjugate to the stress of the system. In this paper, we develop a model for slowly driven granular materials based on the stress ensemble and the idea of a landscape in stress space. The idea of an activated process driven by the angoricity has been shown byBehringer et al (2008)to describe the logarithmic strengthening of granular materials . Just as in the Soft Glassy Rheology (SGR) picture, our model represents the evolution of a small patch of granular material (a mesoscopic region) in a stress-based trap landscape. The angoricity plays the role of the fluctuation temperature in SGR. We determine (a) the constitutive equation, (b) the yield stress, and (c) the distribution of stress dissipated during granular shearing experiments, and compare these predictions to experiments ofHartley & Behringer (2003). Introduction A striking feature of dry granular materials and other athermal systems is that they form force chain networks, in which large forces are distributed inhomogeneously into linear chain-like structures, in response to applied stress (Majmudar & Behringer (2005)). A number of recent experimental studies have visualized and quantified force chain networks using carbon paper (Jaeger et al (1996)) and photo-elastic techniques (Howell & Behringer (1999), Veje et al (1999)). These studies have demonstrated that the characteristics of force chain networks are acutely sensitive to the nature of the prepared state, especially near the jamming transition (Majmudar & Behringer (2005)). For example, in isotropically compressed systems, force chain networks are ramified with only short-ranged spatial correlations of the stress. In contrast, in sheared systems, aligned force chains give rise to long-ranged spatial correlations of the stress in the direction of the shear. Any theory of granular rheology has to, therefore, incorporate the effects of these grain-level structures on the macroscopic, collective response. It has been long realized that classical elasticity theory or Newtonian fluid dynamics is inadequate for describing the response of granular materials for a couple of important reasons (Bouchaud (2003), Cates et al (1998), Blumenfeld (2004)): (1) tensile stresses are completely absent in dry granular materials, and therefore the cohesion of the granular assembly is induced by the applied stress making the zero-stress state ill defined, (2) there is an indeterminacy of the forces at the microscopic level due to friction and disorder, (3) granular materials are athermal and dissipative, therefore there is no established statistical framework that bridges the structure at grain scales to a continuum elasto-plastic theory at large length scales. In addition, granular solids often occur in isostatic states in which the number of degrees of freedom matches the number of constraints (Tkachanko & Witten (1999), Moukarzel (1998)). It has been shown that the critical properties of the isostatic point, not elasticity theory, determine the mechanical response of these marginal solids (Wyart (2005)). A framework, known as soft-glassy rheology has been used widely to understand the rheology of materials such as foams, colloids, and grains (Sollich (1998)). In this work, we use a recently proposed statistical framework, the stress ensemble for describing the response of static granular media (Henkes & Chakraborty (2009), Henkes et al (2007)), to formulate a theory of the response of slowly (quasistatic) sheared granular media and discuss its predictions for granular rheology. Stress Ensemble As shown previously (Henkes & Chakraborty (2009), Ball & Blumenfeld (2002)), the stress field of a a mechanically stable granular material can be fully described by a spatiallyvarying scalar field ψ in two dimensions, and a tensorial field in higher dimensions (Henkes & Chakraborty (2009)). These fields can be used to establish a rigorous conservation principle, valid for grain assemblies in mechanical equilibrium, at the grain level in two dimensions and at a continuum level in higher dimensions (Henkes & Chakraborty (2009)). This conservation principle, in conjunction with a maximum entropy hypothesis (Jaynes (1957)), leads to a generalization of equilibrium thermodynamics to the ensemble of mechanically stable granular states (Henkes & Chakraborty (2009), Henkes et al (2007) where the new conserved tensorial quantity,Σ to be discussed below, plays the role of energy. The microcanonical version of this ensemble is characterized by the complexity, a measure of the number of grain configurations compatible with a given value ofΣ. The assumption of entropy maximization leads to the definition of a temperature-like intensive variable, the angoricity (Henkes & Chakraborty (2009), Blumenfeld & Edwards (2003), which is a tensor. The predictions of this stress ensemble have been remarkably successful in describing results of simulations (Henkes et al (2007), Lois et al (2009)) and experiments in both static (Majmudar & Behringer (2007), Lois et al (2009)) and slowly driven granular media (Behringer et al (2008)). As an example, the stress ensemble framework and the constraint that all contact forces are non-negative has been used to develop a Ginzburg-Landau type action for jammed granular systems in terms of ψ. The theory has been used to predict the spatial correlations of stress in systems subjected to isotropic compression (Henkes & Chakraborty (2009)), and shear (Lois et al (2009)), and to construct a meanfield theory of unjamming under isotropic compression (Henkes & Chakraborty (2009)). In slowly sheared granular systems, the observation of a logarithmic strengthening has been explained by the stress ensemble approach (Behringer et al (2008)). In this paper, we present a detailed development of the theory of slowly sheared granular matter, and make falsifiable predictions. Preliminary tests of the theory, based on comparison to experiments using photo-elastic disks (Hartley & Behringer (2003)) are also presented. (a) Conservation principle and Complexity To understand the mechanical response of granular materials, one needs a theoretical approach that can bridge the gap between microscopic, grain-level quantities and macroscopic, collective properties. Fluctuations are inherently related to the number of micro-scopic states available under a given set of macroscopic parameters. In equilibrium thermodynamics, the microcanonical entropy, or its derivatives in other ensembles, is the measure we use to calculate fluctuations and response. Conservation of energy allows for a rigorous definition of this measure since the states with different energies are not mixed by dynamics. In disordered systems such as spin glasses, the concept of complexity has been useful in formulating a framework for understanding collective properties (Bouchaud et al (1996)). Complexity is a measure of the number of states associated with a free energy minimum. and has proven to be a useful concept for disordered systems that have many metastable states. In mean-field models, these minima are separated by barriers that diverge in the thermodynamic limit, and one can in principle count the number of states unambiguously. Can we identify a physical variable in granular materials, which also have many metastable states (Bouchaud (2003)), which is conserved by natural dynamics and, therefore, leads to a formalism for defining complexity? The lack of a natural dynamics in granular systems makes this a difficult proposition. In mechanically stable states, however, there is a topological conservation law that allows us to proceed to define the analog of complexity. The topological nature implies that a change in this physical variable can be achieved only through rearrangements that involve the boundaries or the whole system. A gedanken experiment serves to illustrate the conservation law. If we draw an imaginary line through a grain assembly and calculate the value of the total normal force being transmitted along this line, then force balance ensures that this value remains unchanged as this imaginary line is translated perpendicular to itself across the assembly (Metzger (2008)). A formal theoretical framework can be formulated using the force-moment ten-sorΣ = R d d rσ(r), whereσ(r) is the local stress tensor. It has been shown (Henkes & Chakraborty (2009), Henkes et al (2007) thatΣ depends only on the boundary conditions of the packing. Thus, the phase space of all mechanically stable configurations can be divided into sectors labeled byΣ. Configurations in different sectors are disconnected under any local dynamics, andΣ plays a role similar to total energy in equilibrium thermodynamics or free-energy minima in spin glasses. Based on some very general principles, such as the factorization of states (Bertin et al (2006)), the conservation ofΣ has been shown to lead to an intensive variable,α = ∂S(Σ) ∂Σ , where S(Σ) is the entropy of the sector labeled byΣ (Henkes & Chakraborty (2009)). The inverse ofα, the angoricity (Edwards & Blumenfeld (2007)) is expected to play the same role as temperature in equilibrium thermodynamics, if processes that create granular assemblies achieve entropy maximization. Comparisons with simulations have demonstrated that a mechanically stable assembly of grains has the same value of angoricity throughout its interior, whileσ(r), the local stress fluctuates. The probability of occurrence of a microscopic state ν within a mechanically stable grain packing, maintained at an angoricity, α, is: P ν = (1/Z)e −α:Σ ν ,(2.1) analogous to the Boltzmann distribution (Henkes & Chakraborty (2009)). The state ν ≡ {r i , f ij }, where r i denote the positions of the grains, and f ij the set of contact forces. A natural question to ask is whether the similarities between temperature and angoricity extend to dynamics of slowly deformed granular systems, situations analogous to dynamics of thermal systems close to thermal equilibrium where stochastic dynamics such as relaxational, Langevin dynamics provides a good description (Goldenfeld (1992)). In the granular context, we are restricting our attention to systems that are close to granular equilibrium defined by Eq. 2.1. As in systems close to thermal equilibrium, we can construct a stochastic, Markov process based on the concept of detailed balance: W ν→ν W ν →ν = P ν P ν (2.2) This is an additional assumption beyond the ones entering the construction of the stress ensemble that leads to the distribution of mechanically stable states, P ν . Stochastic equations, based on detailed-balance conditions lead to the Boltzmann-like distribution, Eq. 2.1, as the time-independent distribution of states (Goldenfeld (1992)). If, however, the stress-landscape (or energy landscape in a thermal ensemble) is such that complete equilibration is not possible then the stochastic equations lead to a rate of escape over a barrier ∆Σ, which is the well-known Kramers rate: e −α:∆Σ (Kramers (1940)). A simple example of a landscape where equilibration is not possible is one that has a single local minimum and a single local maximum. In that situation the system "escapes" over the barrier with a rate prescribed by Kramers but cannot equilibrate. As will become clear from the discussion below, trap models describe systems that escape over barriers with the Kramers rate, however, subsequent sampling of traps is described by a quenched distribution and not by detailed balance. We adopt this form of activated dynamics to analyze the rheology of dense, quasistatic granular flows. Our framework of granular rheology is, therefore, based on a rigorous conservation principle and postulates, similar to the ones adopted in the development of equilibrium thermodynamics (Callen (1957)). These postulates are (a) factorisability of the dynamicsdependent frequency with which each grain configuration is accessed (Henkes & Chakraborty (2009)), which allows for the definition of the angoricity, (b) maximization of entropy, which implies equality of the angoricity across a system in granular equilibrium and a Boltzmann-like distribution (Eq. 2.1) (Henkes & Chakraborty (2009)), and (c) satisfaction of detailed balance by the rates of transition between microscopic states. Trap models and rheology As discussed earlier, the existence of a large number of different microscopic metastable states that are macroscopically equivalent puts granular materials into a wider class of previously well studied systems, including gels, glasses, colloids, emulsions, polymers and foams. A minimal model that encapsulates these features, and leads to a glass transition is the "trap model" (Monthus & Bouchaud (1996)). In this model, one considers a system made of independent subsystems of a certain size ξ, where each subsystem acts self-coherently and independent of the others. Their dynamics involve hopping between different metastable states aided by some kind of fluctuation. This idea of a random walk in a rugged landscape has its roots in the context of glasses (Bouchaud & Georges (1990)). Below, we discuss the trap model and its generalization, which is the framework of soft glassy rheology (SGR) (Sollich (1998)) in some detail. (a) Bouchaud's trap model for glasses Monthus & Bouchaud (1996) constructed a one-element model for glasses. In this model, there exists an energy landscape of traps with various depths E. An element hops between traps when activated, where the fluctuation is assumed to be thermal. At a temperature k B T ≡ 1/β, the probability of being in a trap with depth E at time t evolves according to: ∂ ∂t P(E,t) = −ω 0 e −βE P(E,t) + ω(t)ρ(E) (3.1) where, ω(t) = ω 0 e −βE P(E,t) = ω 0 Z dEP(E,t)e −βE (3.2) The first term on the right-hand-side (rhs) of Eq. 3.1 is the rate of hopping out of a trap, where e −βE is the activation factor and ω 0 a frequency constant. The model assumes that choosing a new trap is independent of history, so that a new trap is chosen from a distribution of trap depths ρ(E) that reflects the underlying disorder in the glassy models. Eq. 3.2 gives the average hopping rate. This rate, multiplied by the distribution of trap depth, Eq. 3.1 gives the probability rate of choosing a new trap. The existence of a glass transition in this model can be demonstrated as follows: assume the distribution function has a simple exponential tail such as: ρ(E) ∝ e −β 0 E , (3.3) where β 0 is a fixed parameter describing the disorder in the inherent energy landscape. The physical justification for having an exponentially decaying ρ(E) is borrowed from systems with quenched random disorder, such as spin glasses, which use extreme value statistics (Monthus & Bouchaud (1996)) (We will later justify applying the same form of distribution to granular materials). Then we can easily solve for the steady-state ( ∂ ∂t P(E,t) = 0) solution: P eq (E) ∝ e βE ρ(E) ∝ e (β 0 −β)E (3.4) Immediately, we see that Eq. 3.4 is non-normalizable for β > β 0 or T < T 0 . This shows that below a temperature T 0 , the system is out of equilibrium; it is non-ergodic and ages by evolving into deeper and deeper traps. Therefore, we call T = T 0 a point of glass transition. (Monthus & Bouchaud (1996)). It should be noted that dynamical equations of the trap model do not lead to a Boltzmann distribution at T since the equations do not obey detailed balance, and specifically, the traps are sampled from a quenched distribution. The escape rate from a trap is, however, determined by the Kramers process. (b) Soft Glassy Rheology In (Sollich (1998)), a simple phenomenological model termed "Soft Glassy Rheology" (or SGR) was proposed to describe the anomalous rheological properties of "soft glassy materials". The model incorporated Bouchaud's glass trap model and an extra degree of freedom, the strain. The model assumes a macroscopic soft glassy material can be subdivided into a large number of mesoscopic regions, each having a linear size ξ. With each mesoscopic region one can then associate an energy E and a strain variable l, which both evolve with time. The number of mesoscopic regions must be made large enough so that ensemble averaging can be performed over them to yield macroscopic properties of soft glassy materials. Similar to Bouchaud's trap model, the SGR model assumes the mesoscopic regions "live" in an energy landscape. This is a mean-field energy landscape in that it is not characterized by a metric, and is defined only by the distribution ρ(E). The new strain variable describes local elastic deformation of the mesoscopic regions, so that l contributes quadratically to the energy E. Since a strain variable was added to the SGR, the model can now describe material under imposed shear strain, also in a mean-field spirit, all mesoscopic regions respond uniformly to externally imposed shear strain. Different from Bouchaud's trap model, thermal fluctuations are unimportant in soft materials (k B T is too small to cause structural rearrangements). In the SGR model, it is the fluctuations in the elastic energy that facilitate structural rearrangements. This fluctuation is determined by a temperature-like quantity x called the "noise level". As a result, the escape rate from a trap becomes e −E/x . The use of x is a mean-field approach to describing all interactions between mesoscopic regions. (c) Incorporating the stress ensemble into the SGR framework The idea of using a temperature-like quantity to replace thermal fluctuations in the SGR model is reminiscent of the stress ensemble for granular materials, where the inverse angoricity, 1/α, plays the role of the noise, x, in SGR. The activated process in the stress ensemble is a result of a coupling between different mesoscopic regions through stress fluctuations. Each mesoscopic region can be viewed as existing in a bath of stress fluctuations, which is characterized by the angoricity, and these fluctuations can lead to activated processes analogous to those occurring in thermal systems. Since the granular material in relevant experiments (Hartley& Behringer (2003)) are sheared setups, and to simplify the model, we will use a scalar model that incorporates only the shear components of the stress through Γ, the deviatoric part ofΣ, and α, the shear component of the inverse-angoricity tensor (referred to as shear-angoricity below), which represents the bath of stress fluctuations that the mesoscopic region is in contact with. To adopt the SGR framework to the stress ensemble, we need to define metastability and escape processes in stress rather than energy space. Since the stress ensemble is based on the premise that there are many states {r i , f ij } with a given Γ, and that these states are broadly sampled even if not with equal weights (Henkes, 2009), we assume that mescoscopic regions of granular assemblies can be in metastable equilibrium characterized by the Boltzmann-like distribution (Eq. 2.1). A dynamics obeying detailed balance (Eq. 2.2) would lead to this distribution. As mentioned earlier, the trap model and SGR, however, describe systems in the presence of quenched disorder which define the distribution barrier heights (or equivalently trap depths) ∆Γ that can be crossed in a finite time. The system can escape from a metastable state represented by a trap. Stochastic equations based on the stress ensemble allows us to adopt the Kramers' escape rate approach. Replacing temperature by the shear angoricity, the escape rate is therefore, e −α∆Γ , which replaces e −βE of Bouchaud's trap model. As in the original trap model framework, the exploration of the traps in stress space, following this escape is assumed to be determined by the intrinsic, quenched-in distribution of trap depths, and this dynamics does not lead to the Boltzmannlike distribution, Eq. 2.1 in the long-time limit. The quenched-in distribution of trap depths determines whether complete equilibrium is attainable in the absence of shearing (Monthus & Bouchaud (1996)). The activated process in the original trap model was thermal, and in SGR the activation is controlled by the noise x. In the scalar version of the stress-ensemble, the activation is in stress space and is controlled by the angoricity α. The disorder in grain packings is represented by an intrinsic distribution of trap depths, just as in SGR or the original trap model. We can now write a generalization of the SGR model to granular materials using the stress ensemble: d dt P (Γ m , Γ,t) = −ω 0 e −α(Γ m −Γ) P (Γ m , Γ,t) + ω(t)ρ (Γ m ) δ(Γ) (3.5) Eq. 3.5 describes the evolution of the probability of finding a mesoscopic region with a "yield stress" Γ m (again, strictly Γ m has the unit of force moment) and an instantaneous stress Γ. The yield stress Γ m is used to define the depth of the trap the system is in, and since Γ is the instantaneous stress of the system, the barrier is ∆Γ = Γ m − Γ. The trap depth E of SGR or the trap model is therefore replaced in the stress ensemble by Γ m . The model accounts for the successive stress buildup and yielding events inside a mesoscopic region. When the instantaneous stress (due to external driving) is less than the yield stress, or Γ < Γ m , the mesoscopic region is considered to be inside a "trap" of depth Γ m . It is possible to yield below the yield stress because of the activated nature of the process. At this point the mesoscopic region "sees" a reduced trap depth (i.e. effective barrier height): Γ m − Γ, and hence can undergo an activated process to "hop" out of the trap with a rate proportional to e −α(Γ m −Γ) . Another process through which the mesoscopic regions can yield is that if no activated process takes place while in the trap (Γ < Γ m ), Γ can increase due to external applied shear up to the point Γ = Γ m where the mesoscopic region yields, hence leaving the trap. This is the only possible process at the limit of zero angoricity (1/α → 0), and is not explicitly included in Eq. 3.5. Either way, after the mesoscopic region leaves a trap, it choose a new trap according the distribution ρ(Γ m ). Fig. 1 shows a schematic of this dynamics of the mesoscopic region in the landscape. The mesoscopic regions are assumed to be large enough to be treated as deforming elastically until a "plastic" yield event occurs. In the trap picture, these plastic events are equated with the mescoscopic region leaving a trap. The grains are driven by a constant shear rate in the experiments (Hartley & Behringer (2003)). In a mean-field approach similar to SGR, we assume this applied shear is transmitted without decay throughout the entire material. Thus each mesoscopic region is driven by the same shear rate. Since the mesoscopic regions deform elastically, the stress Γ increases linearly with time up to the point of yielding: σ = 1 S Γ = kl = kγt (3.6) where σ is the shear stress, S is the area of the mesoscopic region, and k is the elastic constant relating the shear stress to the shear strain, l. The continual process of stress buildup and yielding creates a "sawtooth" pattern in time as show in the bottom of Fig. 1. With a linear dependence of Γ on time, between yield events, we can rewrite Eq. 3.5: ∂ ∂t P (Γ m , Γ,t) = −Skγ ∂ ∂Γ P (Γ m , Γ,t) − ω 0 e −α(Γ m −Γ) P (Γ m , Γ,t) + ω(t)ρ (Γ m ) δ(Γ) (3.7) The full time-differential on the l.h.s of Eq. 3.5 has been converted to a partial time derivative plus an advective term. The delta function in the last term on the r.h.s. of Eq. 3.7 means that immediatly after choosing a new trap, the mesoscopic region is in a stress-free state. See points "3" and "5" in Fig. 1. Ifγ = 0, the model reduces to Bouchaud's trap model in stress space. The steady state distribution, P(Γ m , Γ) ∝ e −α(Γ m −Γ) ρ(Γ m )δ(Γ), which is the equilibrium probability of finding a region with yield stress Γ m . Since there is no imposed shearing, the instantaneous stress is zero. Physically, hopping out of a trap equates to the grains in the mesoscopic region making a re-arrangement. These events are visible in experiments and if analysed with photoelastic techniques, are just buckling or the breaking of force chains. After this re-arrangement, each "trap" in the landscape is given by its depth Γ m . The amount of stress buildup, while still in the trap, is indicated by the height Γ from the bottom of the trap. For "1", the mesoscopic region has just fallen into the trap and it has no stress build up, so it has value of Γ = 0 in the bottom figure. As constant shear stress is applied to the mesoscopic region, its stress starts to build up. In the top figure, "2" is at a height from the bottom of the trap; and in the bottom figure, "2" has increased linearly in value from "1". Next, via the activated process, the mesoscopic region makes a jump from one trap to another. Note the traps shown are intentionally made disjoint, this is to indicate that every trap can be accessed from every other trap and there is no specific connectivity. new force chains form. The newly formed force chains are characterized by a new yield stress Γ m . The distribution of trap depths ρ(Γ m ) describes the disorder in the stress landscape. Theoretically, this distribution can be measured in experiments where no activated stress yielding occurs, or when stress fluctuations exist only locally which leads to the angoricity (1/α) being very small. This is the case when no neighboring stress fluctuation is "felt" within the mesoscopic region. This is realized experimentally (Peidong & Behringer (2009)). In this experiment, shear is applied to a small local region which can be considered a single mesoscopic region. The stress yielding is due entirely to the driving. The distribution of yield stresses in this case is exactly ρ(Γ m ). Generally, in sheared granular systems (e.g. Geng & Behringer (2005)), we can get an idea for the form of ρ(Γ m ) by looking at the tail of the stress distribution (large Γ). This is because large Γ values are accessed only when an activated process has not occurred to make the the system yield. They are also states that sample large Γ m values. Activated dynamics in stress space is not an entirely new concept. It was proposed by Eyring (Glasstone et al (1941) ), and has been applied to analyze the velocity profile in dense granular flows (Pouliquen & Gutfraind (1996)). In our work, the stress-activation is a natural consequence of the ensemble based on angoricity. We can write down a general solution to Eq. 3.7 regardless of the distribution of trap depth ρ(Γ m ). Eq. 3.7 can be simplified with a change of variable: δΓ = Γ(t) − Skγt, which does not explicitly depend on time (away from the points of yielding). This eliminates the advective term in Eq. 3.7 which then becomes: ∂ ∂t P (Γ m , δΓ,t) = −ω 0 e −α(Γ m −(δΓ+Skγt)) P (Γ m , δΓ,t) + ω(t)ρ (Γ m ) δ (δΓ + Skγt) (3.8) A better intuition can be gained by solving for 3.8 while ignoring the second term on its r.h.s., the solution is: P (Γ m , δΓ,t) = P 0 (Γ m , δΓ) exp − ω 0 e αΓ m Z t 0 dt e α(δΓ+Skγt ) (3.9) The solution is just an exponential decay with the time interval replaced by a time integral. This is the "effective time interval" defined in (Sollich (1998)): Z(t,t ; δΓ) ≡ Z t t dt e α(δΓ+Skγt ) (3.10) In essence the behavior of an element in a trap is just that of a decaying process, ∼ e −t/τ . For us, the time is replaced by the effective time interval Z(t, 0; δΓ) that grows much faster than linearly and the mean lifetime is τ = ω 0 −1 e αΓ m . After the stress collapse of the first buildup occurring at t , the element escapes into another trap chosen from the distribution ρ(Γ m ) and starts to undergo its own decay process via the effective time. Therefore the full solution is: P (Γ m , δΓ,t) =P 0 (Γ m , δΓ) exp − ω 0 e αΓ m Z(t, 0; δΓ) (3.11) + Z t 0 dt ω(t )ρ (Γ m ) δ δΓ + Skγt exp − ω 0 e αΓ m Z t,t ; δΓ Theory predictions and comparison to experiments Slow shearing experiments correspond to the long-time or steady state limit of our model. In such a limit we find a solution that is time independent: P (Γ m , Γ) = ρ (Γ m ) exp − ω 0 Skγ α e αΓ m e αΓ − 1 ,(4.1) subject to normalization. We now have a ensemble distribution function of Γ with which we can calculate the constitutive relation by taking the ensemble average of Γ. Using Eq. 3.6 we can write: σ = 1 S Γ P(Γ m ,Γ) = 1 S Z ∞ 0 dΓ Γ Z ∞ 0 dΓ m P (Γ m , Γ) (4.2) We first consider the simplest case of a distribution of trap depths, ρ(Γ m ) = δ(Γ m − Γ 0 ) with Γ 0 > 0. Although not a physical description, this choice will prove to be a fruitful exercise. Eq. 4.2 becomes: σ = 1 α S R ∞ 1 dy y −1 log(y)e −Ay E 1 (A) (4.3) Article submitted to Royal Society where the constant A = ω 0 Skγ α e −αΓ 0 and E 1 is the exponential integral with n = 1. In the interesting limit of A → 0 orγ >> ω 0 e −αΓ 0 Skα, we can expand Eq. 4.3 to O(A 0 ) and obtain: σ = 1 2αS log (γ) + 1 2αS log Skα ω 0 + αΓ 0 − 2γ c (4.4) where γ c is the Euler-Mascheroni constant 0.577. The leading behavior of the constitutive equation has a logarithmic dependence on the shear rate. This agrees qualitatively with experiments (Behringer et al (2008), Hartley & Behringer (2003)). In the opposite limit A → ∞ orγ << ω 0 e −αΓ 0 Skα, we can expand Eq. 4.3 to O(1/A) and obtain: σ = k e αΓ 0 ω 0γ (4.5) This reveals a Newtonian regime with viscosity η = k e αΓ 0 /ω 0 . (a) Constitutive equation for the exponentially decaying distribution of trap depths As discussed in the previous section, an exponentially decaying ρ(Γ m ) is a better description of granular experiments as well as spin-glasses. We use: ρ(Γ m ) = 1 α 0 e −α 0 Γ m (4.6) where α 0 is a constant measuring the disorder in the stress landscape. Inserting Eq. 4.6 into Eq. 4.2 we obtain: σ = 1 αS log(Skγ α/ω 0 ) + 1 αS R ∞ 0 dW W x log(W +ω 0 /Skγα) (W +ω 0 /Skγα) γ (x,W ) R ∞ 0 dW W x (W +ω 0 /Skγα) γ (x,W ) (4.7) where x is given by the ratio: x = α 0 /α, and γ(x,W ) is the lower incomplete gamma function. Again we look in the limit ofγ >> ω 0 / kα and the constitutive equation is approximated to zeroth order as: σ = 2 αS log(γ) + 1 αS log( Skα 2 ω 0 α 0 ) (4.8) This shows that the logarithmic dependence of shear stress on shear rate is independent of the form of the distribution ρ(Γ m ) used. In Fig. 2 we compare this result with the data obtained in 2D Couette granular shearing experiments (Hartley & Behringer (2006)). In the opposite limit,γ << ω 0 / Skα we again obtain a Newtonian regime: σ = k ω 0 x − 1 x − 2γ (4.9) The viscosity diverges at x = 2, the same result is obtained in SGR (Sollich (1998)). (b) Yield Stress Whenγ = 0, the stress has a finite value for α 0 < α or x < 1. This is the yield stress, and it vanishes at the same point as the glassy transition point in Bouchaud's trap model. The yield stress can be calculated by insertingγ = 0 into Eq. 4.2. Performing the Γ integral first in Eq. 4.2, we obtain: σ = 1 αS R ω 0 /Skγ α 0 dz z x−1 G(z) R ω 0 /Skγ α 0 dz z x−1 E 1 (z) , (4.10) where G(z) is the Meijer G-function: G(z\| 1, 1 0, 0, 0 ) = Z ∞ 1 dy log(y) y exp(−z(y − 1)) Whenγ = 0, Eq. 4.10 can be calculated exactly: σ y ≡ σ(γ = 0) = −k x γ c − k x d dx log (Γ(x)) ,(4.11) A plot of the yield stress is shown in Fig. 3 ( c) The distribution of stress drops It is often easier to measure distribution of stress drops in experiments than the stresses themselves, and the distribution provides a more stringent test of theoretical frameworks. While in a trap with depth Γ m , the probability to build up the stress to a value of Γ is given by: (cf Eq. 4.1 ) On the other hand, as the stress grows, it is increasingly likely to fail due to the activated process. The rate of failure is given by: P s (Γ) ∝ exp − ω 0 Skγ α e αΓP f (Γ) ∝ e −α(Γ m −Γ) Then, the probability of building the stress up to Γ and then to fail at this point is given by the product P s × P f . Since the stress falls to zero after a failure event, we can call the magnitude of the stress drop ∆Γ. Its distribution for a given ρ(Γ m ) is proportional to: P(∆Γ) ∝ P s (∆Γ)P f (∆Γ) Also sum over all possible traps: P(∆Γ) ∝ Z dΓ m ρ (Γ m ) P s (∆Γ)P f (∆Γ) (4.12) Using the exponentially decaying distribution ρ(Γ m ) = e −α 0 Γ m , we obtain the normalized distribution: P(∆Γ) = x α ω 0 Skγα −x e α∆Γ e α∆Γ − 1 −1−x γ 1 + x, ω 0 Skγα (e α∆Γ − 1) (4.13) where we see the lower incomplete gamma function again. There are three regimes in which Eq. 4.13 can be simplified. First, using the limit y −β γ(β, y) → 1/β as y → 0, we get: P(∆Γ) = x x + 1 ω 0 Skγ e α∆Γ when ω 0 Skγα (e α∆Γ − 1) → 0 (4.14) The opposite limit for the lower incomplete gamma function is γ(β, y) → (β − 1)! as y → ∞ for which we get: P(∆Γ) = α x 2 (x − 1)! ω 0 Skγα −x e α∆Γ e α∆Γ − 1 −1−x when ω 0 Skγα (e α∆Γ − 1) → ∞ (4.15) Article submitted to Royal Society P(∆Γ) = α x x! ω 0 Skγα −x e −α 0 ∆Γ when ∆Γ → ∞ (4.16) In Eq. 4.16, we recover the exponential distribution of depths. With an analytical form of P(∆Γ), we can compare and fit to data obtained by Hartley & Behringer (2003). The data is fitted to the form Eq. 4.15 and therefore contains two fitting parameters: α and α 0 . Table 1 shows the fitting results. In Fig. 4, four fitting results are plotted and compared with data. As seen from Table 1 b)γ = 0.9972mHz, φ − φ c = 0.0053, α = 0.7, α 0 = 0.2660 , (c)γ = 5.3155mHz, φ − φ c = 0.0088, α = 1.2, α 0 = 0.2131, and (d)γ = 0.0657mHz, φ − φ c = 0.0137, α = 0.4, α 0 = 0.2547. shear rate and the packing fraction. The data fitting gives us insights into the relationship between the shear-angoricity (x = α 0 /α), the shear rate (γ), and the packing fraction φ − φ c . We can see this relation in a plot of x vs.γ in Fig. 5 for different φ−φ c . The first observation one makes is that x is controlled by bothγ, and φ − φ c , implying that the SGR assumption of x being independent of the shear rate does not apply to these measurements in sheared, granular media. In addition, over a decade of φ -φ c values, and three decades ofγ, x seems to be described by a scaling form: x = f + γ \|φ − φ c \| ∆ ,(4.17) with ∆ −0.4, and the scaling function (for φ ≥ φ c ) f + (z) being a decreasing function of its argument, z. The dependence onγ is, however, weaker than the dependence on φ − φ c . Although better statistics are needed to pin down the scaling form, it is intriguing to examine its consequences as such. The scaling itself hints at (φ − φ c = 0,γ = 0) being a critical point, which is consistent with the observations of Olsson & Teitel (2007), and the idea of Point J being a special point in the jamming phase diagram (Liu & Nagel (1998)). At zero shear rate, x becomes independent of φ − φ c , and the scaling form is, therefore, consistent with the packings possessing a yield stress for all φ > φ c . From the perspective of x being the fluctuation temperature, the observation is that stress fluctuations grow as φ approaches φ c from above. Experiments do indicate growing fluctuations as a system approaches the unjamming transition below which it cannot sustain shear (Howell & Behringer (1999). Conclusion A stress-based statistical ensemble has been used in conjunction with the idea of a stress landscape to construct a model for granular rheology. Comparisons of the model to experiments on slowly sheared granular media indicate that the theoretical framework provides a semiquantitative description of the stress-response. The framework is an extension of the Soft Glassy Rheology approach, and the stress ensemble provides a natural definition for the fluctuation temperature. This fluctuation temperature can be obtained by fitting the theoretical predictions to experiments, and the results show that fluctuations are largest at the special point, Point J, where the shear rate goes to zero and the packing fraction approaches the critical value below which one cannot construct a mechanically stable state of grains. In future work, we intend to extend the model to include volume fluctuations since Reynold's dilatancy is one of the signature properties of granular matter (Reynold (1885)). A fundamental premise of the stress ensemble framework is that just as in thermal equilibrium temperature is equalized, for a system in granular equilibrium, the angoricity is the same everywhere inside the system. Comparison to simulations has shown this to be true for a system under pure compression (Henkes et al (2007)). If we assume the same to be true for the shear-angoricity x, then according to Eq. 4.17, the local packing fraction has to adjust to the local shear rate and regions of increased shear rate have lower packing fractions. An interesting question to ask is whether this is the origin of Reynold's dilatancy. Figure 1 . 1Schematic showing dynamics of the model in the stress landscape. The black dots with the numbering (1 thru 6) represent the state of the mesoscopic region at different times. In the top figure, Figure 2 . 2Linear-Log Plot of mean stress in a segment of 2D granular Couette experiment containing roughly 200 particles vs. shear rate. Data, taken from(Hartley & Behringer (2003)), are given for different packing fractions, which are given relative to the critical packing fraction φ c where the system loses all rigidity. Various symbols are: upward triangles: φ = φ c + 0.00035; diamonds: φ = φ c + 0.00141; squares: φ = φ c + 0.00528; solid disks: φ = φ c + 0.00528; downward triangles: φ = φ c + 0.00800; and hollow circles: φ = φ c + 0.01373. Figure 3 . 3Theoretical Plot of the yield stress Eq. 4.11 as a function of x = α 0 /α. It vanishes at x=1. Figure 4 . 4Log-Log plot of experimental distribution of stress drops and its fitting to Eq. 4.15. The experimental and fitting parameters are: (a)γ = 0.3318mHz, φ−φ c = 0.0053, α = 0.70, α 0 = 0.3247, ( Figure 5 . 5Log-log plot of the scaled the relationship (Eq. 4.17). The scaling exponent is found to be: ∆ = −0.4 The data points are: disks: φ − φ c = 0.0004, squares: φ − φ c = 0.0014, diamonds: φ − φ c = 0.0053, upward triangles: φ − φ c = 0.0053, and downward triangles: φ − φ c = 0.0053. Inset: Unscaled log-log plot of x vs.γ. Table 1 . 1Results of fitting Eq. 4.15 to the experimental data ofHartley & Behringer (2003) Finally, the more extreme case of Eq. 4.15 as ∆Γ → ∞:φ − φ c Strain Rateγ (mHz) α α 0 x = α 0 /α 0.0053 0.3318 0.7000 0.32473 0.4639 2.2148 1.0000 0.2339 0.2339 4.6568 1.4000 0.19054 0.1361 8.6491 2.0000 0.1624 0.0812 13.9722 4.0000 0.1244 0.0311 0.0014 0.0390 0.6000 0.34872 0.5812 0.1322 0.6000 0.3279 0.5465 0.1987 0.6000 0.34566 0.5761 0.8308 0.7000 0.3115 0.4450 2.9933 0.9000 0.25857 0.2873 5.2490 1.0000 0.2366 0.2366 6.3203 1.2000 0.21372 0.1781 11.9760 1.3000 0.18083 0.1391 19.6279 4.0000 0.1428 0.0357 21.9567 6.0000 0.1836 0.0306 0.0657 0.5000 0.30135 0.6027 0.5314 0.5000 0.26265 0.5253 2.6607 0.7000 0.24444 0.3492 6.6529 0.8000 0.1928 0.2410 11.9760 1.1000 0.17061 0.1551 19.9606 1.2000 0.17316 0.1443 29.9413 1.2000 0.15372 0.1281 39.9221 1.5000 0.1344 0.0896 0.0088 0.5314 0.5000 0.2375 0.4750 2.6607 0.9000 0.25146 0.2794 3.9848 1.0000 0.2282 0.2282 5.3155 1.2000 0.21312 0.1776 9.9998 1.6000 0.1776 0.1110 13.3068 2.5000 0.163 0.0652 0.0137 0.0657 0.4000 0.25472 0.6368 0.3318 0.5000 0.33945 0.6789 0.6645 0.5000 0.27425 0.5485 1.3299 0.5000 0.26505 0.5301 2.6607 1.3000 0.33384 0.2568 3.3260 1.4000 0.24458 0.1747 6.6529 1.5000 0.2097 0.1398 13.3068 4.0000 0.1776 0.0444 Article submitted to Royal Society We acknowledge many helpful discussions with Mike Cates, R. 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[ "Fermi-LAT Detection of Gamma-ray Pulsars above 10 GeV", "Fermi-LAT Detection of Gamma-ray Pulsars above 10 GeV" ]	[ "P M Saz Parkinson [email protected] \nSanta Cruz Institute for Particle Physics\nUniversity of California\n95064Santa CruzCA\n" ]	[ "Santa Cruz Institute for Particle Physics\nUniversity of California\n95064Santa CruzCA" ]	[]	The Large Area Telescope (LAT) on board the Fermi satellite has detected ∼120 pulsars above 100 MeV. While most γ-ray pulsars have spectra that are well modeled by a power law with an exponential cut-off at around a few GeV, some show significant pulsed high-energy (HE, >10 GeV) emission. I present a study of HE emission from LAT γ-ray pulsars and discuss prospects for the detection of pulsations at very high energies (VHE, >100 GeV) with ground-based instruments. statistics-limited high-energy regime, LAT sensitivity improves faster (i.e. αt) than at lower energies, where backgrounds dominate. Future improvements in reconstruction (e.g. Pass 8) could yield significant increases in effective area at higher energies. Future TeV experiments (e.g. CTA, HAWC) will complement and extend LAT observations in this crucial energy window.	10.1063/1.4772255	[ "https://arxiv.org/pdf/1210.7525v1.pdf" ]	119,260,249	1210.7525	061029a730403e8aed325e17c92a7483a32b1fe5	Fermi-LAT Detection of Gamma-ray Pulsars above 10 GeV 28 Oct 2012 (1991-2000) P M Saz Parkinson [email protected] Santa Cruz Institute for Particle Physics University of California 95064Santa CruzCA Fermi-LAT Detection of Gamma-ray Pulsars above 10 GeV 28 Oct 2012 (1991-2000)arXiv:1210.7525v1 [astro-ph.HE] for the Fermi-LAT Collaboration 1 PULSARS ABOVE 10 GEV: THE EGRET VIEWGamma rays -astronomical observationsGamma-ray telescopesFermi LATPulsars PACS: 9555Ka9585Pw9760Gb The Large Area Telescope (LAT) on board the Fermi satellite has detected ∼120 pulsars above 100 MeV. While most γ-ray pulsars have spectra that are well modeled by a power law with an exponential cut-off at around a few GeV, some show significant pulsed high-energy (HE, >10 GeV) emission. I present a study of HE emission from LAT γ-ray pulsars and discuss prospects for the detection of pulsations at very high energies (VHE, >100 GeV) with ground-based instruments. statistics-limited high-energy regime, LAT sensitivity improves faster (i.e. αt) than at lower energies, where backgrounds dominate. Future improvements in reconstruction (e.g. Pass 8) could yield significant increases in effective area at higher energies. Future TeV experiments (e.g. CTA, HAWC) will complement and extend LAT observations in this crucial energy window. Prior to the launch of Fermi, our best knowledge of the high-energy (HE, >10 GeV) γ-ray sky came from EGRET. Although diffuse emission accounted for the majority (∼1300) of the ∼1500 HE photons detected by EGRET in its 9 years in orbit, a few tens of such photons were coincident with five bright γ-ray pulsars: 10 from the Crab (7 in the peaks), 4 from Vela (all in the peaks), 10 from Geminga (5 in the peaks), 9 from B1706−44 (5 in the peaks), and 2 from B1951+32 (both in the peaks) [1]. FERMI-LAT CATALOGS: PAST AND PRESENT Since its launch in 2008, the Large Area Telescope (LAT [2]) on Fermi has dramatically improved our knowledge of the γ-ray sky. The LAT has produced various catalogs in the last ∼3 years: the Bright Source List (0FGL) [3], using 3 months of data to describe 205 (> 10σ ) γ-ray sources (30 pulsars). The First Pulsar Catalog (1PC) [4], based on 6 months of data, describing 46 γ-ray pulsars. The First/Second LAT Source Catalogs (1FGL [5], 2FGL [6]) using 11/24 months of data, and containing 1451/1873 sources (56/83 pulsars). Two catalogs, using 36 months of data, are currently in preparation: The Fermi-LAT Catalog of Sources above 10 GeV (1FHL [7]) describes the ∼500 "Hard" sources detected by the LAT (25 coincident with pulsars) while the Second LAT Pulsar Catalog (2PC) describes in depth the ∼120 LAT-detected (>100 MeV) γ-ray pulsars [8]. We used 3-year data sets as in 1FHL and 2PC. We first tried to determine how many of the 25 sources from 1FHL associated with LAT γ-ray pulsars show significant pulsations (and can therefore be identified as pulsars). These 25 sources include: 5 EGRET pulsars, 7 young (non-recycled) radio-selected γ-ray pulsars, 10 young (non-recycled) γ-selected pulsars, and 3 millisecond γ-ray pulsars. Using the timing models from 2PC and gtsrcprob, we generated low energy (0.3-10 GeV) normalized 2 weighted light curves (templates). HE histograms were obtained using unweighted Front (Back) events within 0.6 • (1.2 • ) of the known pulsar position, corresponding to ∼r95% of the PSF. For each pulsar, we defined, a priori, an "off-pulse" region, using Bayesian Blocks [9] (See Figure 1, Left), and evaluated the statistical significance of the HE events coming from the "pulsed" region of the light curve. We also selected a subset of the 117 γ-ray pulsars from 2PC which, based on their spectral energy distribution (SED, See Figure 1, Right), appear to emit above 10 GeV but did not meet the criteria for inclusion in 1FHL. These spectrally-selected 2PC pulsars not in the 1FHL catalog are: J0633+0632, J1509-5850, J1747-2958, J1838-0537, J1954+2836, J2017+0603, J2021+4026, J2238+5903, J2302+4442. Our preliminary analysis shows that 10 pulsars with significant pulsed HE emission (including J0007+7303, Crab, J0614-3329, Geminga, Vela, J1028-5819, J1048-5832, J1709-4429, J1809-2332, J2021+3651, and J2032+4127). Several others require a more definitive analysis before a firm detection can be claimed. Figure 2 shows the example of Geminga, where HE pulsed emission is apparent, albeit with a very different pulse shape than what is seen at lower energies. Some of the brightest pulsars (e.g. Geminga, Vela) show pulsed >25 GeV emission (Figure 3), while at >50 GeV, the LAT starts running out of statistics, much like EGRET did at >10 GeV (Figure 4). SUMMARY AND OUTLOOK The LAT has dramatically increased our knowledge of the hitherto barely explored 10-100 GeV region of the γ-ray sky. A new LAT catalog in preparation (1FHL) will contain ∼500 HE sources, of which 25 are coincident with pulsars. In addition, the LAT has detected a large number of >100 MeV γ-ray pulsars (to be described in the upcoming 2PC), some of which show emission above 10 GeV. Future γ-ray pulsars may be discovered (e.g. in blind searches or radio searches of LAT sources), but these will necessarily be fainter than the brightest currently known. Top candidates for VHE pulsations depend on many assumptions and spectral extrapolations from 10 GeV upwards are notoriously unreliable. Thus, empirically speaking, the bright EGRET pulsars (e.g. Geminga, Vela) remain among the best candidates for VHE emission, while some of the newlydiscovered bright LAT radio-quiet γ-ray pulsars (e.g. CTA1) are also very promising. FIGURE 1 . 1Left Preliminary off-pulse definition for PSR J0633+0633 using Bayesian Blocks[9]. Right Preliminary SED of PSR J0614-3329, showing possible evidence for >10 GeV emission. FIGURE 2 . 2Normalized weighted light curve of Geminga in the 0.3-10 GeV energy range (blue, left scale) and unweighted HE light curve (pink, right scale). Weights based on 2PC spectral model. FIGURE 3 . 3Blue curves (left scale), same as Figure 2: normalized 0.3-10 GeV weighted light curve. Pink (unweighted) histogram (right scale) of >25 GeV events: Left Geminga. Right Vela. http://www-glast.stanford.edu/cgi-bin/people The area under the curve equals unity. We use 100 bins, so each bin width is 0.01 units of phase. ACKNOWLEDGMENTSThe Fermi LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT as well as scientific data analysis. These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the National Space Board in Sweden. Additional support from INAF in Italy and CNES in France for science analysis during the operations phase is also gratefully acknowledged. . D J Thompson, arXiv:astro-ph/0412376ApJS. 157D. J. Thompson et al., ApJS 157, 324-334 (2005), arXiv:astro-ph/0412376. . W B Atwood, arXiv:0902.1089ApJ. 697W. B. Atwood et al., ApJ 697, 1071-1102 (2009), arXiv:0902.1089. . A A Abdo, arXiv:0902.1340ApJS. 183A. A. Abdo et al., ApJS 183, 46-66 (2009), arXiv:0902.1340. . A A Abdo, arXiv:0910.1608ApJS. 187A. A. Abdo et al., ApJS 187, 460-494 (2010), arXiv:0910.1608. . A A Abdo, arXiv:1002.2280ApJS. 188A. A. Abdo et al., ApJS 188, 405-436 (2010), arXiv:1002.2280. . P Nolan, arXiv:1108.1435ApJS. 19931P. Nolan et al., ApJS 199, 31 (2012), arXiv:1108.1435. D Paneque, Presentation at the 5th International Symposium on High-Energy Gamma-Ray Astronomy. HeidelbergD. Paneque et al., Presentation at the 5th International Symposium on High-Energy Gamma-Ray Astronomy, Heidelberg, 9-13 July (2012). O Celik, Presentation at the 5th International Symposium on High-Energy Gamma-Ray Astronomy. HeidelbergO. Celik et al., Presentation at the 5th International Symposium on High-Energy Gamma-Ray Astron- omy, Heidelberg, 9-13 July (2012). . J D Scargle, ApJ. 504405J. D. Scargle, ApJ 504, 405 (1998).	[]
[ "Almost Lie structures on an anchored Banach bundle", "Almost Lie structures on an anchored Banach bundle" ]	[ "P Cabau ", "F Pelletier " ]	[]	[]	Under appropriate assumptions, we generalize the concept of linear almost Poisson structures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the "Hilbert snake" as exposed in [PeSa].	10.1016/j.geomphys.2012.06.005	[ "https://arxiv.org/pdf/1111.5908v1.pdf" ]	119,314,540	1111.5908	64eeeba1a90367cde9e02eab2d55f09891a399ee	Almost Lie structures on an anchored Banach bundle 25 Nov 2011 P Cabau F Pelletier Almost Lie structures on an anchored Banach bundle 25 Nov 2011 Under appropriate assumptions, we generalize the concept of linear almost Poisson structures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the "Hilbert snake" as exposed in [PeSa]. Introduction Recent developments about geometric formalism on anchor bundles on a finite dimensional manifold have helped to build a general framework for studying mechanical systems. Essentially, these geometric structures concern, linear almost Poisson structures, almost Lie algebroids and almost differentials (see for example [GLMM], [GLMM], [LMM], [Marl], [Marr], [PoPo] and all references inside these papers). The purpose of this paper is to give a generalization of these geometric structures in the context of Banach anchored bundles. Of course, this framework leads to a lot of obstructions. At first, any local section of a Banach bundle cannot be extended to a global section without some properties of regularity of the typical fiber. So, in the setting of "Banach algebroid", we must impose that the Lie Bracket of sections (defined globally) has a property of "localization" . Indeed, if the Banach manifold is regular, this property is always satisfied, as in finite dimension. However, in the general case, we must impose such a condition (see section 3.3). On the other hand hand, for a Poisson structure on a Banach manifold, we meet the same type of problem but also, if the typical Banach model is not reflexive, we must impose some other conditions (see section 4.1). Finally, the most important obstruction appears in the context of "Lie differential": not only we still meet the problem of localization of sections but, on the opposite of finite dimension, the graded algebra of forms on a Banach space is not generated by elements of degree zero and degree one, so, in general, such a differential is not characterized by its values on elements of these types. However, by imposing appropriate assumptions, we defined the concept of almost Lie bracket, almost Lie algebroid which is a generalization of Lie algebroid on Banach manifold introduced in [Ana] and [Pel]. Now, recall that in finite dimension, on one hand there exists a bijection between Lie algebroid structures on an anchored bundle and Poisson structures on its dual, and a bijection between Lie algebroid structures and Lie differentials (see for instance [Marl] or [GLMM] among many references). In fact, in our context, if the typical fiber of the anchored bundle is not reflexive, we have a bijection with almost Lie algebroid on an anchored bundle but here with "sub almost Poisson structure" on its dual, that is, such a structure is only defined on the set of sections of a "canonical subbundle" of the dual bundle (see subsection 4.2). Of course, in the framework of paracompact Hilbert manifolds, all these obstructions do not exist and we recover the general setting of the finite dimensional context. On the opposite, in the infinite dimensional context, we do not have any bijection between almost Lie algebroid structures and almost Lie differentials even under appropriate assumptions (see subsection 4.3). In the following section, we recall the concept of graded exterior algebra, some classical properties of Banach manifolds under which the mentioned previous assumptions can be avoid. We also precise our notations in local context. The notion of almost Lie algebroid (AL algebroid in short) on a Banach anchored bundle is developed in section 3. The section 4 is devoted to the relation between "sub almost Poisson structure" and AL algebroid in one hand and almost Lie differential and AL algebroid on the other hand. At first we expose the context of sub almost Poisson morphism (sub AP morphism in short) (subsection 4.1). Then we look for the equivalence of AP morphism and AL algebroid structure (subsection 4.2). Finally, under strong appropriate assumption on an almost differential, we can associate an AL algebroid structure (subsection 4.3). As applications of our results, in section 5, we look for an adaptation of classical formalism for mechanical systems: Hamiltonian system, Hamilton-Jacobi equation, Lagrangian system and Euler-Lagrange equation. Finally, in section 6, we illustrate this formalism in the context of the evolution of the "head" of a "Hilbert snake" form the results of [PeSa]. 2 Preliminaries and notations 2.1 Graded exterior algebra on a Banach space Given a Banach space E, we denote by Λ k E * the Banach space of exterior forms of order k of E. More precisely, the set Λ k E * can be identified with the closed subspace of k-multilinear skewsymmetric maps in the Banach space L k (E) of k-multilinear forms on E. This space is the closure of the vector space generated by all exterior products of 1 forms {ξ i1 ∧ · · · ∧ ξ i k , i 1 < · · · < i k }. (for a complete description, see [Ram]). Set ΛE * = l ∞ ( ∞ k=0 Λ k E * ) = {ω = ∞ k=0 ω k with ω k ∈ Λ k E * , sup k \|\|ω k \|\| < ∞}. Then, ΛE * is a Banach space which, provided with the exterior product, is an algebra . If we consider the Banach space E, isometrically embedded in E * * , we can define, in the same way, the vector space Λ k E spanned by all exterior products {u i1 ∧ · · · ∧ u i k , i 1 < · · · < i k }. So, if we set: ΛE = l 1 ( ∞ k=0 Λ k E) = {u = ∞ k=0 u k with u k ∈ Λ k E, ∞ k=0 \|\|u k \|\| < ∞} then, ΛE is a Banach space, which, provided with the exterior product is a Banach algebra and also a graded algebra. Moreover Λ k E * is isomorphic to (Λ k E) * for any k ≥ 0 (see [Ram]). Notice that we have Λ 0 E * = Λ 0 E = R, Λ 1 E = E, and Λ 1 E * = E * . The interior product of ω ∈ Λ k E * by a vector v ∈ E, denoted by i v , is characterized, as usual in the following way: -if k ≤ 0, i v = 0 on Λ k E * -if k = 1, then i v ω = ω, v ∈ R -if k > 1, for v 1 . . . v k−1 ∈ E, then i v ω(v 1 , . . . , v k−1 ) = ω (v, v 1 , . . . , v k−1 ) and then i v ω ∈ Λ k−1 E * . In fact, given any fixed v ∈ E, the interior product i v can be clearly extended to a continuous endomorphism of ΛE * which is a derivation of degree −1 of ΛE * that is i v is send each factor Λ k E * of the graduation into the factor Λ k−1 E * ΛE * . The interior product i P by a multivector P ∈ Λ p E is defined in the following way: -if k < 0, i P = 0 -if k = 0, P ∈ R and for any form α ∈ ΛE * , then i P ω = P ω -if k ≥ 1 and if P is decomposable, i.e. P = v 1 ∧ · · · ∧ v k , we set i v1∧···∧v k = i v1 • · · · • i v k We can extend, by linearity and continuity, the definition of i P , for any P ∈ Λ k E, to the graded algebra ΛE * . For any fixed P ∈ Λ k E we then get an endomorphism i p of degree −p of the graded algebra ΛE * . Let τ : E → M be a Banach bundle of typical fiber E. In this situation, we denote by: -F (E) the algebra of smooth functions on E; -Γ(τ ) the F -module of smooth sections C ∞ of this bundle; -Λ k Γ * (τ ) the F -module of sections of the Banach bundle Λ k E * of typical fiber Λ k E * ; -Λ k Γ(τ ) the F -module of sections of the Banach bundle Λ k E of typical fiber Λ k E; -ΛΓ * (τ ) the F -module of sections of the Banach bundle ΛE * of typical fiber ΛE * ; -ΛΓ(τ ) the F -module of sections of the Banach bundle ΛE of typical fiber ΛE. For any P in Λ k Γ(τ ), the integer k is called the degree of P and we set k = deg P . Notice that Λ 0 Γ(τ ) = F and Λ 1 Γ(τ ) = Γ(τ ). For the exterior product of vectors (resp. of forms) we get a structure of graded exterior algebra on ΛΓ(τ ) (resp. ΛΓ * (τ )) Some classical properties of Banach manifolds First we recall some classical properties of Banach manifolds. The reader can find complete references about all these properties in [KrMi]. Let M be a smooth Banach manifold modeled on the Banach space M. The manifold M is paracompact if and only if the topology of M is metrizable. In particular the Banach space M must be also paracompact. This is always true for any (eventually non separable) Hilbert space. A Banach space which has C k partitions of unity is called C k -paracompact, for k ∈ N ∪ ∞. Any paracompact manifold modeled on a C k -paracompact Banach space has also C k -partitions of unity and so is C k -paracompact. The Banach manifold M is said C k -regular (resp. smooth regular) if for any x ∈ M , there exists an open neighborhood U of x and a C k (resp. smooth) function f : U → R such that f (x) = 1 and the closure of the set {z , f (z) = 0} is contained in U ; such a function is called a bump function. Notice that M is smooth-regular if and only if M is C k -regular for any k ∈ N ∪ ∞. Of course not all Banach spaces are C k -regular for k > 0: for instance, l 1 (Γ), for any set Γ, is not C 1 regular (see [KrMi]). But any C k -regular Banach space is paracompact (for more details about regularity and paracompactness see also [Arn], [Kun], [Llo], [Vand] and [Ion]). Notice that if M is smooth paracompact, then M is smooth regular. Local coordinates in a Banach bundle Consider a Banach bundle (E, τ, M ) and the associated dual bundle (E * , τ * , M ). Fix x ∈ M and consider any open neighborhood U of x such that E U is isomorphic to the trivial bundle U × E, which we always write E U ≡ U × E; we then say that E U is trivialized. Then we also have E * U ≡ U × E * ; T * E * \|E * U ≡ U × E * × M * × E * ; T E * \|E * U ≡ U × E * × M × E * . Here we consider E as a Banach subspace of E * * . Taking into account these equivalences, we get the following coordinates: s = (x, u) on E U ≡ U × E σ = (x, ξ) on E * U ≡ U × E * (s, v) = (s, v 1 , v 2 ) on T E \|EU ≡ U × E × M × E (σ, w) = (σ, w 1 , w 2 ) on T E * \|E * U ≡ U × E * × M × E * (σ, η) = (σ, η 1 , η 2 ) on T * E * \|E * U ≡ U × E * × M * × E * * Local coordinates in basis Suppose that the Banach space M has an unconditional, eventually uncountable, basis, {µ i } i∈I and denote by {µ * i } i∈I the associated weak- * basis of the dual M * (see [FiWo]). So, each z ∈ M can be written in a unique way: x = i∈I x i µ i . Note that we have x i = µ * i (x) . On the other hand, each ω ∈ M * can be "weak- * " written in a unique way: ω ≡ i∈I ω i µ * i which means that, for any u ∈ M, we have: < i∈I ω i µ * i , u >=< ω, u > In fact we have ω i =< ω, µ i >. Consider a chart (U, φ) on M . Via the diffeomorphism φ, we can identify U with an open set of M and so any x ∈ U can be written in a unique way x = i∈I µ * i (x)µ i . We will say that the set of maps {x i := µ * i : U → R} i∈I is the local system of coordinates on U . As the tangent bundle T M \|U is isomorphic to U × M, we denote by { ∂ ∂x i } i∈I the basis of each fiber T z M , for z ∈ U , canonically associated to {µ i } i∈I . So any vector field X on U can be written in a unique way as: X = i∈I X i ∂ ∂x i Moreover, as X can be identified with a map from U to M we have X i = µ * i • X and so each component X i is a smooth function. In the same way, the cotangent bundle T * M \|U is isomorphic to U × M * . We denote by {dx i } i∈I the weak- * basis on each fiber T * z M , for z ∈ U , canonically associated to {µ * i } i∈I . Again each 1-form ω on U can be weak- * written ω ≡ i∈I ω i dx i where of course we have < ω, X >=< i∈I ω i dx i , X > . Again, each component ω i is a smooth function. On the other hand, consider a Banach bundle τ : E → M , and suppose that there exists an unconditional basis {e α } α∈A for E. Consider an open set U ⊂ M which is a chart domain and such that E U ≡ U × E. With the previous properties, we denote again by e α the constant section x → e α in E U . Each section s ∈ Γ(τ U ) can be written as: s = α∈A e * α (s)e α Again each "component" u α = e * α (s) is a smooth function on U . So we have the following local coordinates on E U : • (x, u) = (x i , u α ) on E U if M has also an unconditional basis, on E U the tangent space T s E U is spanned by the basis { ∂ ∂x i } i∈N and { ∂ ∂u α } α∈A where { ∂ ∂u α } α∈A is identified with the basis {e α } α∈A of {s} × E So we have the following local coordinates on T E U • (s, v 1 , v 2 ) = (x i , u α , X i , U α ) In the same way, for the dual bundle τ * : E * → M , on E * U we have the constant sections e * α : x → e * α and any section σ of E * U , we also can write, in a "weak- * " way, σ = w α∈A ξ α e * α and again each component ξ α is a smooth function on U . So we have the following local coordinates on E * U • σ = (x, ξ) = (x i , ξ α ) ("weak- * coordinates" for ξ α ) if M also has an unconditional basis, on E * U , the tangent space T σ E * U is spanned by the basis { ∂ ∂x i } i∈I and "weakly- * " spanned by the basis { ∂ ∂ξ α } α∈A where again { ∂ ∂ξ α } α∈A is identified with the weak- * basis {e * α } α∈A ; So we have the following local coordinates on T E * U • (σ, w 1 , w 2 ) = (x i , ξ α , X i , Ξ α ) Derivations and vector fields Let M be a Banach manifold. Recall that a (global) derivation of F is a R-linear map ∂ : F → F such that: ∂(f g) = f ∂(g) + ∂(f )g We denote by D the vector space of all derivations of F . An operational vector field ∂ at x ∈ M is a derivation of F (U ) for some neighborhood U of x which is compatible with restriction to open V ⊂ U i.e. ∂ induces a unique derivation ∂ V of F (V ) such that ∂(f ) \|V = ∂ V (f \|V ) Let D x M be the vector space of operational vector field at x and DM = ∪ x∈M D x M the set of operational vector fields.. In fact, the canonical projectionp M : DM → M gives rise to a structure of Banach bundle (see [KrMi]). Unlike to the context of finite dimensional manifolds, if M is not smooth regular, then the set of germs at x ∈ M of elements of D can be smaller than D x . On the other hand, any local vector field on M gives rise to an operational vector field but there exist elements of D x which do not induce local vector fields (for more details see [KrMi]); in particular we have T M DM . Almost Banach Lie Algebroid Almost Lie bracket on an anchored Banach bundle Let τ : E → M be a Banach bundle of typical fiber E. We will denote by E x = τ −1 (x) the fiber over x ∈ M . A Banach morphism bundle ρ : E → T M is called an anchor. This morphism induces a map, again denoted by ρ from Γ(τ ) to Γ(M ) defined for any x ∈ M and any section s of E by: ρ(s)(x) = ρ • s(x). We say that (E, τ, M, ρ) is an anchored Banach bundle. Local expressions : In the context of local trivializations (see subsection 2.3), we have: ρ(x, u) ≡ (x, u) → (x, R x (u)) (1) where R : U → L(E, M). Suppose that the Banach spaces M and E have basis. According to subsection 2.4, on any appropriate open U in M , any anchor ρ is locally characterized by a family {ρ i α } i∈I,α∈A of smooth functions such that: ρ(e α ) = i∈I ρ i α ∂ ∂x i (2) Definition 3.1 1. An almost Lie bracket (AL-bracket for short) on an anchored bundle (E, τ, M, ρ) is a bracket [., .] ρ which satisfies the Leibniz property: [s 1 , f s 2 ] ρ = f [s 1 , s 2 ] ρ + (ρ (s 1 )) (f ) s 2 for any f ∈ F and s 1 , s 2 ∈ Γ(τ ). In this situation, (E, τ, M, ρ, [., .] ρ ) is called an almost Lie Banach algebroid (ALalgebroid for short). For any X and Y in G, we have: T x M = E x ⊕ F x . Let π 1 : T M → E be the Banach morphism associated to the projection of T x M onto E x whose kernel is F x . We define [X, Y ] E = π 1 [X, Y ]ρ({X, Y }) = [ρ(X), ρ(Y )] where { , } denotes the Lie algebra bracket on G ( [Bo], [KrMi]). On the trivial bundle M × G, each section can be identified with a map σ : M → G we define a Lie bracket on the set of sections by {{σ, σ ′ }}(x) = {σ(x), σ ′ (x)} + dσ(ξ σ ′ (x) ) − dσ ′ (ξ σ(x) ) We get an anchor Ψ : M ×G → T M by Ψ(x, X) = ξ X (x) It follows that (M ×G, Ψ, M, {{ , }}) has a Banach Lie algebroid structure on M . 3. Let π : N → M be a submersion between Banach manifolds. The subspaces V u N = T u π −1 (x) ⊂ T x,u N , denoted by V N defined a Banach sub-bundle of p N : T N → N called the vertical subbundle. As the Lie bracket of two vertical vector fields is again a vertical vector field, we get a L-algebroid on (V E, τ E \|V E , E). 4. Let θ be a 1-form on a Banach manifold M such that dθ is a weak symplectic form i.e. the canonical map θ ♭ : T M → T * M defined by θ ♭ (X) = i X dθ for any X ∈ T x M is injective (see [OdRa2]). Assume that θ ♭ is closed i.e. This situation precisely occurs on the cotangent bundle T * M of any Banach manifold M where θ is the Liouville 1-form on T * M (see [Lan]) T ♭ x M = θ ♭ (T x M ) is closed in T * x M . If i ♭ : T ♭ M → T * M is the natural inclusion, then we set q ♭ M = q M • i ♭ : T ♭ M → MDefinition 3.3 Let (E i , τ i , M, ρ i , [. , .] ρi ), i = 1, 2, be two AL-algebroids (resp. L-algebroids). A morphism Ψ from (E 1 , τ 1 , M ) to (E 2 , τ 2 , M ) (over Id M ) is called an AL-algebroid morphism (resp. L-algebroid morphism if we have: 1. Ψ • ρ 2 = ρ 1 2. [Ψ(s 1 ), Ψ(s 2 )] ρ2 = Ψ([s 1 , s 2 ] ρ1 ) for any s 1 , s 2 ∈ Γ(τ 1 ) Notice that if (E i , τ i , M, ρ i , [. , .] ρi ), i = 1, 2, are two L-algebroids, any AL-algebroid morphism Ψ from (E 1 , τ 1 , M, ρ 1 , [., .] ρ1 ) to (E 2 , τ 2 , M, ρ 2 , [., .] ρ2 ) induces a Lie algebra morphism from (Γ(τ 1 ), [., .] ρ1 ) to (Γ(τ 2 ), [., .] ρ2 ). In this case, we say that Ψ is a L-algebroid morphism. Classical derivations on an AL-algebroid In this subsection , (E, τ, M, ρ, [., .] ρ ) will be an AL-algebroid or L-algebroid. Lie derivative Given any section s ∈ Γ(τ ), the Lie derivative with respect to s on ΛΓ * (τ ), denoted by L ρ s , is the graded endomorphism, with degree 0, characterized by the following properties : 1. For any function f ∈ Λ 0 Γ(τ ) = F L ρ s (f ) = L ρ•s (f ) = i ρ•s (df ) (L0) where L X denote the usual Lie derivative with respect to the vector field X 2. For any q-form ω ∈ Λ q Γ * (τ ) (where q > 0) (L ρ s ω) (s 1 , . . . , s q ) = L ρ s (ω (s 1 , . . . , s q )) − q i=1 ω s 1 , . . . , s i−1 , [s, s i ] ρ , s i+1 , . . . , s q (Lq) On the other hand, we can also define for any function f ∈ Λ 0 Γ * (τ ) = F the element of Λ 1 Γ * (τ ), denoted d ρ f, by d ρ f = ρ t • df (d0) where ρ t : T * M → E * is the transposed mapping of ρ. The Lie derivative with respect to s commute with d ρ . Almost exterior differential The almost exterior differential on ΛΓ * (τ ), again denoted d ρ , (A-differential for short), is the graded endomorphism of degree 1 characterized by the following properties: 1. For any function f ∈ Λ 0 Γ * (τ ) = F , d ρ f is the element of Λ 1 Γ * (τ ) defined by d ρ f = ρ t • df 2. For any ω in Λ q Γ * (τ ) (q > 0), d ρ ω is the unique element of Λ q+1 Γ * (τ ) such that, for all s 0 , . . . , s q ∈ Γ(τ ), (d ρ ω) (s 0 , . . . , s q ) = q i=0 (−1) i L ρ si (ω (s 0 , . . . , s i , . . . , s q )) + 0≤i<j≤q (−1) i+j ω [s i , s j ] ρ , s 0 , . . . , s i , . . . , s j , . . . , s q We then have the following properties which are obvious or which can be proved as in finite dimension: 1. d ρ (η ∧ ζ) = d ρ (η) ∧ ζ + (−1) k η ∧ d ρ (ζ) for any η ∈ Λ k Γ * (τ ) any ζ ∈ Λ l Γ * (τ ) and any k, l in Z 2. For a L-algebroid, we have d ρ • d ρ = d ρ 2 = 0. In this case we say that d ρ is the exterior differential of the L-algebroid. As in the context of finite dimension (cf. [Ana]), we can prove: Proposition 3.4 Given two AL-algebroids (resp. L-algebroid) (E i , τ i , M, ρ i , [., .] ρi ), i = 1, 2, let d ρi be the associated A-differential (resp. exterior differential). For a bundle morphism Ψ from (E 1 , τ 1 , M ) to (E 2 , τ 2 , M ) (over Id M ), we denote by Ψ * : Λ p Γ * (τ 2 ) → Λ p Γ * (τ 1 ) the induced morphism on p-forms. Then, Ψ is an AL (resp. L)-algebroid morphism if and only if d ρ1 Ψ * (ω) = Ψ * (d ρ2 ω) for any ω ∈ Λ k Γ * (τ 1 ) and any integer k > 0. Recall that, the bracket [d 1 , d 2 ] of derivations d 1 and d 2 of the graded algebra ΛΓ * (τ ) of degree k 1 and k 2 respectively is the derivation d 1 • d 2 − (−1) k1k2 d 2 • d 1 of degree k 1 + k 2 . On the graded algebra ΛΓ * (τ ) with the A-exterior derivation d ρ we have: Proposition 3.5 : For any s 1 and s 2 in Γ(τ ), we have i [s1,s2]ρ (σ) = [[i s1 , d ρ ], i s2 ](σ) for any σ ∈ Γ(τ * ) Proof On one hand, a direct calculation gives the relation [[i s1 , d ρ ], i s2 ](σ) = L ρ s1 (σ(s 2 )) − L ρ s2 (σ(s 1 )) − d ρ (σ)(s 1 , s 2 ) On the other hand according to the definition of d ρ , we get: i [s1,s2]ρ (σ) = σ([s 1 , s 2 ] ρ ) = L ρ s1 (σ(s 2 )) − L ρ s2 (σ(s 1 )) − d ρ (σ)(s 1 , s 2 ). △ Almost Schouten-Nijenhuis bracket An almost Schouten-Nijenhuis bracket (ASN-bracket for short) is an inner composition law in ΛΓ(τ ) (again) denoted by [., .] ρ , characterized by the following properties: 1. [., .] ρ is a bi-derivation of degree −1, i.e. an R−bilinear map such that deg [P, Q] ρ = deg P + deg Q − 1 which fulfills the following property [P, Q ∧ R] ρ = [P, Q] ρ ∧ R + (−1) (deg P +1) deg Q Q ∧ [P, R] ρ 2. For all f, g ∈ Λ 0 Γ(τ ) = F , [f, g] ρ = 0 3. For all s ∈ Λ 1 Γ(τ ) = Γ(τ ), p ∈ Z and Q ∈ Λ p Γ(τ ), [s, Q] ρ = L ρ s Q 4. For all s 1 , s 2 ∈ Λ 1 Γ(τ ) = Γ(τ ), [s 1 , s 2 ] ρ corresponds to the bracket defined on the (A)L- algebroid 5. For all p, q ∈ Z, P ∈ Λ p Γ(τ ), Q ∈ Λ q Γ(τ ) , [P, Q] ρ = (−1) pq [Q, P ] ρ The ASN-bracket [., .] ρ is called Schouten-Nijenhuis bracket (SN-bracket for short) if, for all p, q, r ∈ Z and P ∈ Λ p Γ(τ ), Q ∈ Λ q Γ(τ ) R ∈ Λ r Γ(τ ), the ASN-bracket [., .] ρ satisfies the graded Jacobi identity: (−1) pr [P, Q] ρ , R ρ + (−1) qp [Q, R] ρ , P ρ + (−1) rq [R, P ] ρ , Q ρ = 0 Notice that if we take the canonical L-algebroid (T M, p M , M, Id, [., .]) the associated ASNbracket [., .] Id is the usual Schouten-Nijenhuis bracket on the graded algebra ΛΓ(M ). Locality of an almost Lie bracket In finite dimension it is classical that a an AL-bracket [., .] ρ on an anchored bundle (E, τ, M ρ) respects the sheaf of sections of τ : E → M or, for short, is localizable (see for instance [Marl]), if the following properties are satisfied: (i) for any open set U of M , there exists a unique bracket [., .] U on the space of sections Γ(τ U ) such that, for any s 1 and s 2 in Γ(τ U ), we have: [s 1\|U , s 1\|U ] U = ([s 1 , s 2 ] ρ ) \|U (ii) (compatibility with restriction) if V ⊂ U are open sets, then, [., .] U induces a unique AL- bracket [., .] UV on Γ(τ V ) which coincides with [., .] V (induced by [., .] ρ ). By the same arguments as in finite dimension, when M is smooth regular, we also have: Proposition 3.6 : If M is smooth regular then any AL-bracket [., .] ρ on an anchored bundle (E, τ, M, ρ) is localizable. If M is not smooth regular, we can no more used the arguments used in the proof of Proposition 3.6 . Unfortunately, we have no example of Lie algebroid for which the Lie bracket is not localizable. Note that, according to [KrMi] sections 32.1, 32.4, 33.2 and 35.1, this problem is similar to the problem of localization (in an obvious sense) of global derivations of the module of smooth functions on M or the module of differential forms on M . In [KrMi] and, to our known, more generally in the literature, there exists no example of such derivations which are not localizable. On the other hand, even if M is not regular, the classical Lie bracket of vector fields on M is localizable. So, there always exists an anchored bundle A = T M and a Lie bracket algebroid (T M, Id, M, [; , .]) for which its Lie bracket is localizable. Moreover, in Examples 3.2 we do not assume that M is regular but, nevertheless, these Lie brackets are also localizable Convention 3.7 : in all this paper, from now, we will assume that either M is smooth regular and if M is not regular, then the Lie bracket [ , ] ρ is localizable Remark 3.8 : 1. If M is smoothly paracompact, then M is smooth regular and so any AL-bracket [., .] ρ on an anchored bundle (E, τ, M, ρ) is localizable. On the converse, when M is paracompact, we can define some AL-bracket [., .] ρ "locally": given a locally finite covering {U i , i ∈ I} of M and a smooth partition of unity {θ i , i ∈ I} subordinated to this covering, if [., .] i is any AL-bracket on E Ui then: [., .] ρ = i∈I θ i [., .] i is an AL-bracket on (E, τ, M, ρ). 2. If [., .] ρ satisfies the Jacobi identity, then for any open set U , [., .] U satisfies also the Jacobi identity. In this case, (E U , ρ \|U , U, [., .] U ) is a L-algebroid. Remark 3.9 Let (E, τ, M, ρ, [., .] ρ ) be a L-algebroid. As its bracket is localizable, then D = ρ(E) is a weak distribution on M (i.e. on each "fiber" D x , x ∈ M , we have a Banach structure such that the identity map from D x as Banach space into D x as normed subspace of T x M is continuous see [Pel]) . If the kernel of ρ is complemented in each fiber, then D is integrable (see [Pel]). This situation occurs when ker ρ is finite dimensional or finite codimensional, or when M is a Hilbert manifold. Proof of Proposition 3.6 Let s 1 : M → E be a smooth section of τ : E → M which vanishes an open subset U of M . We first show that , for any other smooth section s 2 of τ : E → M , the bracket [s 1 , s 2 ] ρ vanishes on U . Indeed, choose any point x ∈ U and choose a smooth bump function f : M → R whose support is contained in U and such that f (x) = 1. The section f s 1 is a (global section) which vanishes identically. Therefore, for any other smooth section s 2 we have: 0 = [f s 1 , s 2 ] ρ = −[s 2 , f s 1 ] ρ = −f [s 2 , s 1 ] ρ − df (ρ • s 2 ))s 1 . So at x we have f (x)[s 1 , s 2 ] ρ (x) = df (ρ • s 2 ))s 1 = 0. Since f (x) = 1, we obtain [s 1 , s 2 ] ρ (x) = 0. Now given any open set U in M , we must show that the bracket [ , ] ρ induces an unique bracket [ , ] U on Γ(τ \|U ) such that [s 1\|U , s 1\|U ] U = ([s 1 , s 2 ] ρ ) \|U Choose any x in U and, as before, some bump function f : M → R whose support is contained in U and such that f (x) = 1. Then for any section s 1 and s 2 in Γ(τ \|U ), f s 1 and f s 2 are global sections of τ : E → M . So [f s 1 , f s 2 ] ρ (x) is well defined and from the previous argument, this value do not depends of the choice of the bump function f . Therefore, we can set [s 1 , s 2 ] U (x) = [f s 1 , f s 2 ] ρ (x) for any choice of bump function as previously. It follows clearly from this construction that [ , ] ρ is localizable. △ Local expressions: Let (E, τ, M, ρ, [., .] ρ ) be an AL-algebroid. In the context of local trivializations (subsection 2.3) there exists a field C : U → L(E × E, E) z → C z such that for s(z) = (z, u(z)) and s ′ (z) = (z, u ′ (z)) we have: [s, s ′ ] U (z) = (z, C z (u(z), u ′ (z)))(3) Suppose that the typical fiber E has an unconditional basis. According to subsection 2.4, then for any x ∈ M , there exists an open neighborhood U of x and a set of smooth functions {C γ αβ , α, β, γ ∈ A} on U such that [., .] U is characterized by: [e α , e β ] U = γ∈A C γ αβ e γ(4) More precisely, if s = α∈A s α e α and r = α∈A r α e α , we have: [s, r] U = α,β,γ∈A C γ αβ s α r β e γ + α∈A (dr α (ρ(s))e α − ds α (ρ(r))e α )(5) Note that, the almost exterior differential d ρ on ΛΓ * (τ U ) is also localizable i.e. for any open set U in M , there exists a unique graded derivation d U of degree 1 on ΛΓ * (τ U ), such that (d ρ ω) \|U = d U (ω \|U ) and which is compatible with restriction to open subsets V ⊂ U . So, as for an AL bracket, in the context of local trivializations (subsection 2.3), given local sections s(z) = (z, u(z)) and s ′ (z) = (z, u ′ (z)) of E U , and ω(z) = (z, ξ(z)) any section of E * U , according to (1) and (3) we have d ρ ω(s, s ′ ) =< Dξ(ρ(s)), u ′ > − < Dξ(ρ(s ′ )), u > − < ξ, [s, s ′ ] ρ > =< Dξ(R(u)), u ′ > − < Dξ(R(u ′ )), u > − < ξ, C(u, u ′ ) >(6) where Dξ denotes the differential of the map ξ : U → E * . In the same way, the Lie derivative L ρ s is localizable. For the sake of simplicity, for any open subset U in M , we note [., .] ρ , d ρ and L ρ instead of its restriction ([., .] r ) U , (d ρ ) U ) and L ρ U , to U , respectively Sub almost Poisson morphism Let M be a Banach manifold. An almost Lie bracket (AL-bracket for short) on F is a R-bilinear skew-symmetric pairing {., } on F which satisfies the Leibniz rule, i.e. for any f, g, h ∈ F {f, gh} = g{f, h} + h{f, g}. An almost Poisson morphism (AP-morphism for short) on M is a bundle morphism P : T * M → T M which is skew symmetric according to the duality pairing (i.e. such that < η, P ζ >= − < ζ, P η > for any η, ζ ∈ T * M ). We can associate to such a morphism a R-bilinear skew symmetric pairing {., .} P on Γ * (M ) defined by: {η, ζ} P =< ζ, P η > Moreover, for any f ∈ F we have: {f η, ζ} P =< ζ, f P η >= f {η, ζ} P So, we get on F an almost Lie bracket {., .} P defined by {f, g} P = {df, dg} P For any f ∈ F we can associate a unique vector field grad P (f ) = −P (df ) which is called the almost Hamiltonian vector field of f (A-Hamiltonian gradient for short) Classically, to an AP-morphism P, we can associate a skew-symmetric tensor of type (3, 0) [P, P ] : Γ * (τ ) × Γ * (τ ) → Γ(τ ) defined by: [P, P ] (η, ζ) = P (L P η ζ − L P ζ η + d η, P ζ ) + [P η, P ζ](7) for all η, ζ ∈ Γ * (τ ). As in finite dimension, {., .} P satisfies the Jacobi identity if and only if the the tensor [P, P ] vanishes identically (see for instance [MaMo]). In this case, F has a structure of Lie algebra and (M, {., .} P ) is called a Banach Lie Poisson manifold (P-manifold for short) (see for instance [OdRa1] or [OdRa2]). If the Jacobi identity is not satisfied, we say that we have an almost Banach Lie Poisson manifold (an AP-manifold for short). In this case the vector field grad P (f ) is called the hamiltonian gradient of f and P induces a morphism of Lie algebra between (F , {., .} P ) and (Γ(M ), [., .]) Remark 4.1 In finite dimension, a Poisson manifold is characterized by a bi-vector Λ on a manifold M such that the Schouten-Nijenhuis bracket [Λ, Λ] vanishes identically. On a Banach manifold M , an AL-bracket on F gives rise to an element Λ of ΛT * * M such that Λ(df, dg) = {f, g} Such a bi-vector gives rise to a unique morphism P : T * M → T * * M defined by the relation: < ζ, P η >= Λ(η, ζ). for all η, ζ ∈ Γ * (M ) where T * * M is the bidual tangent bundle of M . To get a Poisson manifold, we need the additional condition: P (T * M ) ⊂ T M (see [OdRa1], [OdRa2]). For examples and more details about P-manifolds the reader can have a look at [OdRa1] or [OdRa2] and some references within these papers. Let q ♭ M : T ♭ M → M be a Banach subbundle of q M : T * M → M . A bundle morphism P : T ♭ M → T M will be called a sub almost Poisson morphism 1 (sub AP-morphism for short) 1 this terminology is chosen in analogy to sub-riemannian structures on a manifold if P is skew-symmetric relatively to the duality pairing i.e. < α, P β >= − < β, P α > for any α, β ∈ T ♭ M . As before, we get a R-bilinear skew-symmetric pairing {., .} on Γ(q ♭ M ). The set F ♭ = {f ∈ F : df ∈ Γ(q ♭ M )} is a sub-algebra of F . Using the same arguments as in [KrMi], section 48, we can see that, as above, P induces on F ♭ an almost Lie bracket which will be again denoted by {., .} P . We will say that (M, F ♭ , {., .} P ) is a sub almost Banach Lie Poisson manifold (sub AP-manifold for short). Of course, when {., .} P satisfies the Jacobi identity, (F ♭ , {., .} P ), has a Lie algebra structure and we say that (F ♭ , {., .} P ) is a sub Banach Lie-Poisson manifold (sub P-manifold for short). Then for any f ∈ F ♭ we can associate a sub almost Hamiltonian gradient grad P (f ) = −P (df ) (sub A-Hamiltonian gradient for short) Of course, to a sub AP-morphism P on M , we can also associate a skew symmetric tensor of type (3, 0): [P, P ] : Γ(q ♭ M ) × Γ(q ♭ M ) → Γ(τ ) using the same definition as in (7), but for α, β ∈ T ♭ M . Again, {., .} P satisfies the Jacobi identity if and only if [P, P ] vanishes identically. In this case, (F ♭ , {., .} P ) has a Lie algebra structure and P induces a Lie algebra morphism from ( F ♭ , {., .} P ) to (Γ(M ), [., .]) Consider two sub AP-morphisms P i on the manifolds M i , i = 1, 2 and a map ψ : M 1 → M 2 . We say that (M 1 , P 1 ) and (M 2 , P 2 ) are ψ-related if we have: P 2 = (T ψ) * • P 1 • T ψ In this case for the associated AL-bracket {., .} Pi on the algebra F ♭ i i = 1, 2 we have: {f • ψ, g • ψ} 2 = ψ • {f, g} 1 for all f, g ∈ F ♭ 1 . Moreover, in this case, P 1 is a sub P-morphism if and only if P 2 is a sub Pmorphism and then ψ gives rise to a Lie algebra morphism between (F ♭ 1 , {., .} 1 ) and (F ♭ 2 , {., .} 2 ). A lot of results about AP-manifolds and P-manifolds can be extended to the context of sub AP-manifolds and sub-P-manifolds when considering the structure of Lie algebra of (F ♭ , {., .} P ). We do not develop these aspects here. Example 4.2 Let ω be a non degenerated 2-form on a manifold M . We denote by ω ♭ : T M → T * M the associated morphism defined by ω ♭ (X) = i X ω for X ∈ T x M . Suppose that T ♭ M = ω ♭ (T M ) is a Banach sub- bundle of T * M . Then ω ♭ is an isomorphism from T M onto T ♭ M . So P = (ω ♭ ) −1 : T ♭ M → T M is a sub AP-morphism. We get a sub P-morphism if and only if ω is closed. In particular, if M is the cotangent bundle T * N of a Banach manifold N , if ω is the 2 fundamental form on T * N , we get a sub Poisson structure on T * N which corresponds to the natural weak symplectic structure on T * N (see Example 3.2 4). Note that we get a Poisson structure on T * N if and only if N is modeled on a reflexive Banach space. Remark 4.3 Consider a sub-Poisson morphism P : T ♭ M → T M , and denote by D = P (T ♭ M ) the associated (weak) distribution on M . If the kernel of P is complemented in each fiber, then D is integrable and each leaf is a weak symplectic manifold (see [Pel]). This situation is always satisfied when ker P is finite dimensional or finite co-dimensional (for instance if P is Fredholm, or injective) , or when M is an Hilbert manifold. Let τ : E → M be a Banach bundle and τ * : E * → M its associated dual Banach bundle. We denote by F (E * ) the set of smooth functions on E * . Any f ∈ F (E * ) is called linear, if the restriction of f to each fiber E * x = τ −1 * (x) is a linear map. Definition 4.4 Let q ♭ : T ♭ E * → E * be a subbundle of q E * : T * E * → E * . Consider a sub AP-morphism P : T ♭ E * → T E * and its associated bracket {., .} P on F ♭ (E * ) ⊂ F (E * ). The sub AL-morphism P on E * is called linear if for any linear functions f and g on E * which belong to F ♭ (E * ) their bracket {f, g} p is linear. Example 4.5 In the context of Example 4.2, if we take E = T M , the sub P-morphism Π = (ω ♭ ) −1 : T ♭ (T * M ) → T (T * M ) is a linear sub P-morphism on T * M . To a given Banach bundle τ : E → M , we will construct a canonical subbundle T ♭ E * on which will be defined all "interesting" sub AP-morphisms (see Proposition 4.7 and subsection 4.2). First of all, for any section s ∈ Γ(τ ), we associate the linear function Φ s on E * defined by Φ s (ξ) =< ξ, s • τ * (ξ) > We then have the following properties: Lemma 4.6 1. The map s → Φ s is linear and injective; we also have Φ f s = (f • τ * )Φ s for any s ∈ Γ(τ ) and f ∈ F In local trivializations (subsection 2.3) we have < dΦ s (σ), w 2 >=< w 2 , s • τ * (σ) > for any w 2 ∈ E * , considered as vertical fiber T σ E * ; if for some f ∈ F , we have d(Φ s + f • τ * )(σ) = 0 then s • τ * (σ) = 0. Proof of Lemma 4.6 The first part is easy and left to the reader. With the previous notations, if w 2 belongs to the vertical part of T σ E * which is {σ} × E * with our notations. We can consider w 2 as an element of E * τ * (σ) . We then have < dΦ s (σ), w 2 >= lim t→0 1/t[< σ + tw 2 , s • τ * (σ) > − < σ, s • τ * (σ) >] =< w 2 , s • τ * (σ) >. Assume that d(Φ s + f • τ * )(σ) = 0. Given any w 2 ∈ E * τ * (σ) . As before, w 2 can be considered as a vector in the vertical part of T σ E * . From the previous relation we get: < dΦ s (σ), w 2 >= − < df (τ * (σ)), T σ τ * (w 2 ) >= 0 which ends the proof. △ Proposition 4.7 The set T ♭ E * = x∈M {(σ, η) ∈ T * σ E * , η = d(Φ s +f •τ * ), s ∈ Γ(τ U ), f ∈ F (U ), U neighbourhood of x} is a well defined subset of T * E * and if q ♭ is the restriction of q E * to T ♭ E * then q ♭ : T ♭ E * → M is a Banach bundle of typical fiber M * ×E. In particular, T ♭ E * = T * E * if and only if E is reflexive. Proof of Proposition 4.7 First notice that, using our notations, at any σ ∈ E * , any vector (σ, η 1 ) ∈ {σ} × M * can be written as η 1 = df (τ * (σ)) for some f ∈ F and moreover, this choice only depends on the value df (τ * (σ)). On the other hand, from Lemma 4.6 part 2, we can identified the restriction of dΦ s (σ) to the vertical part to T σ E * with s • τ * . So, the set {dΦ s (σ) \|{σ}×E * , s ∈ Γ(τ U )} generates the subspace {σ} × E, considered as subspace of the vertical part {σ} × E * * of T * σ E * . Assume that we can write η = d(Φ s1 + f 1 • τ * )(σ) = d(Φ s2 + f 2 • τ * )(σ) . From Lemma 4.6 part 2, we get s 1 • τ * (σ) = s 2 • τ * (σ) and then we must have df 1 (τ * (σ)) = df 2 (τ * (σ)). It follows that the subset T ♭ σ E * = {(σ, η) ∈ T * σ E * , η = d(Φ s + f • τ * ), s ∈ Γ(τ U ), f ∈ F (U ), U neighbourhood of x} is well defined and is a subspace of T * σ E * which, with our notations, is exactly {σ} × M * × E. From the local definition of T ♭ σ E * , it follows that the restriction q ♭ of q E * : T * E * → E * gives rise to a Banach subbundle. Of course, we get T ♭ E * = T * E * if and only if E is reflexive. △ Relation between AP-algebroid and sub AP-morphism Now we are able to give an adaptation to the Banach context of the classical result about equivalence between Poisson structure on E * and structure of AL-algebroid on E (see for example [Marl]). Theorem 4.8 Let P : T ♭ E * → T E * be a linear sub AP-morphism on E * . Then there exists a unique AL-algebroid structure (E, τ, M, ρ, [., .] P ) characterized by: Φ [s1,s2]P = {Φ s1 , Φ s2 } P , for any s 1 , s 2 ∈ Γ(τ ) (8) {Φ s , f • τ * } P = L ρ s (f ) • τ * , for any f ∈ F , s ∈ Γ(τ )(9) Moreover, (E, τ, M, ρ, [., .] P ) is a L-algebroid if and only if P is a sub-Poisson morphism. Conversely, each AL-algebroid structure (E, τ, M, ρ, [., .] ρ ) defines a unique linear bracket [., .] ρ on the sub-ring F ♭ (E * ) which is associated to a unique linear sub AP-morphism on E * which is characterized by relations (8) and (9). Moreover, P is a sub-Poisson morphism on E * if and only if (E, τ, M, ρ, [., .] ρ ) has a L-algebroid structure. Example 4.9 According to Example 4.2 and Example 4.5, as in finite dimension, the canonical sub P-morphism Π induces on the bundle (T M, p M , M ) a structure of L-algebroid which is, in fact, the canonical L-algebroid structure (see Example 3.2 1). Local expressions : In the context of local trivialization (subsection 2.3), recall that T E * U ≡ U × E * × M × E * and T ♭ E * U ≡ U × E * × M * × E. Given an AP-morphism P , recall that locally we have the following characterization (see (1) and (3)) ρ(s) = (., R(u)) for any s(z) = (z, u(z)) [s, s ′ ] ρ = ( , C(u, u ′ ), ) for any s(z) = (z, u(z)) and s ′ (z) = (z, u ′ (z)) so locally we have < (df ′ , u ′ ), P (ω(x))(df, u) > =< ξ, [s, s ′ ] ρ > (x)+ < df ′ , ρ(x, u) > − < df, ρ(x, u ′ ) > =< ξ, C(u, u ′ ) > + < df ′ , R(u) > − < df, R(u ′ ) >(10) for any functions f and f ′ on U , any sections s and s ′ any form ω(z) = (z, ξ(z)). If F is a function in F ♭ (E * U ), in local trivialization, we can write dF = (D 1 F, D 2 F ) on U × E * where D 1 F (resp. D 2 F ) is the partial derivative of F according to the first factor (resp. the second factor). Notice that, D 2 F (σ) belongs to E. So for any F, G ∈ F ♭ (E * U ), their Poisson bracket is given by: {F, G} P (σ) =< D 1 F, R(D 2 G) > (σ) − < D 1 G, R(D 2 F ) > (σ) − < ξ, [D 2 F, D 2 G] ρ > (σ) (11) Suppose that each Banach space E and M has an unconditional basis. According to subsection 2.4 and also local trivializations (subsection 2.3), recall that we have the following local coordinates: • (x, u) = (x i , u α ) on E U • σ = (x, ξ) = (x i , ξ α ) on E * U ("weak-* coordinates" for ξ α ) • on E * U , the tangent space T σ E * U is generated by the basis { ∂ ∂x i } i∈I and "weakly-" generated by the basis { ∂ ∂ξ α } α∈A ; • on E U , according to Lemma 4.6 part 1 and its proof, each fiber T ♭ σ E * is generated by the basis {dx i } i∈I and {e α • τ * } α∈A . Notice that, we have dξ α = e α • τ * . With these notations, any sub AP-morphism P : T ♭ E * → T E * associated to an AL-bracket [., .] ρ (as in Theorem 4.8) are related in the following way: the Lie bracket [., .] P is locally characterized by [e α , e β ] U = γ∈A C γ αβ e γ (see (4)); the anchor ρ can be characterized by ρ(e α ) = i∈I ρ α i ∂ ∂x i (see (2)) the AP-morphism P is characterized by: P (dx i ) = α∈A ρ i α ∂ ∂ξ α P (dξ α ) = − i∈I ρ i α ∂ ∂x i − β,γ∈A C γ αβ ξ γ ∂ ∂ξ β On E * U , the associated algebra of functions F ♭ (E * U ) are functions on E * U which depend on variables (x i ) i∈I and (ξ α ) α∈A . The AL-bracket {., .} P on F ♭ (E * U ) is characterized by: {ξ α , ξ β } P = − γ∈A C γ αβ ξ γ , {x i , ξ α } P = ρ i α , {x i , x j } P = 0 The Poisson Bracket of F, G ∈ F ♭ (E * U ) is given by: {F, G} P = i∈I,α∈A ρ i α ( ∂F ∂x i ∂G ∂ξ α − ∂G ∂x i ∂F ∂ξ α ) − α,β,γ∈A C γ αβ ξ γ ∂F ∂ξ α ∂G ∂ξ β For the first part of the proof of Theorem 4.8, we need the following Lemma: Lemma 4.10 Consider a linear sub AP-morphism P : T ♭ E * → T E * on E * and {., .} p the associated bracket on F ♭ (E * ). 1. for any section s ∈ Γ(τ ) and any f ∈ F the bracket {Φ s , f • τ * } P belongs to F . 2. If f and g belong to F , then {f • τ * , g • τ * } P = 0. Proof of Lemma 4.10 This proof is an adaptation of the proof of the analogous result in finite dimension, (cf [LMM] for instance) Consider any s, s ′ ∈ Γ(τ ), and f ∈ F . Using the Leibniz rule for the bracket {., .} P we get: (12) it follows that Φ s ′ {Φ s , f • τ * } P is a linear function on E * for any s ′ ∈ Γ(τ ). So {Φ s , f • τ * } P must be constant on each fiber and then we get 1. {Φ s , (f • τ * )Φ s ′ } P = (f • τ * ){Φ s , Φ s ′ } P + Φ s ′ {Φ s , f • τ * } P (12) Moreover {Φ s , (f • τ * )Φ s ′ } P is a linear function on E * . Since (f • τ * ){Φ s , Φ s ′ } P is also a linear map, from On the other hand, by same argument, we have {(f • τ * )Φ s ′ , g • τ * } P = (f • τ * ){Φ s ′ , g • τ * } P + Φ s ′ {f • τ * , g • τ * } P So from part 1, we deduce that , Φ s ′ {f • τ * , g • τ * } P belongs to F for any s ′ ∈ Γ(τ ). If follows that we must have {f • τ * , g • τ * } P = 0 and we get 2. △ Proof of Theorem 4.8 (adaptation, in our context, of the proof of [LMM] of the same result in finite dimension) Consider a linear sub AP-morphism P : T ♭ E * → T E * on E * and {., .} P the associated bracket on F ♭ (E * ). As Φ is injective, the pairing [., .] P defined by (8) is well defined and is R-bilinear and skewsymmetric. From Lemma 4.10 part 1 and the Leibniz property of {., .} P it follows that, for some fixed s ∈ Γ(τ ), the map ρ(s) : f → {Φ s , f • τ * } P defines a derivation on F . On the other hand, we have: {Φ s , f • τ * } P =< d(f • τ * ), P (dΦ s ) >=< df, dτ * • P (dΦ s ) > It follows that ρ(s) = dτ * • P (dΦ s ) is a vector field on M . Notice that, from the properties of Φ (Lemma 4.6 part 1), the Leibniz property of {., .} P , and Lemma 4.10 part 2, we have: ρ(f s) = f ρ(s)(13) for any f ∈ F . As P is a bundle morphism, it follows that the bracket {., .} P is localizable, so on one hand the same is true for the bracket [., .] P and on the other hand, from (13) we get that ρ(s) only depends on the value of s at any point x ∈ M so we get a morphism bundle ρ : E → M defined by ρ(u) = ρ(s)(x) where s is any (local) section such that s(x) = u. From (8) and (9), and Lemma 4.6 part 1, for any s 1 , s 2 ∈ Γ(τ ) and f ∈ F we have: Φ [s1,f s2] = {Φ s1 , (f • τ * )Φ s2 } P = (f • τ * ){Φ s1 , Φ s2 } P + Φ s2 L ρ(s1) (f ) • τ * So we get the following relation: [s 1 , f s 2 ] P = f [s 1 , s 2 ] P + ρ(s 1 )(f )s 2 Thus we have proved that (E, τ, M, ρ, [., .] P ) is an AL-algebroid. From the properties of Φ, it follows that if [., .] P satisfies the Jacobi identity, it implies that {., .} P condition is true if and only if P is a Poisson morphism. Conversely, let (E, τ, M, ρ, [., .] ρ ) be an AL-algebroid. Note that, from convention ?? the bracket [ , ] ρ is localizable. We want to associate a linear AP-morphism P : T * E * → T E * such that, according to the first part, the induced AL-algebroid structure on (E, τ, M ) is exactly (E, τ, M, ρ, [., .] ρ ). For this we must have: {Φ s , f • τ * } P = L ρ•s (f ) • τ * and {f • τ * , g • τ * } P = 0 Locally, with the notations of Lemma 4.6, we define P σ : T ♭ σ E * → T σ E * as follows: for η = d(Φ s + f • τ * )(σ), and η ′ = d(Φ ′ s + f ′ • τ * )(σ) we set Λ(σ)(η, η ′ ) = Φ [s,s ′ ]ρ (σ) + L ρ(s) (f ′ ) • τ * (σ) − L ρ(s ′ ) (f ) • τ * (σ)(14) As we have seen that η (resp. η ′ ) only depends on s • τ * (σ) and df • τ * (σ) (resp. s ′ • τ * (σ) and df ′ • τ * (σ)), then it follows that Λ is well defined at σ. Moreover, from its local definition, we get a smooth section of Λ 2 T ♭ E * . It follows that the map η → Λ(σ)(., η) defines a linear map P (σ) : T ♭ σ E * → [T ♭ σ E * ] * directly given by: P σ (η) = Φ [s,.]ρ (σ) + L ρ(s) (.) • τ * (σ) − L ρ(.) (f ) • τ * (σ)(15) Using our notations, we have T ♭ σ E * ≡ {σ} × M * × E and so [T ♭ σ E * ] * ≡ {σ} × M * * × E * . Recall that η = dΦ s + df • τ * . On one hand, any ω ∈ {σ} × M * can be written as ω = dg • τ * (σ) for some g : U ⊂ M → R; so we have < ω, P σ (η) >= Λ σ (η, dg • τ * ) = L ρ(s) (g) • τ * (σ) It follows that P σ (η) \|{σ}×M * belongs to {σ} × M considered as a subspace of {σ} × M * * . On the other hand we have < dΦ s ′ , P σ (η) >= Λ σ (η, dΦ s ′ ) = Λ σ (dΦ s , dΦ s ′ ) + Λ(σ)(df • τ * , dΦ s ′ ) = Φ [s,s ′ ] (σ) − L ρ(s ′ ) (f ) • τ * (σ) It follows that P (σ)(η) \|{σ}×E belongs to {σ} × E * × M. In conclusion, P (σ)(η) belongs to T σ E * ≡ {σ} × M * × E. Notice that the definition (15) of P is in fact local so P is a smooth bundle morphism. Of course as usually, P gives rise to a bracket on F ♭ (E * ) which is exactly given by (14). This bracket is denoted by {., .} P . As P is a bundle morphism, {., .} P is localizable. Moreover, it satisfies the relations {Φ s1 , Φ s2 } P = Φ [s1,s2]ρ =< dΦ s2 , P (dΦ s1 ) >, for any s 1 , s 2 ∈ Γ(τ ) (16) L ρ(s) (f ) • τ * = {Φ s , f • τ * } P =< df • τ * , P (dΦ s ) >, for any f ∈ F , s ∈ Γ(τ ) (17) {f 1 • τ * , f 2 • τ * } P =< df 2 • τ * , P (df 1 ) • τ * >= 0, for any f 1 , f 2 ∈ F .(18) Let F be a smooth linear function on E * which belongs to F ♭ . Fix a point σ = (x, ξ) ∈ E and E * U ≡ U × E * a neighborhood of σ. We denote by d 2 F the partial differential of F relative to E * in the product U × E * . As F is linear, there exists a section S : U → E * * such that d 2 F (ζ) =< S, ζ > for any ζ ∈ E * . But F belongs to F ♭ so dF is a section of T ♭ E * → E * ; then we must have S : U → E and d 2 F = d 2 Φ S . Then, from (16) it follows that P is a linear AP-morphism. On the other hand, the previous relations (16), (17) and (18) mean that the Lie bracket [., .] P induced by P on Γ(τ ) is exactly the original one [., .] ρ . Finally, if [., .] ρ satisfies the Jacobi identity, the previous relation implies that {., .} P also satisfies the Jacobi identity for functions of type Φ s + f • τ * . So it follows that [P, P ] vanishes indentically and then, the Jacobi identity is satisfied for any f, g, h ∈ F ♭ (E * ), which ends the proof. △ Relation between AP-algebroid and A-derivations Recall that an A-exterior differential is a graded derivation δ of degree 1 of ΛΓ * (τ ) which is localizable i.e. for any open set U in M , there exists a unique graded derivation δ U of degree 1 of ΛΓ * (τ U ) such that (δω) \|U = δ U (ω \|U ) which is compatible with restriction to open subsets V ⊂ U and satisfies the following properties: 1. δ(η ∧ ζ) = δ(η) ∧ ζ + (−1) k η ∧ δ(ζ) for any η ∈ Λ k Γ * (τ ) any ζ ∈ Λ l Γ * (τ ) and any k, l ∈ Z 2. For a L-algebroid, we have δ•δ = δ 2 = 0. In this case we say that δ is a exterior differential. We will adapt the classic result obtained in finite dimension: given a derivation δ, one associates a unique bracket [., .] δ on Γ(τ ) with anchor ρ such that the almost exterior derivative d ρ associated is exactly δ. However, in finite dimension, any (local) derivation of F is a vector field, but in infinite Banach context it is not true (see subsection 2.5). So, we must impose another condition on δ to get an analogous result. Moreover, in finite dimension, the exterior algebra ΛΓ * (τ ) is locally generated by its elements of degrees 0 and 1 which is not true in the Banach framework, even when we have Schauder basis (see Remark 4.13). In the context of the Proposition 3.5 first we have: Lemma 4.11 Consider a graded A-derivation δ of degree 1 of ΛΓ * (τ ) which is localizable. For any s 1 and s 2 in Γ(τ ), the bracket [[i s1 , δ], i s2 ] is a derivation of degree −1 and its restriction to Γ(τ * ) can be identified with a section of the bi-dual E * * → M Proof First by construction, the degree of D = [[i s1 , δ], i s2 ] is −1 so D maps Λ 1 Γ(τ * ) into F and Df = 0 for any f ∈ F . It follows that D(f σ) = f Dσ for any σ ∈ Γ(τ * ) and f ∈ F and so the map σ → Dσ is F is linear, which ends the proof. △ Now, in our context, we have 2. [., .] δ is characterized by σ([s 1 , s 2 ] δ ) = δ(σ(s 1 ))(s 2 ) − δ(σ(s 2 ))(s 1 ) − δσ(s 1 , s 2 ), for any σ ∈ Γ(τ * ) (20) In particular (E, τ, M, [., .] δ ) is an AL-algebroid and the almost exterior derivative associated to this structure coincides with δ on Λ 0 Γ * (τ ) = F and Λ 1 Γ * (τ ) = Γ(τ * [., .] δ is obtained from the following results: Φ [s1,f s2] δ (σ) = δ(σ(s 1 ))(f s 2 ) − δ(σ(f s 2 ))(s 1 ) − δσ(s 1 , f s 2 ) = f (δ(σ(s 1 ))(s 2 ) − δ(σ(s 2 ))(s 1 ) − δσ(s 1 , s 2 )) + σ(s 2 )δf (s 1 ) = Φ f [s1,s2] δ (σ) + Φ ρ(s1)(f ) (σ) Now, if δ 2 = 0 by "formal argument" as in finite dimension, used for instance in [Marl], we can prove that [., .] δ satisfies the Jacobi identity. From (19) we obtain that δ = d ρ on F . From Proposition 3.5, we obtain that d ρ = δ on Γ(τ * ) = Λ 1 Γ * (τ ). △ Remark 4.13 1. Note that, in general, if M is not regular, a derivation of the module of smooth functions F on a Banach manifold M is not localizable (see for instance [KrMi] section 35.1) . However, to our known there exists no example of such a derivation which is not localizable. So, in Theorem 4.12, if M is not regular, we must impose that the A-exterior differential on ΛΓ * (τ ) is localizable. 2. In Theorem 4.12, we cannot assert that δ and d ρ coincides on Λ k Γ * (τ ) for k ≥ 2. Indeed, recall that the Banach space Λ k E * is generated by exterior products of 1 forms ξ i1 ∧ · · · ∧ ξ i k . However, even if E * has a Schauder basis {ǫ α } α∈N the family {ǫ α1 ∧ · · · ∧ ǫ α k , α 1 < · · · < α k } could not be a Schauder basis of Λ k E * for k ≥ 2 (this is an unsolved problem see [Ram]). So, for infinite dimensional Banach spaces, locally, the module of local k-forms Λ k Γ * (τ ) is not finitely generated but, moreover we have no "good topology" on Λ k Γ * (τ ) such that each k form ξ can not be locally written as η(x) = i1<···<i k η i1···i k (x)ξ i1 ∧ · · · ∧ ξ i k for some appropriate finite sequences of smooth functions ξ i1 , · · · ξ i k (for topologies on modules of sections see [KrMi] and [Lla]) As a consequence, any two A-derivations which coincide on F (U ) = Λ 0 Γ * (τ U ) and on Γ * (τ U ) = Λ 1 Γ * (τ U ) can be different on Λ k Γ * (τ U ), for k ≥ 2. Once more, unfortunately, we have no example of such a situation. Set of AL structures on an anchored Banach bundle All the essential previous results will be summarized in the next theorem. Let (E, τ, M, ρ) be an anchored Banach bundle. We denote by ALB(τ, ρ) the set of (localizable) AL-brackets on (E, τ, M, ρ) with fixed anchor morphism ρ. We have seen that to any sub AP-morphism on the dual bundle E * is associated an anchor morphism ρ P : E → T M characterized by to s ∈ Γ(τ ) one associates the derivation f → {Φ s , f • τ * } on F . We denote by AP(τ, ρ) the set of sub AP-Poisson morphisms on the dual bundle E * such that the associated anchor morphism is ρ. Theorem 4.14 Let (E, τ, M, ρ) be an anchored Banach bundle. The set ALB(τ, ρ) has a natural structure of affine space in the following sense: given any [., .] E ∈ ALB(τ, ρ) then we have: ALB(τ, ρ) = {[., .] E + D, D ∈ Λ 2 Γ(τ )}. There exists a bijection from ALB(τ, ρ) to AP(τ, ρ) defined in the following way: at any AL-bracket [., .] E one associates the AL-bracket {., .} E on F ♭ (E * ) characterized by Φ [s1,s2]E = {Φ s1 , Φ s2 } E Proof The only thing to prove is the structure of ALB(τ, ρ). Consider two AL-brackets [., .] i , i = 1, 2, on (E, τ, M, ρ). It easy to see that D = [., .] 1 − [., .] 2 is an element of Λ 2 Γ(τ ). The others properties come from Theorem 4.8. △ 5 Mechanical systems on an almost Lie algebroid 5.1 Hamiltonian system and Hamilton-Jacobi equation Let (E, τ, M, ρ, [., .] ρ ) be an AL-algebroid. Denote by P : T ♭ E * → T E * the sub AP-morphism associated to this AL-strucuture and {., .} P the associated AP-bracket on F ♭ (E * ) (see subsection 4.1). Any function h ∈ F ♭ (E * ) is called a Hamiltonian function and the triple (E, {., .} P , h) is called a Hamiltonian system. As we have already seen, to h is associated a vector field grad P (h) = −P (dh) called the sub A-hamiltonian gradient of h. As in this section, the ALalgebroid is fixed, grad P (h) will be denoted by − → h . An integral curve of − → h is called a solution of the Hamiltonian system (E, {., .} P , h) Local expressions : In local trivialization (subsection 2.3), given a Hamiltonian h, in local coordinates (x, ξ) in E U , we denote by D 1 h and D 2 h the partial derivative according to the variable x and ξ respectively. The A-hamiltonian gradient of − → h has components − → h 1 and − → h 2 , on M and E * respectively. So we have the following characterization (see (10)): < df ′ , P ω(x) (dh) >=< df ′ , ρ(D 2 h) > < u ′ , P ω(x) (dh) >=< ξ, C x (D 2 h, u ′ ) > − < D 1 h, R x (u ′ ) >(21) So we get − → h 1 = ρ(D 2 h) < u ′ , − → h 2 >=< ξ, C(D 2 h, u ′ ) > − < D 1 h, R(u ′ ) >(22) When M and E have unconditional basis, we have: − → h = i∈I,α∈A ρ i α [ ∂h ∂ξ α ∂ ∂x i − ∂h ∂x i ∂ ∂ξ α ] − α,β,γ∈A C γ αβ ξ α ∂h ∂ξ β ∂ ∂ξ α So in local coordinates, the integral curves of − → h satisfies the following differential equations: x i = α∈A ρ i α ∂h ∂ξ αξ α = − i∈I ρ i α ∂h ∂x i − β,γ∈A C γ αβ ξ γ ∂h ∂ξ β(23) As in finite dimension (see for instance [LMM]) we have the following result on Hamilton-Jacobi equation Theorem 5.1 Let (E, {., .} P , h) be a Hamiltonian system. Given any section ω ∈ Γ * (τ ) we denote by − → h ω the vector field on M defined by − → h ω (x) = T ω(x) τ * ( − → h (ω(x)) Assume that d ρ ω = 0; then the following properties are equivalent: (i) If c : I → M is an integral curve of the vector field − → h ω , then ω • c : I → E * is a solution of the Hamiltonian system (E, {., .} P , h). (ii) ω satisfies the Hamilton-Jacobi equation i.e. d ρ (h • ω) = 0 Proof of Theorem 5.1 (adaption in our context of the proof of Theorem 4.1 of [LMM]) We will use the local trivializations (subsection 2.3). So ω(z) = (z, ξ(z)) where ξ is a smooth map from U to E * . In this context, L ω (x) = T x ω(ρ(E x )) ⊂ T ω(x) E * ≡ M × E * is the vector space {(ρ(x, v), Dξ(ρ(x, v)), v ∈ E} ⊂ M × E * . Denote by [L ω (x)] 0 ⊂ T ♭ ω(x) E * ≡ M * × E the vector space {η = d(Φ s + f • τ * (ω)) such that < η, α >= 0 for all α ∈ L ω (x)}. In our trivializations we have: [L ω (x)] 0 = {(df, v ′ ) such that < df, ρ(x, v) > + < Dξ(ρ(x, v)), v ′ >= 0 for all v ∈ E} Lemma 5.2 : 1. P ([L ω (x)] 0 ) = L ω (x) if and only if d ρ ω = 0 2. if d ρ ω = 0 then ker P ω (x) ⊂ [L ω (x)] 0 Proof of part 1 In the previous trivializations we have < (df ′ , v ′ ), (ρ(x, v), Dξ(ρ(x, v)) >=< df ′ , ρ(x, v) > + < Dξ(ρ(x, v)), v ′ >(24) So for any (df, v) ∈ [L ω (x)] 0 according to (10) and (24), we have the following equality for any (df ′ , v ′ ) ∈ T ♭ ω(x) E * < df ′ , ρ(x, v) > + < Dξ(ρ(x, v)), v ′ >=< ξ, [s, s ′ ] ρ > (x)+ < df ′ , ρ(x, v) > − < df, ρ(x, v ′ ) > (25) if and only if we have d ρ ω(s, s ′ )(x) = 0, using the expression (6), for any s ′ (z) = (z, v ′ (z)). The proof will be completed if we have (25), for any given local section s : s(z) = (z, u(z)), there exists a function f s such that (df s , v)(x) belongs to [L ω (x)] 0 . Indeed, fix such a section s. We definef s : ρ(E x ) ⊂ T x M ≡ M → R bỹ f s (ρ(x, u)) = − < Dξ(ρ(x, u)), v(x) > for any u ∈ E x . Sof s is a linear form on ρ(E x ). From Hahn-Banach theorem, there exists on E a continuous linear form f s such that f s =f s on ρ(E x ). So, it follows that we have < df s , ρ(x, u) >= − < Dξ(ρ(x, u)), v(x) > for all u ∈ E x i.e. (df s , v(x)) belongs to [L ω (x)] 0 . Proof of part 2 : Consider (df, v) ∈ ker P (ω(x)) ⊂ T ♭ ω(x) E * ≡ M * × E . So, for any (df ′ , v ′ ) ∈ T ♭ ω(x) E * ≡ M * × E, from (10) we have < ξ, [s, s ′ ] ρ > (x)+ < df ′ , ρ(x, v) > − < df, ρ(x, v ′ ) >= 0(26) Under the assumption d ρ ω = 0 the relation (26) is equivalent to: < Dξ(ρ(x, v)), v ′ > − < Dξ(ρ(x, v ′ )), v > + < df ′ , ρ(x, v) > − < df, ρ(x, v ′ ) >= 0(27) So for any given ( x, v ′ ) choose f ′ such that (df ′ , v ′ ) belongs to [L ω (x)] 0 . For this choice, we get < Dξ(ρ(x, v ′ ), v > + < df, ρ(x, v ′ ) >= 0. It follows that (df, v) belongs to [L ω (x)] 0 △ We come back to the proof of Theorem 5.1. The end of this proof follows exactly the same arguments as in Theorem 4.1 of [LMM] with some adaptations. First property (i) is clearly equivalent to (i') T ω( − → h ω (x)) = − → h (ω(x)) We begin by the implication (i')⇒ (ii). With the previous notations, h is a function of the variables (x, ξ) ∈ U × E * . Denote by D 2 h the partial derivative of h according to variable ξ. As dh belongs to T ♭ E * , the differential D 2 h gives rise to a section of E U . The component of − → h ω (x) = P ω(x )(dh) ∈ T ♭ σ(x) E * ≡ M × E * on M is characterized by (see (22)): − → h ω (x) = ρ(D 2 h)(x) . From assumption (i'), we then have − → h (ω(x)) ∈ L ω (x). From Lemma 5.2 part 1, there exists η ∈ [L ω (x)] 0 such that − → h (ω(x)) = P ω(x) (η) So (η − dh)(x) belongs to ker P ω(x) and, using Lemma 5.2 part 2 (dh)(x) also belongs to [L ω (x)] 0 . But < d(h • ω), ρ(s) >=< dh, T x ω • ρ(s) > for any s ∈ E x . As (dh)(x) belongs to [L ω (x)] 0 and T x ω • ρ(s) belongs to L ω (x) we get that < d(h • ω), ρ = 0 >. (ii)⇒ (i'). Under the assumption d ρ ω = 0, if we have < d(h • ω), ρ >= 0, as previously, we can show that (dh)(x) belongs to [L ω (x)] 0 , so − → h (ω(x)) = P ω(x) (dh), belongs to P ω(x) ([L ω (x)] 0 ). But, using Lemma 5.2 part 1, there exists s ∈ E x such that − → h (ω(x)) = T x ω(ρ(s))(28) So we obtain − → h ω (x) = T ω(x) τ * ( − → h (ω(x)) = T x (τ * • ω)(ρ(s)) = ρ(s) Using (28), we finally get T ω( − → h ω (x)) = − → h (ω(x)) △ Lagrangian and Euler-Lagrange equation on an AL-algebroid Given an anchored Banach bundle (E, τ, M, ρ), a semi spray S is a vector field on E such that (see [Ana]): p E • S = Id E where p E : T E → E is the canonical projection; T τ • S = ρ where T τ : T E → T M is the tangent map of τ . On the other hand, a C k -curve (k ≥ 1) c : [a, b] → E is called admissible, if we have T ρ(ċ(t) = ρ(c(t)) for all t ∈ [a, b]. According to [Ana], we have: Proposition 5.3 [Ana] A vector field S on E is a semi spray if and only if all integral curves of S are admissible curves. Among the class of semi sprays the subclass of sprays takes an important place for applications: if we denote by h λ : E → E, the homothety of factor λ > 0 (h λ (u) = λu for any u ∈ E x and any x ∈ M ) a semispray S is a spray if we have S • h λ = λT h λ • S. Local expressions: In local trivializations (subsection 2.3), a semispray can be written in the following way (see [Ana]): S(x, u) = (x, u, R x (u), −2G(x, u))(29) When M and E have unconditional basis, we have: S = α∈A [( i∈I ρ α i ∂ ∂x i ) − 2G α ∂ ∂u α ](30) A Lagrangian on a Banach bundle (E, τ, M ) is a smooth map L : E → R. We say that L is homogenous of degree k if we have: L • h λ = λ k L. The following result is classical (see for instance [AbMa] section 3.5) Lemma 5.4 Let L be a Lagrangian, we denote by L x the restriction of L to the fiber E x . Then the map Λ L : (x, u) → (x, dL x (u)) is a bundle morphism from E to E * We will say that L is regular (resp. strong regular) (resp. hyperregular) if Λ L is an injective morphism (resp. isomorphism) (resp. diffeomorphism). Notice that when L is strong regular then Λ L is a local diffeomorphism. So L is hyperregular if and only if L is strong regular and the restriction of Λ L to each fiber is injective. Denote by h λ : E → E, the homothety of factor λ > 0 (h λ (u) = λu for any u ∈ E x and any x ∈ M ). As in finite dimension, let Θ be the Liouville field on E which is the vector field whose flow is the homothety {h λ } λ∈R . In local trivializations we have Θ(x, u) = (x, u, 0, u) and when M and E have unconditional basis, we have Θ = α∈A u α ∂ ∂u α . We denote by H L the Lagrangian energy associated to L i.e. H L = dL(Θ) − L Given a regular Lagrangian L on an AL-algebroid (E, τ, M, ρ, [., .] ρ ), Λ L is a local diffeomorphism. So for any (x, u) ∈ E, there exists an open neighborhood U × V ⊂ E U of (x, u) such that (Λ L ) \|U×V is a diffeomorphism. Now suppose that E is reflexive. Consider a regular Lagrangian L on E and U × V ⊂ E an open set on which Λ L is a diffeomorphism. Let be the function h L = H L • Λ L −1 on Λ L (U × V ). Then dh L (σ) belongs to T * σ E * on Λ L (U × V ) On U × V we can define the vector field − → L characterized by (Λ L ) * ( − → H L ) = − → h L and which is called the local Euler-Lagrange vector field of L on U × V . In particular, when L is hyperregular, − → L is globally defined and called Euler-Lagrange vector field of L. Theorem 5.5 Consider a regular Lagrangian L on E. 1. Any curve c = (γ, µ) : [a, b] → U × V is an integral curve of the local Euler-Lagrange vector field of L if and only if it is a solution of the Euler-Lagrange equationṡ x = R(u) d dt (D 2 L) = R t (D 1 L) − D 2 L(C( , u)) (31) where R t : U → L(M, E * ) is the field x → (R x ) t of transpose of R x ∈ L(M, E) and where C( , u) denote, for a fixed u, the field of linear maps x → [v → C x (v, u)] (recall that x → C x is a field of bilinear maps) 2. If L is hyperegular, the Euler-Lagrange vector field − → L is a semi-spray. Moreover, if L is homogenous of degree 2 then − → L is a spray. Remark 5.6 If M and E have unconditional basis, then in the associated coordinate systems, the Euler-Equation can be written in the following way: x i = α∈A ρ i α u α d dt ( ∂L ∂u α ) = i∈I ρ i α ∂L ∂x i − β,γ∈A C γ αβ u β ∂L ∂u γ(32) for any i ∈ I. When A and I are finite sets of indexes, (32) is the classical Euler-Lagrange equation on the Lie algebroid (see for instance [GMM]) Proof We again adopt the notations in local trivializations (subsection 2.3) . So L is a function of variable (x, u) and Λ L is the map (x, u) → (x, D 2 L(x, u)). For simplicity, we will denote this map by Λ and for a fixed point (x, u) ∈ E we denote by (x, ξ) = Λ(x, u) The tangent map T Λ of Λ is: Id 0 D 12 L D 22 L (33) So [(T Λ) * ] −1 is Id −D 21 L • (D 22 L) −1 0 (D 22 L) −1(34) On the other hand, we have H L (x, u) = D 2 L(x, u)(u) − L(x, u) so wet get dH L (x, u) = (D 12 L(x, u)(u, ) − D 1 L(x, u)( ), D 22 L(x, u)(u, ))(35) So as dh L = [(T Λ) * ] −1 • dH L , from (34) and (35) we get (33) and (37) D 1 h L (x, ξ) = −D 1 L(x, ξ) and D 2 h L (x, ξ) = u(36)From (21) the AL Hamiltonian − → h L = ([ − → h L ] 1 , [ − → h L ] 2 is characterized by [ − → h L ] 1 (x, ξ) = R x (u), and , [ − → h L ] 2 = −D 2 L(x, ξ) • C x (u, ) + D 1 L(x, ξ) • ρ x (37) Now as − → H L = T Λ( − → h L ) fromwe get [ − → h L ] 1 (x, u) = R x (u) D 22 L([ − → h L ] 2 (x, u)) = −D 12 L(x, u) • R x (u) + D 1 L(x, u) • R x − D 2 L(x, u) • C x ( , u) We can easily see that these last equations are equivalent to the Euler-Lagrange equations △ Riemannian AL-algebroid and mechanical system Let (E, τ, M ) be a Banach bundle. Denote by S 2 T * M → M the Banach bundle of symmetric bilinear form on T M . Recall that a global section g of this bundle is called a riemannian metric on E, if for for any x ∈ M , the bilinear form g x on T x M is positive definite, i.e. g x (u, u) > 0 for any u = 0 To any riemannian metric g on E, is associated a bundle morphism g ♭ : E → E * defined by g ♭ (X)(Y ) = g(X, Y ). Of course, g ♭ is always injective. We say that g is a strong riemannian metric if g ♭ is surjective. Note that, in these conditions, the Banach space E is isomorphic to a Hilbert space, and so E must be reflexive. We will say that the AL-algebroid (E, τ, M, ρ, [., .] ρ ) is a riemannian AL-algebroid if there exists a riemannian metric g on E. In this situation, as in finite dimension, we can consider the Lagrangian system map L : E → R of a mechanical system on E given given by L(s) = 1 2 g(s, s) − V (τ (s))(38) where 1 2 g(s, s) is the "kinetic energy" and V : M → R the "potential energy" of the mechanical system. The associated Lagrangian energy is then: H L (x, u) = 1 2 g(s, s) + V (τ (s)). The Legendre transformation Λ L is g ♭ . Of course, the Lagrangian L is hyperregular, so the Lagrangian field L is well defined. Moreover if V ≡ 0 then L is a spray. Local expressions In local trivialization (cf subsection 2.3), we also denote by x → g x the field of symmetric bilinear maps on E associated to g and x → g ♭ x : E → E * the field of associated isomorphisms. With these notations, we have L(x, u) = 1 2 g x (u, u) − V (x). According to the local expression (29) we can write L(x, u) = (x, u, R(u), −2G(x, u)) where G is characterized by: g ♭ x (G(x, u)) = 1 2 < R t x • D 1 g ♭ x (u), u > − 1 2 < R t x • D 1 g x (u, u), > − < R t x • dV, > − < g ♭ x (u), C x .( , u); > (39) The Euler-Lagrange equation is given by ẋ = R(u) u = 1 2 (g ♭ x ) −1 < R t x • D 1 g ♭ x (u), u > − 1 2 < R t x • D 1 g x (u, u), > − < R t x • dV, . > − < g ♭ x (u), C x ( , u) >(40) In particular, when M and E have unconditional basis , the bilinear map can be written as a matrix g = (g αβ ) α,β∈A and we have according to (39) can be written β∈A g αβ G β = 1 2 β,γ∈A i∈I ∂g αβ ∂x i ρ i γ − 1 2 i∈I ∂g βγ ∂x i ρ i α − δ∈A C δ αβ g δγ u β u γ − i∈I ∂V ∂x i ρ i α(41) and, as in finite dimension, the Euler Lagrange equations have an analogue expression which is left to the reader. Consider a riemannian L-algebroid (E, τ, M, ρ, [., .] ρ ) and g its riemannian metric. If τ ′ : F → M is a Banach subbundle of τ : E → M , we can defined the complemeted Banach subbundle F ⊥ → M whose fiber is F ⊥ x is the orthogonal in E x of F x relatively to the metric g. Let be Π : E → F is the natural morphism projection, we can define an AL-bracket on the set of section of F by: [s 1 , s 2 ] ′ = Π[s 1 , s 2 ] ρ (see Example 3.2 n 0 1). So, if ρ ′ = ρ \|F , for the induced metric g ′ on F induced by g, (F, τ ′ , M, ρ ′ , [., .] ′ ) is a riemannian AL-algebroid. In this context, denote by i F : F → E the canonical inclusion; i * F : E * → F * the dual projection; Π * : E * → F * the dual morphism of Π; P : T * E * → T E * the P-morphism on E associated to [ , ] ρ P ′ : T * F * → T F the AP-morphism associated to [ , ] ′ . The Lagrangian L(x, u) = 1 2 g x (u, u) − V (x) on E induces a Lagrangian L ′ = L • i F on F which is a constrained Lagrangian on E. As in finite dimensional case (see [Marr]), we associate a mechanical system on the AL-algebroid (F, τ ′ , M, ρ ′ , [., .] ′ ) called constrained mechanical system on E which is obtain from the unconstrainted system associated to L on E with the following relations : the Legendre transformation Λ L ′ : T F → T * F satisfies Λ L ′ = i * F • Λ L • i F ; The hamiltonian h L ′ = H L ′ • (Λ L ′ ) −1 on F * is also given by h L ′ = h L • Π * ; The Lagrangian vector field L ′ associated to L ′ is also L ′ = T Π • L • i F . 6 Constrained mechanical system and Hilbert snakes 6.1 The context of an Hilbert snake We will now present the problem of the Hilbert snake and apply the previous results on Riemannian-AL agebroid. The reader can find a complete description of this situation in [PeSa]. In finite dimensional, a snake (of length L) is a (continuous) piecewise C 1 -curve S : [0, L] → R d , arc-length parameterized so that the origin S(0) = 0 ∈ R d . According to [Ro], "charming a snake" is a control problem so that its "head" S(L) describes a given C 1 -curve c : [0, 1] → R d in "minimal way" . More precisely we look for a one parameter family {S t } t∈[0,1] such that S t (L) = c(t) for all t ∈ [0, 1] and such that the family {S t } has an minimal infinitesimal kinematic energy. This problem has the precise following formulation: Each snake S of length L in R d can be given by a piecewise C 0 -curve u : [0, L] → S d−1 so that S(t) = t 0 u(τ )dτ . We look for a 1-parameter family {u t } t∈[0,1] so that the associated family S t of snakes satisfies S t (L) = c(t) for all t ∈ [0, 1] so that the infinitesimal kinematic energy 1 2 L 0 \|\| d dt u t (s)\|\|ds is minimal. A generalization of this problem in the context of an separable Hilbert space is developed in [PeSa]. More precisely, given a separable Hilbert space H we consider the smooth hypersurface S ∞ of element of norm 1. As previously, an Hilbert snake of length L is a continuous piecewise C 1 -curve S : [0, L] → H, arc-length parameterized so that S(0) = 0. Each such snake is again given by a piecewise C 0 -curve u : [0, L] → S ∞ so that S(t) = t 0 u(τ )dτ . Given a fixed partition P of [0, L], the set C L P of such curves will be called the configuration set and carries a natural structure of Banach manifold: when P = {0, L}, the set C L P is an hypersurface of the Banach C([0, L], H) of continuous map from [0, L] to H with the classical norm \|\| \|\| ∞ ; for the general case P = {0 = s 0 , · · · s N = L} then C L P is canonically homemophic to the product [C([0, L], S ∞ ]] N and so we put on C L P the corresponding Banach structure product. Notice that, on each tangent space T u C L P we also have an L 2 product : < v, w > L 2 = L 0 < v(s), w(s) > ds(42) Where < , > is the inner product in H. Of course for the associated norm \|\| \|\| L 2 , the normed space (T u C L P , \|\| \|\| L 2 ) is not complete. To any "configuration" u ∈ C L P is naturally associated the "end map" E(u) = L 0 u(s)ds. This map is smooth and its kernel has a canonical complemented subspace which is the orthogonal of ker T u E in T u C L P according to the inner product (42). We then get a closed distribution D on C L P . As in finite dimension, for a one parameter family {u t } t∈[0,1] the associated family S t of snakes satisfies S t (L) = c(t) for all t ∈ [0, 1] so that the infinitesimal kinematic energy 1 2 L 0 \|\| d dt u t (s)\|\|ds is minimal, if c(t) has a "lift "c in C L P which is tangent to D, called an "horizontal lift". So the problem for the head of the Hilbert snake to join an initial state x 0 to a final state x 0 can be transformed in the following "accessibiliy problem" : Given a initial (resp; final) configuration u 0 (resp. u 1 ) in C L P , so that E(u i ) = x i , i = 1, 2, find a piecewise C 1 horizontal curve γ : [0, T ] → C L P (i. e. γ is tangent to D) and which joins u 0 to u 1 . Given any configuration u ∈ C L P we look for the accessibility set A(u) of all configurations v ∈ C L P which can be joined from u by piecewise C 1 horizontal curves. In the context of finite dimension, in [Ro], using arguments about the action of the Moëbus group on C L P , it can be shown that A(u) is the maximal integral manifold of a finite dimensional distribution on C L P . Unfortunately, in the context of Hilbert space, the same argument does not work. Moreover, as we are in the context of infinite dimension for S ∞ , we cannot hope to get a finite dimensional distribution whose maximal integral manifolds is A(u). However, we can construct a canonical distributionD modelled on Hilbert space, which is integrable and so that the accessibility set A(u) is a dense subset of the maximal integral manifold through u ofD ( [PeSa]Theorem 4.1). Moreover this distribution is minimal in some natural sense. In fact, when H is finite dimensional,D is exactly the finite distribution in [Ro] whose leaves are the accessibility sets. AL -algebroid structure To any Hilbert basis {e i , i ∈ N}, we can associate a family of global vector fields {E i , i ∈ N} on C L P which generates D and the anounced distributionD is the Hilbert distribution generated by {E i , [E j , E l ], i, j, l ∈ N}. As E is not a submersion everywhere, it follows that D is not a subbundle of T C L P . On one hand if Λ = {(i, j) ∈ N 2 , i < j}, we can consider the Hilbert space G = l 2 (N) ⊕ l 2 (Λ) Now, given any Hilbert basis {e i , i ∈ N} of H we define an anchored bundle (C L P × G, ρ, C L P ) by ρ(σ, ξ) = i∈N σ i E i + (j,l)∈Λ ξ jl [E j , E l ] Of course ρ is well defined surjective and do not depend of the choice of the Hilbert basis. On the other hand, the Lie bracket of vector fields of the family {E i , [E j , E l ], i, j, l ∈ N} satisfies the following relations: (Lemma 4.3 [PeSa] ) [E i , E j ](u) =< e j , u > E i (u)− < e i , u > E j (u) for any u ∈ C L P and any i, j ∈ N; [E i [E j , E k ]] = δ ij E k − δ ik Ej for any i, j, k ∈ N [[E i , E j ], [E k , E l ]] = δ il [E j , E k ] + δ jk [E i , E l ] − δ ik [E j , E l ] − δ jl [E i , E k ] for any i, j, k, l ∈ N. So, on G we define a Lie algebra structure in the following way: let be (ǫ i ) i∈N (resp. (ǫ ij ) (i,j)∈Λ the canonical basis of (l 2 (N) (resp. (l 2 (Λ)); on G we define a Lie algebra structure in the following way: let be (ǫ i ) i∈N (resp. (ǫ ij ) (i,j)∈Λ the canonical basis of (l 2 (N) (resp. (l 2 (Λ)); according to the previous relations, we then define: [ǫ i , ǫ j ] = ω ij , for all i, j ∈ N [ǫ i , ω jk ] = δ ij ǫ k − δ ik ǫ j , for all i ∈ N and (j, k) ∈ Λ [ω ij , ω kl ] = δ il ω jk + δ jk ω il − δ ik ω jl − δ jl ω ik , for all (i, j)(kl) ∈ Λ. For any σ = σ i α i , σ ′ = σ ′ j ǫ j in l 2 (N) and ξ = ξ ij ω ij , η = η kl ω kl in l 2 (Λ), naturally we can define: [σ, σ ′ ] = i,j∈N σ i σ ′ j [ǫ i , ǫ j ] [σ, η] = i∈N,(k,l)∈Λ σ i η kl [ǫ i , ω kl ] [ξ, η] = (i,j)∈Λ,(k,l)∈Λ ξ ij η kl [ω ij , ω kl ]. Coming back to the anchored bundle (C L P × G, ρ, C L P ), each section ϕ of the trivial bundle C L P × G → C L P can be identified with a map ϕ : C L P → G. So, on the set Γ(G) of section of this trivial bundle we can defined a Lie bracket by: [ϕ, ϕ ′ ](u) = [ϕ(u), ϕ ′ (u)] + dϕ(ρ(u, ϕ ′ (u)) − dϕ ′ (ρu, ϕ(u)) It follows that (C L P × G, ρ, C L P , [ , ]) is a Banach Lie algebroid structure on C L P In G let be π : G → l 2 (N) the canonical projection whose kernel is l 2 (Λ) and denote again by π : C L P × G → C L P × l 2 (N) the associated projection bundle. Again any section of the trivial bundle C L P × l 2 (N) → C L P can be identified with a map from C L P to l 2 (N). Of course the set Γ(l 1 (N)) of such sections is contained in Γ(G). So, as in Example 3.2 1, on Γ(l 2 (N)), we can define an almost Banach Lie bracket by: [[ϕ, ϕ ′ ]](u) = π([ϕ, ϕ ′ ](u)). So, if we denote by θ the restriction of ρ to l 2 (N) × C L P we get an almost Banach Lie algebroid structure (C L P × l 2 (N), θ, C L P , [ , ]) on C L P . On the other hand, the distributionD is a weak Hilbert integrable distribution, on C L P , this means that, for any u ∈ C L P , there exists an Hilbert manifold N and a smooth injective map f : N → C L P such that (see [Pel]): u belongs to f (N ) , T x f : T x N → T f (x) C L P is injective , T x f (T x N ) =D f (x) for any x ∈ N We say that N is an integral manifold through u Given such integral manifold which maximal (for the inclusion), the pull back f * {C L P × G} and f * {C L P × l 2 (N)} can be identified with N × G and N × l 2 (N) respectively; Then, ρ (resp. θ) induces an anchor ρ N : N × G → T N (resp. Proposition 6.1 Let be N a maximal integral manifold ofD and fix some u ∈ N . Then we have the following properties 1. The set Σ(E) at which E : C L P → H is not a submersion is a weak manifold of C L P which is diffeomorphic to the projective space P ∞ of H. Its complementary R(E) is an open dense set of C L P . Moreover, N is a maximal integral manifold ofD. (1) Assume that u ∈ Σ(E) then N = Σ(E). Let be L v the 1-codimensional Hilbert subspace [ker θ N ] ⊥ v ⊂ {v} × l 2 (N) for any v ∈ N . Then L = ∪ v∈N L v is a 1-codimensional Hilbert subbundle of N × l 2 (N) and the restriction ψ N of θ N to L is an isomorphism onto T N and we have D \|N = T N . (2) Assume that u ∈ R(E). Then N is contained in R(E). Let be V u the Hilbert subspace of H generated by the set {u(t) − u(0), t ∈ [0, L]} and choose an Hilbert basis {e ′ a , a ∈ A} (resp e ′ b , b ∈ B}) of [V u ] ⊥ (resp. V u ). If Λ u is the set of pair (i, j) ∈ Λ such that that i or j do not belongs to A, then N is an Hilbert manifold modeled on l 2 (N) ⊕ l 2 (Λ u ) and is contained in R(E) . Let be L v the orthogonal of ker[ρ] v ⊂ {v} × G. Then L = ∪ v∈N L v is a Hilbert subbundle of N × G 2 which contains N × l 2 (N) and the restriction of ψ N of Ψ N to L is an isomorphism on T N Moreover, L contains N × l 2 (N ) and the restriction of θ N to N × l 2 (N ) is an isomorphism on D \|N So, on N , if we denote by [ , ] the usual Lie bracket and p N : T N → N the tangent bundle, we have the canonical L algebroid (T N, p N , N, Id N , [ , ]). When u ∈ R(E, on N , the distribution F = D \|N is an hilbert subbundle p ′ : F → N of p N : T N → N . When u ∈ Σ(E) we have D \|N = T N and so again we can consider p ′ : F = D \|N → N as an Hilbert subbundle of T N Constrained mechanical system Let be x and y two states of the head of the Hilbert snake which can be joined by an optimal curve (in the sense of subsection 6.1) and consider the set Ω(x, y) the set of optimal curves c which joins x to y. Classically if c ∈ Ω(x, y) is defined on [0, T ] its kinematic energy is E(c) = 1 2 T 0 \|\|ċ(t)\|\| 2 dt Let be E(x, y) = inf c∈Ω(x,y) E(c). Assume that there existsc ∈ Ω(x, y) such that E(c) = E(x, y), thenc will be called an optimal minimizing curve which joins x to y. Of course, such a curve is also an optimal minimizing curve between any pair of its points. So we can look for the existence of optimal minimizing curve which begin at a given original state x of the head of the considered snake. On one hand, each c ∈ Ω(x, y) has an horizontal lift γ in C L P , and γ lies in N As L → N is an Hilbert subbundle of N × G → N , we have a natural riemannian metric g on L. From Proposition 6.1, the isomorphism ψ N gives rise to a riemannian metric -again denoted by g -on T N and induces a riemannian metric g ′ on F = D \|N . Note that the inner product induces by g ′ on each fiber D u , u ∈ N , is exactly the inner product induced by ρ u : D u → T u H ≡ H of the canonical one on H. So we can defined -as in subsection 5.2 -an AL bracket [ , , ] ′ on F and we get an AL algebroid (F, p ′ , i F , N, [ , ] ′ ) Now, coming back to our problem of optimal minimizing curve on H. To each c ∈ Ω(x, y) has an horizontal lift γ in C L P , and γ lies in N for such a lift γ be an of some optimal curve c : [0, T ] → H, we have then E(c) = 1 2 T 0 \|\|γ\|\| 2 dt. Finally, It follows that ifc is an optimal minimizing curve which joins x and y, then the associated liftγ in N is an extremal of the Lagrangian L ′ : F → R: L ′ (v, σ) = 1 2 \|\|σ\|\| 2 on the AL algebroid (F, p ′ , i F , N, [ , ] ′ ). So we get a constrained mechanical system on the natural riemannian algebroid (T N, p, N, Id N , [ , ]). It follows that suchγ is a solution of the Euler-Lagrange equation of L ′ . We will now give the differential system satisfied by of such extremals in local coordinates. Fix some u ∈ R(E). According to Proposition 6.1 N is modelled on l 2 (N) ⊕ l 2 (Λ u ). So we have a local coordinates (σ, ξ) = {(σ i ) i∈ν , (ξ jl ) (j,l)∈Λu }. We denote by { ∂ ∂σ i , ∂ ∂ξ jl } the local Hilbert basis of T N associated to this coordinates system. With these notations, we have: E i = ∂ ∂σ i + (i,l)∈Λu σ l ∂ ∂ξ il − (l,i)∈Λu σ l ∂ ∂ξ li So from (40) and (41), the components (σ i , ξ jl ) of an extremal is given by σ i = 0, i ∈ N ξ jl =σ jσl , (j, l) ∈ Λ u Now, for u ∈ Σ(E), if we consider a basis (e i ) i∈N of H such that u = ±e 1 , then N is modelled on e ⊥ 1 (see Proposition 6.1). So, in local coordinates {σ i , i > 1} the component of an extremal satisfiesσ i = 0, i > 1 2. A Lie bracket (L-bracket for short) on an anchored bundle (E, τ, M, ρ) is an AL-bracket[., .] ρ which satisfies the Jacobi identity: for all s 1 , s 2 , s 3 ∈ Γ(τ ),J(s 1 , s 2 , s 3 ) = [s 1 , [s 2 , s 3 ]] + [s 2 , [s 3 , s 1 ]] + [s 3 , [s 1 , s 2 ]] = 0 In this case (E, τ, M, ρ, [., .] ρ ) is called a Lie Banach algebroid (L-algebroid for short) When (E, τ, M, ρ, [., .] ρ ) is a L-algebroid, then the bracket [., .] ρ induces on Γ(τ ) a Lie algebra structure. In this case ρ : (Γ(τ ), [., .] ρ ) → (Γ(M ), [., .]) is a Lie algebra morphism. Examples 3.2 1. Let (E, τ, M ) be a Banach subbundle of (T M, p M , M ) which is complemented, i.e. there exists a Banach subbundle (F, p, M ) of (T M, p M , M ) such that where[., .] is the usual Lie bracket on vector fields. Then(E, τ, M, ρ, [., .] E ) is an AL-algebroid where the anchor is the natural inclusion ρ of E in T M . Notice that (E, τ, M, ρ, [., .] E )is a L-algebroid if and only if (E, τ, M ) is involutive. In particular (T M, p M , M, Id, [., .]) is a L-algebroid. 2. Consider a smooth right action ψ : M × G → M of a connected Banach Lie group G over a Banach manifold M . Denote by G the Lie algebra of G. We have a natural morphism ρ from the trivial Banach bundle M × G into T M which is defined by ρ(x, X) = T (x,e) ψ(0, X) the restriction of q M : T * M → M . Then (T ♭ M, q ♭ M , M ) is a Banach subbundle of (T * M, q M , M ). Denote by Π : T ♭ M → T M the morphism (θ ♭ ) −1 . We define a structure of L-algebroid on the anchored bundle (T ♭ M, q ♭ M , M, Π) by setting [η, ζ] ♭ = θ ♭ ([Πη, Πζ]) where [., .] is the usual Lie bracket of the vector fields Πη and Πζ. So (T * M, q M , M, [., .] ♭ ) is a L-algebroid. Of course [Λ, Λ] = 0 if and only if [P, P ] = 0. Let τ : E → M be a Banach bundle and τ * : E * → M its dual bundle. Consider a localizable graded derivation δ of degree 1 of the graded algebra ΛΓ * (τ ). Assume that for any s 1 and s 2 in Γ(τ ), the bracket [[i s1 , δ], i s2 ] in restriction to Γ(τ * ) can be identified with a section of E ⊂ E * * → M Then δ defines a unique bracket [., .] δ on Γ(τ ) and there exists a unique morphism ρ : E → M such that:1. for any function f ∈ F and any section s ∈ Γ(τ ) we have ρ(s)(f ) =< δf, s >(19) θ N : N × l 2 (N) → T N ). The barcket [[ , ]] and(resp. the almost bracket [[ , ]]) induces a bracket (resp. an almost bracket) again denoted [[ , ]].So we get also a Banach Lie algebroid (N × G, ρ N , N, [[ ; ]]) and an almost Banach Lie algebroid structure (N × l 2 (N), θ N , N, [[ , ]]) on N . Now recall the following result of [PeSa] : for any open U in M . On the other hand, as δf is a 1-form on E, for any smooth function h defined on an open U we have: < δf \|U , hs >= h < δf \|U , s >; it follows that ρ(s) only depends on the value of s at each point. So, we get a bundle morphism from E to T M . Taking into account the definition of the map s → Φ s in subsection 4.1, the LHS of (20) is exactly Φ [s1,s2] δ (σ). From our assumption, there is a section which we can denote by [s 1 , s 2 ] δ of τ : E → M such that Φ [s1,s2] δ = [[i s1 , δ], i s2 (σ)]. As s → Φ s is injective, [s 1 , s 2 ] δ is well defined. Moreover, using again the injectivity of Φ the Leibniz property for). Moreover, (E, τ, M, [., .] δ ) is a L-algebroid if and only if δ 2 = 0 Proof. From (19) and our assumption, we get a linear map ρ : Γ(τ ) → Γ(M ) which gives rise to a linear map ρ U : Γ(τ U ) → Γ(U ) Notice that, according to the proof of Proposition 3.5 the RHS of (20) is exactly [[i s1 , δ], i s2 (σ)]. Almost Lie algebroid defined by a sub AP-morphism or an A-derivationThe notion of sub almost Poisson morphism is a generalization, in the context of Banach manifolds, of Poisson morphism or equivalently Lie Poisson structure (see[OdRa2]). 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[ "WEYL GROUP ACTION ON WEIGHT ZERO MIRKOVIĆ-VILONEN BASIS AND EQUIVARIANT MULTIPLICITIES", "WEYL GROUP ACTION ON WEIGHT ZERO MIRKOVIĆ-VILONEN BASIS AND EQUIVARIANT MULTIPLICITIES" ]	[ "Dinakar Muthiah " ]	[]	[]	We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities of Mirković-Vilonen cycles. We prove it for small coweights in type A. In this case, using work of Braverman, Gaitsgory and Vybornov, we show that the Mirković-Vilonen basis agrees with the Springer basis. We rephrase this in terms of equivariant multiplicities using work of Joseph and Hotta. We also have analogous results for Ginzburg's Lagrangian construction of sl n representations.	10.1016/j.aim.2021.107793	[ "https://arxiv.org/pdf/1811.04524v1.pdf" ]	119,695,397	1811.04524	20b98b79f39b6e0be6e5972dbc15990c53cb17b1	WEYL GROUP ACTION ON WEIGHT ZERO MIRKOVIĆ-VILONEN BASIS AND EQUIVARIANT MULTIPLICITIES 12 Nov 2018 Dinakar Muthiah WEYL GROUP ACTION ON WEIGHT ZERO MIRKOVIĆ-VILONEN BASIS AND EQUIVARIANT MULTIPLICITIES 12 Nov 2018 We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities of Mirković-Vilonen cycles. We prove it for small coweights in type A. In this case, using work of Braverman, Gaitsgory and Vybornov, we show that the Mirković-Vilonen basis agrees with the Springer basis. We rephrase this in terms of equivariant multiplicities using work of Joseph and Hotta. We also have analogous results for Ginzburg's Lagrangian construction of sl n representations. Introduction The celebrated Geometric Satake equivalence of Mirković and Vilonen [MV07] provides a geometric realization of the category of representations of a reductive group. Their work is a foundational result in the geometric Langlands program, and they work with arbitrary coefficients, so their work is intimately connected with modular representation theory. Finally, of most relevance to us, Mirković and Vilonen construct a basis in every irreducible representation (we work with Ccoefficients) indexed by the so-called Mirković-Vilonen (MV) cycles. The combinatorial study of such cycles via the theory of MV polytopes has progressed quite far (see e.g. [Kam10]), but understanding of the MV basis remains fairly murky (see [Bau12] for some general progress in this direction and [BGV07] for a type A description that plays a critical role in this paper). Even when compared with other geometric constructions of representations, many of which involve difficult intersection theory, the Geometric Satake equivalence is more mysterious. Unlike many other geometric constructions, the action is not via an explicit description of Chevalley generators. Rather it follows from abstract facts about Tannakian categories. Vasserot [Vas02] has succeeded in describing the action of the Chevalley generators in general, but even his description has some mystery: the E generators are constructed via a recipe involving the first Chern class of an ample line bundle, but the construction of the F-generators involves an appeal to the sl 2 -action of the Hard Lefschetz theorem. 1.0.1. Main conjecture and theorem. Let us briefly recall some details about the Mirković-Vilonen basis. Fix a complex reductive group G. For each dominant coweight λ of G (the coweight λ is a weight for the Langlands dual group G ∨ ), one can construct a variety Gr λ . Under the Geometric Satake equivalence the intersection homology of Gr λ is canonically identified with the irreducible module V(λ) of G ∨ with highest weight λ. Additionally, for any coweight µ (not necessarily dominant) Mirković and Vilonen consider an infinite-dimensional variety denoted S µ . They show that under Geometric Satake the µ-weight space V(λ) µ is canonically identified with the top dimensional compactly supported cohomology H top c (Gr λ ∩ S µ ). We will be interested in the case of µ = 0, which corresponds to the zero weight space V(λ) 0 . We have an action of the Weyl group W on this zero weight space, which is realized as H top c (Gr λ ∩ S 0 ). It will be more convenient to consider the W-action on the dual space H top (Gr λ ∩ S 0 ), which is the top-degree Borel-Moore homology. A basis of H top (Gr λ ∩S 0 ) is given by fundamental classes of the irreducible components of Gr λ ∩S 0 . The space Gr λ ∩ S 0 is invariant under a maximal torus T of G, and it contains a unique fixed point 0. Therefore, for each irreducible component Z it makes sense to speak about e T 0 (Z), which is the T -equivariant multiplicity of Z at 0. The T -equivariant multiplicity exactly gives the contribution at 0 when expressing the fundamental class [Z] using the localization theorem (see §2.3.2). We have e T 0 (Z) ∈ Frac(H • T (pt)), which is the fraction field of the T -equivariant cohomology of a point. Observe that the Weyl group W acts on Frac(H • T (pt)). A main goal of ours is to advance the following conjecture. for each Z ∈ Irr(Gr λ ∩ S 0 ) is W-equivariant. Our main theorem is the following. Theorem 1.4 (Theorem 7.12 in the main text). Conjecture 1.1 is true for G = SL d and λ dω 1 . 1.1. Description of our work. Our primary method for making contact with the MV basis is work [BGV07] of Braverman, Vybornov, and Gaitsgory. Their work, in type A, realizes the MV basis via an action on the top cohomology of big Spaltenstein varieties (see §2.2.2 for the terminology). Their construction proceeds by realizing Schur-Weyl duality and symmetric (GL n , GL n )-duality geometrically. This action on big Spaltenstein varieties was first considered in [BG99] by Braverman and Gaitsgory. We proceed by carefully analyzing this construction. 1.1.1. Warmup with Ginzburg's construction. The main result of the original work of Braverman and Gaitsgory [BG99] is an explicit realization of Ginzburg's Lagrangian construction of sl n representations on little Spaltenstein varieties (see §2.2.1 for the terminology). Specifically, it differs from the action on big Spaltenstein varieties by a sheaf-theoretic Fourier transform. The Braverman-Gaitsgory construction gives a sheaf-theoretic interpretation of the Ginzburg action (which is a priori convolution-theoretic) on little Spaltenstein varieties. We consider the case of small representations (for SL d , these are exactly those with highest weight dω 1 ) and their zero weight spaces. In this case, the relevant little Spaltenstein varieties are exactly Springer fibers. We show that the action on the zero weight space is exactly given by the Springer action (Theorem 4.36). We prove this by an explicit computation in equivariant Borel-Moore homology. 1.1.2. The main work. Our first task is to analyze the Braverman-Gaitsgory construction. Their construction is via sheaf theory and Schur-Weyl duality. We show that the construction can be realized by certain explicit symmetrization formulas, and we show (Theorem 5.64) that the Braverman-Gaitsgory action is realized via convolution by explicit correspondences. Our work described above provides a convolution-theoretic construction of the Braverman-Gaitsgory action on big Spaltenstein varieties (analogous to how Braverman and Gaitsgory provide a sheaf-theoretic construction of Ginzburg's convolution-theoretic action on little Spaltenstein varieties). We then focus on the case of small representations and their zero weight spaces. In this case, the relevant big Spaltenstein varieties are exactly Springer components (in terms of reduced scheme structure, big and little coincide in this case). We show that the action on the zero weight space is exactly given by the Springer action tensored with the sign character (Theorem 5.78). We prove this by an explicit computation in equivariant Borel-Moore homology, which is exactly analogous to the computation for little Spaltenstein varieties. Finally, to phrase our main result, we need to move from Springer components to the so-called orbital varieties. Joseph conjectured [Jos84] and Hotta proved [Hot84] that the Springer representation can be realized using orbital varieties and equivariant multiplicities. However, to use the Joseph-Hotta construction, we need to verify that two a priori different ways of identifying MV cycles and orbital varieties agree (one is due to Braverman, Vybornov, and Gaitsgory, and the comes from a classical construction of Lusztig). We show this is true by carefully unwinding the work of Braverman, Vybornov, and Gaitsgory and comparing it with the usual correspondence between Springer components and orbital varieties. This is the contents of §6 and specifically Theorem 6.28. 1.1.3. A remark about existing work comparing Geometric Satake and Springer theory. The relationship between weight-zero spaces of small representations and Springer theory has been studied [Ree98], and in particular the relationship between Springer theory and Geometric Satake has been studied by works of Achar, Henderson, and Riche [AH13,AHR15]. However, those works do not make any mention of how the Mirković-Vilonen basis or of how equivariant multiplicities fit into their story. Nonetheless, we suspect there should a way to see our results from their perspective. As they work with small representations in general, perhaps one can use their point of view to extend the main theorem to small representations in general. Finally, we should mention Mautner's work [Mau14], which geometrically realizes Schur-Weyl duality on the affine Grassmannian. This is implicit in the Braverman-Gaitsgory-Vybornov construction. 1.2. Acknowledgements. I am grateful to Joel Kamnitzer, Allen Knutson, Ivan Mirković, Alexei Oblomkov, and Alex Weekes for many enlightening conversations. I also thank Joel Kamnitzer, Eric Sommers, and Hiraku Nakajima for their comments on an earlier version of this manuscript. Notation and preliminaries Given vectors v 1 , . . . , v ℓ is a vector space, we will write v 1 , . . . , v ℓ for their span. Let Z be a complex algebraic variety, and let d 0 be an integer. We write Irr(Z) for the set of irreducible components of Z, and we write Irr d (Z) for the set of d-dimensional irreducible components of Z. We will write H • (Z) to denote the Borel-Moore homology of Z with complex coefficients. If d = dim Z, then H 2d (Z) is the top degree non-vanishing homology group, and it is canonical isomorphic to the formal span of Irr d (Z). We will write H top (Z) = H 2d (Z) in this situation. If we have an algebraic torus A acting on Z, we will also consider the equivariant Borel-Moore homology 2.1. Some type A representation theory. For any positive integer k, we will write [k] = {1, . . . , k}. We fix integers n 1 and d 1 throughout. Let: H A • (ZP n,d = {(d 1 , . . . , d n ) \| d i 0 and d 1 + · · · + d n = d} (2.1) We call P n,d the set of (n-step) compositions of d. We will write P ++ n,d ⊆ P n,d for the set of (d 1 , . . . , d n ) ∈ P n,d such that d 1 · · · d n . So P ++ n,d denotes the n-step partitions of d (with some parts allowed to be size zero). Recall the usual notation λ ⊢ d that denotes that λ is a partition of d (with no restriction on the number of parts); observe that equivalently we can write λ ∈ P ++ d,d using our notation. We will often consider the case n = d. For each d ∈ P n,d , we will form the subgroup S d = S d 1 × · · · × S d n consisting of all permutations of S d that preserve the discrete flag ∅ ⊆ [d 1 ] ⊆ [d 1 + d 2 ] ⊆ · · · ⊆ [d 1 + · · · + d n ] = [d]. 2.1.2. General/special linear groups. We will write G = GL d (C) for the general linear group, and we will write g d for the Lie algebra of G. We will also consider the special linear group SL d . Let v 1 , . . . , v d denote the standard basis for GL(C d ). This defines a maximal torus of G. Let T be corresponding maximal torus of SL d . We identify S d as the Weyl group of G and as the Weyl group of SL d . The distinction between these Weyl groups is slight, but it will be clear from context which copy of S d is relevant. For each d ∈ P n,d , consider the partial flag Notice that P 1 d is the standard Borel subgroup of G consisting of invertible upper triangular matrices. The Lie algebra p 1 d consists of upper triangular matrices, u 1 d are strictly upper triangular matrices, and u − 1 d are strictly lower triangular matrices. We will often write B = P 1 d , b = p 1 d , u = u 1 d , and u − = u − 1 d . Write E = C n , and let θ 1 , . . . , θ n denote the standard basis of E. For each i ∈ [n], let E i denote the span of θ i . Then we can write: ∅ ⊆ v 1 , . . . v d 1 ⊆ v 1 , . . . v d 1 +d 2 ⊆ · · · ⊆ v 1 , . . . v d 1 +···+d n = C d .E = E 1 ⊕ · · · ⊕ E n (2.2) Some constructions we consider will only depend on this decomposition into lines (and not the individual basis vectors), and this notation will help to indicate this. For each d = (d 1 , . . . , d n ) ∈ P n,d , let E d = E ⊗d 1 1 ⊗ · · · ⊗ E ⊗d n n (2.3) which is a distinguished line in E ⊗d . We will consider the general and special linear groups G = GL(E) = GL n and SL(E) = SL n along with their Lie algebras gl n and sl n . The decomposition (2.2) determines a maximal torus D of G. 2.1.3. Some basic facts about Schur-Weyl duality and irreducible modules. Consider the d-fold tensor product E ⊗d . We have a S d × GL(E) action on E ⊗d . For us Schur-Weyl duality is the statement that, as a S d × GL(E)-module, E ⊗d decomposes as E ⊗d = λ∈P ++ n,d S(λ t ) ⊗ V(λ) (2.4) where S(λ t ) is a Specht module, and V(λ) is an irreducible GL(E)-module. The decomposition implies that we can construct V(λ) as: V(λ) = Hom S d (S(λ t ), E ⊗d ) (2.5) Notice that this decomposition also holds for the SL(E), and we can explicitly identify the highest weights for each irreducible representation GL(E). Let λ = (λ 1 , · · · , λ n ) ∈ P ++ n,d , then we have V(λ) = V((λ 1 − λ 2 )ω 1 + · · · + (λ n−1 − λ n )ω n−1 ) as SL(E)-modules where {ω 1 , . . . , ω n−1 } are the fundamental weights for SL(E). We will abuse notation and also write λ for the SL(E)-weight (λ 1 − λ 2 )ω 1 + · · · + (λ n−1 − λ n )ω n−1 . 2.1.4. Weyl group action on weight zero spaces. For each a ∈ [n − 1], define: T a = exp(E a ) exp(−F a ) exp(E a ) ∈ SL d (2.6) where E a , F a ∈ sl n are the a-th simple Chevalley generators. These operators realize the simple reflections in the Weyl group S n = N(T )/T . In any integrable sl n -module, the right hand side of (??) acts the same way as a finite sum in the enveloping algebra U(sl n ) given by expanding the exponentials and truncating sufficiently high powers. The operators T a satisfy the braid relations, but do not square to the identity element. However, the operators T a do preserve the zero weight space of any integrable module and furthermore they do square to the identity as on the zero weight space. Therefore, we obtain explicit formulas for the Weyl group action on zero weight spaces. 2.1.5. Small representations. We will focus on the case of n = d. The dominant SL d weights satisfying λ dω 1 are exactly the dominant weights corresponding to λ ⊢ d. Following a standard terminology, we call corresponding representations V(λ) small representations of sl d . Looking at non-vanishing weight spaces, one immediately sees that T a ≡ 1 − E a F a − F a E a + 1 2 E a F 2 a E a (2.7) as operators on the zero weight space of small representations. One can further simplify formula (2.7) using sl 2 relations. Notice that E a F a and F a E a act the same way on the weight zero space. So we can write (2.7) as 1 − 2E a F a + 1 2 E a F a E a F a . We can further write the inner F a E a as E a F a + 2 because that operator is acting on a −2-weight space for the a-th root sl 2 . Finally noting that F 2 a acts by zero on this zero weight space, we conclude that: T a ≡ 1 − E a F a (2.8) The author thanks Joel Kamnitzer for alerting him to this simplification. 2.2. Some type A geometry. For each d = (d 1 , . . . , d n ) ∈ P n,d , we can form the partial flag variety: F d = F • = (F 1 , . . . , F n ) \| 0 = F 0 ⊆ F 1 ⊆ · · · ⊆ F n = C d where dim F i /F i−1 = d i (2.9) We define: F n,d = d∈P n,d F d (2.10) Notice that for F 1 d is exactly the complete flag variety consisting of complete flags in C d . We identify F d = G/P d (2.11) but note that there are d = d ′ with P d = P d ′ , so F d = F d ′ .N n,d = {x ∈ N d \| x n = 0} (2.12) Notice that N n,d = O (n,n,...,n,r) where d = kn + r for 0 r n − 1 and n occurs k times in (n, n, . . . , n, r). Define: N n,d = (x, F • ) ∈ N n,d × F n,d \| xF i ⊆ F i−1 for i ∈ [n] (2.13) Corresponding to (2.10), we have a connected component decomposition: N n,d = d∈P n,d N d (2.14) For each d ∈ P n,d , we have an identification: T * F d = N d (2.15) Under (2.11), we also can identify N d = G × P d u d (2.16) where the notation G × P d u d denotes quotient of G × u d under the diagonal action of P d (acting on the right for the G-factor). For each d ∈ P n,d , we have the natural map µ d : N d → N n,d . The map µ d is a resolution of singularities. Taking disjoint union, we obtain a map µ n,d : N n,d → N n,d . It is known that µ n,d is semi-small. Notice that N d,d = N d and that the map µ 1 d : N 1 d → N d (2.17) is usual Springer resolution. Fix d ∈ P n,d . For x ∈ O λ ⊆ N n,d , we define N x d = µ −1 d (xg n,d = (x, F • ) ∈ g d × F n,d \| xF i ⊆ F i for i ∈ [n] (2.18) Notice the difference from (2.13). Corresponding to (2.10), we have: g n,d = d∈P n,d g d (2.19) For d ∈ P n,d , we have g d = G × P d p d (2.20) from which we see that g d is connected and non-singular with dim g d = dim g d = d 2 . For each d ∈ P n,d , we have the natural map p d : g d → g d . Taking disjoint union, we obtain a map p n,d : g n,d → g d . It is known that p n,d is small. Notice that the map µ 1 d : g 1 d → g d (2.21) is usual Grothendieck-Springer alteration. Fix d ∈ P n,d . For x ∈ O λ ⊆ N d ⊆ g d , we define g x d = p −1 d (x). We call g x d a big Spaltenstein variety. It is known that dim g x d dim F d − 1 2 dim O λ . Unlike little Spaltenstein varieties, big Spaltenstein varieties need not be equidimensional. Observe that for x ∈ N d that the natural inclusion N x 1 d ֒→ g x 1 d induces an isomorphism at the level reduced schemes; in particular, they are homeomorphic in the analytic topology. Because we are only concerned with topological notions, we will consider all the spaces above with their induced reduced scheme structure. In particular, for us N x 1 d = g x 1 d . In this case, little and big Spaltenstein varieties coincide, and we will call them Springer fibers following the usual terminology. Springer's Weyl group action. Recall that there is an S d -action on H • (F 1 d ). There are two natural choices for such an action, which differ by tensoring with the sign representation. We fix the choice where H 0 (F 1 d ) is the trivial S d module. Fix λ ⊢ d and x ∈ O λ ⊆ N d . Let d λ = dim F 1 d − 1 2 dim O λ . Then we have that the Springer fiber N x 1 d has dimension d λ . We have a natural closed embedding N x 1 d ֒→ F 1 d that induces a map H 2d λ ( N x 1 d ) ֒→ H • (F 1 d ) (2.22) that is an injection [CG97, Theorem 6.5.2] (in fact, the entire homology H • ( N x 1 d ) embeds into H • (F 1 d ) in type A, but we will not need this harder fact). The subspace H 2d λ ( N x 1 d ) is preserved under the S d -action, and we have an isomorphism H 2d λ ( N x 1 d ) ∼ = S(λ) of S d -modules. As explained in loc. cit., this is exactly the action coming from Springer theory. We will call this the Springer action on H 2d λ ( N x 1 d ). Convolution in equivariant Borel-Moore homology. The main method that we will use to study representations is that of convolution in Borel-Moore homology as developed in [CG97, Chapter 2]. The method developed there does not explicitly involve equivariant Borel-Moore homology, but as usual equivariant Borel-Moore homology is constructed using finite-dimension approximations of the classifying space (see e.g. [Lus88,EG98]), so the constructions from the non-equivariant situation carry over. We will not recall the full details of convolution construction but will recall briefly how to compute using convolution in A-equivariant Borel-Moore homology where A be a linear algebraic torus, i.e. a finite product of G m 's. In our applications, we will have A = T × G m . Our primary computational tool will be the localization theorem. We note that we will only apply the localization theorem to algebraic cycles, so we will make use of Brion's [Bri97] formulas for localization in equivariant Chow groups, which upon applying cycle-class map, will give us the corresponding formulas in equivariant Borel-Moore homology. 2.3.1. Euler classes and equivariant multiplicities. Let Z be a variety with an A-action so that Z A is finite and that all fixed points z ∈ Z A are non-degenerate, which is to say that the zero weight does not appear in the tangent space T z Z. This non-degeneracy condition is immediately verified if we can A-equivariantly embed Z into a non-singular variety with finitely many A-fixed points; this will be the case for all varieties we consider. If z ∈ Z A is a non-singular point of of Z, then we will define the A-equivariant Euler class of Z at z, denoted eu z (Z), to be the product of the A-weights of the tangent space T z (Z) counted with multiplicity. A special case of this is an A-representation M with M A = 0. In this case, the Euler class eu 0 (M) is simply the product of the weights of M counted with multiplicity. In this case, we will drop the subscript 0 and simply write eu(M) for the Euler class of M at 0. If z ∈ Z A is a possibly singular point of Z, then there is a natural generalization of the Euler class called the A-equivariant multiplicity of z at Z denoted e A z (Z). We refer to [Bri97,§4.2] for the precise definition. This was also considered earlier by Joseph [Jos84] and Rossmann [Ros89]. The notion is closely related to the notion of multidegree in commutative algebra (see e.g. [?, ]); in particular, it is computable by the methods of computational commutative algebra. The relevant facts for us are the localization theorem (Theorem 2.25 below), which in fact characterizes the equivariant multiplicities, and the fact that if z ∈ Z A is a non-singular point of Z then e A z (Z) = 1 eu z (Z) . 2.3.2. Equivariant multiplicities and the localization theorem. Let S = H • T (pt) denote the Aequivariant cohomology of a point, and let Q denote the fraction field of S. Let Z be an A-variety as above with finitely many A-fixed points, all of which are non-degenerate. Let i : Z A ֒→ Z denote the closed embedding of the fixed points. We have a proper pushforward map H A • (Z A ) → H A • (Z) (2.23) that is an S-module map. Explicitly, H A • (Z A ) is a free S-module spanned by the equivariant fundamental classes [z] of fixed points z ∈ Z A . We can tensor up with Q to obtain a map: Q ⊗ S H A • (Z A ) → Q ⊗ S H A • (Z) (2.24) Let Y ⊆ Z be a A-invariant closed subvariety. Then we can consider the A-equivariant funda- mental class [Y] ∈ H A • (Z). We can also consider [Y] ∈ Q ⊗ S H A • (Z). Then we have the following formula (see e.g. [Bri97, Corollary 4.2]), which is the version of the localization theorem that is necessary for our purposes. Theorem 2.25. As classes in Q ⊗ S H A • (Z), we have: [Y] = z∈Z A e z (Y) · [z] (2.26) 2.3.3. Computing convolution at fixed points. Convolution involves three operations: smooth pullback, refined intersection, and proper pushforward. We will recall how these operations behave for fundamental classes of fixed points. Using (2.26), we will be able to calculate for more general A-invariant fundamental classes. Proper pushforward is easy to understand: the pushforward of a fixed point is a fixed point. Smooth pullback is also fairly straightforward: the smooth pullback of a fixed point is the fundamental class of the fiber, which is non-singular by assumption. Furthermore, these operations commute with the action of S. The most subtle part of convolution is the operation of refined intersection. Let X be a nonsingular A-variety; for simplicity, let us suppose X has finitely many A-fixed points. Let Z 1 , Z 2 ⊆ X be A-invariant closed subvarieties, and let Z 1 ∩ Z 2 be their intersection. Let m = dim X, and i, j ∈ Z. Then we can consider the refined intersection pairing ∩ : H i (Z 1 ) ⊗ H j (Z 2 ) → H i+j−2m (Z 1 ∩ Z 2 ) (2.27) relative to the ambient space X. We can also consider this A-equivariantly and at fixed points. The result is the following commutative diagram: H A i (Z A 1 ) ⊗ H A j (Z A 2 ) H A i (Z 1 ) ⊗ H A j (Z 2 ) H i (Z 1 ) ⊗ H j (Z 2 ) H A i+j−2m (Z A 1 ∩ Z A 2 ) H A i+j−2m (Z 1 ∩ Z 2 ) H i+j−2m (Z 1 ∩ Z 2 ) ∩ ∩ ∩ (2.28) All the maps in the left square commute with the action of H • A (pt), so to calculate H A i (Z A 1 ) ⊗ H A j (Z A 2 ) ∩ → H A i+j−2m (Z A 1 ∩ Z A 2 ) it suffices to calculate the intersection [z 1 ] ∩ [z 2 ] for z 1 ∈ Z A 1 and z 2 ∈ Z A 2 . This intersection is given by the excess intersection formula (see e.g [CG97, Proposition 2.6.47] and [Ful84, Corollary 6.3]): [z 1 ] ∩ [z 2 ] = eu z (X) · [z] if z = z 1 = z 2 0 if z 1 = z 2 (2.29) 3. Some geometric sl n -representations We will recall three geometric constructions of sl n -modules: Ginzburg's Lagrangian construction, Mirković and Vilonen's Geometric Satake equivalence, and Braverman and Gaitsgory's construction via Schur-Weyl duality and Springer theory. All three constructions give rise to geometrically defined bases. Savage [Sav06] has shown that Ginzburg's basis agrees with the basis arising in the quiver variety construction of Nakajima [Nak98]. Braverman, Gaitsgory, and Vybornov have shown that the Braverman-Gaitsgory basis coincides with the Mirković-Vilonen basis in type A. 3.1. Ginzburg's construction of sl n -representations. We recall Ginzburg's Lagrangian construction of sl n -representation ( [Gin91] and [CG97, Chapter 4]). For any pair d ′ , d ∈ P n,d , we can form the "Steinberg" variety N d ′ × N n,d N d , which is equal to the union of conormal bundles to the (diagonal) G-orbits on F d ′ × F d . Let Y d ′ ,d denote the unique minimal (and hence closed) G-orbit on F d ′ × F d , and let Z d ′ ,d denote the total space of the conormal bundle to Y d ′ ,d inside F d ′ × F d . Similarly to (2.11) and (2.16), we have Y d ′ ,d = G/(P d ′ ∩ P d ) (3.1) and: Z d ′ ,d = G × P d ′ ∩P d (u d ′ + u d ) (3.2) For any x ∈ N n,d , we have an action by convolution: [Z d ′ ,d ] ⋆ − : H • ( N x d ) → H •+2 dim N x d ′ −2 dim N x d ( N x d ′ ) (3.3) Notice that Z d ′ ,d is invariant under T × G m , where the G m factor acts by scaling cotangent fibers. Therefore, we can consider [Z d ′ ,d ] ∈ H T ×G m • ( N d ′ × N n,d N d ). In the special case of x = 0, N 0 d = F d and N x d ′ = F d ′ are T ×G m invariant, so we can consider the convolution action T ×G m -equivariantly [Z d ′ ,d ] ⋆ − : H T ×G m • (F d ) → H T ×G m •+2 dim F d ′ −2 dim F d (F d ′ ) (3.4) which recovers (3.3) by setting T × G m -equivariant parameters equal to 0. 3.1.1. Chevalley generators. Given d = (d 1 , . . . , d n ) ∈ P n,d and a ∈ [n − 1], we can form e a (d) = (d 1 , . . . , d a−1 , d a + 1, d a+1 − 1, d a+2 , . . . , d n ) if d a+1 1 ∇ otherwise (3.5) where ∇ is the "ghost" composition. Similarly, we form: f a (d) = (d 1 , . . . , d a−1 , d a − 1, d a+1 + 1, d a+2 , . . . , d n ) if d a 1 ∇ otherwise (3.6) For each a ∈ [n − 1], define E a = d∈P n,d : e a d =∇ [Z e a d,d ] (3.7) and F a = d∈P n,d : e a d =∇ (−1) sgn a (d) T * Y d − a ,d F d − a × F d (3.8) where for d = (d 1 , . . . , d k ), sgn a (d) = d a+1 − d a + 1. Remark 3.9. We refer to [Sav06, Remark 3.6] for an explanation of the sign in (3.8). There are many other possible sign choices that would work. For example, Vasserot [Vas93] picks signs in a way that correspond to orienting the fundamental class of cotangent bundles via their symplectic structure. This does not necessarily agree with the orientations coming from their complex structure, but it leads to nicer formulas. For any x ∈ N n,d , we consider E a and F a operating on d∈P n,d H • ( N x d ) by convolution; note that these operators do not preserve the grading. However, these operators do preserve the topdegree homology, and we have the following theorem of Ginzburg. Theorem 3.10 ([Gin91, CG97]). Let λ ∈ P ++ n,d , and let x ∈ O λ t ⊆ N n,d . The operators {E a , F a } a∈[n−1] define an sl n -action on d∈P n,d H • ( N x d ). As an sl n -module, d∈P n,d H top ( N x d ) is irreducible and is isomorphic to V(λ t ). Geometric Satake and the Mirković-Vilonen basis. Write O = C[[t]] and K = C((t)). Let G be a complex reductive algebraic group with a fixed pair of opposite Borels B and B − . Let U and U − be the unipotent radicals of the Borels, and let T = B ∩ B − be the corresponding maximal torus. We form the affine Grassmannian Gr = Gr G = G(K)/G(O), which has the structure of an projective ind-scheme of ind-finite-type. For each coweight µ of the torus T , there is a corresponding element µ ∈ Gr. The U(K)-orbits on Gr are indexed by coweights and are precisely the sets S µ = U(K)µ for coweights µ. The G(O)-orbits on Gr are indexed by dominant coweights λ and are precisely the sets Gr λ = G(O)λ for dominant coweights λ. 3.2.1. The Geometric Satake action and Mirković-Vilonen cycles. One considers the Satake category Sat consisting of G(O)-equivariant perverse sheaves (we will only consider sheaves with complex coefficients) on Gr G . The category Sat has a symmetric monoidal structure given by convolution of sheaves. Let G ∨ denote the Langlands dual group of G, and let Rep(G ∨ ) denote the category of finitedimensional G ∨ -representations. The celebrated Geometric Satake equivalence is the following explicit description of the Satake category. The Mirković-Vilonen proof of Geometric Satake provides finer information by geometrically realizing the decomposition of G ∨ -representations into weight spaces. Let λ be a dominant coweight and let µ be any coweight. The relevant fact for us is that there is a canonical isomorphism: V(λ) µ = H 2·ht(λ−µ) c (Gr λ ∩ S µ ) (3.13) Here ht(λ − µ) denotes the height of the coweight λ − µ, and H 2·ht(λ−µ) c denotes compactly supported cohomology. Furthermore, Mirković and Vilonen prove that Gr λ ∩ S µ is equidimensional of dimension ht(λ − µ). The elements of Irr(Gr λ ∩ S µ ) are called Mirković-Vilonen (MV) cycles (of weight (λ, µ)). We see therefore that the weight space V(λ) µ has a basis (the Mirković-Vilonen (MV) basis) indexed by the set of MV cycles of weight (λ, µ). Specifics in type A. We will be interested in the case of G = GL n and G = SL n . The group GL n is its own Langlands dual group, and there is a natural bijection between weights and coweights: both sets are naturally in bijection with Z n . For SL n , the Langlands dual group is PGL n , and there are more weights than coweights. We can identify the set of coweights with a subset of weights: the coweights are identified precisely with the root lattice. We will use these identifications freely. The affine Grassmannian Gr GL n is a disjoint union of connected components Gr d GL n . A partition λ ∈ P ++ n,d gives rise to a dominant coweight of λ of GL n . The corresponding point λ of the affine Grassmannian lies in Gr d GL n . Multiplication by the scalar matrix t induces isomorphisms Gr d GL n ∼ → Gr d+n GL n . The map t induces an auto-equivalence of the Satake category. Under Geometric Satake, this auto-equivalence corresponds precisely to tensoring with the determinant character of the Langlands dual group. Because the determinant is trivial on SL n (viewed as a subgroup of the Langlands dual group), the auto-equivalence commutes with the SL n -action coming from Geometric Satake. It also commutes with the Mirković-Vilonen construction of weight spaces for the maximal torus of SL n (but not for GL n ). The affine Grassmannian Gr SL n is connected, and the inclusion SL n ֒→ GL n induces a homeomorphism between Gr SL n and Gr 0 GL n . So we can realize the Satake category for SL n as a subcategory of the GL n Satake category. We will be primarily interested in the case when n = d, i.e. we consider partitions λ ∈ P ++ d,d . In this case, we will shift the emphasis of our notation slightly and write SL d instead of SL n (of course n = d in this case). Recall that we can consider partitions in P ++ d,d as weights for SL d . In this case the weights lie in the root lattice and are also coweights. The weights that appear are precisely the weights that are less than or equal to dω 1 in the dominance order. However, by our above discussion, such partitions naturally correspond to points in the d-th connected component Gr d GL d . Multiplication by t identifies Gr 0 GL d with Gr d GL d . So although we will primarily interested in the representations and MV basis arising in the SL d Satake category, we will sometimes equivalently work in Gr 0 GL d and Gr d GL d . 3.3. The Braverman-Gaitsgory action. Braverman, Gaitsgory, and Vybornov [BGV07] have shown that the Mirković-Vilonen basis in type A can be realized via an action of sl n on the cohomology of big Spaltenstein varieties. This action was first constructed by Braverman and Gaitsgory in [BG99]. 3.3.1. The construction. Let E be as in section 2.1, i.e. E is an n-dimensional vector space equipped with a decomposition E = E 1 ⊕ · · · ⊕ E n into lines. Recall that this decomposition specifies a maximal torus D in GL(E). Recall the map p n,d : g n,d → g d . Following Braverman and Gaitsgory [BG99], one defines a sheaf K n,d E on g n,d by declaring: K n,d E g d = E d [d 2 ] (3.14) Note that K n,d E carries an action of D by construction. The degree shift ensures that K n,d E is a perverse sheaf. We also consider the usual Grothendieck-Springer alteration p 1 d : g 1 d → g d , and we define the Grothendieck sheaf: Gro = (p 1 d ) * C g 1 d [d 2 ] (3.15) The Grothendieck sheaf carries an action of S d , so it makes sense to form (Gro ⊗ E ⊗d ) S d , which by construction will carry a GL(E)-action. Braverman and Gaitsgory prove the following. Theorem 3.16 ([BG99, §2.6]). There is a natural D-equivariant isomorphism: (Gro ⊗ E ⊗d ) S d ∼ → (p n,d ) * K n,d E (3.17) One can therefore transport the GL(E)-action to (p n,d ) ). Similarly because p n,d is a proper map, by taking * -stalks we have GL(E)-action on d∈P n,d H • ( g x d ). Again cohomological degree is preserved, and we have a GL(E)-action on: * K n,d E . Let λ ⊢ d, let x ∈ O λ ⊂ N d ⊂ g d ,d∈P n,d H 2d λ ( g x d ) (3.19) If λ ∈ P ++ n,d , this module is isomorphic to V(λ). Otherwise, this module is 0 (see e.g. [BM83, Corollary 3.5(a)]). Again, H 2d λ ( g x d ) has a basis indexed by Irr d λ ( g x d ). Braverman, Vybornov, and Gaitsgory have proved the following. Theorem 3.20. [BGV07] Under the GL(E)-isomorphism V(λ) = d∈P n,d H 2d λ ( g x d ) (3.21) the basis d∈P n,d Irr d λ ( g x d ) coincides with the Mirković-Vilonen basis. For our calculations, we will focus on the GL(E)-action (3.18) on Borel-Moore homology. In particular, we will show that the action of Chevalley generators is given by convolution by explicit correspondences. The action on cohomology will then be obtained via a straightforward duality procedure (see Proposition 5.22 below). Ginzburg's action and weight-zero spaces for small representations We will consider the case of n = d for the remainder of this section. Fix λ ∈ P ++ d,d , i.e., λ is a partition of d. Let x ∈ O λ t ⊂ N d . Recall Ginzburg's sl n -action on d∈P d,d H top ( N x d ) (4.1) that realizes the irreducible module V(λ). Recall the composition 1 d = (1, 1, . . . , 1) ∈ P d,d . The space H top ( N x 1 d ) is precisely the zero weight space of V(λ) as an sl n -module (warning: this is not true for the usual gl n -module structure where the center acts non-trivially). Therefore, the Weyl group S d acts on H top ( N x 1 d ). Notice however, that H top ( N x 1 d ) is exactly the top homology of a Springer fiber. Therefore, the Weyl group S d also acts H top ( N x 1 d ) via Springer theory. In this section we will compare these actions and show that they are equal. Remark 4.2. It is well known that Springer's Weyl group action H top ( N x 1 d ) and the Weyl group action on the zero weight space of V(λ) both realize the Specht module S(λ t ) (e.g. [Gut73] and [Kos76]). So by comparing the actions, we are essentially comparing the two natural bases. 4.1. Action on zero weight space. For each a ∈ [d − 1], recall that T a ≡ 1 − E a F a (4.3) as operators on H top ( N x 1 d ). 4.1.1. Computing the action of T a via convolution. We need to compute the operator: E a F a : H top ( N x 1 d ) → H top ( N x 1 d ) (4.4) The operator F a : H top ( N x 1 d ) → H top ( N x f a 1 d ) (4.5) is given by convolution by: −[Z f a 1 d ,1 d ] = x∈S d /(S fa1 d ∩S 1 d ) x 1 eu 0 (u − f a 1 d + u − 1 d )eu 0 ((u f a 1 d ∩ u 1 d ) ⊗ C h ) [xS f a 1 d , xS 1 d ] (4.6) Similarly, E a : H top ( N x f a 1 d ) → H top ( N x 1 d ) (4.7) is given by convolution by: [Z 1 d , f a 1 d ] = x∈S d /(S 1 d ∩S fa1 d ) x 1 eu 0 (u − 1 d + u − f a 1 d )eu 0 ((u 1 d ∩ u f a 1 d ) ⊗ C h ) [xS 1 d , xS f a 1 d ] (4.8) Consider the triple product N 1 d × N f a 1 d × N 1 d (4.9) We have projection maps p 12 : N 1 d × N f a 1 d × N 1 d → N 1 d × N f a 1 d (4.10) p 23 : N 1 d × N f a 1 d × N 1 d → N f a 1 d × N 1 d (4.11) p 13 : N 1 d × N f a 1 d × N 1 d → N 1 d × N 1 d (4.12) Let us compute the convolution: [Z 1 d , f a 1 d ] ⋆ [Z f a 1 d ,1 d ] = (p 13 ) * p * 12 ([Z 1 d , f a 1 d ]) ∩ p * 23 ([X f a 1 d ,1 d ]) (4.13) First compute p * 23 ([Z f a 1 d ,1 d ]) = w∈S d x∈S d /(S fa1 d ∩S 1 d ) (4.14) w 1 eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) x 1 eu 0 (u − f a 1 d + u − 1 d )eu 0 ((u f a 1 d ∩ u 1 d ) ⊗ C h ) [wS 1 d , xS e a 1 d , xS 1 d ] = (4.15) w∈S d x∈S d w 1 eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) x 1 eu 0 (u − 1 d )eu 0 (u f a 1 d ⊗ C h ) [wS 1 d , xS e a 1 d , xS 1 d ] (4.16) as u − 1 d = u − e a 1 d + u − 1 d , u f a 1 d = u f a 1 d ∩ u 1 d , and #S f a 1 d ∩ S 1 d = 1. Similarly, p * 12 ([Z 1 d , f a 1 d ]) = v∈S d y∈S d /(S fa1 d ∩S 1 d ) (4.17) y 1 eu 0 (u − f a 1 d + u − 1 d )eu 0 ((u f a 1 d ∩ u 1 d ) ⊗ C h ) v 1 eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) [yS 1 d , yS e a 1 d , vS 1 d ] = (4.18) v∈S d y∈S d y 1 eu 0 (u − 1 d )eu 0 (u f a 1 d ⊗ C h ) v 1 eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) [yS 1 d , yS e a 1 d , vS 1 d ] (4.19) When we compute the refined intersection p * 12 ([Z 1 d , f a 1 d ])∩p * 23 ([X f a 1 d ,1 d ]), we need w = y, x = v, and xS f a 1 d = yS f a 1 d . Because S f a 1 d = {1 , s a }, we must have y = xs a or x = y. So we have: p * 12 ([Z 1 d , f a 1 d ]) ∩ p * 23 ([Z f a 1 d ,1 d ]) = (4.20) x∈S d x eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) x    eu 0 (u − f a 1 d )eu 0 (u f a 1 d ⊗ C h ) eu 0 (u − 1 d )eu 0 (u f a 1 d ⊗ C h ) 2    × (4.21) x eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) [xS 1 d , xS f a 1 d , xS 1 d ] (4.22) + xs a eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) xs a eu 0 (u − f a 1 d )eu 0 (u f a 1 d ⊗ C h ) eu 0 (u − 1 d )eu 0 (u f a 1 d ⊗ C h ) × (4.23) x eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) eu 0 (u − 1 d ) 2 eu 0 (u f a 1 d ⊗ C h )eu 0 (u 1 d ⊗ C h ) [xs a S 1 d , xs a S f a 1 d , xS 1 d ] (4.24) Therefore: (p 13 ) * p * 12 ([Z 1 d , f a 1 d ]) ∩ p * 23 ([X f a 1 d ,1 d ]) = (4.25) x∈S d x eu 0 (u − f a 1 d ) eu 0 (u − 1 d ) x eu 0 (u − 1 d )eu 0 (u f a 1 d ⊗ C h ) [xS 1 d , xS 1 d ] (4.26) + xs a eu 0 (u − f a 1 d ) eu 0 (u − 1 d ) x 1 eu 0 (u − 1 d )eu 0 (u f a 1 d ⊗ C h ) [xs a S 1 d , xS 1 d ] = (4.27) x∈S d x α a + h −α a x 1 eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) [xS 1 d , xS 1 d ] (4.28) + x(α a + h) xs a (−α a ) x 1 eu 0 (u − 1 d )eu 0 (u 1 d ⊗ C h ) [xs a S 1 d , xS 1 d ] (4.29) For u ∈ S d , we therefore compute: where ∂ a is just notation for the usual Bernstein-Gelfand-Gelfand operator. [Z 1 d , f a 1 d ] ⋆ [Z f Specializing h = 0, we have: [Z 1 d , f a 1 d ] ⋆ [Z f a 1 d ,1 d ] ⋆ [u]\| h=0 = (s a − 1)[u] (4.34) Therefore we have computed E a F a ≡ 1 − s a , and we can calculate the Weyl group action to be: T a ≡ s a (4.35) We know that this action on H • (F 1 d ) restricts to the Springer action on H top ( N x 1 d ), therefore we have proved the following. 5.1.1. Invariants vs. coinvariants. Let G be a group, and let V be a G-representation. Then we can form the G-invariants V G , which comes via a canonical map V G ֒→ V. Similarly, we have the Gcoinvariants V G , which we can realize as the quotient of V by the span of {gv − v \| g ∈ G and v ∈ V}. We always have the canonical map V G → V G defined as the composition: V G ֒→ V ։ V G (5.1) For us G is a finite group, and we work over a field where #G is invertible (namely C). In this case, the map V G → V G is an isomorphism, and we have explicit inverse V G → V G given by: [v] → 1 #G g∈G gv (5.2) In the discussion below, the more natural object will sometimes be coinvariants, and we will use map (5.2) to canonically identification coinvariants with invariants, 5.1.2. Schur-Weyl duality and taking invariants. Let V be an S d -module. Consider the S d × GL(E)-module V ⊗ E ⊗d . We will explicitly describe a T -equivariant isomorphism: d∈P n,d V S d ⊗ E d = (V ⊗ E ⊗d ) S d (5.3) To describe this map, we need to define for each d ∈ P n,d and injective map V S d ⊗ E d ֒→ (V ⊗ E ⊗d ) S d (5.4) Note that we have a natural injection V S d ⊗ E d ֒→ V ⊗ E d ֒→ V ⊗ E ⊗d . But the image of this map does not lie in the S d -invariants. Instead we will further map to the coinvariants (V ⊗ E ⊗ ) S d and then map to the invariants (V ⊗ E ⊗ ) S d via the canonical section (5.2). Thus we have defined an explicit map realizing (5.3). The action of Chevalley generators. Observer that GL(E) acts on the right hand side of (5.3). Therefore, we can transport this action to the left hand side of (5.3), and below we will write formulas for the action of Chevalley generators. For each i ∈ [n], choose a basis vector θ i ∈ E i . For each d ∈ P n,d , let θ d be the corresponding basis vector in E d . We need to make this choice to define Chevalley generators. For each a ∈ [n−1], we define operators E a and F a by E a (θ a+1 ) = θ a (5.5) E a (θ i ) = 0 otherwise (5.6) and: F a (θ a ) = θ a+1 (5.7) F a (θ i ) = 0 otherwise (5.8) Let d ∈ P n,d , and consider the case when e a (d) = ∇. The operator E a acts sending the dweight space of (V ⊗ E ⊗d ) S d to its e a (d)-weight space. Transporting structure via (5.3), we get an operator E a : V S d ⊗ E d → V S e a d ⊗ E e a d . Because we have trivialized the lines E d and E e a d , we therefore get an operator E a : V S d → V S e a d . Similarly, we can construct operators F a . The following is an explicit description of these operators. Proposition 5.9. Let d = (d 1 , . . . , d n ) ∈ P n,d , and consider the case when e a d = ∇. The operator E a : V S d → V S e a d is given by the symmetrization operator d a+1 · Ψ e a d,d : V S d → V S e a d (5.10) where Ψ e a d,d defined by: Ψ e a d,d (v) = 1 #S e a d w∈S ead w(v) (5.11) In the case when f a d = ∇, the operator F a : (d 1 + · · · + d a + 1, d 1 + · · · + d a + j)θ e a d (5.13) V S d → V S f a d is given by d a · Ψ f a d, where (d 1 + · · · + d a + 1, d 1 + · · · + d a + j) is the transposition. So E a v corresponds to: 1 #S d x∈S d d a+1 j=1 x(v) ⊗ x((d 1 + · · · + d a + 1, d 1 + · · · + d a + j)θ e a d ) (5.14) Note that for each j, the transposition (d 1 + · · · + d a + 1, d 1 + · · · + d a + j) ∈ S d . So we can rewrite as: 1 #S d x∈S d d a+1 j=1 x((d 1 + · · · + d a + 1, d 1 + · · · + d a + j)v) ⊗ x((d 1 + · · · + d a + 1, d 1 + · · · + d a + j)θ e a d ) (5.15) = d a+1 #S d x∈S d x(v) ⊗ x(θ e a d ) (5.16) We can write: d a+1 #S d x∈S d x(v) ⊗ x(θ e a d ) = d a+1 #S e a d #S d x∈S d w∈S ead x(w(v)) ⊗ x(w(θ e a d )) (5.17) Because θ e a d is S e a d -invariant, this is equal to: as an element of V S ead . d a+1 #S d x∈S d x(Ψ e a d,d (v)) ⊗ x(θ e a d )( 5.1.4. Dual construction. Above we started with an S d -module V and produced a GL(E) module. We can also consider the dual S d -module V * and run the above procedure to obtain an GL(E)module. As S d -modules are self-dual, we know that the resulting module is the same as an abstract GL(E)-module. We will describe explicitly the Chevalley generators. For each d ∈ P ++ n,d , the d-weight space is the invariant space (V * ) S d , which is canonically isomorphic to the dual of the coinvariant space (V S d ) * . Using (5.2), we can identify this with the dual of the invariant space (V S d ) * . Therefore, we obtain Chevalley generators for each a ∈ [n − 1] E a : (V S d ) * → (V S ead ) * (5.20) and F a : (V S d ) * → (V S fad ) * (5.21) under the appropriate assumptions that e a d = ∇ and f a d = ∇. Unwinding Proposition 5.9 and using (5.2) to identify invariants and coinvariants, we obtain the following proposition. Proposition 5.22. Let a ∈ [n − 1] and d = (d 1 , . . . , d n ) ∈ P ++ n,d . In the case when e a d = ∇, the Chevalley generator E a (5.20) is the adjoint of the map: d a+1 d a + 1 F a : V S ead → V S d (5.23) When f a d = ∇, the Chevalley generator F a (5.21) is the adjoint of the map: d a d a+1 + 1 E a : V S fad → V S d (5.24) 5.2. Realizing Chevalley generators in the Braverman-Gaitsgory construction. Recall that a key step in the Braverman-Gaitsgory construction is Theorem 3.16, which we recall is the isomorphism: (Gro ⊗ E ⊗d ) S d ∼ → (p n,d ) * K n,d E (5.25) Roughly speaking, (5.25) is a sheaf-theoretic version of (5.3). Below we will explicitly describe this isomorphism in detail. 5.2.1. Explicit description of (5.25). Write π d,1 d : g 1 d → g d (5.26) for the natural map. Notice that π d,1 d is proper. Recall that K n,d E g d = E d [d 2 ] (5.27) by definition. Because both g 1 d and g d are both smooth of dimension d 2 , we have: E d [d 2 ] = π ! d,1 d E d [d 2 ] (5.28) By the (π d,1 d ) ! , π ! d,1 d -adjunction, we obtain a map (π d,1 d ) ! E d [d 2 ] → E d [d 2 ] (5.29) of sheaves on g d . Because π d,1 d is proper, we equivalently have a map: (π d,1 d ) * E d [d 2 ] → E d [d 2 ] (5.30) Applying (p d ) * , we obtain a map Gro ⊗ E d → (p d ) * E d [d 2 ] (5.31) which on fibers exactly corresponds to the proper pushforward in Borel-Moore homology from the fibers of p 1 d to the fibers of p d . Summing over d ∈ P n,d , we obtain a map: d∈P n,d Gro ⊗ E d → (p n,d ) * K n,d E (5.32) Notice that for each d, we have the map: Gro S d ⊗ E d → Gro ⊗ E d (5.33) which sums to a map: d∈P n,d Gro S d ⊗ E d → d∈P n,d Gro ⊗ E d (5.34) We can consider the composed map: d∈P n,d Gro S d ⊗ E d → d∈P n,d Gro ⊗ E d → (p n,d ) * K n,d E (5.35) By the smallness of the maps p n,d and the p 1 d , all the sheaves in (5.35) are Goresky-MacPherson extensions of their restrictions to the open set g rs d of regular semisimple elements. On the regular semi-simple locus, the composed map (5.35) is an isomorphism by the discussion in [BG99, §2.6]. Therefore, by the Perverse Continuation Principle, the composed map (5.35) is an isomorphism on all of g d . Recall, that by (5.4), we have a canonical isomorphism: d∈P n,d Gro S d ⊗ E d ∼ → (Gro ⊗ E ⊗d ) S d (5.36) Noting that (Gro ⊗ E ⊗d ) S d has a GL(E)-action, we can transport structure to obtain a GL(E)action on d∈P n,d ( g x 1 d ) to H • ( g x d ) . Therefore, two maps in (5.35), induce the maps: H • ( g x 1 d ) S d ֒→ H • ( g x 1 d ) → H • ( g x d ) (5.37) As the composed map in (5.35) is an isomorphism, the composed map H • ( g x 1 d ) S d ∼ → H • ( g x d ) (5.38) is an isomorphism. Summing over all d and tensoring with E d we have an isomorphism: d∈P n,d H • ( g x 1 d ) S d ⊗ E d ∼ → d∈P n,d H • ( g x d ) ⊗ E d (5.39) The subrepresentation d∈P n,d H 2d λ ( g x 1 d ) S d ⊗ E d ∼ → d∈P n,d H 2d λ ( g x d ) ⊗ E d (5.40) is isomorphic to the irreducible GL(E)-module V(λ). The d λ -dimensional irreducible components of g x d form a basis of H 2d λ ( g x d ) . Note that g 0 1 d = F 1 d , and we have an inclusion g x 1 d ֒→ F 1 d . By the argument in the proof of Theorem 6.5.2(b) in Chriss-Ginzburg, the map H • ( g x 1 d ) → H • (F 1 d ) is S d -equivariant. Therefore, the map d∈P n,d H • ( g x 1 d ) S d ⊗ E d → d∈P n,d H • (F 1 d ) S d ⊗ E d (5.41) is GL(E)-equivariant. Similarly, the map d∈P n,d H • ( g x d ) ⊗ E d → d∈P n,d H • (F d ) ⊗ E d (5.42) is GL(E)-equivariant. By the irreducibility of V(λ), the map d∈P n,d H 2d λ ( g x d ) ⊗ E d ֒→ d∈P n,d H • (F d ) ⊗ E d (5.43) is injective (see the proof of Theorem 6.5.2(a) in Chriss-Ginzburg). We summarize this discussion as the following proposition. 1). Then all the maps in the following commutative square are GL(E)-equivariant. Proposition 5.44. Let λ ⊢ d, and let x ∈ O λ . Let d λ = dim F 1 d − 1 2 dim O λ = λ i (i −d∈P n,d H 2d λ ( g x 1 d ) S d ⊗ E d d∈P n,d H • (F 1 d ) S d ⊗ E d d∈P n,d H 2d λ ( g x d ) ⊗ E d d∈P n,d H • (F d ) ⊗ E d ∼ ∼ (5.45) The horizontal maps are inclusions, and the vertical maps are isomorphisms. Chevalley generators. We follow the discussion and notation in section 5.1. We have our basis element θ d ∈ E d for each d ∈ P n,d . Let d ∈ P n,d . Consider a ∈ [n − 1] so that e a d = ∇. Then we have the operator E a = d a+1 Ψ e a d,d : H • (F 1 d ) S d → H • (F 1 d ) S ead . We have a commutative diagram: H • (F 1 d ) S d H • (F 1 d ) S ead H • (F d ) H • (F e a d ) ∼ E a ∼ E a (5.46) The S d -action on H • (F 1 d ) is given by restricting the S d -action on H T • (F 1 d ), so formula (5.11) defining Ψ e a d,d has a natural equivariant lift to an operator: Ψ e a d,d : H • (F 1 d ) S d → H • (F 1 d ) S ead (5.47) Therefore the commutative square (5.46) has a natural T -equivariant lift: H T • (F 1 d ) S d H T • (F 1 d ) S ead H T • (F d ) H T • (F e a d ) ∼ E a ∼ E a (5.48) The S d -action on H T • (F 1 d ) commutes with the action of H T • (pt), and we therefore conclude that all the maps in (5.48) are equivariant for the action of H T • (pt). We will compute the how the operators E a act on torus fixed points, which by the localization formula, will determine the operators. Let w ∈ S d , and let 1 #S d x∈S d [wx] ∈ H • (F 1 d ) S d .Z d ′ ,d N d ′ × N n,d N d Y d ′ ,d F d ′ × F d (5.58) Similarly, on the " g"-side, we can form the "Steinberg" variety g d ′ × g d g d . Analagous to Z e a d,d , we can form the subvariety X d ′ ,d that is defined as the following fiber product: X d ′ ,d g d ′ × g d g d Y d ′ ,d F d ′ × F d (5.59) We have Y d ′ ,d = G/P d ′ ∩ P d (5.60) and: Z d ′ ,d = G × P d ′ ∩P d (u d ∩ u d ′ ) (5.61) Similarly, we have: X d ′ ,d = G × P d ′ ∩P d (p d ∩ p d ′ ) (5.62) From this, we compute [X d ′ ,d ] = x∈S d /(S d ′ ∩S d ) x 1 eu 0 (u − d ′ + u − d )eu 0 ((p d ′ ∩ p d ) ⊗ C h ) [xS d ′ , xS d ] (5.63) as classes in localized equivariant homology. Theorem 5.64. Let d ∈ P n,d , and let c ∈ H • (F d ). Let a ∈ [n − 1]. When e a d = ∇, we have: [X e a d,d ] ⋆ c = E a (c) (5.65) When f a d = ∇, we have: [X f a d,d ] ⋆ c = F a (c) (5.66) Proof. We will prove (5.65). Equation (5.66) is similar. We will show that (??) holds Tequivariantly. Initially, we will work T × G m -equivariantly. By the localization, it suffices to consider the case when c = [wS d ] is a torus-fixed point. We compute: 5.3. Action on weight zero spaces for small representations. We will consider the analogue of the situation in § 4 for the Braverman-Gaitsgory action. Namely we will consider the case of n = d for the remainder of this section. Fix λ ⊢ d. Let x ∈ O λ ⊂ N d . We have an sl d -action on [X e a d,d ] ⋆ [wS d ] = y∈S d /S ead∩S d wy eu 0 (u − d )eu 0 (p d ⊗ C h ) eu 0 (u − e a d + u − d )eu 0 ((p e a d ∩ p d ) ⊗ C h ) [d∈P d,d H 2d λ ( g x d ) (5.73) that realizes the irreducible module V(λ). The space H 2d λ ( g x 1 d ) is precisely the zero weight space for the sl d -module. Recall that the operator T a acts as 1 − E a F a . The following computation is a straightforward variation of the computation in §4.1.1, which we will omit to save space. Proof. The statement in Borel-Moore homology is exactly what we have shown in the discussion above. To obtain the statement in cohomology, we use Proposition 5.22 to compute that the Weyl group action on H 2d λ ( g x 1 d ) is exactly the dual action. Recall also that there is SL d -equivariant embedding j : N d ֒→ Gr (6.1) originally due to Lusztig [Lus81] and defined as follows. Given x ∈ N d , we can form the element 1 − t −1 x ∈ SL d [t −1 ] , which we map to Gr by acting on the unit point. The image of (6.1) is precisely Gr nω 1 ∩ Gr 0 . Under (6.1), the image of O λ is Gr λ ∩ Gr 0 , and the image of u ⊂ N d is S 0 . Therefore, j induces an isomorphism: j λ : O λ ∩ u ∼ → Gr λ ∩ S 0 (6.2) Therefore a fortiori we have a bijection j λ : Irr(O λ ∩ u) ∼ → Irr(Gr λ ∩ S 0 ) (6.3) The irreducible components of O λ ∩ u are usually called orbital varieties. Remark 6.4. Usually, one defines orbital varieties as closures in u of irreducible components of O λ ∩u. However, using the map j λ , we can identify these with irreducible components of Gr λ ∩ S 0 . The work of Mirković and Vilonen shows that the irreducible components of Gr λ ∩ S 0 coincide with the irreducible components of Gr λ ∩ S 0 . Applying j −1 λ , we see that orbital varieties are precisely the irreducible components of O λ ∩ u. We do not know if this holds more generally in other types (where the comparison map j λ to the affine Grassmannian does not exist). 6.1.1. Identifying orbital varieties with Springer components. Let us briefly recall how one identifies orbital varieties with Springer components. Recall that we have the Springer resolution µ 1 d : N 1 d → N d . Let x ∈ O λ . Because the centralizer of x in G = GL d is connected, we have a natural bijection between Irr( N 1 d ) and Irr((µ 1 d ) −1 (O λ )). One can also realize (µ 1 d ) −1 (O λ ) = G × B (O λ ∩ u) (6.5) Again we have a natural bijection between Irr(G × B (O λ ∩ u)) and Irr(O λ ∩ u). Finally, we have a bijection Irr(O λ ∩ u) between Irr(O λ ∩ u). To summarize, we have constructed a bijection s λ : Irr( N 1 d ) ∼ → Irr(O λ ∩ u) (6.6) 6.2. The Braverman-Gaitsgory-Vybornov construction and the second identification of MV cycles and orbital varieties. In [BGV07], Braverman, Gaitsgory, and Vybornov (to be referred to as "the authors" for the remainder of this section for brevity) construct a bijection between MV cycles and big Spaltenstein components. In the special case of λ ∈ P ++ d,d , the big Spaltenstein components are usual Springer components. Combining this with the bijection s λ between Springer components and orbital varieties, we obtain another bijection: β λ : Irr(O λ ∩ u) ∼ → Irr(Gr λ ∩ S 0 ) (6.7) We will show that this bijection agrees with the bijection j λ constructed above. To aid the reader, we will following the notation in [BGV07, § §1,2] closely. We will refer the reader to their paper for background on the lattice model of GL n affine Grassmannians. The authors define E and V are two distinct copies of C n that we canonically identify. The two vector spaces play different roles hence the reason for distinguishing them. Unfortunately, the vector space called V in [BGV07] corresponds to the vector space E in [BG99]. This corresponds to the vector space we have been calling E elsewhere in the paper. We will be interested in the case of d = n and µ = 1 d = (1, . . . , 1) according to their notation. given by sending a nilpotent matrix x ∈ N d to the lattice: (1 − xt −1 ) −1 t −1 M 0 (6.9) Recall that because x is a nilpotent d × d matrix, we have: (1 − xt −1 ) −1 = 1 + t −1 x + · · · + t −(n−1) x n−1 (6.10) There is also the space Conv j E : N 1 d ֒→ Conv 1 d ,− (Gr E ) (6.11) sending a pair (x, 0 = F 0 ⊂ F 1 ⊂ · · · ⊂ F d = E) of nilpotent operator x and an invariant flag F 0 ⊂ F 1 ⊂ · · · ⊂ F d = E to the sequence of lattices (M 1 , . . . , M d ) defined by: M i = M 0 ⊕ (1 − t −1 x) −1 t −1 F i (6.12) The authors construct a Cartesian square of stacks Conv 1 d ,− (Gr E ) N 1 d /GL d Gr d,− E N d /GL d (6.13) that they use to compare the Geometric Satake action to the Braverman-Gaitsgory action we have considered above. This can be extended to the following two Cartesian squares N 1 d Conv 1 d ,− (Gr E ) N 1 d /GL d N d Gr d,− E N d /GL d j E j E (6.14) where the composed horizontal maps are the tautological quotient maps. We have projections π E : P loc → Gr d,− E and π V : P loc → Gr d,+ V . We define the space N loc to be the set of triples (x, y, g) where x, y ∈ N d , g ∈ GL d , and y = gxg −1 . Then we have the projections π 1 , π 2 : N loc → N d . Notice also that we have N loc ∼ = GL d ×u under the map (x, y, g) → (y, g). We define a map b : N loc → P loc (6.18) by sending (x, y, g) to (j E (x), j V (y), α) where α is the isomorphism α : (1 − t −1 x) −1 t −1 M 0 /M 0 ∼ → M ′ 0 /(1 − t −1 y)tM ′ 0 (6.19) defined by α((1 − t −1 x) −1 t −1 v) = gv (6.20) for all v ∈ M 0 . Here we identify M 0 and M ′ 0 . Then it is easy to see that b is an open embedding and that the following diagram is Cartesian: N loc P loc N d × N d Gr d,− E × Gr d,+ V π 1 ×π 2 b π E ×π V j E ×j V (6.21) 6.3. Identifying MV cycles with Springer components. Let P loc (1 d ) = π −1 v (∩S 1 d ). Note that j −1 V (S 1 d ) = u. Furthermore, the authors construct a factorization of the map π E : P loc (1 d ) → Gr d,− E by a map: . Then diagram (6.23) restricts to a diagram: π Conv E : P loc (1 d ) → Conv 1 d ,− (Gr E ) (P λ loc (1 d ) Gr λ V ∩ S 1 d Conv 1 d ,− (Gr E ) λ,• (6.25) The authors show that the arrows in (6.25) are smooth with all fibers non-empty. Therefore, the set of irreducible components for each of the three spaces in (6.25) are identified. The irreducible components of Gr λ V ∩ S 1 d are exactly MV cycles of weight (λ, 1 d ), and the irreducible components of Conv 1 d ,− (Gr E ) λ,• are identified with components of a Springer fiber of type λ. Combining this with the map α λ identifying Springer components with orbital varieties and the map identifying Gr d V with Gr SL d , we obtain a bijection: Corresponding to the isomorphism between (6.23) and (6.24), the restriction of (6.25) is G × (O λ ∩ u) O λ ∩ u G × B (O λ ∩ u) (6.27) Notice that we therefore obtain a bijection between Irr(G × B (O λ ∩ u)) and Irr(O λ ∩ u), which is exactly the same as the bijection s λ considered above. Again using the comparison between Gr d V and Gr SL d , we have proved the following. Theorem 6.28. The two bijections j λ and β λ comparing Irr(Gr λ ∩ S 0 ) and Irr(O λ ∩ u) agree. The Joseph-Hotta construction of Springer representations and a conjecture about Weyl group actions in general Because H top c (Gr λ ∩ S 0 ) is identified with V(λ) 0 under Geometric Satake, we obtain a S d -action on H top c (Gr λ ∩ S 0 ). On the other hand we have a bijection between Gr λ ∩ S 0 and and the irreducible components of any Springer fiber N x 1 d where x ∈ O λ . A priori there are two such bijections, but by Theorem 6.28 the two bijections agree. Therefore we also have a S d -action on H top c (Gr λ ∩ S 0 ) coming from Springer theory. By Theorem 5.78 these two actions differ by tensoring with the sign representation of S d . Joseph conjectured [Jos84], and Hotta [Hot84] later proved that one can construct the Springer action directly on the span of orbital varieties in terms of equivariant multiplicities. The data of these equivariant multiplicities is recorded by the so called Joseph polynomials. This will allow us to rephrase Theorem 5.78 in a way that makes sense for arbitrary types and arbitrary dominant coweights λ. Therefore we can state a general conjecture that our work shows is true in type A and for λ dω 1 . 7.1. Phrasing Joseph-Hotta construction. We will phrase Joseph-Hotta construction in a way that is will make sense for MV cycles in general. Recall that the Borel-Moore homology H top (O λ ∩u) is the vector space dual of H top c (O λ ∩u). Furthermore, H top (O λ ∩u) has a basis indexed by orbital varieties Irr(O λ ∩ u). Each orbital variety is T -invariant and therefore has an equivariant fundamental class in H T top (O λ ∩ u). We have a proper pushforward map H T • (0) → H T • (Gr λ ∩ S 0 ) (7.1) that becomes an isomorphism after tensoring with the fraction field of H • T (pt). In particular, for each irreducible component Z ∈ Irr(O λ ∩ u), we can consider the T -equivariant multiplicity e T 0 (Z) of Z at the fixed point 0. We refer the reader to the book of Borho, Brylinski, and MacPherson [BBM89], especially chapter 4, for a thorough description of this story. Remark 7.6. Because u is smooth, one can write e T 0 (Z) = J Z eu 0 (u) (7.7) where J Z ∈ H • T (pt). In commutative algebra language, the polynomial J Z is essentially the Tequivariant multidegree of Z embedded in u. In this specific setting it is called the Joseph polynomial of the orbital variety Z. Usually the above theorem is phrased in terms of Joseph polynomials. As eu 0 (u) transforms under the sign character, the analogue of Theorem 7.2 stated with Joseph polynomials in place of equivariant multiplicities does not require tensoring with sign representation. 7.2. Rephrasing Theorem 5.78 in terms of equivariant multiplicities. Let G be a reductive group as in §. Let λ be a dominant coweight that lies in the coroot lattice. We can consider the representation V(λ) of the dual group G ∨ and the zero weight space V(λ) 0 . The Weyl groups of G and G ∨ are canonically identified; write W for this group. Then we have the identification: V(λ) 0 ∼ = H top c (Gr λ ∩ S 0 ) (7.8) Therefore, we have a W-action on H top c (Gr λ ∩ S 0 ). Taking vector space dual we obtain a W-action on H top (Gr λ ∩ S 0 ). We can now state the main conjecture of this paper. Conjecture 7.9. The map H top (Gr λ ∩ S 0 ) → Frac(H • T (pt)) (7.10) defined by [Z] → e T 0 (Z) (7.11) for each MV cycle Z and extending linearly is W-equivariant. Combining Theorem 5.78 and Theorem 7.2, we obtain the following. Theorem 7.12. Conjecture 7.9 is true for SL d and λ dω 1 . Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan E-mail address: [email protected] Conjecture 1 . 1 ( 11Conjecture 7.9 in the main text). The mapH top (Gr λ ∩ S 0 ) → Frac(H • T (pt)) Theorem 3.11 ([Gin95,MV07]). There is an equivalence of symmetric monoidal categoriesSat ∼ → Rep(G ∨ ) (3.12)such that the forgetful functor Rep(G ∨ ) → Vect corresponds to the hypercohomology functor H • : Sat → Vect. Theorem 4 . 36 . 436Let λ ⊢ d, x ∈ O λ ⊂ N d .Then the Weyl group action on H top ( N x 1 d ) arising from the Ginzburg construction agrees with the Weyl group action coming from Springer theory. Remark 4 . 37 . 437Consider the usual Steinberg variety Z = N 1 d × N d N 1 d . The irreducible components of Z are naturally indexed by the Weyl group S d ; for each w ∈ S d , let Z w be the corresponding component. The above calculation shows that[Z s a ] = [Z 1 d , f a 1 d ] ⋆ [Z f a 1 d ,1 d ](4.38) as classes in H T ×G m • (Z). Formula (4.25) shows that the classes agree after tensoring with the fraction field of H T ×G m • (pt), but by (4.30) we can conclude that they agree prior to tensoring because the action of H T ×G m • (Z) on H T ×G m • (F 1 d ) is known to be faithful. 5. Further analysis of the Braverman-Gaitsgory action 5.1. Some preliminaries on Schur-Weyl duality. 6 . 6Mirković-Vilonen cycles and orbital varieties 6.1. Lusztig's map and the first bijection between Mirković-Vilonen cycles and orbital varieties. In this section we will use the result of Braverman, Gaitsgory, and Vybornov (Theorem 3.20) and our Theorem 5.78 to compare Weyl group actions on MV cycles and orbital varieties. Let us write Gr for the affine Grassmannian of SL d . Let Gr 0 be the big cell of SL d , that is, the SL d [t −1 ]-orbit of the unit point. Let λ dω 1 be a dominant weight. Recall that we can consider λ as a partition of d. Then we have the corresponding SL d (O)-orbit closure Gr λ and the U(K)orbit S 0 . Following Mirković and Vilonen, we can identify the compactly supported cohomology H top c (Gr λ ∩ S 0 ) with identified with the 0-weight space of V(λ). 6. 2 . 1 . 21Gr E . The Gr E is the affine Grassmannian for GL(E), and Gr d,− E consists of lattices M containing the standard lattice M 0 with dim M/M 0 = d. We have Gr −dω 1 E where −dω 1 = (−d, 0, . . . , 0). Then we have an open embedding j E : N d ֒→ Gr d, 1 d ,− (Gr E ) which consists of sequences of lattices (M 1 , . . . , M d ) with M i−1 ⊂ M i and dim M i /M i−1 = 1. We have an open embedding (after taking reduced scheme structures) 6. 2 . 2 . 22Gr V .The Gr V is the affine Grassmannian for GL(V), and Gr d,+ V consists of lattices M ′ contained in the standard lattice M ′ 0 with dim M ′ 0 /M ′ = d. We have Gr d,+ V = Gr dω 1 V where dω 1 = (d,0, . . . , 0). Then we have an open embedding j V : N d ֒→ Gr d V (6.15) given by sending a nilpotent matrix x ∈ N d to the lattice: . P loc . The authors define P loc to be the space of triples (M, M ′ , α) where M ∈ Gr d,− E , M ′ ∈ Gr d,+ V , and α is a C[[t]]-equivariant isomorphism: β λ : Irr(O λ ∩ u) ∼ → Irr(Gr λ ∩ S 0 ) (6.26) ). This equivariant setting will be discussed in more detail below. Similarly, let us write H • (Z) for the cohomology of Z and H• c (Z) for the compactly supported cohomology. Recall that H • c (Z) is canonically the dual vector space of H • (Z). Therefore, the highest degree non-vanishing cohomology H top c (Z) is identified with the space of complex-valued functions on the set of top-dimensional irreducible components of Z. In particular, H top c (Z) has a basis indexed by the top-dimensional irreducible components of Z. Let us write 1 d = (1, . . . , 1) ∈ P ++ d,d for the d-step composition of d consisting of all 1's. 2.1.1. Symmetric groups. Let S d denote the set of permutations of the set [d]. Let λ ⊢ d, and let S(λ) denote the corresponding Specht module of S d . Over C, the set {S(λ) \| λ ⊢ d} is an enumeration of the irreducible representations of S d . Let P d be the parabolic subgroup of G consisting of all elements that preserve this flag. The Weyl group of the Levi factor of P d is naturally identified with S d . Write p d for the Lie algebra of P d , and write u d for the Lie algebra of the unipotent radical of P d . Write u − d for the image of u d under the Chevalley involution, so g d = p d ⊕ u − d . Similarly, write p − d for the image of p d under the Chevalley involution. and let i x : {x} ֒→ g d denote the inclusion of x. Using the basis of E, we can identify the !-stalk i ! x (p n,d ) * K n,dE with the Borel-Moore homology d∈P n,d H • ( g x d ). By (3.17) we have a GL(E)-action on d∈P n,d H • ( g x d ). Because g n,d is equidimensional, this action preserves homological degrees. In particular, d∈P n,d H 2d λ ( g x d ) (3.18) caries a GL(E)-action. Because dim g x d d λ , H 2d λ ( g x d ) has a basis indexed by Irr d λ ( g x d d . dProof. We do the case of E a . The case of F a is essentially the same. Let v ∈ V S d .Under (5.3), this maps to the S d -invariant: 1 #S d x∈S d x(v) ⊗ x(θ d ) (5.12) Writing out d = (d 1 , . . . , d n ), we calculate that E a (θ d ) = d a+1 j=1 This is given by the usual Springer S d -action on H • ( g x 1 d ) (see e.g. [Jan04, Ch. 13]). Recall that the map (5.32) corresponds to the proper pushforward from H •Gro S d ⊗ E d 5.2.2. The GL(E)-action on stalks. Let λ ⊢ d. Let O λ ⊂ g d be the corresponding nilpotent orbit and let x ∈ O λ . The !-stalks of Gro and (p d ) * E d [d 2 ] are equal to the Borel-Moore homologies H • ( g x 1 d ) and H • ( g x d ) respectively. The S d -action on Gro gives rise to an S d -action H • ( g x 1 d ). Under proper pushforward to H • (F d ): Remark 5.57. Analagous formulas for the action of Chevalley generators on the cohomology of little Spaltenstein varieties have been given by Brundan, Ostrik, and Vasserot [Bru08, BO11, Vas93]. 5.2.4. Chevalley generators as correspondences. Let d, d ′ ∈ P n,d . Recall that we can form the "Steinberg" variety N d ′ × N n,d N d . The component Z d ′ ,d , which we initially defined as the conormal bundle of the subvariety Y d ′ ,d ⊆ F d ′ × F d , can also be defined as the following fiber product:1 #S d x∈S d [wx] → [wS d ] (5.49) Under E a : H • (F 1 d ) S d → H • (F 1 d ) S ead , by Proposition 5.9: 1 #S d x∈S d [wx] → d a+1 #S d #S e a d x∈S d y∈S ead [wxy] (5.50) Under proper pushforward to H • (F e a d ): d a+1 #S d #S e a d x∈S d y∈S ead [wxy] → d a+1 #S d x∈S d [wxS e a d ] (5.51) Therefore, we have the following. Proposition 5.52. Let d ∈ P n,d , and let wS d ∈ S d /S d . Let a ∈ [n − 1]. When e a d = ∇, the operator E a : H • (F d ) → H • (F e a d ) (5.53) satisfies: E a ([wS d ]) = d a+1 #S d x∈S d [wxS e a d ] (5.54) Similarly, when f a d = ∇, the operator F a : H • (F d ) → H • (F f a d ) (5.55) satisfies: F a ([wS d ]) = d a #S d x∈S d [wxS f a d ] (5.56) Proposition 5.74. Let a ∈ [d−1], as operators on H T ×G m• ( g 0 1 d ) convolution by [X 1 d , e a 1 d ]⋆[X e a 1 d ,1 d ] is equal to: s a + 1 + h∂ a (5.75) Further specializing h = 0 and restricting attention to the top homology of g x 1 d we have E a F a = s a + 1 (5.76) as operators on H 2d λ ( g x 1 d ) We therefore calculate: T a ≡ −s a (5.77) Theorem 5.78. Fix λ ⊢ d. Let x ∈ O λ ⊂ N d . The Braverman-Gaitsgory Weyl group actions on the zero weight spaces H 2d λ ( g x 1 d ) and H 2d λ ( g x 1 d ) are equal to the Springer actions tensored with the sign representation. 6.22)One can check that the image of π Conv E is contained in the image of j E . Let us write Conv 1 d ,− (Gr E ) • for the image.We have a diagram:P loc (1 d ) Gr d,+ V ∩ S 1 d Conv 1 d ,− (Gr E ) • Let N loc (1 d ). Under the isomorphism N loc ∼ = GL d × u, we have N loc (1 d ) ∼ = GL d × u.One can check that diagram (6.23) is isomorphic to the diagram via the obvious isomorphisms.Let λ ∈ P ++ d,d . Then define P λ loc (1 d ) = π −1 E (Gr λ V ∩ S 1 d . Let Conv 1 d ,− (Gr E ) λ,• be the image of P λ loc (1 d ) under the map π Conv(6.23) G × u u G × B u (6.24) E Theorem 7.2 ([Hot84]). The map H top (O λ ∩ u) → H • for each orbital variety Z and extending linearly is injective. Furthermore, its image is a Weyl group invariant subspace of H • T (pt). x ∈ O λ . Under the bijection s λ : Irr(O λ ∩ u) the Weyl group action induced by (7.3) action coincides with the Springer action on H top ( N x 1 d ) tensored with the sign character.T (pt) (7.3) defined by [Z] → e T 0 (Z) (7.4) Let ∼ → Irr( N x 1 d ) (7.5) Geometric Satake, Springer correspondence and small representations. N Pramod, Anthony Achar, Henderson, MR 3131493Selecta Math. (N.S.). 194Pramod N. Achar and Anthony Henderson, Geometric Satake, Springer correspondence and small repre- sentations, Selecta Math. (N.S.) 19 (2013), no. 4, 949-986. 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[ "OPERADIC CATEGORIES AND DÉCALAGE", "OPERADIC CATEGORIES AND DÉCALAGE", "OPERADIC CATEGORIES AND DÉCALAGE", "OPERADIC CATEGORIES AND DÉCALAGE" ]	[ "Richard Garner ", "Joachim Kock ", "Mark Weber ", "Richard Garner ", "Joachim Kock ", "Mark Weber " ]	[]	[]	Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres"-also objects of the same category-subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the décalage comonad D on small categories. A simple case involves unary operadic categoriesones wherein each map has exactly one abstract fibre-which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monadD induced on Cat D by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a "modified décalage" comonad Dm on the arrow category Cat 2 .	10.1016/j.aim.2020.107440	[ "https://arxiv.org/pdf/1812.01750v1.pdf" ]	119,309,458	1812.01750	015913212fd1e81b285b95b9e78f74cbc42ba697	OPERADIC CATEGORIES AND DÉCALAGE 4 Dec 2018 Richard Garner Joachim Kock Mark Weber OPERADIC CATEGORIES AND DÉCALAGE 4 Dec 2018 Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres"-also objects of the same category-subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the décalage comonad D on small categories. A simple case involves unary operadic categoriesones wherein each map has exactly one abstract fibre-which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monadD induced on Cat D by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a "modified décalage" comonad Dm on the arrow category Cat 2 . . Introduction Operads originated in algebraic topology, first appearing in Boardman and Vogt [ ] under the name "category of operators in standard form", with the modern name and modern definition being provided by May in [ ]. They quickly caught on, with applications subsequently being found not only in topology, but also in algebra, geometry, physics and beyond; see [ ] for an overview. As the use of operads has grown, it has proven useful to recast the definition: rather than explicitly listing the data and axioms, one may re-express them in various more abstract ways [ , , ], each of which points towards a range of practically useful generalisations of the original notion. This has led to a rich profusion of operad-like structures, and various authors have proposed unifying frameworks to bring some order to this proliferation. One such framework is that of operadic categories [ ], introduced by Batanin and Markl to specify certain kinds of generalised operad necessary for their proof of the duoidal Deligne conjecture. An operadic category is a combinatorial object which specifies a flavour of operad; an "algebra" for an operadic category is an operad of that flavour. Such an operad will, in turn, have its own algebras, but this extra layer will not concern us here. As the name suggests, operadic categories are categories, but endowed with extra structure of a somewhat delicate nature. This structure seems to invite attempts at reconfiguration, so as better to link it to other parts of the mathematical landscape. One such reconfiguration was given by Lack [ ], who drew a tight correspondence between operadic categories and the skew-monoidal categories of Szlachányi [ ], which in recent years have figured prominently in categorical quantum algebra and work of the Australian school of category theorists. The present paper gives another reconfiguration of the definition of operadic category, which links it to the (upper) décalage construction. While primarily an operation on simplicial sets, décalage may also-via the nerve functor-be seen as an operation on categories; namely, that which takes a category to the disjoint union of its slices: D(C) = X∈C C/X . There are two main aspects to the tight relationship between operadic categories and décalage. To explain these, we must first recall the data for an operadic category. These are: a small category C with a chosen terminal object in each connected component; a cardinality functor \|-\| : C → S into the category of finite ordinals and arbitrary mappings; and an operation assigning to every f : Y → X in C and i ∈ \|X\| an "abstract fibre" f −1 (i) ∈ C, functorially in Y . The first connection between operadic categories and décalage arises from the fact that the décalage construction on categories underlies a comonad D on Cat, whose coalgebras may be identified, as in Proposition below, with categories endowed with a choice of terminal object in each connected component. In particular, each operadic category is a coalgebra for the décalage comonad. The second connection arises through the functorial assignation of abstract fibres f → f −1 (i) in an operadic category. Functoriality says that, for fixed X ∈ C and i ∈ \|X\|, this assignation is the action on objects of a functor ϕ X,i : C/X → C, so that the totality of the abstract fibres can be expressed via a single functor ( . ) ϕ : X∈C,i∈\|X\| C/X → C . The domain of this functor is clearly related to the décalage of C, and in due course, we will explain it in terms of a modified décalage construction on categories endowed with a functor to S. However, there is a special case where no modification is necessary. We call an operadic category unary if each \|X\| is a singleton; in this case, the domain of ( . ) is precisely the décalage D(C), so that the fibres of a unary operadic category are encoded in a single functor D(C) → C. So, for a unary operadic category C, we have on the one hand, that C is a D-coalgebra; and on the other, that C is endowed with a map D(C) → C. To reconcile these apparently distinct facts, we apply a general observation: any comonad C on a category A induces a monadC on the category of C-coalgebras A C , namely, the monad generated by the forgetful-cofree adjunction A C A. In the case of the décalage comonad, we induce a décalage monadD on Cat D ; and the axioms of a unary operadic category turn out to be captured precisely by the requirement that the map D(C) → C giving abstract fibres should endow the D-coalgebra C withD-algebra structure in Cat D . Our first main result is thus: Theorem. The category of algebras for the décalage monadD on Cat D is isomorphic to the category of unary operadic categories. In order to remove the qualifier "unary" from this theorem and accommodate the "multi" aspect of the general definition, we will need, as anticipated above, to adjust the décalage construction. Rather than the décalage comonad D on Cat, we will consider a modified décalage comonad D m on the arrow category Cat 2 whose action on objects is given by ( . ) E P − → C → Y ∈E E/Y Σ Y ∈C P/Y − −−−−−− → Y ∈E C/P Y . To relate this to operadic categories, we consider those objects of Cat 2 which are obtained to within isomorphism as the canonical projection P C : E C → C from the category of elements of a functor \|-\| : C → S. These objects span a full subcategory of Cat 2 which is equivalent to the lax slice category Cat//S; they are moreover closed under the action of D m , which thus restricts back to a comonad on Cat//S. Now by following the same trajectory as the unary case, starting from this comonad on Cat//S, we already come very close to characterising operadic categories. The first point to make is that an object (C, \|-\| : C → S) ∈ Cat//S is a D mcoalgebra just when C has chosen terminal objects in each connected component, and \|-\| sends each of these to 1 ∈ S. Since these are among the requirements for an operadic category, every operadic category gives rise to a D m -coalgebra. Like before, we can ask what it means to equip such a D m -coalgebra with algebra structure for the induced monadD m on (Cat//S) Dm . The action of this monad on (C, \|-\|) is given by the category X∈C,i∈\|X\| C/X of ( . ), endowed with a suitable functor to S; therefore, the basic datum ofD m -algebra structure is a functor of the same form as ( . ). This seems promising, but what we find is: Theorem. The category of algebras for the modified décalage monadD m on (Cat//S) Dm is isomorphic to the category of lax-operadic categories. Here, a lax-operadic category is a new notion, which generalises that of operadic category by replacing the assertion of equalities \|f \| −1 (i) = f −1 (i) on cardinalities of abstract fibres with a collection of coherent functions \|f \| −1 (i) → f −1 (i) . Since it is not yet clear that this extra generality has any practical merit, our final objective is to find a version of the above result which removes the qualifier "lax". The source of the laxity is easy to pinpoint. AD m -algebra structure is given by a map D m (C, \|-\|) → (C, \|-\|) in Cat//S, whose data involves not only a functor ( . ), but also a natural transformation relating the functors to S. The components of this natural transformation are the comparison functions \|f \| −1 (i) → f −1 (i) , so that the genuine operadic categories correspond to thoseD m -algebras whose structure map is given by a strictly commuting triangle over S. However, we cannot simply restrict the modified décalage comonad D m from the lax slice category Cat//S back to the strict slice category Cat/S, and then proceed as before. The problem is that D m does not restrict, since the counit maps ε C : D m (C) → C in Cat//S involve triangles which are genuinely lax-commutative. On the other hand, it turns out that we can restrict the lifted monadD m on (Cat//S) Dm back to the subcategory (Cat D )/S on the strictly commuting triangles. Having done so, our final result quickly follows: Theorem. The category of algebras for the modified décalage monadD m on Cat D /S is isomorphic to the category of operadic categories. The rest of this article will fill in the details of the above sketch. The plan is quite simple. In Section , we recall Batanin and Markl's definition of operadic category [ ]; then in Section we recall the décalage construction and establish the first of the two links with the notion of operadic category. In Section , we prove our first main theorem, characterising unary operadic categories in terms of décalage. Section is devoted to describing the modified décalage construction required to capture general operadic categories. Finally, in Sections and , we prove our second and third theorems, giving the characterisations of lax-operadic categories and, finally, of operadic categories themselves. . Operadic categories We begin with some necessary preliminaries. We say that a category C is endowed with local terminal objects if each connected component of C is provided with a chosen terminal object; we write uX for the chosen terminal in the connected component of X ∈ C and τ X : X → uX for the unique map. We write S for the category whose objects are the sets n = {1, . . . , n} for n ∈ N and whose maps are arbitrary functions. Note that S has a unique terminal object 1 which we use to endow S with local terminal objects; we may also sometimes write the unique element of 1 as * rather than 1. Given ϕ : m → n in S and i ∈ n, there is a unique monotone injection ( . ) ε ϕ,i : ϕ −1 (i) → m in S whose image is { j ∈ m : ϕ(j) = i }; we call the object ϕ −1 (i) the fibre of ϕ at i. Given also ψ : → m in S, we write ψ ϕ i for the unique map of S rendering ( . ) (ϕψ) −1 (i) ψ ϕ i G G ε ϕψ,i ϕ −1 (i) ε ϕ,i ψ G G m commutative, and call it the fibre map of ψ with respect to ϕ at i. The Batanin-Markl notion of operadic category which we now reproduce can be seen as specifying a category with formal notions of fibre and fibre map. The fibres of a map need not be subobjects of the domain as in the case of S, but the axioms will ensure that they retain many important properties of fibres in S. Definition . [ ] An operadic category is given by the following data: (D1) A category C endowed with local terminal objects; (D2) A cardinality functor \|-\| : C → S; (D3) For each object X ∈ C and each i ∈ \|X\| a fibre functor ϕ X,i : C/X → C whose action on objects and morphisms we denote as follows: Y f G G X → f −1 (i) Z g G G f g ) ) Y f Ñ Ñ X → g f i : (f g) −1 (i) → f −1 (i) , referring to the object f −1 (i) as the fibre of f at i, and the morphism g f i : (f g) −1 (i) → f −1 (i) as the fibre map of g with respect to f at i; all subject to the following axioms, where in (A5), we write εj for the image of j ∈ \|f \| −1 (i) under the map ε \|f \|,i : \|f \| −1 (i) → \|Y \| of ( . ): (A1) If X is a local terminal then \|X\| = 1; (A2) For all X ∈ C and i ∈ \|X\|, the object (1 X ) −1 (i) is chosen terminal; (A3) For all f ∈ C/X and i ∈ \|X\|, one has \|f −1 (i)\| = \|f \| −1 (i), while for all g : f g → f in C/X and i ∈ \|X\|, one has \|g f i \| = \|g\| \|f \| i ; (A4) For X ∈ C, one has τ −1 X ( * ) = X, and for f : Y → X, one has f τ X * = f ; (A5) For g : f g → f in C/X, i ∈ \|X\| and j ∈ \|f \| −1 (i), one has that (g f i ) −1 (j) = g −1 (εj), and given also h : f gh → f g in C/X, one has (h f g i ) g f i j = h g εj . A functor F : C → C between operadic categories is called an operadic functor if it strictly preserves local terminal objects, strictly commutes with the cardinality functors to S, and preserves fibres and fibre maps in the sense that F (f −1 (i)) = (F f ) −1 (i) and F (g f i ) = (F g) F f i for all g : f g → f in C/X and i ∈ \|X\|. We write OpCat for the category of operadic categories and operadic functors. The preceding definitions are exactly those of [ ] with only some minor notational changes for clarity. The most substantial of these is that we make explicit the use of the monotone injections ( . ) in the axiom (A5), whereas in [ ] this is left implicit. In light of this, let us spend a moment doing the necessary type-checking to see that this axiom makes sense. Intuitively, the first clause of (A5) identifies the fibres of the fibre maps of a map, with the fibres of that map. Therein we have g f i : (f g) −1 (i) → f −1 (i), and j ∈ \|f \| −1 (i) = \|f −1 (i)\|, so that one can consider the object (g f i ) −1 (j). On the other hand, we have εj ∈ \|Y \| and g : Z → Y and so can equally consider the object g −1 (εj); now the first part of (A5) states that these two are equal. As for the second part of (A5), this says that the fibre maps of the fibre maps of a map, are themselves fibre maps of that map. In this case, functoriality of the fibre functor ϕ X,i : C/X → C implies that we have an equality (f gh) −1 (i) (gh) f i − −−− → f −1 (i) = (f gh) −1 (i) h f g i −−→ (f g) −1 (i) g f i − − → f −1 (i) and again we have j ∈ \|f \| −1 (i) = f −1 (i) . It follows that the fibre map of h f g i with respect to g f i at j is given as to the left in: (h f g i ) g f i j : ((gh) f i ) −1 (j) → (g f i ) −1 (j) h g εj : (gh) −1 (εj) → g −1 (εj) . On the other hand, one could just consider the fibre map of h with respect to g at εj ∈ \|Y \|, as to the right. The first part of (A5) assures us that the domains and codomains of these maps coincide, and now the second part asserts that the maps themselves are equal. Example . The most basic example of an operadic category is S itself. The choice of local terminals is the unique one, the cardinality functor is the identity, and the action of the fibre functors is defined as in ( . ) and ( . ). Many more examples of operadic categories are discussed in [ ]; we give here two new examples inspired by probability theory. Example . Let C be the category of finite sub-probability spaces. Its objects are lists r = (r 1 , . . . , r n ) where each r i ∈ [0, 1] and Σ i r i 1; its maps ϕ : (s 1 , . . . , s m ) → (r 1 , . . . , r n ) are maps ϕ : m → n of S such that r i = Σ j∈ϕ −1 (i) s i . There is an obvious cardinality functor \|-\| : C → S, and a unique choice of local terminals: indeed, C is a coproduct of categories C = Σ r∈[0,1] C r where C r has the unique terminal object (r). For the abstract fibres, given ϕ : s → r in C and i ∈ \|r\|, we define ϕ −1 (i) to be (s ε1 , . . . , s εk ) where k = \|ϕ\| −1 (i) and ε = ε \|ϕ\|,i : \|ϕ\| −1 (i) → \|s\| is as in ( . ); finally, fibre maps in C are as in S. Example . Let C 1 be the category of finite probability spaces, i.e., the connected component of (1) in the category C of the previous example. This subcategory of C is not a sub-operadic category; however, it bears a different operadic category structure which describes disintegration of finite probability measures. We begin with the cardinality functor \|-\| 1 : C 1 → S. For any r = (r 1 , . . . , r n ) ∈ C 1 , we let (r p 1 , . . . , r p k ) be the sublist of r obtained by deleting all zeroes, and now take \|r\| 1 = k. Given a map ϕ : s → r in C 1 , where s has sublist (s q 1 , . . . , s q ) of non-zero entries, we determine \|ϕ\| 1 : \|s\| 1 → \|r\| 1 by requiring that \|ϕ\|(q j ) = p \|ϕ\| 1 (j) ; i.e., \|ϕ\| 1 is the restriction of \|ϕ\| to the indices of non-zero entries. To define the C 1 -fibres, we employ the normalisation of a non-zero subprobability space r = (r 1 , . . . , r n ) ∈ C \ C 0 ; this is the probability space r ∈ C 1 with r = (r 1 /Σ i r i , . . . , r n /Σ i r i ). Now given ϕ : s → r in C 1 and i ∈ \|r\| 1 , we define the C 1 -fibre ϕ −1 (i) to be the normalisation of the ith non-zero C-fibre ϕ −1 (p i ). Note that we cannot normalise a C-fibre ϕ −1 (j) for which r j = 0; this is why we had to remove such j in defining the cardinality functor \|-\| 1 . Finally, given also ψ : t → s in C 1 , we define the C 1 -fibre map ψ ϕ i to have underlying S-map ψ ϕ p i . . Décalage In this section, we recall the décalage comonad on Cat, characterise its category of coalgebras, and explain how this links up with the notion of operadic category. Throughout, "décalage" will always mean upper décalage. The décalage comonad on Cat can be obtained as a restriction of Illusie's décalage comonad [ ] on [∆ op , Set], the category of simplicial sets. This is, in turn, obtained from the monad T = (T, η, µ) on the category ∆ of non-empty finite ordinals and monotone maps given by freely adjoining a top element. In terms of the usual presentation of ∆ in terms of "coface" maps δ i and "codegeneracy" maps σ j , this monad is given by the data: Set]; this is the décalage comonad. The classical nerve functor N : Cat → [∆ op , Set] exhibits the category of small categories as equivalent to a full subcategory of simplicial sets. The simplicial sets in this full subcategory happen to be closed under the action of the décalage comonad, which thereby restricts to a comonad D on Cat. The underlying endofunctor D of this comonad sends a category C to the coproduct of its slices: T [n] = [n + 1], T δ i = δ i , T σ j = σ j , η [n] = δ n+1 and µ [n] = σ n+1 . It follows that T op is a comonad on ∆ op , so that precomposition with T op is a comonad on [∆ op ,( . ) D(C) = X∈C C/X ; the counit ε C : D(C) → C is the copairing of the domain projections C/X → C from the slices (i.e., the map induced from the family of domain projections by the universal property of coproduct); while the comultiplication δ C : D(C) → DD(C), which is a functor δ C : X∈C C/X → f ∈D(C) D(C)/f , sends the X-summand to the 1 X -summand via the isomorphism C/X → D(C)/1 X . We now characterise the category of coalgebras for the décalage comonad as the category Cat t whose objects are small categories endowed with local terminal objects, and whose morphisms are functors which preserve chosen local terminals. Proposition . The category Cat D of D-coalgebras is isomorphic to Cat t over Cat. Under this isomorphism, the D-coalgebra structure on C ∈ Cat t is given by the functor τ : C → D(C) which takes X ∈ C to τ X : X → uX ∈ D(C). Proof. It suffices to show that the forgetful functor U : Cat t → Cat is strictly comonadic, and that the induced comonad is isomorphic to D. Towards the first of these, it is clear that U strictly creates limits and is faithful, and so by the Beck theorem will be strictly comonadic so long as it has a right adjoint. We can endow the category D(C) with the chosen terminal object 1 X in each connected component C/X, so making it into an object of Cat t ; we claim this gives the value at C of the desired right adjoint. Thus, for any B ∈ Cat t and functor F : B → C, we must exhibit a unique factorisation ( . ) F = B G − → D(C) ε C − − → C where G strictly preserves chosen local terminals. Such a G must send each object X ∈ C to an object of D(C) with domain projection F X. In particular, each chosen terminal uX of B must be sent to a chosen terminal of D(C) with domain F uX, and so we must have G(uX) = 1 F uX . Furthermore, such a G, if it exists, must send each map f : Y → X of B to a map in D(C) as to the left in: F Y F f G G GY 3 3 F X GX } } F Y F τ X G G GX 3 3 F uX . 1 F uX } } • F uX In particular, taking f = τ X yields the commuting triangle to the right, so that on objects we must have GX = (F τ X : F X → F uX). So G is unique if it exists; but it easy to see that defining G in this way does indeed yield a map G : B → D(C) in Cat t preserving chosen terminals and factorising ( . ) as required. So U : Cat t → Cat has a right adjoint R, and by strict comonadicity, Cat t is isomorphic to the category of U R-coalgebras. By construction, the underlying functor and counit of U R are equal to D and ε, while the comultiplication at C is the unique factorisation ( . ) of F = 1 D(C) : D(C) → D(C) through a map in Cat t . As δ C : D(C) → DD(C) is easily seen to be such a factorisation, we conclude that D = U R and so Cat t ∼ = Cat D as required. To motivate the developments which will follow, we now establish a first link between operadic categories and décalage, by showing how the data and axioms for an operadic category can be partially re-expressed in terms of structure in Cat t ∼ = Cat D . Of course, (D1) asserts that C is an object in Cat t , whereupon axiom (A1) asserts that the cardinality functor \|-\| : C → S is a map therein. Similarly, axiom (A2) states that each functor ϕ X,i : C/X → C is a map of Cat t , where we take the chosen (local) terminal object in C/X to be the identity 1 X . To express (A3), we define for each X ∈ C and i ∈ \|X\| a cardinality functor \|-\| X,i : C/X → S as the composite of \|-\|/X : C/X → S/\|X\| with the fibre functor ϕ \|X\|,i : S/\|X\| → S of the operadic category S; thus, on objects, \|f \| X,i = \|f \| −1 (i). Now (A3) asserts that the following diagram commutes for all X ∈ C, i ∈ \|X\|: ( . ) C/X ϕ X,i G G \|-\| X,i 1 1 C . \|-\| S We may express all of the above more compactly as follows. For any object \|-\| C : C → S of Cat t /S, we write D m (C) for the category Σ X∈C,i∈\|X\| C/X, seen as an object of Cat t by choosing each identity map as a local terminal, and write \|-\| Dm(C) : D m (C) → S for the copairing of the maps \|-\| X,i : C/X → S. Now to give the data (D1)-(D3) and axioms (A1)-(A3) for an operadic category is to give an object (C, \|-\| C ) of Cat t /S and a map ϕ : (D m (C), \|-\| Dm(C) ) → (C, \|-\| C ). It remains to account for axioms (A4) and (A5). In fact, it turns out that the assignation (C, \|-\| C ) → (D m (C), \|-\| Dm(C) ) is the action on objects of a monadD m on the category Cat t /S, and that the remaining axioms are just those needed for ϕ : (D m (C), \|-\| Dm(C) ) → (C, \|-\| C ) to endow (C, \|-\|) withD m -algebra structure. While we could verify this straight away in a hands-on fashion, we prefer to give an argument which justifies the constructions in terms of a deeper link to the décalage construction. In the end, the claimed monad structure onD m will be exhibited in Definition below, and the characterisation of its algebras as operadic categories given in Theorem . . Characterising unary operadic categories The characterisation of general operadic categories in terms of décalage will require a modification of the décalage construction, to be introduced in Section below. As a warm-up for this, we consider the case of unary operadic categories, for which the usual décalage will suffice. Definition . An operadic category is unary if \|X\| = 1 for all X ∈ C. We write OpCat 1 for the category of unary operadic categories and operadic functors. Example . For any category C, the category D(C) = X∈C C/X is a unary operadic category. The chosen local terminals are the identity maps, and the unique fibre of a map g : f g → f is the object g. Given another map h : f gh → f g, the fibre map of h with respect to g at * is taken to be h : gh → g. Example . If C is a pointed category with a chosen zero object and chosen kernels, we can attempt to impose a unary operadic structure as follows: the chosen (local) terminal is the zero object; the unique fibre of a map f : Y → X is its kernel; and the fibre map of g : Z → Y with respect to f is the restriction g\| ker f g : ker f g → ker f . However, whether these data satify the required axioms is sensitive to the choice of kernels. For instance, if g : Z → Y and f : Y → X, then the chosen kernel of g, though always isomorphic to the chosen kernel of g\| ker f g : ker f g → ker f , need not be equal to it as required by axiom (A5). Often, there is an appropriate choice of kernels; for example if C is Set * or Ab or k-Vect or Ch(R-Mod), then we can take the kernel of any identity map to be the chosen zero object, and the kernel of any other map to be given by the usual subset formula; this yields the necessary axioms for a unary operadic category. Yet even for a C where we cannot choose kernels appropriately, we can always consider the equivalent category Pt(C op , Set * ) rep of representable zero-preserving functors to Set * , and endow this with unary operadic structure given pointwise as in Set * . Note that this structure need not transport back to an operadic structure on C, since the notion of operadic category is not invariant under equivalence (in the terminology of [ ] it is not flexible). In the unary case, we can effectively ignore the cardinality functor down to S; so on repeating the analysis at the end of the preceding section, we find that the data and first three axioms for a unary operadic category C are encoded precisely by a map D(C) → C in Cat t . To complete this analysis, we will show that the assignation C → D(C) underlies a monad on Cat t whose category of algebras is isomorphic to OpCat 1 . The monad structure arises as follows. Definition . The décalage monadD = (D, η, µ) on Cat t ∼ = Cat D is the monad induced by the forgetful-cofree adjunction Cat D Cat. Since the proof of Proposition furnishes us with an explicit description of the forgetful-cofree adjunction Cat D Cat, we can read off from it the following description of the décalage monad: (i) The underlying functorD on objects sends C to X∈C C/X endowed with the local terminal objects 1 X ∈ C/X; while on morphisms, it sends F : C → C to the functor which maps the X-summand of X∈C C/X to the F X-summand of Y ∈C C /Y via F/X : C/X → C /F X; (ii) The unit map η C : C →D(C) is defined on objects by η C (X) = τ X : X → uX and on morphisms by η C (f : Y → X) = f : τ Y → τ X ; (iii) The multiplication map µ C :DD(C) →D(C), which is given by a functor f ∈D(C) D(C)/f → X∈C C/X, sends the summand indexed by f : Y → X to the summand indexed by Y via the isomorphism D(C)/f → C/Y . Using this description, we can now prove our first main theorem. Theorem . The category of algebras for the décalage monadD on Cat t ∼ = Cat D is isomorphic to the category OpCat 1 of unary operadic categories. Proof. We have already argued that the data and first three axioms for a unary operadic category C are encapsulated by giving the object C ∈ Cat t together with the map ϕ :D(C) → C in Cat t obtained as the copairing of the fibre functors ϕ X, * : C/X → C. Given this, we can read off from Definition that (A4) asserts precisely the unit axiom ϕ • η C = 1 C , and that (A5) asserts the multiplication axiom ϕ • µ C = ϕ •D(ϕ) :DD(C) → C. SoD-algebras in Cat t are in bijection with unary operadic categories; the corresponding bijection on maps is direct. Using this result, we may obtain a further description of unary operadic categories which, though not necessary for the subsequent results of this paper, is nonetheless enlightening. We observed above that the décalage comonad on Lemma . The comparison functor [∆ op , Set] → ([∆ op , Set] D )D sending a simplicial set X to D(X) with its canonicalD-algebra structure, is an equivalence of categories. Proof. The functor part of the comonad D on [∆ op , Set] is given by precomposition with T op : ∆ op → ∆ op , and so is cocontinuous. Thus, for the forgetful-cofree adjunction [∆ op , Set] D U D G G o o G D [∆ op , Set] the functor G D is again cocontinuous. Moreover, U D is conservative, and it is easy to see that U D G D = D is conservative-since the set of 0-simplices of a simplicial set is the splitting of an idempotent on the set of 1-simplices-so that G D is also conservative. Thus by the Beck monadicity theorem G D is monadic, and so the comparison functor [∆ op , Set] → ([∆ op , Set] D )D is an equivalence. Combining this with the characterisation of the essential image of ( . ) yields: Corollary . The category OpCat 1 of unary operadic categories is isomorphic to the full (reflective) subcategory of [∆ op , Set] on those simplicial sets C for which D(C) satisfies the Segal condition. Explicitly, the simplicial set C giving the "undecking" of a unary operadic category C has as 0-simplices, the chosen terminal objects of C, and as (n + 1)simplices the n-simplices of the nerve of C. The faces of a 1-simplex X are ϕ(1 X ) X − − → uX where we write ϕ(f ) for the unique fibre f −1 ( * ) of a map f : Y → X of C. The faces of a 2-simplex f ∈ C(Y, X) are given by ϕ(1 X ) X ) ) ϕ(1 Y ) ϕ(f ) e e Y G G f uX ; while the faces of a 3-simplex (g, f ) ∈ C(Z, Y ) × C(Y, X) are given by ϕ(1 Y ) g Y @ @ ϕ(f ) G G ϕ(1 X ) f X ' ' ϕ(1 Z ) ϕ(g) g g Z G G uX ϕ(1 Y ) ϕ(f ) G G g f * ϕ(1 X ) f g X ' ' ϕ(1 Z ) ϕ(f g) T T ϕ(g) g g Z G G uX . The degeneracies are easily written down, and the remaining data is determined by coskeletality. Note that D(C) is the nerve of C, which satisfies the Segal condition. Conversely, if C is a simplicial set for which D(C) satisfies the Segal condition, then D(C) ∼ = N(C) for a category C, and by working backwards through the above description we may read off the operadic structure on C. Remark . The condition on a simplicial set X that D(X) should satisfy the Segal condition gives half of the axioms for a discrete decomposition space [ ]. (Decomposition spaces are also known as 2-Segal spaces [ ].) In particular, for any discrete decomposition space X : ∆ op → Set, its décalage is a unary operadic category, generalising Example . For example, there is a discrete decomposition space X of (combinatorialists') graphs, wherein X n is the set of graphs with a map from the set of vertices to n. The corresponding unary operadic category has graphs as objects; a map is the opposite of a full inclusion of graphs, and the fibre of such a map is the induced graph on the complementary set of vertices. In fact, the remaining axioms for a discrete decomposition space X can be expressed in terms of the associated unary operadic category C: they say precisely that the fibre functor ϕ : D(C) → C is a discrete opfibration. This establishes a link with Lack's [ ], which characterises operadic categories with object set O in terms of certain left-normal skew monoidal [ ] structures on Set/O, and provides conditions for these skew structures to be genuinely monoidal; in the unary case, the necessary condition is, again, that ϕ be a discrete opfibration. In the following result, the equivalence between (i) and (ii) is thus due to Lack; we omit the proof, since the result is not needed elsewhere in this paper. Theorem. Let C be a unary operadic category. The following are equivalent: (i) The fibre functor ϕ : D(C) → C is a discrete opfibration; (ii) The associated skew monoidal structure on Set/ob C is genuinely monoidal; (iii) The "undecking" C is a discrete decomposition space. In fact (cf. [ , Remark . ]) the left-normal skew monoidal structures induced by unary operadic categories are precisely those whose tensor preserves colimits in each variable; these can be identified with skew monoidales in the monoidal bicategory Span, and in this case Lack's characterisation reduces to one given by Andrianopoulos [ ]. Under this identification, the unary operadic categories satisfying the equivalent conditions of the above theorem correspond to genuine monoidales in Span: in the language of [ ], this monoidale is the incidence algebra of the corresponding discrete decomposition space. Remark . The equivalence of Corollary is also interesting in the other direction. If C is a unary operadic category derived from a category with a zero object and kernels, as in Example , then the associated simplicial set is a discrete version of Waldhausen's S • construction. . Modified décalage We now wish to expand on Theorem to give a characterisation of general operadic categories in terms of décalage. As explained in the introduction, the key to this will be a comonad D m on the arrow category Cat 2 given on objects by ( . ) E P − → C → Y ∈E E/Y Σ Y ∈E P/Y − −−−−−− → Y ∈E C/P Y , which we call modified décalage. In this section, we describe this comonad, and show that it restricts back to the the lax slice category Cat//S, identified with the full subcategory of Cat 2 on the discrete opfibrations with finite fibres. While we could describe the comonad D m and its coalgebras by hand, we prefer in the spirit of the rest of the paper to obtain it by way of more general considerations. The key is the following construction on a functor P : E → C. It begins by decomposing E and C into their connected components: E = y∈Y E y and C = x∈X C x . Now for each y ∈ Y , the restriction of P to E y must factor through a single connected component C f y of C. If we write P y : E y → C f y for this factorisation, then summing the P y 's over all y ∈ Y yields the first map L P in a factorisation: ( . ) y∈Y E y P G G L P 8 8 x∈X C x y∈Y C f y R P V V whose second map R P maps the y-summand to the f y-summand via 1 Cfy . Let us call a functor π 0 -bijective if, like L P , the induced function on connected components is invertible, and π 0 -cartesian if, like R P , it maps each connected component of its domain bijectively onto a connected component of its codomain. As these two classes of functors are easily seen to be orthogonal, we have a factorisation system (π 0 -bijective, π 0 -cartesian) on Cat; and so by [ , Theorem . ] we have: Lemma . The full subcategory π 0 -Bij of Cat 2 whose objects are the the π 0bijective functors is a coreflective subcategory. The counit of the coreflection at P is given by the morphism (1, R P ) : L P → P in Cat 2 . Remark . Whenever H : T → B is a Grothendieck fibration, there is a factorisation system on T whose left and right classes are, respectively, the maps inverted by H, and the cartesian maps with respect to H. The above factorisation system arises in this way from the connected components functor π 0 : Cat → Set. Now, if the P : E → C of ( . ) is a strictly local-terminal-preserving functor between categories endowed with local terminal objects, then there is a unique way of endowing the interposing y C f y with local terminal objects such that both L P and R P preserve them strictly. It follows that the (π 0 -bijective, π 0 -cartesian) factorisation system on Cat lifts to Cat t , and so again by [ , Theorem . ]: Lemma . The full subcategory π 0 -Bij t of (Cat t ) 2 whose objects are the π 0bijective functors is a coreflective subcategory. The counit of the coreflection at P is given by the morphism (1, R P ) : L P → P in (Cat t ) 2 . Remark . The lifting of the (π 0 -bijective, π 0 -cartesian) factorisation system from Cat to Cat t is in fact also the lifting of the comprehensive factorisation system [ ], whose classes are the final functors and the discrete fibrations. So the category π 0 -Bij t is equally the full subcategory of (Cat t ) 2 on the final functors. Now, if we let L and L t denote the idempotent comonads on Cat 2 and (Cat t ) 2 corresponding to the coreflective subcategories of the last two lemmas, then it is evident from their explicit descriptions that L t is a lifting-in the sense of [ ]-of L along the strictly comonadic (Cat t ) 2 → Cat 2 . It follows by the proposition in § of ibid. that the composite adjunction ( . ) π 0 -Bij t G G o o (Cat t ) 2 G G o o Cat 2 is also strictly comonadic. Thus, if we define the modified décalage comonad D m to be the comonad generated by this adjunction, then we have: Proposition . The category (Cat 2 ) Dm of D m -coalgebras is isomorphic over Cat 2 to the full subcategory π 0 -Bij t of (Cat t ) 2 on the π 0 -bijective functors. By combining Proposition and Lemma , we see that the cofree functor Cat 2 → (Cat 2 ) Dm sends the object P : E → C of Cat 2 to the object ( . ) D m (P ) = Y ∈E E/Y Σ Y ∈E P/Y − −−−−−− → Y ∈E C/P Y endowed in domain and codomain with the respective local terminals 1 Y and 1 P Y for each Y ∈ E. Furthermore, the counit at P of the adjunction ( . ) is the map D m (P ) → P of Cat 2 whose two components Y E/Y → E and Y C/P Y → C are given by the appropriate copairings of slice projections. We now show that the comonad D m on Cat 2 restricts to the lax slice category Cat//S. The objects of this category are pairs of a small category C and a functor \|-\| C : C → S, while morphisms (C, \|-\| C ) → (C , \|-\| C ) are pairs of a functor F and natural transformation ν fitting into a diagram: ( . ) C F G G \|-\| C 2 2 ν + 3 C . \|-\| C S To embed Cat//S into Cat 2 , we use the category of elements construction. For a functor Q : C → Set, its category of elements el(Q) has objects given by pairs (X ∈ C, i ∈ QX), and maps (Y, j) → (X, i) given by maps f ∈ C(Y, X) with (Qf )(j) = i. Associated to the category of elements we have a discrete opfibration π Q : el(Q) → C sending (X, i) to X; recall that a functor P : E → C is a discrete opfibration if, for every Y ∈ E and f : P Y → X in C, there is a unique mapf : Y →X with Pf = f . In particular, to each (C, \|-\| C ) ∈ Cat//S we can associate the discrete opfibration P C : E C → C obtained as the projection from the category of elements of \|-\| C : C → S → Set. Proposition . The assignation (C, \|-\| C ) → (P C : E C → C) is the action on objects of a fully faithful functor Υ : Cat//S → Cat 2 . Its essential image comprises the discrete opfibrations with finite fibres, and choosing an isomorphism with an object in the image amounts to endowing each of these fibres with a linear order. While this result is well known, we prove it for the sake of self-containedness. Proof. If (C, \|-\| C ) and (C , \|-\| C ) are objects of Cat//S, then a map P C → P C of Cat 2 is a commutative square E C G G G P C E C P C C F G G C . Commutativity forces G(X, i) = (F X, ν X (i)) for suitable ν X (i) ∈ \|F X\| C , so yielding functions ν X : \|X\| C → \|F X\| C , which by applying G to morphisms we see are natural in X. So every map P C → P C arises from a lax triangle ( . ), and it is easy to see that any such triangle induces a map P C → P C in this manner. So Υ is well defined and fully faithful. As for its essential image, it is well known (and easily proved) that H : E → C is a discrete opfibration just when it is isomorphic over C to π Q : el(Q) → C for some functor Q : C → Set. In this case, H will have finite fibres just when π Q does so, which happens just when each Q(B) is finite. But such a Q may always be replaced by an isomorphic one which factors through S ⊆ Set, and so the discrete opfibration H has finite fibres just when it is in the essential image of Υ. Finally, the fibre of P C : E C → C over X ∈ C is the set {(X, i) : i ∈ \|X\| C } which inherits a linear order from \|X\| C . So any specified isomorphism H ∼ = P C induces by transport of structure a linear order on each fibre of H. Conversely, given a linear order on the fibres of H, we may reconstruct an isomorphism with P C by requiring each map on fibres to be a monotone isomorphism. We now show that the modified décalage comonad D m on Cat 2 restricts back to a comonad on Cat//S. Proposition . The essential image of Υ : Cat//S → Cat 2 is closed under the action of modified décalage, which thus restricts to a comonad D m on Cat//S. The category of coalgebras (Cat//S) Dm is isomorphic to the lax slice Cat t //S. Proof. Given (C, \|-\|) in Cat//S, applying D m to the corresponding P C : E C → C in Cat 2 yields by ( . ) the functor ( . ) X∈C,i∈\|X\| E C /(X, i) Σ X,i P C /(X,i) − −−−−−−−− → X∈C,i∈\|X\| C/X . We must show this is a discrete opfibration with finite fibres. Since functors of this kind are closed under coproducts, it suffices to show that each P C /(X, i) : E C /(X, i) → C/X is a discrete opfibration with finite fibres. It is a discrete opfibration since it is a slice of the discrete opfibration P C ; as for the fibres, given f : Y → X in C/X, the objects over it in E C /(X, i) are maps of E C of the form f : (Y, j) → (X, i), which are indexed by the finite set { j ∈ \|Y \| : \|f \|(j) = i }. It follows that D m restricts back to a comonad on Cat//S, and the corresponding category of coalgebras fits into a pullback (Cat//S) Dm G G U (Cat 2 ) Dm U Cat//S Υ G G Cat 2 . Now given (C, \|-\|) ∈ Cat//S, endowing its image P C : E C → C under Υ with D m -coalgebra structure means, first of all, endowing C with local terminal objects. Having done this, we must endow E C with local terminals such that P C preserves them, and it is easy to see that the unique way of doing this is by choosing the set {(X, i) : X is local terminal in C, i ∈ \|X\|}. Finally, to assert that P C is π 0 -bijective, there must be a unique (X, i) over each chosen local terminal of C, which is to say that \|X\| = 1 for each local terminal of C. So objects of (Cat//S) Dm are in bijection with those of Cat t //S. The argument on maps is similar and left to the reader. . Characterising lax-operadic categories In this section, we take the procedure employed in Section for the décalage comonad on Cat-considering its category of coalgebras, then the monad induced on the category of coalgebras, and then the algebras for that monad-and apply it to the modified décalage comonad on Cat//S. By doing so, we come very close to obtaining a characterisation of operadic categories. What we in fact characterise are instances of the more general notion of lax-operadic category. These generalise operadic categories by replacing the fact of the commutativity of the triangles ( . ) by the data of coherent 2-cells filling these triangles. Definition . A lax-operadic category is given by the following data, which augment those of an operadic category by the addition of (D4): (D1) A category C endowed with local terminal objects; We now begin our abstract rederivation of lax-operadic categories in terms of modified décalage. Recall that in Proposition , we exhibited the category of coalgebras for the modified décalage comonad on Cat//S as isomorphic to the lax slice category Cat t //S. Thus we are justified in giving: Definition . The modified décalage monadD m on Cat t //S ∼ = (Cat//S) Dm is the monad induced by the forgetful-cofree adjunction (Cat//S) Dm Cat//S. Towards a concrete description of the modified décalage monad, we note that the sets {j ∈ \|Y \| : \|f \|(j) = i} giving the fibres of ( . ) inherit linear orders from \|Y \|, so that we may use the last clause of Proposition to obtain a particular instantiation of the forgetful-cofree adjunction for the modified décalage comonad on Cat//S. The cofree functor Cat//S → (Cat//S) Dm sends an object (C, \|-\|) to the object (D m (C), \|-\| Dm(C) ), where D m (C) = Σ X∈C,i∈\|X\| C/X is the codomain of ( . ), with the chosen terminal 1 X in the connected component indexed by (X, i), and where \|-\| Dm(C) : D m (C) → S is defined on objects and morphisms by ( . ) (X ∈ C, i ∈ \|X\|, f : Y → X) → \|f \| −1 (i) (X, i, f g) g − → (X, i, f ) → \|f g\| −1 (i) \|g\| \|f \| i −−−→ \|f \| −1 (i) . The counit at (C, \|-\|) of the forgetful-cofree adjunction is given by a lax triangle ( . ) D m (C) E C G G \|-\| Dm(C) 3 3 ε C + 3 C \|-\| } } S wherein the functor E C : Σ X∈C,i∈\|X\| C/X → C is the copairing of the slice projections, and the natural transformation ε C has component at (X, i, f ) given by the map ε \|f \|,i : \|f \| −1 (i) → \|Y \| of ( . ). We now use this to read off a description of the modified décalage monad on (Cat//S) Dm ∼ = Cat t //S. (i) The underlying functorD m : Cat t //S → Cat t //S is given on objects by (C, \|-\|) → (D m (C), \|-\| Dm(C) ) as above, and on morphisms by: C F G G \|-\| C 3 3 ν + 3 C \|-\| C } } → D m (C) Dm(F ) G G \|-\| Dm(C) 3 3 Dm(ν) + 3 D m (C ) . \|-\| Dm(C ) } } S S Here D m (F ) has action on objects (X, i, f ) → (F X, ν X (i), F f ) and action on maps inherited from F ; while the component of D m (ν) at an object (X, i, f ) is the unique map rendering commutative the square \|f \| −1 C (i) Dm(ν) (X,i,f ) G G ε \|f \|,i \|F f \| −1 C (ν X (i)) ε \|F f \|,ν X (i) \|X\| C ν Y G G \|F X\| C . (ii) The unit η C : (C, \|-\|) →D m (C, \|-\|) is a strictly commuting triangle, whose upper edge is the functor C → D m (C) sending X to (uX, 1, τ X ) and sending f : Y → X to f : (uX, 1, τ X ) → (uY, 1, τ Y ). (iii) The multiplication µ C :D mDm (C, \|-\|) →D m (C, \|-\|) is also a strictly commuting triangle, whose upper edge is the functor ( . ) (X,i,f )∈DmC,j∈\|f \| −1 (i) D m C/(X, i, f ) → X∈C,i∈\|X\| C/X defined as follows. Since a typical map in D m C is of the form g : (X, i, f g) → (X, i, f ), an object of the domain of ( . ) comprises the data of ( . ) X ∈ C, i ∈ \|X\|, f : Y → X, j ∈ \|f \| −1 (i), g : Z → Y while each morphism is of the form h : (X, i, f, j, gh) → (X, i, f, j, g). In these terms, we can define the functor ( . ) on objects and morphisms by ( . ) (X, i, f, j, g) → (Y, εj, g) (X, i, f, j, gh) h − → (X, i, f, j, g) → (Y, εj, gh) h − → (Y, εj, g) , where, like before, we write εj for ε \|f \|,i (j). Using this description, we can now give our second main result. Theorem . The category of algebras for the modified décalage monadD m on Cat t //S ∼ = (Cat//S) Dm is isomorphic to the category LaxOpCat of lax-operadic categories. Proof. The data (D1)-(D2) and axiom (A1) specify exactly an object (C, \|-\|) in Cat t //S. Giving the fibre functors (D3) is equivalent to giving a single functor ϕ : D m (C) → C, and the relabelling maps of (D4) give the components of a natural transformation ( . ) D m (C) ϕ G G \|-\| Dm(C) 1 1 γ + 3 C \|-\| S whose naturality is then asserted by (A3-lax). Since axiom (A2) asserts that ϕ in ( . ) is a map of Cat t , we conclude that giving the data for a lax-operadic category plus the first three axioms is the same as giving an object (C, \|-\|) of (Cat//S) Dm endowed with a morphism (ϕ, γ) :D m (C, \|-\|) → (C, \|-\|). It is not hard to see that (A4) is equivalent to (ϕ, γ) satisfying the the unit axiom (ϕ, γ) • (η (C,\|-\|) , 1) = (1 C , 1 \|-\| ) for aD m -algebra; we claim, finally, that (A5lax) asserts the multiplication axiom given by the equality of pastings: ( . ) D m D m C \|-\| DmDm(C) 5 5 Dmϕ G G D m C ϕ G G Dm(γ) + 3 γ + 3 \|-\| DmC C \|-\| { { S = D m D m C \|-\| DmDm(C) µ C G G D m C ϕ G G \|-\| DmC = γ + 3 C . \|-\| { { S OPERADIC CATEGORIES AND DÉCALAGE Now, the functors across the top of ( . ) act on a typical object ( . ) of D m D m C by the respective assignations: (X, i, f, j, g) → (f −1 (i), γj, g f i ) → (g f i ) −1 (γj) and (X, i, f, j, g) → (Y, εj, g) → g −1 (εj) , whose equality is precisely the first clause of (A5-lax). On the other hand, at this same object ( . ), the components of the two composite natural transformations in ( . ) are given by the two sides of the left square of ( . )-whose equality is the second clause of (A5-lax). Finally, the actions on a map h : (X, i, f, j, gh) → (X, i, f, j, g) of D m D m C of the functors across the top of ( . ) are given by h → h f g i → (h f g i ) g f i γj and h → h → h g εj , whose equality is precisely the final clause of (A5-lax). This proves thatD malgebras in (Cat//S) Dm correspond bijectively with lax-operadic categories. A similar argument verifies the same for the maps between them, and we leave this to the reader. . Characterising operadic categories There is not much left to do to get from the preceding result to our main result, characterising genuine operadic categories in terms of décalage. If we define a morphism of Cat t //S as in ( . ) to be strict whenever the natural transformation ν therein is an identity, then it is immediate from the preceding proof that: Proposition . Under the isomomorphism of Theorem , aD m -algebra corresponds to an operadic category just when its algebra structure map in Cat t //S is strict; while aD m -algebra morphism corresponds to an operadic functor just when its underlying map in Cat t //S is strict. At this point, it is not possible to restrict the modified décalage comonad D m on Cat//S back to the strict slice category Cat/S, and obtain operadic categories as algebras for the induced monadD m on (Cat/S) Dm . The reason for this, as noted in the introduction, is simply that modified décalage D m does not restrict from Cat//S to Cat/S, since its counit maps ( . ) are only lax triangles. However, the modified décalage monadD m on (Cat//S) Dm ∼ = Cat t //S does interact well with strictness: inspection of the description following Definition shows that the functorD m preserves strictness of triangles, and that each unit and multiplication component is a strict triangle. We are thus justified in giving: Definition . The modified décalage monadD m on Cat t /S is the restriction to Cat t /S of the modified décalage monad Cat t //S. And so, from Theorem and Proposition , our main result immediately follows: Theorem . The category of algebras for the modified décalage monadD m on Cat t /S is isomorphic to the category OpCat of operadic categories. Date: th December . Mathematics Subject Classification. Primary: D , C , C . The authors acknowledge, with gratitude, the following grant funding. Garner was supported by Australian Research Council grants DP and FT ; Kock by grants MTM --P (AEI/FEDER, UE) of Spain and -SGR-of Catalonia; and Weber by Czech Science Foundation grant GA CR P / /G . arXiv:1812.01750v1 [math.CT] 4 Dec 2018 Cat is the restriction along the full inclusion N : Cat → [∆ op , Set] of the décalage comonad on simplicial sets. It follows that we have a full inclusion( . ) OpCat ∼ = − → (Cat D )D (N D )D − −−− → ([∆ op , Set] D )D(where we re-use the notation D andD for the décalage comonad on [∆ op , Set] and the induced monad on [∆ op , Set] D ) whose essential image comprises just thoseD-algebras in [∆ op , Set] D whose underlying simplicial set satisfies the Segal condition. On the other hand, we have a straightforward characterisation of the category ([∆ op , Set] D )D: (D2) A cardinality functor \|-\| : C → S; (D3) For all X ∈ C and i ∈ \|X\| a fibre functor ϕ X,i : C/X → C notated as before; (D4) For each f : Y → X in C and i ∈ \|X\|, a relabelling functionThese data are subject to the following axioms, which are as for an operadic category, except that (A3) and and (A5) are suitably modified to take account of the relabelling functions of (D4). In stating (A5-lax), we write γj and εj for theFor all X ∈ C and i ∈ \|X\|, the object (1 X ) −1 (i) is chosen terminal; (A3-lax) For all g : f g → f in C/X and i ∈ \|X\|, the fibre map is compatible with relabelling, in the sense that \|gFor X ∈ C, one has τ −1 X ( * ) = X, and for f : Y → X, one has f τ X * = f ; (A5-lax) For g : f g → f in C/X, i ∈ \|X\| and j ∈ \|f \| −1 (i) one has that (g f i ) −1 (γj) = g −1 (εj) and that the square left below commutes:whereγ f g,i is the unique map making the square right above commute.Given moreover h : f gh → f g in C/X, one has (h f g i )A strictly local-terminal-preserving functor F : C → C between lax-operadic categories is called a lax-operadic functor if it comes endowed with a natural family of relabelling functions ν X : \|X\| → \|F X\|, which are compatible with fibre functors in the sense of rendering commutative each diagram of the form:We write LaxOpCat for the category of lax-operadic categories and lax-operadic functors.It is perhaps worth type-checking the display in (A5-lax) to see that it makes sense. In the left square, the left edge is well-defined simply by computing cardinalities of fibres; while the right edge is well-defined by the equality (g f i ) −1 (γj) = g −1 (εj) asserted directly beforehand. In the right square, for the factorisationγ f g,i to exist, we must know that γ f g,i maps each k ∈ \|f g\| −1 (i) with \|g\|	[]
[ "Gravitational distortion on photon state at the vicinity of the Earth", "Gravitational distortion on photon state at the vicinity of the Earth" ]	[ "Qasem Exirifard \nDepartment of Physics\nUniversity of Ottawa\n25 Templeton StK1N 6N5OttawaOntarioCanada\n", "Ebrahim Karimi \nDepartment of Physics\nUniversity of Ottawa\n25 Templeton StK1N 6N5OttawaOntarioCanada\n" ]	[ "Department of Physics\nUniversity of Ottawa\n25 Templeton StK1N 6N5OttawaOntarioCanada", "Department of Physics\nUniversity of Ottawa\n25 Templeton StK1N 6N5OttawaOntarioCanada" ]	[]	As a photon propagates along a null geodesic, the space-time curvature around the geodesic distorts its wave function. We utilise the Fermi coordinates adapted to a general null geodesic, and derive the equation for interaction between the Riemann tensor and the photon wave function. The equation is solved by being mapped to a time-dependent Schrödinger equation in (2 + 1) dimensions. The results show that as a Gaussian time-bin wavepacket with a narrow bandwidth travels over a null geodesic, it gains an extra phase that is a function of the Riemann tensor evaluated and integrated over the propagation trajectory. This extra phase is calculated for communication between satellites around the Earth, and is shown to be measurable by current technology.	10.1103/physrevd.105.084016	[ "https://arxiv.org/pdf/2110.13990v2.pdf" ]	239,998,321	2110.13990	2ab49a1e497dfd70fa228d41e345d9cd5111b85d	Gravitational distortion on photon state at the vicinity of the Earth Qasem Exirifard Department of Physics University of Ottawa 25 Templeton StK1N 6N5OttawaOntarioCanada Ebrahim Karimi Department of Physics University of Ottawa 25 Templeton StK1N 6N5OttawaOntarioCanada Gravitational distortion on photon state at the vicinity of the Earth As a photon propagates along a null geodesic, the space-time curvature around the geodesic distorts its wave function. We utilise the Fermi coordinates adapted to a general null geodesic, and derive the equation for interaction between the Riemann tensor and the photon wave function. The equation is solved by being mapped to a time-dependent Schrödinger equation in (2 + 1) dimensions. The results show that as a Gaussian time-bin wavepacket with a narrow bandwidth travels over a null geodesic, it gains an extra phase that is a function of the Riemann tensor evaluated and integrated over the propagation trajectory. This extra phase is calculated for communication between satellites around the Earth, and is shown to be measurable by current technology. I. INTRODUCTION Einstein gravity is a geometric theory for gravity wherein energy/mass distribution curves its surrounding space-time geometry and particles propagate along the geodesics of the curved geometry. The light bending by a gravitational source, manifesting that photons propagate along null geodesics, was first observed by Eddington et al. [1] with 30% precision, and the effect has now been measured with the precision of 0.01% [2][3][4][5]. In light bending as well as all other observed effects of Einstein gravity, the photon is treated as a point-like particle [6]. The photon, however, is governed by the rules of quantum mechanics where particle-wave duality is manifest. As a photon moves along the geodesic, its quantum wave function interacts with the curvature of the space-time geometry around the geodesic and gets distorted. Whenever the photon wave function is used as an information carrier [7][8][9][10][11], the distortion affects the communication channel and may introduce errors. The current race to establish a quantum network in space [12][13][14] may need to take into account how the curvature of the space-time geometry affects the wave function of the photon as it moves along a geodesic. Approximations, where all the multi-polar modes are neglected, are used to calculate the Green function for the propagation, which resulted in a measurable distortion [15][16][17][18]. Jonsson et al. [19] has kept the first few multi-polar modes to calculate the distortion for a photon scattered from a black hole. Though none of these methods return the exact distortion, they highlight that the distortion is a substantial effect. In Ref. [20], the Fermi coordinates along a geodesic is adapted, and a distortion of measurable magnitude is reported for communication between the Earth and the International Space Station. However, turbulence due to the Earth's atmosphere adds noise, and may not allow measurement of the effect. Here, we aim to calculate the gravitational distortion for communication per- * Electronic address: [email protected] † Electronic address: [email protected] formed between two satellites around the Earth where atmospheric effects are absent. The paper is organised as follows: Section II considers a photon that propagates along a null geodesic in a general curved space-time geometry. The photon is approximated to a time-bin wave packet with a very narrow frequency line. The Fermicoordinates adapted to a null geodesic are utilised to calculate the interaction between the photon's wave function and the curvature of the space-time geometry around the geodesic. We show that as the photon propagates along the null geodesic, it gains an extra phase that is a function of the Riemann tensor evaluated and integrated over the null geodesic. This extra phase, which depends on the space-time geometry, is calculated for communication between satellites near the Earth. The results are shown in Section III, and Section IV provides the discussion on how to measure this extra phase. II. PREDICTING A GENERAL GEOMETRIC PHASE The photon moves on the null geodesic γ. We adapt the Fermi frame wherein the metric on the geodesic coincides to the Minkowski metric, and Levi-Cevita symbol vanishes too: g µν \| γ = η µν ,(1)Γ µ νη \| γ = 0.(2) We represent the coordinates in the Fermi frame by (x ± , x 1 , x 2 ) where x ± = 1 √ 2 (x 3 ± ct) are the Dirac lightcone coordinates while x + is tangent to γ [21]. The metric around the geodesic in the Fermi coordinates up to quadratic order in the transverse coordinates is given by [22]: ds 2 = 2dx + dx − + δ ab dx a dx b − R +ā+b xāxb(dx + ) 2 − 4 3 R +bāc xbxc(dx + dxā) − 1 3 Rācbdxbxc(dxādxb) + O(xāxbxc) ,(3) where x a = (x 1 , x 2 ), and xā = (x − , x a ), and all the curvature components (R +ā+b , R +ācd and Rābcd) are arXiv:2110.13990v2 [gr-qc] 23 Mar 2022 evaluated on γ, and Einstein's notation is used wherein twice appearance of an index variable in a single term means summation over that index, and δ ab is the Kronecker delta. We approximate the space-time geometry around the Earth to the Schwarzschild space-time geometry. Therefore, the effective Lagrangian of a massless point particle propagating on a null geodesic can be given by, L = − 1 − m r ṫ 2 +ṙ 2 1 − m r + r 2 θ 2 + sin 2 θφ 2 ,(4) where dot presents variation with respect to the affine parameter on the geodesic, m = (2G N M ⊕ )/c 2 , and M ⊕ is the mass of the Earth. We choose the units such that the speed of light in vacuum is set to 1, i.e. c = 1, and m = 1. Due to the spherical symmetry, without loss of generality, we can choose the equatorial plane θ = π/2 andθ = 0 to describe any given geodesic at all time. The cyclic variables of ϕ and t lead to invariant quantities: r 2φ = l, (1 − 1/r)ṫ = E, where l and E are constant values. We consider null geodesics reaching the asymptotic infinity, and set E = 1. Due to the form of the Lagrangian, its Legendre transformation, which is the Lagrangian itself, is invariant. We consider a null geodesic, and set L = 0 that gives \|ṙ\| = 1 − 1 r 2 1 − 1 r l 2 . Letê t ,ê r ,ê ϕ andê θ represent the normalized unit vectors in t, r, θ, ϕ coordinates. The normalized unit vectors in the Fermi coordinates can be chosen aŝ e + = f √ 2 (+ê t +ṙê r + rfφê ϕ ),(5a)e − = 1 √ 2f (−ê t +ṙê r + rfφê ϕ ) ,(5b)e 1 = −rfφê r +ṙê ϕ ,ê 2 =ê θ ,(5c) where f = 1 − 1/r. The components of the Riemann tensor in the Fermi coordinates should be computed by the tensor transformation law [23]: R αβγδ = R µ ν σ τ (ê α ) µ (ê β ) ν (ê γ ) σ (ê δ ) τ .(6) Utilizing the abstract method employed in [20] identifies the non-vanishing components of the Riemann tensor in the Fermi frame R +−+− = 3l 2 (r − 1) 2r 6 − 1 r 3 ,(7a)R +2+2 = −R +1+1 = 3l 2 4r 5 ,(7b)R +−+1 = 3 √ 2l 2r 4ṙ .(7c) Next, we consider the electromagnetic potential A µ , whose field strength is given by F µν = ∂ µ A ν −∂ ν A µ . The dynamics of the electromagnetic potential in a curved space-time geometry endowed with metric g µν around the geodesic, and is given by, Γ[A µ ] = − 1 4 d 4 x g µµ νν F µν F µ ν ,(8a)g µµ νν = 1 2 − det g (g µµ g νν − g µν g νµ ) . (8b) Here, g µν is the inverse of the metric, and det g is its determinant. Note that g µµ νν has all the symmetries of F µν F µ ν under exchange of its indices. The functional variation of the action with respect to the gauge field gives its equation of motion, i.e., ∂ µ g µµ νν F µ ν = 0 ,(9) which we would like to solve for a photon that travels along a null geodesic. We choose the Fermi coordinates adapted to the null geodesic, Eq. (3), to describe the space-time at the vicinity of the geodesic where the components of the Riemann tensor are given in Eq. (7). At the vicinity of the Earth, the components of the Riemann tensor are minimal. We, therefore, treat them as a perturbation. We introduce ε as the systematic parameter of the perturbation. In other words, we add a factor of ε to all terms in Eq. (3) where the components of the Riemann tensor are present. At the end of the computation, we set ε = 1. The ε-perturbation to the electromagnetic potential and g µµ νν follow: A µ = A (0) µ + εA (1) µ + O(ε 2 ) ,(10a)g µµ νν = g (0)µµ νν + εg (1)µµ νν + O(ε 2 ) . (10b) Equation (9) at the leading order in ε can be simplified to (0) A (0)µ + ∂ µ ∂ ν A (0)ν = 0.(11) Henceforth η µν is utilized to move up or down the indices, i.e., A (0)ν = η νλ A (0) λ , ∂ µ = η µν ∂ ν , where η µν represents the Minkowski metric in Dirac coordinates. We choose the Lorentz gauge, ∂ ν A (0)ν = 0, which simplifies the equation for A (0) to (0) A (0) µ = 2∂ + ∂ − + ∇ 2 ⊥ A (0) µ = 0 where ∇ 2 ⊥ is the Laplace operator in x 1 and x 2 direc- tions: ∇ 2 ⊥ = ∂ 2 1 + ∂ 2 2 . Utilizing the Fourier expansion of the gauge field in terms of the variable x − , i.e., A (0) µ = dωf (0) µ (ω, x + , x a )e iωx − leads to, (2iω∂ + + ∇ 2 ⊥ )f (0) µ = 0,(12) which is referred to as the paraxial Helmholtz equation. Note that due to the definition ofê − in Eq. (5b), the gravitational red-shift is already encoded in the Fourier expansion of A µ . We refer to f (0) µ (w, x + , x a ) as the structure function of the photon with frequency ω in mode µ. The structure mode can be expanded in terms of the Hermite-Gaussian or Laguerre-Gaussian modes. For the purpose of communication, we are interested in a field configuration that can be understood as a perturbation modulated over a frequency that holds: \|∂ + f (0) ν \| ω\|f (0) ν \|, \|∂ − f (0) ν \| ω\|f (0) ν \| (13) which is the same as the paraxial approximation in optics. Employing Eq. (13) in the Lorenz gauge condition yields: ωf (0) + + ∂ + f (0) − + ∂ 1 f (0) 1 + ∂ 2 f (0) 2 = 0.(14) The paraxial approximation conditions, i.e. Eq. (13), imply the following perturbative solutions: f (0) + = 0 , (15a) ∂ + f (0) − + ∂ a f (0) a = 0 ,(15b) where ∂ a f (0) a = ∂ 1 f (0) 1 + ∂ 2 f (0) 2 is used. We solve Eq. (15b) for f (0) − . This leaves f (0) a as the physical modes, which can be perceived as the distribution of the photon's polarization. This means that each polarization of photon that we choose to represent by Ψ, satisfies Eq. (12). The paraxial wave equation, Eq. (12), can be rewritten as − 1 2ω ∇ 2 ⊥ Ψ = i∂ + Ψ, which is the Schrödinger equation for a particle with a rest mass of ω in"2+1" dimensions where x + plays the role of time. Utilizing Eqs. (10) and (11) in Eq. (9) yields the equation of motion for A (1) , (0) A (1)µ = −∂ ν g (1)µµ νν F (0) µ ν ,(16) where Lorenz gauge condition is assumed on A (1) too. The propagation of a photon in a smooth space-time geometry holds \|∂ λ g (1)µν \| ω\|g (1)µν \|. Therefore, the derivative of the components of the metric on the right hand side of Eq. (16) can be neglected, and thus we have (0) A (1)µ = −g (1)µµ νν ∂ ν F (0) µ ν . We express F (0) µ ν in terms of A (0) µ : (0) A (1)µ = −g (1)µµ νν (∂ ν ∂ µ A ν − ∂ ν ∂ ν A µ ) ,(17) The paraxial approximation expressed in Eq. (13) implies that the dominant term on the right hand side of Eq. (17) is the one that ∂ 2 − acts on A (0) . Keeping only the dominant term results (0) A (1)µ = g (1)−−µα − g (1)−αµ− ∂ 2 − A (0) α .(18)(0) A (1) + = 0, (19a) (0) A (1) a = −g (1)−− ∂ 2 − A (0) a .(19b) g (1)−− can be expressed in terms of the components of the Riemann tensor, and thus we have, (0) A (1) i = −R +ā+b xāxb∂ 2 − A (0) i .(20) We would like to consider the Fourier transformation of A (1) i with respect to the variable x − , i.e., A (1) i = dωf (1) i (ω, x + , x a )e iωx − , where f (1) i is the correction to the structure function of mode i. Utilizing the Fourier transformations in Eq. (20) yields, (2iω∂ + + ∇ 2 ⊥ )f (1) i = −R +−+− ∂ 2 ω + 2iR +−+−a x a ∂ ω + ω 2 R +a+b x a x b ω 2 f (0) i .(21) We observe that different physical modes, i.e., different i, are not coupled at the sub-leading order. Therefore, without loosing generality, we consider one physical mode, and we set i = 1. However, all the results that we will calculate, will be equally valid for i = 2. We consider f (0) 1 in the form of f (0) 1 = A(ω)f mn (ω, x + , x a ), where f mn is the Hermite-Gaussian mode, and A(ω) is the amplitude that we choose as a normal distribution around ω = ω 0 with the width of σ for the first polarization of photon: A(ω) = 1 √ σπ 1 4 exp − (ω−ω0) 2 2σ 2 . We consider a time-bin wavepacket with a narrow bandwidth such that the first term on the right hand side of Eq. (21) is the dominant term. Since σ is very small, we can utilise f (0) 1 ≈ A(ω)f mn (ω 0 , x + , x a ) . This approximation allows us to neglect ∂ ω f mn in derivatives of f (0) 1 with respect to ω. Size of the wave packet is identified by two parameters. Its size perpendicular to its trajectory is given by the width of the beam while its size in direction of propagation is proportional to c σ . We assume c σ is much larger than the width of package, and the wavepacket is extended in the direction of propagation where the first term on the right hand side of Eq. (21), becomes the dominant term. Keeping only the dominant term yields (2iω∂ + + ∇ 2 ⊥ )f (1) 1 = −R +−+− ω 2 (ω − ω 0 ) 2 σ 4 + ω(4ω 0 − 5ω) σ 2 f (0) 1 ,(22) where 2f (0) 1 on the right hand side of Eq. (22) is also neglected. Equation (22) can be perceived as the perturbation of 2iω∂ + + ∇ 2 ⊥ Ψ = 2εωV (x + )Ψ,(23) where Ψ = f (0) 1 + εf (1) 1 + O(ε 2 ),(24a)V (x + ) = − ω(ω − ω 0 ) 2 2σ 4 + 4ω 0 − 5ω 2σ 2 R +−+− (x + ).(24b) Equation (23) can be rewritten as, i∂ + Ψ = − 1 2ω ∇ 2 + εV (x + ) Ψ,(25) which is the Schrödinger equation for a particle with a mass "ω" in (2+1) dimensions with a time-dependent potential where the potential is only a function of time. For any Ψ (0) that satisfies i∂ + Ψ (0) = − 1 2ω ∇ 2 Ψ (0) , the perturbative solution to Eq. (25) is Ψ = Ψ (0) (1 + εχ(x + )) where χ = −i x + 0 dτ V (τ ) . This implies that the correction to the structure function, f 1 , is expressed in term of the structure function, f 1 , i.e., f(0)1 = if (0) ω(ω − ω 0 ) 2 2σ 4 + 4ω 0 − 5ω 2σ 2 G,(1) where G, the geometrical factor, is given by G = x + 0 dτ R +−+− (τ ).(27) Here, τ is the affine parameter on the geodesic and x + = 0 is the wave-packet initial plane. Recalling that f 1 and A (1) 0 , allows us to integrate over ω, and obtain the electromagnetic field: A 1 = √ 2σπ 1 4 e − (σx − ) 2 2 +iω0x − f mn (ω 0 , x + , x a ) × 1 − i ω 0 G 2 (x − ) 2 ,(28) where only the dominant term is kept, and we set ε = 1 (note that ε is a dummy parameter to systematically track the perturbation). We could have chosen i = 2 to obtain the same expression for the second polarization, i.e., A 2 . We, therefore, observe that as a Gaussian timebin wave-packet, with sharp width of σ around frequency of ω 0 , travels over the geodesic, and it gains an extra geometric phase that is given by, χ g = − ω 0 G 2 (x − ) 2 ,(29) where G is the integration of "+ − +−" component of the Riemann tensor evaluated on the geodesic, as defined in Eq. (27). Equation (29) is in accord with [20] wherein the equations are solved by a different method. Let it be emphasized that χ g is the change in the phase of a Gaussian beam with the width of σ. There exist some difficulties associated with measuring χ g at the far tail (\|σx − \| ≥ 5) of the Gaussian beam because the amplitude decreases exponentially, and it would not be easy to generate a Gaussian beam whose far tail remains Gaussian too. To avoid these problems, we suggest to measure χ g around the peak of the Gaussian beam, or equivalently for \|σx − \| 1. In doing so, it is convenient to re-express χ g to χ g = − ω 0 G 2σ 2 (σx − ) 2 ,(30) and note that χ g is measured for \|σx − \| 1. Equation (30) explicitly shows that the maximum measurable value of χ g depends on σ. Figure 1 depicts the amplitude, the initial phase and the change in the phase in term of σx − for −1.5 ≤ σx − ≤ 1.5. III. GEOMETRIC PHASE FOR COMMUNICATION BETWEEN SATELLITES Let us consider a communication link where Alice (sender) and Bob (receiver) are on different satellites located at radii of r = a and b from the centre of the Earth, respectively, where a ≤ b. In this section, we assume that Alice and Bob are stationary with respect to the standard spherical coordinates of the Schwarzschild geometry. In next section, we utilise relativistic Doppler shift to generalise the result to the case that Alice and Bob are not stationary. We use α to represent the angular separation of the two satellites; considering the lines from satellites to the centre of Earth, α is the angle between these two lines. First case, as shown in Fig. 2a forṙ ≥ 0: The "+ − +−" component of the Riemann tensor evaluated on the geodesic is given in Eq. (7a) which for large r can be approximated to: R +−+− = 3l 2 2r 5 − 1 r 3 . Applying the same approximation onṙ holdsṙ = 1 − l 2 r 2 . The geometrical factor defined in Eq. (27) then is given by G = b a dṙ r R +−+− = √ b 2 − l 2 2b 3 − √ a 2 − l 2 2a 3 .(31) Here l is the minimum of the distance between the centre of the Earth and the line or the extrapolation of the line connecting Alice and Bob, i.e., l = ab sin α √ a 2 +b 2 −2ab cos α . The geometrical phase, Eq. (30), is then given by, fig. 2b,ṙ can be negative in some parts of the geodesic. For the geometrical factor defined in (27), therefore, we can write: χ g = + ω 0 cm ⊕ (b 2 − a 2 ) 4(σab) 2 (a − b cos α) 2 a 2 + b 2 − 2ab cos α σx − c 2 ,(32)G = dτ R +−+− = − l a dr \|ṙ\| R +−+− + b l dr \|ṙ\| R +−+− = 2 a l dr \|ṙ\| R +−+− + b a dr \|ṙ\| R +−+−(33) that leads to χ g = − ω 0 cm ⊕ (b 2 + a 2 ) 4(σab) 2 (a − b cos α) 2 a 2 + b 2 − 2ab cos α ( σx − c ) 2 . (34) IV. ON MEASURING THE GEOMETRIC PHASE NEAR EARTH In the following, we would like to evaluate the geometrical phase for a set of parameters to see if the geometrical phase can be detected in communication between two satellites around the Earth. In so doing, we first would like to generalise the result of the previous section to the case that Alice and Bob are not stationary. Alice at position of r a prepares a time-bin Gaussian pulse with the mean frequency of ω A and line-width of σ A . The pulse that Alice produces in Alice's rest frame is given by: A Alice = A 0 Alice e − (σ A x − ) 2 2 +iω A x − .(35) Alice moves with velocity of v A with respect to the local Riemann coordinates at r a , which is stationary with respect to the standard spherical coordinates in the Schwarzschild geometry. In the local Riemann coordinates at r a , since the source of the pulse moves with velocity of v, the pulse at the event of its generation is given by, A a = A 0 a e − (σa x − ) 2 2 +iωax − .(36) where ω a = ∆ va ω A , σ a = ∆ va σ A , and ∆ va stands for the relativistic Doppler shift. We should still transform this pulse to the Fermi coordinates. Noticing the factor of f and 1/f in the right-hand side of Eq. (5a) and (5b), the pulse in the Fermi coordinates, at the event of its generation, can be derived from the pulse in the local Riemann coordinates by scaling the frequencies: A 1 = A 0 e − (σx − ) 2 2 +iω0x − .(37) where ω 0 = ∆ va f (r a )ω A , σ = ∆ va f (r a )σ A .(38) Here f = 1 − 1/r, see the text after Eq. (5c). As the pulse moves toward Bob's geodesic, it gains an extra geometric phase. At the time of its detection, the pulse in the Fermi coordinates is given by, A 1 = A 0 e − (σx − ) 2 2 +iω0x − exp(−iχ g ),(39) where χ g is given in Eq. (30). Bob is moving with velocity v B with respect to the local Riemann coordinates at r b . The pulse that Bob observes can be obtained by transforming the beam from the Fermi coordinates to the local Riemann coordinates, and then to the Bob's rest frame. Bob in his rest frame observes A Bob = A 0 e − (σ B x − ) 2 2 +iω B x − exp(−iχ g ),(40) where ω B = f (r a ) f (r b ) ∆ va ∆ v b ω A , σ B = f (r a ) f (r b ) ∆ va ∆ v b σ A . (41) Here, ∆( v a )∆( v b ) accounts for relativistic Doppler shift, while f (r a )/f (r b ) describes the gravitational red-shift. Equations (38) and (41) can be utilised to re-express the geometric phase by: χ g = − ω B f (r b )G 2∆ v b σ 2 B (σ B x − ) 2 ,(42) where (38) and (41) are used to express ω 0 in Eq. (30) in term of ω B . We notice that Bob can interpret χ g as a time-dependent phase modulated over the Gaussian time-bin wave packet with the mean frequency ω B and line-width of σ B [24]. So, Bob can measure it. For terrestrial satellites with velocities less than 10 4 mph, \|∆ v − 1\| ≤ 10 −5 . For satellites around the Earth, \|f (r b ) − 1\| ≤ 10 −9 . So in measuring the geometric phase with a precision larger than 0.001 percent, the Doppler and gravitational effects can be neglected, and the geometric phase that Bob observes can be approximated to χ g = − ωG 2σ 2 (σx − ) 2(43) where ω and σ respectively respectively represent the mean frequency and the line-width. It is worth noting that in our notation, a plane-wave in x + -direction is expressed as e iωx − = e i ω √ 2 (x 3 −ct) . In optics, however, a plane-wave in x 3 -direction is represented by e iω(x 3 −ct) . Thus, what we define as a frequency is √ 2 times the notation in optics. The geometrical phase presented in Eq. (32) in the standard optic notation is χ s g1 = f 1 (α) y 2 , where y = σ(x 3 − ct) c , (44a) f 1 (α) = √ 2ν 0 cm ⊕ (b 2 − a 2 ) 16π(σab) 2 (a − b cos α) 2 a 2 + b 2 − 2ab cosα ,(44b) where ω 0 is replaced with √ 2ν s 0 /2π, instead of ν 0 /2π, and σ is changed to √ 2σ. The geometrical factor presented in Eq. (34) in the standard optical notation reads χ s g2 = f 2 (α) y 2 , where f 2 (α) = − √ 2ν 0 cm ⊕ (b 2 + a 2 ) 16π(σab) 2 (a − b cos α) 2 a 2 + b 2 − 2ab cos α ,(45) and y is defined in Eq. (44a). In order to understand how the geometric factor depends on the Newton Gravitational constant, Heisenberg constant and speed of light in vacuum, we treat the photon as a particle entity with an energy of E 0 = ν 0 and variance of ∆E 0 = σ. The geometric phase, then, can be expressed by, χ g = l 2 pf (a, b, α) × (M ⊕ c 2 )E 0 (∆E 0 ) 2 ,(46) where l 2 p = G N c 3 is the Planck's length, whilef (a, b, α) encodes the geodesic's details and has the unit dimension of the inverse length squared, and (M⊕c 2 )E0 (∆E0) 2 encodes properties of the pulse. Near the Earth, l 2 pf can be estimated to be at the order of 10 −84 . The factor of (M⊕c 2 )E0 (∆E0) 2 can become arbitrary large in the limit of ∆E 0 → 0, but this divergence points to the break of the perturbative methods in calculating the geometric phase and demands non-perturbative derivation of the geometric phase. The ultra-stable lasers with bandwidth of 5 mHz at 194 THz reported in [25][26][27] gives rise to (M⊕c 2 )E0 (∆E0) 2 at the order of 10 94 . We, however, notice that σ should satisfy some other conditions: Here, δχ is calculated by perturbative methods, so a value should be chosen for σ that results in \|χ g \| ≤ 1. The length of the wave packet in the direction of the propagation is given by c σ , and the employed method has assumed that the Riemann tensor is constant within the wave packet. We also have implicitly assumed that the whole of the wave packet propagates in the space before its detection. The choice of bandwidth of 5 mHz at 194 THz violates these conditions. However, bandwidth at a few kHz satisfies these conditions and (as shown below) leads to a measurable value of χ g . In order to consistently neglect the effect of atmosphere on the geometrical factor, let us consider communication between satellites in space, and choose a = 7, 000 km and b = 7, 500 km, respectively. We notice that commercial portable continuous lasers with a line-width of 1 Hz at We do not want the signal to enter the atmosphere below the altitude of 400 km, where the air density is about 10 −12 kg/m 3 . Thus, the gravitational effects are a couple of orders larger than the atmosphere's diffraction. This constrains α to \|α\| ≤ 0.675 , where the saturation occurs when the line connecting the satellites is tangent to the orbit with a radius of 6800 km. Figure 3 depicts the value of the geometrical phase divided by y 2 for the allowed range of α. The dependency on α is very non-trivial. For the chosen parameters, the value of χ[α]/y 2 ranges from 0.05 to −0.187. This means that the geometrical phase at y = 1, depending on the value of α, varies in the range of 0.05 to −0.187 Radians which can be measured. One may choose smaller values of σ to obtain a larger geometrical phase. It, however, should be noted that the geometrical phase is calculated by perturbative methods. So the choices leading to a large geometric factor can not be supported by perturbative methods developed in this work. The choice of σ = 3.16 kHz for ν 0 = 2.87×10 15 Hz, and a = 7, 000 km and b = 7, 500 km as depicted in Figure 3 leads to a measurable geometric phase consistent with a perturbative calculation. V. CONCLUSIONS As the photon's wave-function travels along a null geodesic, it interacts with the Riemann tensor around the geodesic. The interaction distorts the photon's wavefunction. Here, the Fermi coordinates along the null geodesic have been utilized. The equations for the U (1) gauge field theory in the Fermi coordinates have been calculated. The equation for the interaction between the Riemann tensor and the photon's wave-function has been derived and mapped to a time-dependent Schrödinger equation in (2 + 1) dimensions. It has been shown that as a Gaussian time-bin wave-packet, with a sharp width of σ around the frequency of ω 0 travels over the null geodesic, it gains an extra geometric phase given by χ g = − ω0G 2σ 2 (σx − ) 2 , where G is "+ − +−" component of the Riemann tensor in the Fermi coordinates evaluated on and integrated over the null geodesic: G = R +−+− (x + )dx + , where x + represents the coordinate of Fermi frame tangent to the central null geodesic, and the integration is performed from the event of generation of the pulse to the event of its detection. The space-time geometry outside the Earth has been approximated by the Schwarzschild space-time geometry. The geometrical phase has been calculated for a signal sent between two satellites, located at radii of a and b, respectively. The current commercial ultrastable continuous-wave lasers (wavelength of 657 nm and σ = 3.16 kHz) have been utilized to calculate the geometrical phase between satellites at radii 7, 000 km and 7, 500 km. It has been shown that for the chosen range of the parameters, the geometrical phase within the peak of the Gaussian pulse varies from 0.05 to −0.187 Radians, as depicted in Fig. 3. This illustrates that the predicted geometrical phase can be measured by the currently available commercial devices. The geometrical phase calculated in the current work is consequent of applying quantum field theory in curved space-time geometry. The three paradigms of special relativity, general relativity, and quantum mechanics are equally important in this derivation. It, therefore, is a prediction of how Einstein's gravity "talks" to the quantum realm. Hence, measuring this phase will provide the first experimental datum on if and how gravity affects the quantum realm. FIG. 1 : 1The initial amplitude (shown in blue), the initial phase which chosen to be zero (shown in red), and the geometric phase multiplied by − σ 2 ω 0 G (shown in purple) in term of σx − . FIG. 2 : 2the entire journey of the signalṙ > 0. (b) At the beginningṙ < 0 thenṙ > 0. Alice at radius a, at one instant, sends a signal toward Bob at radius b with a < b. The signal propagates along a null geodesic that can be approximated to a straight line. Notice that l is the minimum distance between the centre of the Earth and the line, or the extrapolation of the line, connecting Alice and Bob. α is the angle between lines connecting Alice and Bob to the centre of the Earth. Alice and Bob's trajectories are not fixed. There exist two scenarios depicted above. In the first scenario,ṙ ≥ 0, while in the second caseṙ can be negative. where the Schwarzschild radius of the Earth m ⊕ = 2GM⊕ c 2 = 8.87 millimeters, and c is recovered. Equation (32) for α = 0 coincides to the result reported in [20] for radial communication between the Earth and the International Space Station. Second case as shown in Fig. 2b: For α ≥ arccos a b as depicted in FIG. 3 : 3The geometrical phase is a time-dependent phase quadratic in y = σ(x 3 −ct) c modulated on a time-bin Gaussian wave-packet. The coefficient of the quadratic term depends on the distance between two satellites and their apparent angle as seen from the centre of the Earth. This figure depicts the dependency of the coefficient of the quadratic term in the geometrical factor for satellites at a = 7, 000 km and b = 7, 500 km, and for Gaussian time-bin communication performed with ν0 = 2.87 × 10 15 Hz, σ = 3.16 kHz . The vertical axis is the coefficient of the quadratic term in the geometric factor χg y 2 ; the horizontal axis is the apparent angle. the wavelength of 657 nm exists [28]. These can be used to construct a sharp Gaussian time-bin with line-width of σ = 3.16 kHz for ν 0 = 2.87 × 10 15 Hz. Using these numerical values simplify f 1 and f 2 to: f 1 (α) = 0.055 × \|7.5 cos α − 7\| √ 105.25 − 105 cos α , (47a) f 2 (α) = − 0.802 × \|7.5 cos α − 7\| √ 105.25 − 105 cos α . Due to gauge symmetry and the chosen Lorentz gauge, we choose to solve Eq. (18) for µ = −, 1, 2. Substituting Eq. (15) into Eq. (18) results in AcknowledgmentsThis work was supported by the High Throughput and Secure Networks Challenge Program at the National Research Council of Canada, the Canada Research Chairs (CRC) and Canada First Research Excellence Fund (CFREF) Program, and Joint Centre for Extreme Photonics (JCEP). We thank Felix Hufnagel for proofreading the paper. Testing relativity from the 1919 eclipse-a question of bias. Daniel Kennefick, Physics Today. 6237Daniel Kennefick. Testing relativity from the 1919 eclipse-a question of bias. Physics Today, 62:37, 2009. Herring. 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[ "Direct Constraints on Minimal Supersymmetry from Fermi-LAT Observations of the Dwarf Galaxy Segue 1", "Direct Constraints on Minimal Supersymmetry from Fermi-LAT Observations of the Dwarf Galaxy Segue 1" ]	[ "Pat Scott \nOskar Klein Centre for Cosmoparticle Physics\nDepartment of Physics\nDepartment of Physics\nOskar Klein Centre for Cosmoparticle Physics\nStockholm University AlbaNova\nSE-10691StockholmSweden\n", "Jan Conrad [email protected] \nDepartment of Physics\nOskar Klein Centre for Cosmoparticle Physics\nStockholm University AlbaNova\nSE-10691StockholmSweden\n", "Joakim Edsjö \nDepartment of Physics\nOskar Klein Centre for Cosmoparticle Physics\nStockholm University AlbaNova\nSE-10691StockholmSweden\n", "Lars Bergström \nLaboratoire de Physique Thorique et Astroparticules\nStockholm University AlbaNova\nSE-10691StockholmSweden\n", "Christian Farnier [email protected] \nDepartment of Physics\nOskar Klein Centre for Cosmoparticle Physics\nCNRS/IN2P3\nUniversité Montpellier II\nCC 70, Place Eugne BataillonF-34095, Cedex 5MontpellierFrance\n", "Yashar Akrami [email protected]:0909.3300v2[astro-ph.co] \nStockholm University AlbaNova\nSE-10691StockholmSweden\n" ]	[ "Oskar Klein Centre for Cosmoparticle Physics\nDepartment of Physics\nDepartment of Physics\nOskar Klein Centre for Cosmoparticle Physics\nStockholm University AlbaNova\nSE-10691StockholmSweden", "Department of Physics\nOskar Klein Centre for Cosmoparticle Physics\nStockholm University AlbaNova\nSE-10691StockholmSweden", "Department of Physics\nOskar Klein Centre for Cosmoparticle Physics\nStockholm University AlbaNova\nSE-10691StockholmSweden", "Laboratoire de Physique Thorique et Astroparticules\nStockholm University AlbaNova\nSE-10691StockholmSweden", "Department of Physics\nOskar Klein Centre for Cosmoparticle Physics\nCNRS/IN2P3\nUniversité Montpellier II\nCC 70, Place Eugne BataillonF-34095, Cedex 5MontpellierFrance", "Stockholm University AlbaNova\nSE-10691StockholmSweden" ]	[]	The dwarf galaxy Segue 1 is one of the most promising targets for the indirect detection of dark matter. Here we examine what constraints 9 months of Fermi -LAT gamma-ray observations of Segue 1 place upon the Constrained Minimal Supersymmetric Standard Model (CMSSM), with the lightest neutralino as the dark matter particle. We use nested sampling to explore the CMSSM parameter space, simultaneously fitting other relevant constraints from accelerator bounds, the relic density, electroweak precision observables, the anomalous magnetic moment of the muon and B-physics. We include spectral and spatial fits to the Fermi observations, a full treatment of the instrumental response and its related uncertainty, and detailed background models. We also perform an extrapolation to 5 years of observations, assuming no signal is observed from Segue 1 in that time. Results marginally disfavour models with low neutralino masses and high annihilation cross-sections. Virtually all of these models are however already disfavoured by existing experimental or relic density constraints.	10.1088/1475-7516/2010/01/031	[ "https://arxiv.org/pdf/0909.3300v2.pdf" ]	5,837,912	0909.3300	79af1f5c20ebbfff178ce29458f8edc0b5841c7c	Direct Constraints on Minimal Supersymmetry from Fermi-LAT Observations of the Dwarf Galaxy Segue 1 15 Dec 2009 Pat Scott Oskar Klein Centre for Cosmoparticle Physics Department of Physics Department of Physics Oskar Klein Centre for Cosmoparticle Physics Stockholm University AlbaNova SE-10691StockholmSweden Jan Conrad [email protected] Department of Physics Oskar Klein Centre for Cosmoparticle Physics Stockholm University AlbaNova SE-10691StockholmSweden Joakim Edsjö Department of Physics Oskar Klein Centre for Cosmoparticle Physics Stockholm University AlbaNova SE-10691StockholmSweden Lars Bergström Laboratoire de Physique Thorique et Astroparticules Stockholm University AlbaNova SE-10691StockholmSweden Christian Farnier [email protected] Department of Physics Oskar Klein Centre for Cosmoparticle Physics CNRS/IN2P3 Université Montpellier II CC 70, Place Eugne BataillonF-34095, Cedex 5MontpellierFrance Yashar Akrami [email protected]:0909.3300v2[astro-ph.co] Stockholm University AlbaNova SE-10691StockholmSweden Direct Constraints on Minimal Supersymmetry from Fermi-LAT Observations of the Dwarf Galaxy Segue 1 15 Dec 2009Preprint typeset in JHEP style -HYPER VERSIONdwarfs galaxiesdark matter theorysupersymmetry and cosmologygamma ray theory The dwarf galaxy Segue 1 is one of the most promising targets for the indirect detection of dark matter. Here we examine what constraints 9 months of Fermi -LAT gamma-ray observations of Segue 1 place upon the Constrained Minimal Supersymmetric Standard Model (CMSSM), with the lightest neutralino as the dark matter particle. We use nested sampling to explore the CMSSM parameter space, simultaneously fitting other relevant constraints from accelerator bounds, the relic density, electroweak precision observables, the anomalous magnetic moment of the muon and B-physics. We include spectral and spatial fits to the Fermi observations, a full treatment of the instrumental response and its related uncertainty, and detailed background models. We also perform an extrapolation to 5 years of observations, assuming no signal is observed from Segue 1 in that time. Results marginally disfavour models with low neutralino masses and high annihilation cross-sections. Virtually all of these models are however already disfavoured by existing experimental or relic density constraints. Introduction The identity of dark matter is one of the most compelling problems facing modern physics. A wealth of viable theoretical candidates have been put forward (see e.g. [1,2,3,4]), with the majority based on extensions to the standard model (SM) of particle physics. One of the more durable suggestions is that dark matter consists of weakly-interacting massive particles (WIMPs), thermally produced in the early universe and therefore naturally present in approximately the right cosmological abundance. Models of supersymmetry (SUSY) where R-parity is conserved provide a prototypical WIMP candidate in the lightest neutralino. Low-energy SUSY is also highly attractive because it generically solves the SM hierarchy problem whilst simultaneously providing a favourable framework for gauge-coupling unification and electroweak symmetry breaking [5,6]. Because the neutralino is a Majorana particle, its self-annihilation opens a potential channel for discovery via the observation of annihilation products like photons, hadrons and leptons. Self-annihilation rates are proportional to the square of the particle density, so any environment with a high density of dark matter is a good prospective target. In practice, expected backgrounds from different targets strongly influence their suitability for such indirect detection. Dwarf spheroidal galaxies have recently emerged as leading targets for gamma-ray detection of dark matter [7,8,9,10,11], thanks to their high mass-to-light ratios [7,12,13,14] and small expected astrophysical backgrounds. Segue 1 is probably the most promising object in this respect [15,16], due to its extreme dark matter domination (M/L ≈ 1320), relative proximity (23 kpc) and high latitude [14]. As with other dwarf galaxies, constraining the density profile of dark matter in Segue 1 is difficult; being small and faint, very few stars are available to act as kinematic tracers of the gravitational potential. Its spatial superimposition upon the leading arm of the Sagittarius stream [17] complicates matters further, as do the partially degenerate impacts of dark matter, bulk rotation and magnetic fields upon the stellar velocity dispersion [18]. Indeed, the status of Segue 1 as a dwarf galaxy rather than a star cluster, and therefore its domination by dark matter, have been called into question [17,19]. We will assume here that it is indeed a galaxy, an assertion strongly supported by further recent (but as yet unpublished) spectroscopic data [20]. These new data should also significantly reduce the uncertainty associated with the density profile of dark matter within Segue 1. As of the time of writing, the best available estimate of this profile comes from Markov-Chain Monte Carlo (MCMC) scans of halo parameters and corresponding solutions to the Jeans equation, based on line-of-sight velocities of 24 stars in Segue 1 [15]. The Large Area Telescope (LAT; [21]), aboard the Fermi satellite, is a high-energy, pair-conversion gamma-ray space telescope. The LAT is designed to operate predominantly in survey mode, and has been doing so since August 4, 2008. With its energy range (20 MeV to over 300 GeV) and high spatial and spectral resolution (∆E/E ≈ 12%, pointspread function < 0.1 • at 100 GeV), the LAT is well-suited to gamma-ray searches for dark matter annihilation. A major undertaking within the LAT collaboration has been to try to discover or place limits upon theories of dark matter using Fermi observations of Milky Way dwarf galaxies and satellites, the Galactic centre, the Galactic halo and extragalactic sources [22,23,24,25,26]. The detector design also facilitates direct observation of cosmicray electrons [27], another possibly relevant channel for dark matter indirect detection. By any measure, SUSY is an extensive and highly developed addition to the SM, giving rise to a wealth of potential experimental signatures beyond dark matter. Any explanation of dark matter as a neutralino must therefore satisfy a host of other phenomenological constraints. Even within the minimal supersymmetric extension of the SM (the MSSM), the low-energy phenomenology of the theory strongly depends upon the particular parameterisation employed in the soft SUSY-breaking sector, and the specific values of the chosen parameters. Given that the most general soft SUSY-breaking Lagrangian in the MSSM has over a hundred free parameters, one must choose some reduced parameterisation in order to make any progress in fitting experimental data. One approach is to employ a low-energy effective Lagrangian for the soft breaking terms, with various parameters set to zero or made equal for computational convenience (and in order to avoid experimental constraints on e.g. flavour-changing neutral currents). The alternative is to choose a specific breaking scheme, such as gravity mediation in minimal supergravity (mSUGRA) [28], gauge mediation (GMSB) [29] or anomaly mediation (AMSB) [30], where a small number of breaking parameters are defined and unified at some high energy, and the masses and couplings are run down to low energy using the renormalisation group equations (RGEs) in order to obtain phenomenological predictions. In this paper we focus on the Constrained MSSM (CMSSM) as a convenient example of one such high-energy parameterisation. This scheme is defined at the gauge coupling unification scale (∼ 10 16 GeV) in terms of 4 free continuous parameters and one sign: {m 0 , m 1 2 , A 0 , tan β, sgn µ}. (1.1) Here m 0 is the universal scalar mass, m 1 2 the gaugino mass parameter, A 0 the trilinear coupling between Higgs bosons, squarks and sleptons, tan β the ratio of vacuum expectation values of up-type and down-type Higgs bosons, and sgn µ the sign of the Higgs mixing parameter in the superpotential. We choose µ to be positive throughout this paper. The magnitude of µ is set by the requirement that SUSY breaking radiatively induces electroweak symmetry breaking; in this sense the CMSSM differs slightly from mSUGRA (where electroweak symmetry breaking is not strictly part of the definition and tan β is swapped for the parameter B -see e.g. [5]), but for nearly all intents and purposes the two can be considered equivalent. The CMSSM possesses a number of distinct regions where the relic density of the lightest neutralino matches the observed dark matter abundance (see e.g. [31,32] and references therein). The majority of the CMSSM parameter space results in too high a relic density; regions producing the correct amount of dark matter are those where some channel of neutralino destruction is especially efficient. Until recently, most analyses focused on the so-called bulk region at low m 0 and m 1 2 , where neutralino annihilation proceeds efficiently by exchange of light sleptons. This region is now mostly ruled out by collider limits on sparticle masses and difficulty in meeting Higgs mass limits when both m 0 and m 1 2 are small. The stau coannihilation region occurs at low m 0 , where the stau is almost degenerate in mass with the lightest neutralino. Here the correct relic density is achieved via co-annihilations between the two sparticles rather than any increase in the neutralino self-annihilation cross-section. A similar situation occurs in the stop coannihilation region, which exists at large negative A 0 . The stau coannihilation region is still viable, but stop coannihilation is disfavoured by low-energy experiments and Higgs constraints. In the focus point region at large m 0 , the lightest neutralino picks up a significant Higgsino component, opening new annihilation channels and boosting certain coannihilations. Finally, small 'funnels' of parameter space exist where neutralino annihilation can be increased by a mass resonance with one of the MSSM Higgs particles (i.e. where the Higgs in question has roughly twice the mass of the lightest neutralino). The focus point and funnel regions are still allowed by present experimental constraints. Scanning of MSSM parameter spaces is nowadays a highly developed art. Starting from simple grid and random scans [33,34,35,36,37] within slices of the mSUGRA parameter space, efforts expanded to MCMC searches of the full CMSSM/mSUGRA space [38], later also including the most important SM uncertainties [39,32]. As nested sampling [40,41] has come to replace the MCMC as the scanning technique of choice, hope has risen that MSSM scans might now finally be globally convergent [42,43]. New results with genetic algorithms [44], however, suggest that current scanning techniques may yet have some distance to go in this respect. Some authors have begun to focus on higherdimensional low-energy effective MSSM parameterisations [43,45,46], which provide for a broader range of phenomenological consequences but are almost impossible to scan effectively without sophisticated algorithms and substantial supercomputing resources. Ex-plorations of SUSY-breaking schemes beyond mSUGRA have also been carried out lately using similar parameter scans [47,48,49,50], as have investigations of next-to-minimal SUGRA [51,52,53]. SUSY scans are generally either based on the Bayesian posterior probability [39,42,47,43], the direct use of the frequentist likelihood [36,32,49] (usually by a χ 2 analysis), or a simple 'in-or-out' approach to individual points being permitted by experimental data [33,34,35,38]. MSSM scans have thus far focused on the constraints provided by particle experiments and the dark matter relic density determined from the microwave background, sometimes to produce corresponding predictions for astronomical observations (e.g. [15,54,55,56]). To our knowledge, none have so far included actual constraints from searches for annihilating dark matter; this is no doubt because such constraints have only recently come within a reasonable distance of model predictions. In this paper, we include the first 9 months of the search for dark matter annihilation in Segue 1 with Fermi in explicit CMSSM parameter scans. We use spectrally and spatially resolved photon counts observed by the LAT to directly assess the likelihood of the different regions in the CMSSM parameter space, then combine these with laboratory and cosmological data to perform global fits to the model parameters. We also provide a predicted impact on the parameter space after 5 years of observations. In Sect. 2 we describe our analysis techniques, before presenting results in Sect. 3 and conclusions in Sect. 4. Analysis Gamma-rays from neutralino annihilation in dwarf galaxies The expected differential gamma-ray flux per unit solid angle from a source of neutralino annihilations is (see e.g. [57]) dΦ dEdΩ = 1 + BF 8πm 2 χ f dN γ f dE σ f v l.o.s. ρ 2 χ (l)dl. (2.1) Here m χ is the neutralino mass, BF is the boost factor due to any unresolved substructure in the source, f labels different annihilation final states, dN γ f /dE is the differential photon yield from any particular final state, σ f is the cross-section for annihilation into that state, v is the relative velocity between neutralinos, and the integral runs over the line of sight to the source. In the absence of any bound states (i.e. Sommerfeld enhancements), massive neutralinos move so slowly that they can effectively be considered to collide at rest, allowing σ f v to be replaced with the velocity-averaged term in the zero-velocity limit, σ f v 0 . Three main channels contribute to the spectrum of neutralino annihilation. Through loop processes, annihilation can proceed directly into two photons [58,59] dN γ γγ dE = 2δ(E − m χ ), (2.2) or into a Z boson and a photon [60] dN γ Zγ dE = δ(E − m χ + m 2 Z 4m χ ),(2.3) giving a monochromatic gamma-ray line. A hard spectrum can also be produced by the so-called internal bremsstrahlung (consisting of final-state radiation and virtual internal bremsstrahlung), generated when a photon is emitted from a virtual particle participating in the annihilation diagram [61]. Finally, continuum gamma-rays can be produced by annihilation into quarks, leptons and heavy gauge bosons (including the Z from the Zγ line), which subsequently decay via π 0 to softer photons. The cross-sections and resultant spectral yields for each of these processes are directly calculable from the SUSY parameters which define a point in e.g. the CMSSM parameter space (after appropriate RGE running). We use DarkSUSY [62] for this calculation. The integral and boost factor in Eq. 2.1 are determined by the dark matter distribution in the astrophysical source. We use the Einasto profile [63] ρ(r) = ρ s exp −2n r r s 1 n − 1 (2.4) to describe the average dark matter content of Segue 1, where n is the Einasto index and r s and ρ s are the scale radius and density, respectively. This profile is somewhat more conservative than the traditional NFW [64], in the sense that it is less steep in the central regions, leading to generally better agreement with observations of various dark matter halos [63,65,66]. It is also slightly more dense at intermediate radii. The adopted form of the density profile actually makes little overall difference to the expected flux. This is because in general, dwarf galaxies will appear either as point sources or very close to pointlike to the LAT, meaning that observations mostly probe the full halo rather than just the central cusp. We use the best-fit values of the scale radius and scale density found by Martinez et al. [15] in their recent fits to stellar kinematic data (r s = 0.07 kpc, ρ s = 3.8 GeV cm −3 ). Since Martinez et al. found no preference for a particular Einasto index, we adopt the central value considered in their scans, n = 3.3. We note that the fits were not only influenced by the kinematic data, but also by a theoretical prior imposed by assuming the same correlation between r s and ρ s as seen in subhalo populations of theoretical Nbody simulations of cold dark matter structure formation. Whilst this presents no real problem, it is encouraging to see that additional kinematic data [20] largely dominate the prior in more recent fits. The same authors performed an extensive investigation of the possible substructure boosts in Segue 1, showing that all BF values between 0 and ∼70 are compatible with kinematic data and small-scale structure predictions within the CMSSM, with the most likely value depending strongly on the particular model employed for the concentration-mass relation. We therefore employ two indicative values for the boost factor: a rather pessimistic case, BF = 1, and an optimistic case, BF = 50. It is important to note that BF = 0 has a very low probability in the results of Martinez et al. Observations and instrumental considerations We considered photon events observed in a 10 square degree, stereographically-projected section of the sky centred on Segue 1 (RA, Dec. = 151.763 • , 16.074 • [14]). We applied cuts on event zenith angles (θ < 105 • ), energies (100 MeV < E < 300 GeV) and identifications (only 'diffuse class' events -see [21]). All data were processed using the same reconstruction algorithms and instrument response functions (IRFs) as the publicly-released first-year data. Counts and corresponding exposures were placed into 64 × 64 spatial and 14 logarithmic energy bins. The resultant energy-integrated map of photon counts is shown in Fig. 1. The Fermi -LAT IRFs consist of the effective area, point-spread function (PSF) and energy dispersion. We factored the effective area of the telescope into our calculations of the exposure for each bin of observed photon counts, using the the standard analysis tools available from HEASARC, specifically ScienceTools 9.11 1 . We convolved our modelled gamma-ray fluxes with the PSF and energy dispersion of the LAT using the publiclyavailable Fortran90 library FLATlib [67], which was designed specifically for performing this task quickly enough to be useful in MSSM scans. Full Fermi -LAT IRFs are defined not only as a function of photon energy, but impact angle with respect to the telescope zenith (and even azimuthal angle, though the dependence is weak). FLATlib achieves its fast convolution by averaging the IRFs over impact angles, allowing the integral over the PSF to be cast as a true convolution and performed by fast spectral methods. The energy integral cannot be performed in a similar way, because all three IRFs remain energy-dependent. FLATlib performs this integral explicitly, using a fast importance-sampling technique which utilises the rough resemblance of the energy dispersion function to a Gaussian. For the sake of computational speed, we truncated the PSF at a width of 3.2 • in the scans we present in this paper. With the present IRF set, this is well beyond the LAT's 95% containment resolution at e.g. 100 GeV (≈ 0.3 • ) or even 1 GeV (≈ 2 • ). Likelihoods from Segue 1 The expected spatial extent of Segue 1 in the gamma-ray sky, if it shines with dark matter annihilation, is comparable to the width of the LAT PSF. This puts Segue 1 on the borderline between a predicted point source and a predicted extended source. For every set of CMSSM parameters, we computed model spectra at each pixel in the inner 6 × 6 square shown in red in Fig. 1. We took care to explicitly integrate the density profile over the innermost 2 × 2 pixels as a whole, so as to correctly capture the contribution of the very centre of the galaxy (located at their vertex). We compared the predicted spectra with the observed ones in each of the 36 pixels to obtain a likelihood based on 504 data points, which we then included in the total likelihood for that point in our CMSSM scan. We chose only to include the inner 36 pixels in the CMSSM likelihood simply because these are the only pixels where there is a predicted signal at any significant level. All modelled spectra explicitly included contributions from gamma-ray lines, internal bremsstrahlung and continuum radiation. To properly model the observed event counts in the region around Segue 1, we also took the Galactic and isotropic diffuse emissions into account. We used a preliminary form of the GALPROP fit to the emission observed by Fermi [68] to describe the Galactic diffuse emission. The contribution of the isotropic diffuse emission, presumably originating from extragalactic sources, is much weaker and depends on the Galactic diffuse model adopted. To describe this, we adopted an isotropic power law model with index −2.1, derived from EGRET observations by Sreekumar et al. [69]. The models recommended by the LAT team were updated recently, and released to the Fermi Science Support Centre. At the ∼50 • latitude of Segue 1, the differences between the old and new models are not important for this analysis. The normalisations for both backgrounds were set to the best-fit values obtained in the preliminary 9-month LAT dwarf upper-limit analysis [24,70], based on the full 10 square degree region of interest rather than just the inner 36 pixels included in our likelihoods. No sources were detected in this region in the first 9 months of LAT operation. Because of the very low statistics ob- served in LAT photon counts towards Segue 1, a χ 2 estimation of the likelihood is inappropriate in this case. We calculated the likelihood using a binned Poissonian measure 5) or, recast in the more familiar minus loglikelihood form (analogous to half the χ 2 ), L = j θ n j j e −θ j n j ! ,(2.− ln L = j [θ j + ln(n j !) − n j ln(θ j )] . (2.6) Here n j and θ j are the observed and predicted number of counts respectively, in the jth bin. This prescription clearly accounts for statistical errors by definition, but including systematic errors is less obvious. To do so, one can marginalise over an assumed probability density function (PDF) of a systematic error in a semi Bayesian manner, treating it as a nuisance parameter. If we consider a systematic error that has the impact of consistently rescaling the observed number of counts as n j → n j (i.e. a constant percentage systematic error \|1 − \|), and assume a Gaussian form with width σ for the PDF of , the marginalised log-likelihood is (see e.g. [71]) − ln L = − j ln        1 √ 2πσ ∞ 0 ( θ j ) n j e − θ j exp − 1 2 1− σ 2 n j ! d        (2.7) = − j ln θ n j √ 2πσ n j ! ∞ 0 n j exp − θ j − 1 2 1 − σ 2 d . (2.8) The integral is only analytically soluble for θ j < σ −2 , which is not generally true when dealing with small statistics; we performed it numerically for each likelihood evaluation. We included estimated systematic errors from the LAT effective area (f ) and our modelled spectra (τ ) by combining them in quadrature, i.e. σ (E j ) = f (E j ) 2 + τ 2 . Note the explicit energy dependence of f ; for the present IRF set, f (E j ) ranges from 10% at 100 MeV, to 5% at 562 MeV, to 20% at 10 GeV. We interpolated between these values linearly, and assumed the edge values outside this range. We tuned the importance sampling algorithm used by FLATlib using slower, more accurate standard numerical integration schemes, choosing a sampling efficiency for our specific problem that would introduce an overall systematic theoretical error τ of no more than 5% in the normalisation of flux predictions. Other systematic errors are no doubt also present in the theoretical predictions, but we expect the term from the fast integration to dominate. Extrapolation to 5 years of observations To make predictions about the impact of 5 years of LAT observations, we explicitly assume that no excess events will have been observed after this time. There is no correct way to rescale Poissonian counts to longer timescales, so the Poissonian likelihood above cannot be used when extrapolating to longer observing times. We instead set the 'observed' number of photons equal to the number predicted by the background model, using rescaled 9-month exposures. This prescription also avoids the erroneous shifts which confidence intervals based on Poissonian statistics can sometimes experience due to a downward statistical fluctuation of the background. In this case the observed counts become a continuous instead of a discrete variable, so the problem of small statistics disappears. The appropriate likelihood measure is then once more the χ 2 χ 2 = j (Φ model,j − Φ observed,j ) 2 σ 2 j = j θ j −n j E j 2 σ 2 model,j + σ 2 observed,j ,(2.9) where Φ model,j and Φ observed,j are the predicted and observed fluxes, σ model,j and σ observed,j are their standard deviations, and E j is the exposure. The exposure is itself the product of the effective area and observing time. The standard deviation of the predicted flux can be estimated as simply the product of the predicted flux and the percentage systematic theoretical uncertainty τ (5% in our case -see above), σ model,j = τ Φ model,j . The standard deviation in the observed flux can be estimated from the standard deviation of the observed counts σ n j , and the uncertainty on the exposure σ E j , giving σ 2 observed,j = n j E j 2 σ 2 n j n 2 j + σ 2 E j E 2 j . (2.10) Since the underlying physical process is still Poissonian, the best estimate of σ n j is in fact σ n j = θ j . Furthermore, since the uncertainty in the observing time is negligible, σ E j can be estimated as simply the percentage systematic error of the effective area f (E j ) times the actual exposure, σ E j = f (E j )E j . We then have σ 2 observed,j = n j E j 2 θ j n 2 j + f (E j ) 2 (2.11) = Φ model,j E j + Φ 2 observed,j f (E j ) 2 ,(2.12) giving χ 2 = j (Φ model,j − Φ observed,j ) 2 Φ model,j E j + Φ 2 observed,j f (E j ) 2 + τ 2 Φ 2 model,j . (2.13) We hasten to point out that constraints based on this extrapolation are probably overly conservative, as we assume the same background rejection, systematic errors and background model for both the 9-month analysis and the 5-year extrapolation. Our overall understanding of the instrument will improve over time, as will our understanding of the background as Fermi accumulates better statistics on the Galactic diffuse and extragalactic components, leading to correspondingly better constraints on the annihilation cross-section. Kinematic constraints upon the dark matter density profile of Segue 1 will also improve in time [20,25], which may impact constraints on CMSSM parameters. CMSSM scans We scanned the CMSSM parameter space using a modified version of SuperBayeS 1.35 [42], employing the MultiNest [41] nested sampling algorithm with 4000 live points. In the plots we show, all parameters except those shown on figure axes have been marginalised over in some way. In the case of the frequentist profile likelihood, this is simply a matter of maximising the likelihood in the other dimensions of the parameter space. In the case of the Bayesian posterior, the total posterior (prior times likelihood) is integrated over the other dimensions of the space (for a review see e.g. [72]). Because we are somewhat more interested in the prior-independent profile likelihood than the marginalised posterior 2 , we prefer linear priors on the CMSSM parameters because they are flat relative to the likelihood, causing the sampling algorithm to proceed strictly according to the frequentist likelihood function. The effects of alternative priors have already been discussed in detail for previous CMSSM scans [42]. We used DarkSUSY 5.04 for the relic density and indirect detection computations. This allowed us to calculate internal bremsstrahlung spectra, and improved the continuum spectrum and relic density calculations. We also improved the interface between SuperBayeS and DarkSUSY, most notably pertaining to the energies at which some particle masses were defined. Apart from the Fermi data, the experimental data and nuisance parameters which we included in scans were identical to those in [42] and [44]. SM nuisance parameters were the top and bottom quark masses and strong and electromagnetic coupling constants. Experimental data were precision electroweak measurements of SM parameters from the Large Electron-Positron collider (LEP), the relic density from 5-year Wilkinson Microwave Anisotropy Probe (WMAP) fits (Ω DM h 2 = 0.1099 ± 0.0062 [73]), LEP constraints on sparticle masses, LEP constraints on the Higgs mass, the anomalous magnetic moment of the muon (g − 2), theB s − B s mass difference, and branching fractions of rare processes b → sγ,B u → ντ − andB s → µ + µ − . Details can be found in [42]. In our chosen configuration, completing the integration over the LAT IRFs for a given point in the CMSSM parameter space required a similar order of magnitude in processing time as a relic density calculation. Since the relic density computation is the main bottleneck in MSSM scans, this meant that scans took roughly twice as much total processor time to complete as a standard SuperBayeS run. One advantage of FLATlib, however, is that it can employ the multi-threaded version of the FFTW library [74], allowing the IRF integration to be performed with a considerably greater degree of parallelisation than the present relic density routines. Results and Discussion Fits to Fermi data only In Fig. 2 we show results of scans where the likelihood function only included Fermi data, LEP measurements of nuisance parameters and the requirements of physicality (the absence of tachyons, that the neutralino is the lightest SUSY particle, and that electroweak symmetry breaking is induced by SUSY breaking). Preferred values of the neutralino self-annihilation cross-section and mass are shown for scans including 9 months of data, scans including the extrapolation to 5 years of data, and a control case without any Fermi data. Preferred regions are also given for both the pessimistic and optimistic boost factors discussed in Sect. 2.1. One apparent feature of Fig. 2 is the lack of viable models with large annihilation cross sections for large neutralino masses. This feature is present simply because the annihilation cross section goes as m −2 χ , causing it to fall off at higher masses. Given the absence of any observed signal from Segue 1, Fermi data clearly disfavours models with the highest cross-sections and lowest masses. This is expected, since higher cross-sections and lower masses lead to a larger predicted signal. That constraints are best at lower neutralino masses is also consistent with the falling sensitivity of the LAT with energy above about 50 GeV, and the reduced source statistics at higher energies. The improvement in constraints when moving from the current 9 months of data to the 5-year predictions is also roughly what would be expected from a √ t improvement in sensitivity. This shows that the two different likelihood estimators we employ give consistent results (we also checked this explicitly for 9 months of data, finding very good agreement). Predictably, the adopted boost factor plays a large role in determining the extent of constraints brought to bear on the CMSSM by Segue 1. In the most pessimistic scenario, 9 months of LAT observations have no impact on confidence regions, as all disfavoured crosssections are larger than allowed by physicality arguments. In the most optimistic scenario, the data disfavours all models with cross-sections greater than ∼ 3 × 10 −25 cm 3 s −1 . Improvements in constraints when moving from BF = 1 to BF = 50 are consistent with the factor of 51/2 improvement in sensitivity expected from Eq. 2.1, as the most pessimistic constraints lie above the extent of contours in the upper middle panel of Fig. 2. Extrapolating to 5 years of observations, all values above 10 −25 cm 3 s −1 would be disfavoured, as would a region extending down below 10 −26 cm 3 s −1 at the lowest masses. Once again, we caution the reader that this extrapolation does not take into account systematic improvements in the background and dark matter profile modelling after 5 years, nor in the LAT reconstruction algorithms (see Sect. 2.3.1). For comparison, in Fig. 2 we also show the previously-presented, preliminary 95% confidence level upper limit from 9 months of LAT observations [70]. This limit was derived assuming annihilation proceeds only into bb pairs. Apart from the obvious difference in overall strategy (upper limits from an assumed final state versus inclusion in explicit model scans), our analysis differs from the upper limit one in a number of ways. The upper limit was derived assuming a point source for Segue 1, whereas we perform spatial fits; the upper limit is based upon an NFW rather than Einasto density profile, and does not include systematic errors nor a treatment of the energy dispersion. Nonetheless, the areas disfavoured in our scans are broadly consistent with the 9 month upper limit, a positive comment on the reliability of both analyses. Our corresponding exclusions do however occur at somewhat higher cross-sections than in the upper limit analysis (i.e. our exclusion region is above both the extent of coloured contours and the black line in the upper middle panel of Fig. 2). This is to be expected, as our ability to exclude models is degraded relative to the upper limit analysis by properly accounting for the systematic error in the effective area. Because this error is energy-dependent, our exclusions also have a slightly different energy-dependence than the 95% upper limit. It should be noted that the degree of substructure apparent in the confidence regions of Fig. 2 is unlikely to be physical, and is indeed probably something of an artefact of the scanning technique (i.e. 'scanning noise'). In the absence of any constraint on the annihilation cross-section from the relic density, the vast majority of points providing a good fit to the included data lie at much lower cross-sections. This prompts the scanning algorithm to concentrate its efforts there, leaving the region in which we are most interested somewhat poorly sampled. From a Bayesian point of view, one would say that when the relic density is not included, this region sits well above the most likely annihilation crosssections in the CMSSM, so is not meant to be very well sampled by the nested sampling technique. Because only a small number of models are disfavoured by including just Segue data in the likelihood function, there is little overall impact on the favoured values of m 0 , m 1 2 , A 0 and tan β beyond what is allowed purely on physicality grounds. We will show confidence regions from global fits only for these parameters. Global fits In Fig. 3 we show the result of including the relic density constraint from the WMAP 5-year data, along with all other experimental bounds. The effect is to favour models populating Left: with no constraining experimental data except measurements of SM nuisance parameters and physicality requirements. Middle: constraints provided by 9 months of Fermi data on Segue 1, under the most pessimistic (top) and optimistic (bottom) assumptions about the substructure boost factor. Right: projected constraints after 5 years of Fermi observations. Colours indicate 68% (yellow) and 95% (red) confidence regions. The preliminary 95% confidence level upper limit on the annihilation cross-section from 9 months of Fermi data, assuming 100% WIMP annihilation into bb [70], is given for comparison (black curve). two distinct regions: a broad strip around the canonical WIMP annihilation cross-section at 3 × 10 −26 cm 3 s −1 , and a low-mass region at smaller cross-sections, corresponding to models where stau co-annihilations reduce the relic density to the observed level despite the very low self-annihilation rates. The models disfavoured by Fermi observations of Segue 1 in Fig. 2 are here already strongly disfavoured by the relic density constraint, so the additional data from the LAT appears to have little impact upon the preferred crosssections and masses. A slight reduction in the profile likelihoods of the lowest mass, highest cross-section corner of the preferred region appears to be present in the extrapolation to 5 years of data. The best-fit point is however rather different in the 9-month scan as compared to scans without Fermi data, or with 5 years of mock observations (where we assumed that no excess above background will be seen in 5 years). In the 9-month scan, the best fit occurs in the focus point region, at a high annihilation cross-section and a low neutralino mass ( σv = 1.8 × 10 −26 cm 3 s −1 , m χ = 95 GeV), whereas the best fits in the other cases are for stau coannihilation models. This difference appears to be the result of a very small statistical excess above the modelled background in the 9-month data. Because the corresponding confidence regions are not substantially altered despite the movement of the best fit, the excess would appear to be consistent with observational (statistical) noise. Given the range of Fermi 's sensitivity, it is thus not at all surprising that the best-fit would appear at this location, falling right on the edge of the instrument's sensitivity. This point may however be an interesting one to watch as statistics improve. It is instructive to note the difference in how the co-annihilation region is represented in Fig. 3 by the profile likelihood and the marginalised posterior. Because the range of CMSSM parameters spanned by the co-annihilation region is quite narrow (i.e. finetuned), the total number of points in this region found by the scans is not particularly high, leading to a relatively low posterior PDF. This is despite the fact that very good fits can be found with a reasonably broad range of neutralino masses and cross-sections in this region, as evidenced by its size in the profile likelihood plots. In this sense, the Bayesian posterior PDF can be seen to penalise the co-annihilation region to a certain degree for being fine-tuned. Whether this is a desirable characteristic or not is of course a matter of opinion. It is, however, important to recognise that such information is only accessible by comparing the posterior PDF and the profile likelihood; the information in their combination is greater than the sum of the parts. A natural question to ask might be whether more interesting constraints could be obtained from Segue 1 by allowing the neutralino to be a sub-dominant component of dark matter. Unfortunately, this generally does not add a lot to the discussion when considering constraints from indirect detection with gamma-rays. Even though the relic density is essentially inversely proportional to the annihilation cross-section, in mixed dark matter scenarios the density of neutralinos in Segue 1 becomes directly proportional to the relic density. The expected signal is then increased due to the larger annihilation crosssections permitted by sub-dominant relic densities, but reduced by the reduction in signal due to the reduced galactic densities. The net result is a reduction in the expected signal, since the flux (Eq. 2.1) depends upon the first power of the annihilation cross-section, but the square of the density. Thus for a decrease in the relic density such that Ω χ → Ω χ /X, the flux is modified as Φ → X/X 2 Φ = Φ/X. The result is that the favoured cross-sections move to higher values, but the constraints from Segue 1 move even further, providing less constraining power than when the neutralino is assumed to be the only component of dark matter. This argument of course may not hold for points in the parameter space where the relic density is not strictly inversely proportional to the annihilation cross-section, such as strong co-annihilation or resonant annihilation scenarios. The former certainly are not probed by the Segue 1 observations in any case, since they lie at very low annihilation crosssections. In principle though, highly fine-tuned points in the latter scenario could slightly modify the impact of the Segue constraints in subdominant situations. As discussed below however, our scans do not uncover a significant number of models where such a mechanism occurs. In Fig. 4 we investigate whether variations in the dark matter profile of Segue 1, within the errors of Martinez et al. [15], might also produce more interesting constraints. Here we again take an Einasto profile (Eq. 2.4), but instead use parameters corresponding to the most dark-matter-rich profile allowed at ∼2σ (r s = 10 pc, ρ s = 70 GeV cm −3 ). The corresponding constraints on annihilation cross-sections are indeed stronger than in Fig. 3, but are still largely dominated by the relic density. This is not surprising, as even though the scale density is a factor of 18 higher in this case, the smaller scale radius means that the higher density occurs at a smaller radius. In this sense the two parameters are partially degenerate; because Fermi probes essentially the whole dwarf halo (as Segue 1 should appear almost as a point source), and the total mass of Segue 1 is not substantially altered by the change in halo parameters, the corresponding constraints are not massively improved. The constraints coming from 9 months of data can be seen to cluster more tightly around the best-fit point at low mass and high cross-section, but not to the point where significant parts of the rest of the parameter space are excluded. This is consistent with our assertion above that any excess can be explained in terms of statistical fluctuations. The preferred CMSSM parameter regions including all constraints are shown in Fig. 5. Given the marginal impact of Segue 1 observations on scans including the relic density, it is not surprising that the regions are very similar to those shown in [42], even when using the extrapolation to 5 years of observations. The stau co-annihilation region is clearly visible at low m 0 and m 1 co-annihilation region in the m 0 -m 1 2 plane. The high-probability region at low tan β in the A 0 -tan β plane favoured by the co-annihilation region shows up as a much smoother peak in our scans than in some previous works [39,32,42]. We suspect that this is due to our use of the upgraded version of DarkSUSY for the relic density calculation. The 'funnel' region, where resonant annihilation can become important at very low m 1 2 , does not show up in our scans here. This is unsurprising, as the nested sampling algorithm is designed to sample according to the total posterior mass, and the linear prior places a very small scanning weight upon such fine-tuned regions at low mass. Nested sampling routines only find this region when using logarithmic priors on m 0 and m 1 2 [42], though normal MCMC scans can find it a little more easily (e.g. [39,32]). On the other hand, standard MCMCs and nested sampling implemented with logarithmic priors sample the focus point region less densely, causing them to sometimes miss the highest-likelihood points important for a profile likelihood analysis. These difficulties are typical consequences using scanning algorithms designed for Bayesian analyses to compute the frequentist profile likelihood; a more promising path for frequentist scans appears to be to use genetic algorithms [44]. Using genetic algorithms, it seems possible to find all regions in a prior-independent way, but the ability to effectively map their surroundings and produce reliable confidence regions lags behind other techniques. Some recent MCMC scans [49,50] have not found large focus-point regions which fit all experimental constraints well, leading the authors to claim that the co-annihilation region is favoured by present data. In these cases, the reduced likelihood in the focus point region relative to the co-annihilation region was almost entirely due to the fact that it is virtually impossible to produce a good fit to the muon g − 2 with large values of m 0 in the CMSSM. Using nested sampling with linear priors however, and the physics and likelihood routines within SuperBayes, one can find points in the focus point region where this effect is essentially offset by a correspondingly better fit to other observables [42]. Conclusions We have incorporated fits to 9 months of Fermi -LAT observations of the dwarf galaxy Segue 1 into explicit global CMSSM parameter scans. We included gamma-ray lines, internal bremsstrahlung and secondary decay, as well as detailed characterisations of the detector response, its uncertainties and the observed background. We have also presented scans illustrating the estimated impact of a non-observation of dark matter annihilation in Segue 1 after 5 years of LAT operation. The LAT data disfavour a small number of physically-viable CMSSM models with low neutralino masses and high annihilation cross-sections, but results depend strongly upon the assumed substructure boost factor in Segue 1. Such models are already strongly disfavoured by relic density constraints. Extrapolating to 5 years of operation and assuming the most optimistic boost factor presently allowed by astronomical data, the absence of any annihilation signal from Segue 1 would disfavour all models with cross sections higher than 10 −25 cm 3 s −1 , as well as a number at low mass with cross-sections as low as 10 −26 cm 3 s −1 . Even at this level however, the CMSSM models disfavoured by Fermi would already be essentially excluded by existing data from the microwave background and terrestrial experiments. Acknowledgments We are grateful to our colleagues in the Fermi -LAT Dark Matter & New Physics group for many helpful comments and discussions, and to Riccardo Rando for advice on the LAT IRFs. We also thank Roberto Trotta for comments on an earlier version of the manuscript. Through DarkSUSY, FLATlib and SuperBayeS we also drew upon a number of other publicly-available scientific codes, including CUBPACK [75], FeynHiggs [76], SL-HALib [77], SoftSUSY [78] and WCSLIB [79]. PS, JC, JE, LB and YA are grateful to the Swedish Research Council (VR) for financial support. JC is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. A number of agencies and institutes have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariatà l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organisation (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the and the Centre National d'Études Spatiales in France. Figure 1 : 1Photon counts observed by Fermi in the region around Segue 1 during the first 9 months of LAT operation in all-sky survey mode. Counts are integrated over all energies between 100 MeV and 300 GeV. The red cross shows the exact location of the centre of Segue 1, and the red box shows the region included in our likelihood calculations. Figure 2 : 2Neutralino self-annihilation cross-sections in the CMSSM, in the zero-velocity limit. Figure 3 : 3Annihilation cross-sections in the CMSSM which fit all experimental constraints, assuming the neutralino to be the dominant component of dark matter. Favoured regions are as implied by existing experimental data only (left), and with the addition of 9 months of Segue 1 observations by Fermi (middle). We also show the extrapolated impact of a non-observation of Segue 1 after 5 years (right). Upper plots show profile likelihoods (where yellow and red indicate 68% and 95% confidence regions respectively), while lower plots show marginalised posterior PDFs (where solid blue contours give 68% and 95% credible regions). Solid dots indicate posterior means, whereas crosses indicate best-fit points. Figure 4 : 4Annihilation cross-sections in the CMSSM which fit all experimental constraints, assuming a 'maximally dense' dark matter halo profile for Segue 1. In this case, the halo scale radius and density were chosen ∼2σ away from the best-fit values derived from stellar kinematic data. Here we again assume the neutralino to be the dominant component of dark matter. Favoured regions are as implied by 9 months of Segue 1 observations by Fermi (left), and extrapolations to 5 years of data assuming no signal from Segue 1 (right). Shadings and markings are as perFig. 3. 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[ "A Numerical Study of the Performance of a Quantum Adiabatic Evolution Algorithm for Satisfiability", "A Numerical Study of the Performance of a Quantum Adiabatic Evolution Algorithm for Satisfiability" ]	[ "Edward Farhi [email protected] \nCenter for Theoretical Physics\nDepartment of Mathematics Northeastern University Boston\nMassachusetts Institute of Technology Cambridge\n02139, 02115MA, MA\n", "Jeffrey Goldstone [email protected] \nCenter for Theoretical Physics\nDepartment of Mathematics Northeastern University Boston\nMassachusetts Institute of Technology Cambridge\n02139, 02115MA, MA\n", "Sam Gutmann \nCenter for Theoretical Physics\nDepartment of Mathematics Northeastern University Boston\nMassachusetts Institute of Technology Cambridge\n02139, 02115MA, MA\n" ]	[ "Center for Theoretical Physics\nDepartment of Mathematics Northeastern University Boston\nMassachusetts Institute of Technology Cambridge\n02139, 02115MA, MA", "Center for Theoretical Physics\nDepartment of Mathematics Northeastern University Boston\nMassachusetts Institute of Technology Cambridge\n02139, 02115MA, MA", "Center for Theoretical Physics\nDepartment of Mathematics Northeastern University Boston\nMassachusetts Institute of Technology Cambridge\n02139, 02115MA, MA" ]	[]	Quantum computation by adiabatic evolution, as described in quant-ph/0001106, will solve satisfiability problems if the running time is long enough. In certain special cases (that are classically easy) we know that the quantum algorithm requires a running time that grows as a polynomial in the number of bits. In this paper we present numerical results on randomly generated instances of an NP-complete problem and of a problem that can be solved classically in polynomial time. We simulate a quantum computer (of up to 16 qubits) by integrating the Schrödinger equation on a conventional computer. For both problems considered, for the set of instances studied, the required running time appears to grow slowly as a function of the number of bits.	null	[ "https://arxiv.org/pdf/quant-ph/0007071v1.pdf" ]	15,242,433	quant-ph/0007071	8fb335895f0ad49552c7d903aba15f056b8e3735	A Numerical Study of the Performance of a Quantum Adiabatic Evolution Algorithm for Satisfiability Jul 2000 Edward Farhi [email protected] Center for Theoretical Physics Department of Mathematics Northeastern University Boston Massachusetts Institute of Technology Cambridge 02139, 02115MA, MA Jeffrey Goldstone [email protected] Center for Theoretical Physics Department of Mathematics Northeastern University Boston Massachusetts Institute of Technology Cambridge 02139, 02115MA, MA Sam Gutmann Center for Theoretical Physics Department of Mathematics Northeastern University Boston Massachusetts Institute of Technology Cambridge 02139, 02115MA, MA A Numerical Study of the Performance of a Quantum Adiabatic Evolution Algorithm for Satisfiability Jul 2000arXiv:quant-ph/0007071v1 19 MIT-CTP #3006 Quantum computation by adiabatic evolution, as described in quant-ph/0001106, will solve satisfiability problems if the running time is long enough. In certain special cases (that are classically easy) we know that the quantum algorithm requires a running time that grows as a polynomial in the number of bits. In this paper we present numerical results on randomly generated instances of an NP-complete problem and of a problem that can be solved classically in polynomial time. We simulate a quantum computer (of up to 16 qubits) by integrating the Schrödinger equation on a conventional computer. For both problems considered, for the set of instances studied, the required running time appears to grow slowly as a function of the number of bits. Introduction A quantum algorithm for the satisfiability problem was presented in [1]. This algorithm is based on quantum adiabatic evolution. If a state \|ψ(t) evolves according to the Schrödinger equation with a slowly varying Hamiltonian H(t) and \|ψ(0) is the ground state of H(0), then \|ψ(t) will stay close to the instantaneous ground state of H(t). The Hamiltonian H(t) used in the algorithm is designed so that the ground state of H(0) is easy to construct and the ground state of H(T ) encodes the solution to the instance of satisfiability. A crucial question is how large must the running time, T , be to achieve an acceptable probability of success. In this paper we simulate an n-qubit continuous time quantum computer by numerically integrating the Schrödinger equation in a 2 n -dimensional Hilbert space. We randomly generate difficult instances of an NP-complete problem and study how large T must be as a function of the number of bits n. For 7 ≤ n ≤ 16, we find that the required T grows modestly with n since the data is well fit by a quadratic in n. A quantum algorithm based on adiabatic evolution An n-bit instance of satisfiability is a Boolean formula with M clauses C 1 ∧ C 2 ∧ · · · ∧ C M (2.1) where each clause C a is True or False depending on the values of some subset of the n bits. The task is to discover if one (or more) of the 2 n possible assignments of the values of the n bits satisfies all of the clauses, that is, makes formula (2.1) True. We consider two restricted versions of satisfiability, both of which are restricted versions of a problem called "exact cover". In the first, EC3, each clause involves only three bits. The clause always has the same form. The clause is True if and only if one of the three bits is a 1 and the other two are 0, so there are three satisfying assignments out of the eight possible values of the three bits. The second, EC2, has the restriction that each clause involves only two bits. In this case the clause is True if and only if the two bits have the value 01 or 10, so there are two satisfying assignments out of the four possible values for the two bits. We pick these two examples because EC2 is classically solvable in polynomial time whereas EC3 is NP-complete and no polynomial time algorithm is known. It is interesting to see how the quantum algorithm treats these two classically very different problems. To understand the quantum algorithm we first recall the content of the adiabatic theorem. We are given a HamiltonianH(s) that depends smoothly on the parameter s where s varies from 0 to 1. Suppose that for each value of s,H(s) has a unique lowest energy state, the ground state \|g; s . That is,H (s)\|g; s = E 0 (s)\|g; s (2.2) where E 0 (s) is strictly less than any of the other eigenvalues ofH(s). Introduce a time scale T and define a time-dependent Hamiltonian H(t) by H(t) =H(t/T ) (2.3) where t varies from 0 to T . As T gets bigger, H(t) becomes more slowly varying. Let \|ψ(t) obey the Schrödinger equation i d dt \|ψ(t) = H(t)\|ψ(t) (2.4) with \|ψ(0) = \|g; s = 0 . (2.5) That is, at t = 0, \|ψ(0) is the ground state of H(0). The adiabatic theorem tells us that lim T →∞ \| g; s = 1\|ψ(T ) \| = 1 . (2.6) This means that for T large enough, \|ψ(T ) is (up to a phase) very close to the ground state of H(T ). Eq. (2.6) only holds if the gap between the ground state energy, E 0 (s) ofH(s), and the next highest energy, E 1 (s) ofH(s), is strictly greater than zero, that is, E 1 (s)−E 0 (s) > 0 for 0 ≤ s ≤ 1. We will also discuss cases where at s = 1 there are multiple ground states. In this situation the evolution ends very close to the ground state subspace. The idea behind the quantum algorithm is the following. Given an instance of the satisfiability problem it is straightforward to construct a Hamiltonian, H P , whose ground state corresponds to an assignment of the bits that violates the least number of clauses. Although it is easy to construct H P , finding its ground state may be computationally challenging. However we can construct a Hamiltonian H B whose ground state we explicitly know. Now letH (s) = (1 − s)H B + sH P (2.7) and accordingly H(t) = (1 − t/T )H B + (t/T )H P (2.8) for some fixed T . We start our quantum system in the known ground state of H B and then evolve according to Eq. (2.4) for time T . Suppose that the instance of satisfiability that gave rise to H P has a unique satisfying assignment. If T is large enough, then \|ψ(T ) will be very close to the ground state of H P . Measuring \|ψ(T ) will, with high probability, produce the satisfying assignment. In general this algorithm will produce an assignment that violates the minimum number of clauses in Eq. (2.1). Restricting to instances with a unique satisfying assignment appears to pick out difficult instances and simplifies the analysis of our algorithm. More explicitly, given an n-bit instance of satisfiability, we work in an n-qubit Hilbert space that is spanned by the 2 n basis vectors \|z 1 \|z 2 · · · \|z n where z i = 0, 1 and \|z i is an eigenstate of the z component of the i th spin, 1 2 (1 − σ (i) z )\|z i = z i \|z i where each σ (i) z = 1 0 0 −1 . (2.9) We also need the eigenstates of the x component of the i th spin, \|x i = 0 = 1 √ 2 (\|z i = 0 + \|z i = 1 ) and \|x i = 1 = 1 √ 2 (\|z i = 0 − \|z i = 1 ) (2.10) that obey 1 2 (1 − σ (i) x ) \|x i = x = x \|x i = x where each σ (i) x = 0 1 1 0 . (2.11) For concreteness imagine that each clause in formula (2.1) involves 3 bits. Let i C , j C and k C be the 3 bits associated with clause C. For each clause C define an "energy" function h C (z i C , z j C , z k C ) = 0, if (z i C , z j C , z k C ) satisfies clause C 1, if (z i C , z j C , z k C ) violates clause C . (2.12) We immediately turn these into quantum operators H P , C (\|z 1 \|z 2 · · · \|z n ) = h C (z i C , z j C , z k C )\|z 1 \|z 2 · · · \|z n (2.13) and define H P = C H P , C . (2.14) By construction H P is nonnegative, that is, ψ\|H P \|ψ ≥ 0 for all \|ψ and H P \|ψ = 0 if and only if \|ψ is a superposition of states of the form \|z 1 \|z 2 · · · \|z n where the bit string z 1 z 2 . . . z n satisfies all of the clauses. In this context seeing if formula (2.1) has a satisfying assignment is accomplished by finding the ground state of H P . If formula (2.1) has no satisfying assignment, the ground state (or states) of H P correspond to the assignment (or assignments) that violate the least number of clauses. H P given by Eq. (2.14) is the problem Hamiltonian whose ground state we seek when we run the quantum algorithm. For the beginning Hamiltonian H B , first define H (i) B = 1 2 (1 − σ (i) x ). (2.15) For each clause C, involving bits i C , j C , and k C , let H B , C = H (i C ) B + H (j C ) B + H (k C ) B . (2.16) The beginning Hamiltonian is given by H B = C H B , C . (2.17) The ground state of H B is \|x 1 = 0 \|x 2 = 0 · · · \|x n = 0 = 1 2 n/2 z 1 z 2 · · · zn \|z 1 \|z 2 · · · \|z n (2.18) which we will take to be the initial state when we run the quantum algorithm. The Hamiltonian that governs the evolution of the quantum system is given by Eq. (2.8) with H P specified in Eq. (2.14) and H B specified in Eq. (2.17). Note that H(t) is a sum of terms, each of which is associated with one of the clauses in (2.1), H(t) = C H C (t) (2.19) where H C (t) = (1 − t/T )H B , C +(t/T )H P , C . (2.20) Each H C (t) involves only the bits associated with clause C and therefore H(t) is a sum of terms each of which involves a fixed number of bits. For a given problem such as EC3 or EC2 we must specify the running time as a function of the number of bits, T (n). Since the state \|ψ(T ) is not exactly the ground state of H P we must also specify R(n), the number of times we repeat the quantum evolution in order to have a desired probability of success. This paper can be viewed as an attempt to determine T (n) and R(n) by numerical methods. We now summarize the ingredients of the quantum adiabatic evolution algorithm. Given a problem and an n-bit instance, we assume we know the instance-independent running time T (n) and repetition number R(n). For the instance and given T (n), Choosing instances For our numerical study we randomly generate instances of the two problems under study, EC3 and EC2. Focus first on EC3. For a decision problem, it suffices to produce a satisfying assignment when one or more exists. For now, we restrict our attention to instances with a unique satisfying assignment. We believe that instances with only one satisfying assignment include most of the difficult instances for our algorithm. In fact as we will see later, our algorithm runs faster on instances with more than one satisfying assignment so the restriction to a unique satisfying assignment appears to restrict us to the most difficult cases. With the number of bits fixed to be n, we generate instances of EC3 as follows. We pick three bits at random, uniformly over the integers from 1 to n. (The bits must all be different.) We then have a formula with one exact cover clause. We calculate the number of satisfying assignments. We then add another clause by picking a new set of three bits. Again we calculate the number of satisfying assignments. We continue adding clauses until the number of satisfying assignments is reduced to one or zero. If there is one satisfying assignment we accept the instance. If there are none we reject the instance and start again. Using this procedure the number of clauses is not a fixed function of the number of bits but rather varies from instance to instance. For EC3 we find that the number of clauses is typically close to the number of bits. We follow a similar procedure for EC2. When we add a clause we randomly specify which two bits are involved in the clause. Again we repeat this procedure until there are two satisfying assignments (or none in which case we discard the instance). We stop at two satisfying assignments because EC2 has a bit negation symmetry. If w 1 w 2 . . . w n is a satisfying assignment so isw 1w2 . . .w n and accordingly there are no instances with a single satisfying assignment. For EC2 the number of clauses is typically close to the number of bits. We know that EC2 is classically computationally simple but of course there is no guarantee that quantum adiabatic evolution will work well on EC2. We choose instances of EC2 with two satisfying assignments to parallel as closely as possible our study of EC3. Numerical simulation We are exploring the quantum adiabatic evolution algorithm by numerically simulating a perfectly functioning quantum computer. The quantum computer takes an initial state \|ψ(0) , given by Eq. If the number of bits is n, the dimension of the Hilbert space is 2 n . This exponential growth in required space is the well-known impediment to simulating a quantum computer with many bits. For our modest numerical investigation, using Macs running MATLAB for a few hundred hours, we can only explore out to 16 bits. We integrate the Schrödinger equation using a variable step size Runge-Kutta algorithm, checking accuracy by keeping the norm of the state equal to unity to one part in a thousand. Since the number of bits is modest, we can always explicitly determine the ground state (or ground states) of H P . Given \|ψ(T ) and the ground state (or states) of H P we can calculate the probability that a measurement of z 1 , z 2 , . . . , z n will give a satisfying assignment by taking the sum of the squares of the inner products of \|ψ(T ) with \|w 1 \|w 2 · · · \|w n where the bit strings w 1 w 2 . . . w n are the satisfying assignments. The median time to achieve probability 1/8 Our goal in this paper is to explore the running time T (n) and the repetition number R(n) that will give a successful algorithm. To this end we first determine the typical running time needed to achieve a fixed probability of success for a randomly generated instance with n bits for 7 ≤ n ≤ 15. In particular we determine the median time required to achieve a success probability of 1/8. Since this is a numerical study we actually hunt for a time that produces a probability between 0.12 and 0.13. For each n between 7 and 15, for both EC3 and EC2, we find the median of 50 randomly generated instances. In Figure 1 the circles represent the data for EC3 and the solid curve is a quadratic fit to the data. At each number of bits the times required to reach probability 1/8 range from roughly half the median to twice the median. For this range of n, a quadratic, or even linear fit, is clearly adequate. The exponential 1.618 (1.215) n is also a good fit. In the next section we show a situation where an exponential fit to the data is required for the same range of n. We know of one anomalous instance (discovered by Daniel Preda, outside of the data collected for this paper) with 11 bits and a time to achieve probability 1/8 of close to 300. However, at T = 14.57, which is the value of the quadratic fit in Figure 1 at n = 11, the probability of success for this instance is already 0.0606. Because this probability is not anomalously low, the algorithm proposed in Section 7 will have no difficulty with this instance. In Figure 2 the circles represent the data for EC2 and the solid curve is a linear fit to the data. Here the maximum time required to reach probability 1/8, for each number of bits, is roughly 20% greater than the median. Destroying the bit structure of the Hamiltonian The Hamiltonian H(t) is a sum of terms H C (t) each of which involves only the few bits mentioned in clause C; see Eq. (2.19). Each H C (t) is associated with a subspace of the Hilbert space corresponding to the satisfying assignments of clause C, that is, the space spanned by the ground states of H P , C ; see Eq. (2.13). Quantum adiabatic evolution (for T big enough) yields a state in the intersection of the subspaces associated with all of the clauses. Our intuition is that the bit structure of these subspaces is crucial to the success of the quantum algorithm. To test this intuition we destroy the bit structure of the Hamiltonian and run the algorithm again. Specifically consider the classical energy function that counts the number of violated clauses h(z) = C h C (z i C , z j C , z k C ) (6.1) where z = z 1 z 2 . . . z n and h C is given in Eq. (2.12). From Eq. (2.13) and Eq. (2.14) we have that H P (\|z 1 \|z 2 · · · \|z n ) = h(z)\|z 1 \|z 2 · · · \|z n . (6.2) Now let h SCRAM (z) = h(Π(z)) (6.3) where Π is a random permutation of the integers {0, 1, 2, . . . , 2 n − 1}. Note that Π is not a permutation of the bits but rather a random scrambling of all 2 n of the z's. Let H SCRAM , P (\|z 1 \|z 2 · · · \|z n ) = h SCRAM (z) \|z 1 \|z 2 · · · \|z n (6. 4) and accordingly H SCRAM (t) = (1 − t/T ) H B + (t/T ) H SCRAM,P . (6.5) The spectrum of H SCRAM,P is identical to that of H P but the relationship between the eigenvalues and the values of z has been scrambled. Accordingly the spectrum of H SCRAM (t) is not equal to the spectrum of H(t) except at t = 0 and t = T . If H P has a unique ground state so does H SCRAM,P and for T large enough we expect (again by the adiabatic theorem) that quantum evolution by H SCRAM (t) will bring us to the ground state of H SCRAM,P . Once the bit structure has been destroyed, finding the minimum of h SCRAM (z) is essentially the problem of minimizing an arbitrary function defined on {0, 1, 2, . . . , 2 n −1}. Solving this problem requires exponential time even on a quantum computer [2,3]. To confirm this we generated 100 instances of EC2 for each of n = 7, 8, 9, 10, 11, 12. For each instance we generate a random permutation Π and quantum evolve with H SCRAM (t) for time T . In Figure 3 we show the median time T required to achieve a success probability of 1/8. The linear behavior of the data on the log plot indicates an exponential growth as a function of n and the solid line represents 0.0689 (1.7565) n . In contrast with the data in Figures 1 and 2, the data in Figure 3 cannot be well fit by a quadratic. 7 Probabilities of success at a proposed running time Figures 1 and 2, at least by comparison with Figure 3, indicate that the median times required to achieve probability 1/8, for EC3 and EC2 with 7 ≤ n ≤ 15, grow modestly with n. Thus the fitted medians, for EC3 and EC2, are reasonable candidates for the running times T (n) of our algorithm for these two problems. Automatically, with these run times, our algorithm will achieve a success probability of at least 1/8 for about half of all instances. Our goal now is to explore how low the success probability can get at these run times. To this end we generate 100 new instances of EC3 and EC2 for each n. Now n runs from 7 to 16 for EC3 and from 7 to 15 for EC2 and T (n) is given by the fit to the data shown in Figures 1 and 2. The fact that the smallest probability does not decrease with n was not anticipated. This means that, at least for the range of the number of bits considered, the number of repetitions R(n) can be taken to be constant with n to achieve a fixed desired probability of success. In Figure 5 the data for EC2 is presented. Here the run time T (n) is given by the linear fit to the data in Figure 2. Again the median probability at each value of n is close to 1/8 as is expected. However, even for this classically easy problem we had no guarantee that the worst case probability would not decrease with n. In fact it does not appear to decrease at all. In order to show that the running time T (n) must increase with n to produce a successful algorithm, we explore the success probabilities obtained when using an n-independent running time. More specifically for the EC3 instances used to generate Figure 4 we run the algorithm for n = 7, 8, . . . , 14 for a constant value of T , the one previously used for n = 7. In Figure 6 the log plot shows that the median success probability decreases exponentially, which means that the number of repetitions R(n) would have to grow exponentially to achieve a fixed probability of success. In Section 6 we gave evidence that the bit structure is crucial to the success of the quantum adiabatic evolution algorithm. We make this point again by taking the instances of EC3, for n = 7, 8, . . . , 14, and running the algorithm with T (n) given by the fit in Figure 1 but with H(t) replaced by H SCRAM (t). In Figure 7, the median probability of success is seen to decrease exponentially with n. This helps confirm our intuition that the quantum adiabatic evolution algorithm takes advantage of the bit structure inherent in the problem. 8 Instances of EC3 with more than one satisfying assignment All of the EC3 data presented up to this point was generated from instances with unique satisfying assignments. Now we explore EC3 instances with more than one satisfying assignment. As in Section 3 clauses are added at random but now instances are accepted as soon as the number of satisfying assignments is 6, 7, 8, or 9. If adding a clause reduces the number of satisfying assignments from more than 9 to less than 6, the instance is rejected. We do this with 100 instances for 10, 11, 12, and 13 bits and run at the same times T (n) used for instances with a unique satisfying assignment to generate Figure 4. In Figure 8 we show the median probability, the smallest probability, and the 10 th smallest for these instances. At the running times used, the median probability for instances with unique satisfying assignments is close to 1/8 (for any number of bits). For the instances with multiple satisfying assignments the medians are about 1/3 and the worst case has a probability of about 1/8. This substantiates our intuition that instances with unique satisfying assignments are generally the most difficult for the quantum adiabatic algorithm. At the running times explored in this paper transitions out of the instantaneous ground state are not uncommon. In the case of a unique satisfying assignment such a transition (assuming no transition back) leads to a final state that does not correspond to the satisfying assignment. In the case of multiple satisfying assignments such transitions may lead to states that are headed towards the subspace spanned by the satisfying assignments. This is why the success probabilities are typically higher when there is more than one satisfying assignment. Discussion We have presented numerical evidence that a quantum adiabatic evolution algorithm can solve instances of satisfiability in a time that grows slowly as a function of the number of bits. Here we have worked out to 16 bits, but with more computing power instances with higher numbers of bits can be studied. This algorithm operates in continuous time. The algorithm can be written as a product of few-qubit unitary operators [1] where the number of factors in the product is of order T (n) 2 times a polynomial in n. However, understanding the idea behind the algorithm is obscure in the conventional quantum computing paradigm. (A quantum algorithm for satisfiability that is explicitly within the ordinary paradigm is presented in [4].) The algorithm studied in this paper works by having the quantum system stay close to the ground state of the time-dependent Hamiltonian that governs the evolution of the system. We imagine that protecting a device that remains in its ground state from decohering effects may be easier than protecting a device that requires the manipulation of excited states. (2.18), and evolves it according to the Schrödinger equation (2.4), for a time T , to produce \|ψ(T ) . The Hamiltonian H(t) is given by Eq. (2.8) with H P and H B determined by the instances of satisfiability being studied. Figure 1 : 1Exact Cover with three-bit clauses. Each circle is the median time to achieve probability between 0.12 and 0.13 for 50 instances. The curve is a quadratic fit to the data. Figure 2 : 2Exact Cover with two-bit clauses. Each circle is the median time to achieve probability between 0.12 and 0.13 for 50 instances. The curve is a linear fit to the data. Figure 3 : 3Exact Cover with two-bit clauses and a scrambled problem Hamiltonian. Each circle is the median time to achieve probability between 0.12 and 0.13 for 100 instances. The solid line on this log plot represents an exponential fit to the data. Figure 4 : 4Exact Cover with three-bit clauses. Probability of success at the proposed running time. Circles represent the medians of 100 instances at each number of bits. Triangles are the 10 th lowest probabilities and X's are the lowest. Figure 4 4displays the results for EC3. For each value of n from 7 to 16 we show the median probability of success at T (n) as well as the smallest of the 100 probabilities and the 10 th smallest. It is no surprise that the median probability is close to 1/8 for all values of n. Figure 5 : 5Exact Cover with two-bit clauses. Probability of success at the proposed running time. Circles represent the medians of 100 instances at each number of bits. Triangles are the 10 th lowest probabilities and X's are the lowest. Figure 6 : 6Exact Cover with three-bit clauses running at T = 5.82 for all numbers of bits. Circles represent the median probability of 100 instances. The straight line behavior on the log plot shows an exponential decrease in success probability. Figure 7 : 7Exact Cover with three-bit clauses with a scrambled problem Hamiltonian. The running time is given by the quadratic fit shown inFigure 1. The log plot shows an exponential decrease in success probability. Figure 8 : 8Exact Cover with three-bit clauses and 6, 7, 8, or 9 satisfying assignments. Circles represent the medians of 100 instances at each number of bits. Triangles are the 10 th lowest probabilities and X's are the lowest. Compare withFigure 4. 1 . 1Construct the time-dependent Hamiltonian H(t) given by Eq. (2.8) with H B and H P Measure z 1 , z 2 , . . . , z n in the state \|ψ(T ) and check if the bit string z 1 z 2 . . . z n satisfies all clauses. 5. Repeat R(n) times.given by Eq. (2.14) and Eq. (2.17). 2. Start the quantum system in the state \|ψ(0) given by Eq. (2.18). 3. Evolve according to Eq. (2.4) for a time T = T (n) to arrive at \|ψ(T ) . 4. AcknowledgmentsThis work was supported in part by the Department of Energy under cooperative agreement DE-FC02-94ER40818. E.F. thanks the participants at the Aspen Center for Physics meeting on Quantum Information and Computation (June 2000) for many helpful discussions. We thank Mehran Kardar, Joshua Lapan, Seth Lloyd, Andrew Lundgren, and Daniel Preda for valuable input. Quantum Computation by Adiabatic Evolution. E Farhi, J Goldstone, S Gutmann, M Sipser, quant-ph/0001106E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, "Quantum Computation by Adi- abatic Evolution", quant-ph/0001106. Strengths and Weaknesses of Quantum Computing. C H Bennett, E Bernstein, G Brassard, U V Vazirani, quant-ph/9701001SIAM J. Comput. 261510C.H. Bennett, E. Bernstein, G. Brassard, and U.V. Vazirani, "Strengths and Weak- nesses of Quantum Computing", quant-ph/9701001, SIAM J. Comput. 26, 1510 (1997). An Analog Analogue of a Digital Quantum Computation", quant-ph/9612026. E Farhi, S Gutmann, Phys. Rev. A. 572403E. Farhi and S. Gutmann, "An Analog Analogue of a Digital Quantum Computation", quant-ph/9612026; Phys. Rev. A 57, 2403 (1998). Quantum search heuristics. T Hogg, Phys. Rev. A. 6152311T. Hogg, "Quantum search heuristics", Phys. Rev. A 61, 052311 (2000); Quantum Optimization. T Hogg, D Portnov, quant-ph/0006090T. Hogg and D. Portnov, "Quantum Optimization", quant-ph/0006090.	[]
[ "Vacuum fluctuations of a massless spin-1 2 field around multiple cosmic strings", "Vacuum fluctuations of a massless spin-1 2 field around multiple cosmic strings" ]	[ "A N Aliev ", "M Hortaçsu \nFaculty of Sciences and Letters\nDepartment of Physics\nITU\n80626Maslak, IstanbulTurkey\n", "N Özdemir \nFaculty of Sciences and Letters\nDepartment of Physics\nITU\n80626Maslak, IstanbulTurkey\n", "\nTÜBITAK\nResearch Institute for Basic Sciences\n41470GebzeTurkey\n" ]	[ "Faculty of Sciences and Letters\nDepartment of Physics\nITU\n80626Maslak, IstanbulTurkey", "Faculty of Sciences and Letters\nDepartment of Physics\nITU\n80626Maslak, IstanbulTurkey", "TÜBITAK\nResearch Institute for Basic Sciences\n41470GebzeTurkey" ]	[]	We study the interaction of a massless quantized spinor field with the gravitational field of N parallel static cosmic strings by using a perturbative approach. We show that the presence of more than one cosmic string gives rise to an additional contribution to the energy density of vacuum fluctuations, thereby leading to a vacuum force of attraction between two parallel cosmic strings.	10.1088/0264-9381/14/12/008	[ "https://arxiv.org/pdf/gr-qc/9803044v1.pdf" ]	18,165,725	gr-qc/9803044	a04b66f739622dfe4a81d116d66bfcc48b72a543	Vacuum fluctuations of a massless spin-1 2 field around multiple cosmic strings arXiv:gr-qc/9803044v1 12 Mar 1998 A N Aliev M Hortaçsu Faculty of Sciences and Letters Department of Physics ITU 80626Maslak, IstanbulTurkey N Özdemir Faculty of Sciences and Letters Department of Physics ITU 80626Maslak, IstanbulTurkey TÜBITAK Research Institute for Basic Sciences 41470GebzeTurkey Vacuum fluctuations of a massless spin-1 2 field around multiple cosmic strings arXiv:gr-qc/9803044v1 12 Mar 1998 We study the interaction of a massless quantized spinor field with the gravitational field of N parallel static cosmic strings by using a perturbative approach. We show that the presence of more than one cosmic string gives rise to an additional contribution to the energy density of vacuum fluctuations, thereby leading to a vacuum force of attraction between two parallel cosmic strings. We study the interaction of a massless quantized spinor field with the gravitational field of N parallel static cosmic strings by using a perturbative approach. We show that the presence of more than one cosmic string gives rise to an additional contribution to the energy density of vacuum fluctuations, thereby leading to a vacuum force of attraction between two parallel cosmic strings. Introduction Cosmic strings predicted in the framework of various gauge theories with spontaneously broken symmetries could have been created at cosmological phase transitions in the early universe [1,2]. Since cosmic strings are produced at very large energy scales, one might expect a highly curved spacetime around them. However in the case of a static and straight-line cosmic string there exists a very simple exact solution of the Einstein field equations, that describes a locally flat conical spacetime around the string [3]. Furthermore, it has been shown that one may also construct an appropriate exact solution of the Einstein equations for a snapping cosmic string, which serves as a source for spherical impulsive gravitational waves [4]. Although the locally flat structure of the spacetime implies that straightline cosmic strings will not exert any local gravitational force on surrounding particles, the particles "interact" with the global conical structure. It gives rise to the distinctive gravitational effects, such as lensing of distant objects, conical bremsstrahlung, etc. [5,6]. On the other hand, it is well known that the non-trivial topological structures restrict the modes of quantized fields propagating in locally flat spacetimes, thereby providing the appearance of vacuum boundary effects [7]. One may, therefore, consider the conical spacetime around static and straight-line cosmic strings as an attractive model for investigation the influence of gravitational field on the behaviour of quantized fields. In particular, such investigations for a massless quantized fields of different spins, propagating in the spacetime of a static cosmic string have been carried out in [8][9][10]. The propagation of a massless quantized scalar field in the spacetime of a snapping cosmic string was studied in [11]. Recently in [12] we calculated the effects of vacuum fluctuations of a massless quantized electromagnetic field propagating in the spacetime of multiple cosmic strings represented by N parallel static (fixed), straight-line strings. It has been shown that the presence of more than one cosmic string provides an additional contribution to the energy density of vacuum fluctuations, which results in the Casimir-like force of attraction between two parallel cosmic strings. The aim of the present paper is to extend the calculations of [12] to the case of a massless quantized spinor field. First, for an instructive purpose, we shall rederive the expression for the vacuum expectation value of the energy-momentum tensor of a massless spin-1 2 field around a single cosmic string. For the case of multiple cosmic strings we shall adopt a perturbative approach, in which the gravitational field of the strings is treated as small metric perturbations about the Minkowskian spacetime. We construct, at first-order in metric perturbations, the Feynman propagator for the spinor field and evaluate the energy density arising from vacuum fluctuations. We shall also show that at second-order perturbations of the metric, the effect of vacuum polarization of the spinor field gives rise to a force of attraction between two parallel static and straight-line cosmic strings. Throughout the paper we use geometrical units, in which G = c = 1 andh ≈ 2.612 × 10 −66 cm 2 . A single cosmic string The metric of a static and straight-line cosmic string lying along the z-axis of the cylindrical coordinate system is given by the interval ds 2 = dt 2 − dz 2 − dr 2 − b 2 r 2 dθ 2(1) where b = 1 − 4µ, and µ is the linear mass density of the string. This metric is locally Minkowskian that is readily seen by passing to a new angular coordinate ϕ → b θ. We have ds 2 = dt 2 − dz 2 − dr 2 − r 2 dϕ 2(2) However the global structure of this metric is a conical, as for the new azimuthal angle ϕ we have the following range 0 ≤ ϕ < 2π(1 − 4µ). The presence of such a "deficit" in the azimuthal angle provides boundary conditions for quantized fields in the metric (2), which result in Casimir-like distortions of the spectrum of vacuum fluctuations. The effect of vacuum fluctuations for a massless spin- 1 2 field has been calculated in [9] by using the method of Green's functions. Here we shall briefly reproduce this result using a different approach, involving the summation of the exact field modes. The propagation of a massless spin-1 2 field in curved space-time is governed by the Dirac equation i γ µ ∇ µ ψ = 0(3) where the γ µ matrices satisfy the anticommutation relations {γ µ , γ ν } = 2 g µν .(4) The spinor covariant derivative operator is defined as ∇ µ = ∂ µ − Γ µ where Γ µ is the spinor connection. Let us consider a tetrad of vector field e µ a , satisfying the relations η ab = g µν e µ a e ν b , e µ a e ν b η ab = g µν(5) For the sake of further convenience we shall use the Newman-Penrose tetrad of null vectors [13] e µ a = {l µ , n µ , m µ , m * µ } which satisfy the normalization conditions l µ n µ = 1 and m µ m * µ = −1 . The symbol * denotes a complex conjugation. It is clear that with the choice of the tetrad (6) the symmetric constant matrices η ab and η ab have the only nonvanishing components η 12 = η 21 = −η 34 = −η 34 = 1, η ab = η ab . Since the ordinary constant Dirac matrices γ a satisfy the condition {γ a , γ b } = 2 η ab ,(7) then the coordinate-dependent matrices γ µ obeying the relation (4) will be defined through the above introduced tetrad e µ a by γ µ = e µ a γ a(8) while the spinor connection Γ µ is defined as Γ µ = 1 4 γ a γ b e ν a ; µ e b ν(9) where the semicolon denotes the covariant differantation with respect to the metric g µν . The explicite form of the γ µ matrices written in terms of the null vectors is γ µ = √ 2      0 0 n µ −m * µ 0 0 −m µ l µ l µ m * µ 0 0 m µ n µ 0 0     (10) Turning to the cosmic string metric (2) one may choose the following null vectors l µ = 1 √ 2 (1, −1, 0, 0) n µ = 1 √ 2 (1, 1, 0, 0) m µ = 1 √ 2 (0, 0, −1, − i r ) m * µ = 1 √ 2 (0, 0, −1, i r )(11) Let us now consider the four component spinor ψ =      ψ 1 ψ 2 ψ 3 ψ 4     (12) then the Dirac equation (3) projected onto the null vectors (11) is decomposed into the two independent pairs of equations D ψ 1 + (δ * + β) ψ 2 = 0 ∆ ψ 2 + (δ + β) ψ 1 = 0 (13) and ∆ ψ 3 − (δ * + β) ψ 4 = 0 D ψ 4 − (δ + β) ψ 3 = 0 (14) where β = 1 2 m * µ ; ν m µ m ν = − 1 2 √ 2 r and D = l µ ∂ µ = 1 √ 2 (∂ t − ∂ z ) ∆ = n µ ∂ µ = 1 √ 2 (∂ t + ∂ z ) δ = m µ ∂ µ = − 1 √ 2 (∂ r + i r ∂ ϕ )(15) are the directional derivative operators. Combining the equations (13) and (14) in an appropriate way we obtain the decoupled set of equations ✷ s ψ s = 0(16) where we have introduced the spin-weighted operator ✷ s = ∂ 2 t − ∂ 2 z − 1 r ∂ r (r∂ r ) − 1 r 2 ∂ 2 ϕ − 2is r 2 ∂ ϕ + s 2 r 2 with the spin-weight s = ± 1 2 and the spin-weighted modes ψ (s=+ 1 2 ) = ψ 2 ψ 4 , ψ (s=− 1 2 ) = ψ 1 ψ 3 The equations (16) may be solved by the seperation of variables and the regular at r = 0 solutions have the form ψ s ∼ e −i(ωt−kz−mνϕ) J \|mν+s\| (p r)(17) where J(p r) is a Bessel function and p 2 = ω 2 − k 2 , ν = 1/b. Since the transformation properties of spinors implies that a 2 π rotation of a spinor changes its sign [14], the single-valuedness of the solutions (17) requires that m = n − 1 2 , or m = n + 1 2 with n = 0, ±1 ± 2... . Having in hand these modes one can easily construct the complete sets of positive-and negativefrequency solutions u smkp and v smkp of the Dirac equation, which satisfy the normalization conditions u s ′ m ′ k ′ p ′ (x) γ 0 u s m k p (x) √ −g d 3 x = v s ′ m ′ k ′ p ′ (x) γ 0 v s m k p (x) √ −g d 3 x = δ ss ′ δ mm ′ δ(k − k ′ ) δ(p − p ′ ) √ p p ′ (18) where u = u † γ 0 and v = v † γ 0 are the adjoint spinors. The canonical quantization are performed in a usual way [7], by expanding the field operatorψ (x) on the complete sets of positive and negative frequency modeŝ ψ (x) = s,m ∞ −∞ dk ∞ 0 dp p [â smkp u smkp (x) +b † smkp v smkp (x)](19) whereâ smkp represents annihilation operator for particles andb † smkp is the creation operator for antiparticles. It should be noted that as the spacetime is flat everywhere outside cosmic strings one can define a vacuum state by choosing positive frequency modes with respect to the timelike Killing vector ∂/∂t of the flat spacetime. The energy-momentum tensor for spin-1 2 has the form T µν (x) = i 2 [ ψ γ (µ ∇ ν) ψ − (∇ (µ ψ) γ ν) ψ ](20) The vacuum expectation value of this quantity can be evaluated by representing it as a bilinear function of the fields and performing a renormalization procedure at the coincidence points x = x ′ . Substituting the expansion (19) into the equation (20) we obtain that its vacuum averaged < T 00 (x) > component are reduced to the form < T 00 (x) >= lim x→x ′ (∂ 2 t − ∂ t ∂ t ′ ) U(x, x ′ ) 2ω(21) where the two-points function U(x, x ′ ) is U(x, x ′ ) 2ω = ν 8π 2 m e iνm(ϕ−ϕ ′ ) ∞ 0 dp p ∞ −∞ dk e −i √ k 2 +p 2 τ + ik ζ √ k 2 + p 2 [J \|mν+ 1 2 \| (p r)J \|mν+ 1 2 \| (p r ′ ) + J \|mν− 1 2 \| (p r)J \|mν− 1 2 \| (p r ′ )] (22) and τ = t − t ′ , ζ = z − z ′ . The integrals over k and p in this expression are evaluated by means of the corresponding formulae given in [17]. As a result we have U(x, x ′ ) 2ω = ν 8π 2 r r ′ 1 √ u 2 − 1 ∞ n=−∞ e −iν (n− 1 2 )(ϕ−ϕ ′ ) ξ −\|ν(n− 1 2 )+ 1 2 \| + ξ −\|ν(n− 1 2 )− 1 2 \| (23) where u = r 2 + r ′2 + ζ 2 − τ 2 2 rr ′ , ξ = u + √ u 2 − 1 and we have taken m = n − 1 2 . The renormalization of the function U(x, x ′ ) is achieved by substracting from this expression its pure Minkowskian value, i.e at ν = 1. Since the equation (21) involves only the derivatives over t and t ′ one can put ϕ = ϕ ′ , z = z ′ and r = r ′ . Then the evaluation of the sum in (23) is significantly simplified and we have ∞ n=−∞ ξ −\|ν(n− 1 2 )+ 1 2 \| + ξ −\|ν(n− 1 2 )− 1 2 \| = 2 ξ − ν 2 (ξ − 1 2 + ξ 1 2 ) 1 − ξ −ν(24) The further calculation of the renormalized quantity U(x, x ′ ) becomes straightforward and substituting the result into the equation (21) we arrive at the following expression for the energy density of vacuum fluctuations < T 00 (x) >= −h 2880 1 π 2 r 4 (ν 2 − 1) (7ν 2 + 17)(25) This result is in agreement with that of given in [9]. The other components of < T µν (x) > can be found by using the symmetry properties of the space-time (1) along with the conditions T µ µ = 0 and T µν ; ν = 0. Multiple cosmic strings If one makes a transformation of the radial coordinate r → r b i /b, then the metric (1) can be transformed into the the following form ds 2 = dt 2 − dz 2 − e −2Λ(x,y) (dx 2 + dy 2 ) (26) where Λ(x, y) = N i=1 4 µ i ln r i r i = [(x − α i ) 2 + (y − β i ) 2 ] 1/2(27) It turns out that this metric is the exact solution of the Einstein equations [15], describing the space-time around N parallel static cosmic strings, passing through the points x i = (α i , β i ). The corresponding tetrad of null vectors for the metric (26) can be chosen as l µ = 1 √ 2 (1, −1, 0, 0) n µ = 1 √ 2 (1, 1, 0, 0) m µ = 1 √ 2 e Λ (0, 0, −1, −i) m * µ = − 1 √ 2 e Λ (0, 0, 1 − i) (28) Using these vectors in the equations (8) and (9) we find that γ 0 (x) = γ 0 , γ 3 (x) = γ 3 , γ 1 (x) = e Λ γ 1 , γ 2 (x) = e Λ γ 2 ,(29) where γ 0 , γ 1 , γ 2 , γ 3 are ordinary constant Dirac matrices, and the only non-vanishing components of the spinor connection Γ µ are Γ 1 = − 1 2 γ 1 γ 2 e Λ ∂ y Λ Γ 2 = − 1 2 γ 2 γ 1 e Λ ∂ x Λ(30) Now we shall evaluate the vacuum expectation value of the energy-momentum tensor (20) in the metric (26). We start with the expression < T µν (x) >= −h 4 lim x→ x ′ Tr [γ µ (∇ ν − ∇ ν ′ ) + γ ν (∇ µ − ∇ µ ′ )]S F (x, x ′ ) (31) where S F (x, x ′ ) = −i < 0\| T ψ (x ′ ) ψ(x) \|0 >(32) is the Feynman propagator which obeys the equation iγ µ (∂ µ − Γ µ )S F (x, x ′ ) = 1 √ −g δ 4 (x − x ′ )(33) The substitution of the relations (29) and (30) into this equation enables us to cast it in the form iγ a ∂ a S F (x, x ′ ) = δ 4 (x − x ′ ) + V S F (x, x ′ ) (34) where γ a once again denotes the flat space-time Dirac matrices, V = i [(1 − e −2Λ )γ A ∂ A + (1 − e −Λ )γ α ∂ α + 1 2 e −Λ γ α ∂ α Λ](35) and the index A takes the values (0, 3) ≡ (t, z), while α = (1, 2) ≡ (x, y). In order to construct the solution of equation (34) we use a perturbative approach. For this purpose, we assume that the linear mass densities of the cosmic strings are sufficiently small, (µ i << 1), which is indeed the case for realistic cosmic strings (µ i ≈ 10 −6 ) Then the metric (26) can be expanded in powers of µ i , about a fixed flat background, and the potential (35) can be considered as a small perturbing term in the equation (34). This approximation allows us to write the solution of equation (34) as perturbation series S F (x, x ′ ) = S (0) F + S (0) F V S (0) F + S (0) F V S (0) F V S (0) F + · · ·(36) The zeroth order free Feynman propagator is defined as S (0) F (x, x ′ ) = d 4 k (2π) 4 γk k 2 e −ik(x−x ′ )(37) where γk = γ a k a . The successive terms in the expansion (36) correspond to the higher order in µ i contributions to the free propagator. Expanding the equation (35) in powers of µ i and taking it into account in (36) we obtain that the first order corrections to the free Feynman propagator are given by S (1) F (x, x ′ ) = 1 64π 6 d 4 k δ(k 0 ) δ(k 3 ) Λ(k) e −ikx ′ d 4 q q 2 (q − k) 2 γq [ γ A q A + γq − 1 2 γk ] γ(q − k) e −iq(x−x ′ )(38) In order to evaluate the finite vacuum expectation value of the energymomentum tensor(31) one needs to regularize it, for which we shall use the dimensional regularization procedure [16]. The latter gives to the the vanishing value for the zeroth order propagator [7], however, substituting the first-order propagator (38) into the expression (31), for its < T (1) 00 (x) > component at the coincidence limit x → x ′ we find < T (1) 00 (x) >= ih 64π 6 Tr d 4 k δ(k 0 ) δ(k 3 ) Λ(k) e −ikx ′ d 4 q q 2 (q − k) 2 γ 0 q 0 γq [ γ A q A + γq − 1 2 γk ] γ(q − k)(39) In the framework of the dimensional regularization procedure the integral over q is calculated, by performing an analytical continuation to d dimensional space and then the result is expanded about d = 4 − ǫ, (ǫ → 0) [14]. Having done all these we obtain < T (1) 00 (x) >= −h 240π 3 Γ(−1 + ǫ 2 ) N i=1 µ i d 2 k (k 2 ) 1− ǫ 2 e −ikr i(40) In obtaining of this expression we have used the explicit form for the Fourier component of the function Λ(x, y) Λ(k) = 8π N i=1 µ i e ik x i k 2(41) It is important to stress that the expression (40) is finite, as the divergent terms are exactly cancelled by one another. Indeed, using the integral [12] ∞ 0 d d k(k 2 ) ν e −ikR = π d/2 2 2ν+d R 2ν+d Γ(ν + d/2) Γ(−ν)(42) in Eq.(40), in which d = 2 − ǫ, and ν = 1 − ǫ 2 , we finally obtain the following result < T (1) 00 (x) >= −h 15π 2 N i=1 µ i r 4 i(43) It is seen from this expression that at first order in µ i , the contributions to the energy density of vacuum fluctuations of spin-1 2 field around N parallel static cosmic strings are linearly summed. One can easily see that in the case of a single cosmic string the expression (43) coincides with (25) taken at µ << 1. Vacuum force between two parallel cosmic strings Let us now proceed to the second-order metric contributions to the vacuum expectation value of the energy-momentum tensor (31). In analogy with the case of electromagnetic fluctuations [12] we shall show that the energy density of vacuum fluctuations of a massless spinor field involves a term which depends upon the seperation distance between cosmic strings, therefore, produces an attractive force between the strings. Using the expressions (35) and(37) in the expansion (36) we find that the second-order metric corrections to the free Feynman propagator have the form S (2) F (x, x) = 1 256π 8 d 4 k δ(k 0 )δ(k 3 ) Λ(k)e −ik x ′ d 4 p δ(p 0 )δ(p 3 ) Λ(p)e −ip x ′ d 4 q q 2 γq (γ A q A + γq − 1 2 γk) γ(q − k) (q − k) 2 (q − k − p) 2 [γ B q B + γ(q − k − p) + 1 2 γp] γ(q − k − p) e −iq (x−x ′ )(44) Substituting this expression into the < T 00 > component of Eq. (31) at the coincidence limit x = x ′ , we calculate the traces of γ matrices using the wellknown theorems [14], then taking into account the relation (41) we arrive at the expression < T (2) 00 (x) >= ih 4π 6 N i=1 µ i N j=1 µ j d 4 k k 2 δ(k 0 ) δ(k 3 ) e −ik (x−x i ) d 4 p p 2 δ(p 0 ) δ(p 3 ) e −ip (x−x j ) d 4 q q 2 q 2 0 (q − k) 2 (q − k − p) 2 N (q, k, p) (45) where N (q, k, p) = 4(q A q A )[4 q A q A + (q−k) 2 + (k+p) 2 −kp]+ 2(q 2 − 2 kq) (kq− 2q 2 ) − k 2 (2 q 2 + 2 kq− p 2 − 2 k 2 )− (kp) (q 2 − 3 k 2 + 2 kq)+ 2 (pq) (q−k) (3q−k) We note that in this expression we are interested only in contributions to the vacuum energy density which are proportional to the products of the linear mass densities of different cosmic strings. It is clear that the latter describes the energy density of vacuum interaction between the strings. As for the remaining contributions, they form higher order corrections to the vacuum energy density given by (43). However first we need to regularize the expression (45), for which we again use the dimensional regularization procedure. We evaluate the integral over q by performing an analytical continuation to d = 4 − ǫ dimensions using the scheme, described in [14]. After some straightforward algebra, we arrive at the following result < T (2) 00 (x) >= −h 4π 4 i=1 N µ i N j=1 µ j d 2 k k 2 e −ik (x−x i ) d 2 p p 2 e −ip (x−x j ) 1 0 dz 1 1−z 1 0 dz 2 [40 Γ(−2 + ǫ 2 ) B 2− ǫ 2 + C 1 Γ(−1 + ǫ 2 ) B 1− ǫ 2 + C 2 Γ( ǫ 2 ) B − ǫ 2 ](46) where we have introduced C 1 = −12Q (Q − k − p) + 4 (k 2 + p 2 + 1 8 kp ) C 2 = Q 2 (−2 Q 2 + 5 kQ − k 2 − 1 2 kp + 3pQ ) − kQ ( k 2 + kp + 2kQ + 4pQ ) + 1 2 k 2 ( p 2 + 2k 2 + 3kp + 2pQ ) and Q α = k α z 1 + (k α + p α ) z 2 B = k 2 z 1 (1 − z 1 ) + (k + p) 2 z 2 (1 − z 2 ) − 2k (k + p) z 1 z 2 The energy of vacuum fluctuations per unit length of the strings may be evaluated by means of the formula E = d 2 x √ −g < T 00 (x) >(47) which at the second order metric perturbations takes the following form E = dxdy < T (2) 00 (x) > − 2Λ(x) < T (1) 00 (x) >(48) For the sake of certainty let us consider two parallel cosmic strings. Using the relations (40) and (46) in the equation (48) we first carry out the integration over x and y, then the calculations of remaining integrals become straightforward and keeping only the terms involving the product of the linear mass densities of the cosmic strings we find E int = −h 15π 2 µ 1 µ 2 Γ( ǫ 2 ) d 2 k (k 2 ) − ǫ 2 e −ika .(49) It should be stressed that this quantity is finite as the involved divergent at ǫ → 0 terms are compensated by one another. Indeed taking the double integration over k using the integral (42) we obtain the expression E int = − 4h 15π µ 1 µ 2 a 2(50) where a is the seperation distance between the cosmic strings. It is clear that the presence of this energy, will produce an attractive force per unit length of the strings given by F = − 8h 15π µ 1 µ 2 a 3(51) As we have already mentioned above the static and straight-line cosmic strings do not exert any local gravitational force on surrounding matter. Here we have shown that the propagation of a massless quantized spinor field in the spacetime of more than one cosmic string induces a force of attraction (51) between two cosmic strings, which falls of as the third power of the separation distance. The reason for this is the restriction of the modes of quantized field by the multiconical structure of the spacetime around the cosmic strings. Unlike the case of a single cosmic string, it is difficult to construct the exact modes of the field equations in the metric of multiple cosmic strings, so we have used a perturbative approach along with the dimensional regularization procedure. We note that the expression (51) coincides with the corresponding result for a massless scalar field [18], while 2 times smaller than the result for a massless vector field [12]. 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[ "ON THE RESTRICTED PROJECTIVE OBJECTS IN THE AFFINE CATEGORY O AT THE CRITICAL LEVEL", "ON THE RESTRICTED PROJECTIVE OBJECTS IN THE AFFINE CATEGORY O AT THE CRITICAL LEVEL" ]	[ "Peter Fiebig " ]	[]	[]	This article gives both an overview and supplements the articles [AF08] and [AF09] on the critical level category O over an affine Kac-Moody algebra. In particular, we study the restricted projective objects and review the restricted reciprocity and linkage principles.	10.1090/conm/565/11178	[ "https://arxiv.org/pdf/1103.1540v1.pdf" ]	117,446,878	1103.1540	5ab33582c0676927cad460ee5a6927830f48f7af	ON THE RESTRICTED PROJECTIVE OBJECTS IN THE AFFINE CATEGORY O AT THE CRITICAL LEVEL 8 Mar 2011 Peter Fiebig ON THE RESTRICTED PROJECTIVE OBJECTS IN THE AFFINE CATEGORY O AT THE CRITICAL LEVEL 8 Mar 2011 This article gives both an overview and supplements the articles [AF08] and [AF09] on the critical level category O over an affine Kac-Moody algebra. In particular, we study the restricted projective objects and review the restricted reciprocity and linkage principles. Introduction The main objective of this paper is to introduce to and to complement the results of the papers [AF08] and [AF09] on the critical level representation theory of affine Kac-Moody algebras that provide the first steps in a research project, joint with Tomoyuki Arakawa, whose main motivation is the determination of the critical level simple highest weight characters. There are at least two (essentially different) approaches to character problems in Lie theory. The first (and slightly more classical), is due to Beilinson and Bernstein and utilizes a localization functor, i.e. a functor that realizes representations of a Lie algebra as D-modules on a convenient algebraic variety. This functor was initially introduced in [BB81] in order to determine the characters of simple highest weight representations of semisimple Lie algebras. Later, Kashiwara and Tanisaki used a similar functor in the case of symmetrizable Kac-Moody algebras (cf. [KT00]). For modular Lie algebras, i.e. Lie algebras over a field of positive characteristic, a version of the localization functor is one of the main ingredients in the work of Bezrukavnikov et al. (cf. [BMR08]). Recently, Frenkel and Gaitsgory used it in their formulation of the local geometric Langlands conjectures and their study of the critical level representation theory of affine Kac-Moody algebras (cf. [FG06]). The second approach goes back to Soergel (cf. [S90]). Here, the main idea is to link the representation theory to the topology of an algebraic * partially supported by a grant of the Landesstiftung Baden-Württemberg and by the DFG-Schwerpunkt 1388. variety (most notably to the category of perverse sheaves) by an intermediate "combinatorial" category. These combinatorial categories often have a slightly artificial flavour, examples of which are categories of Soergel bimodules, of sheaves on moment graphs and the highly complicated category studied in [AJS94]. However, it turned out that they can also play a significant role outside their originial habitat, for example in knot theory or for p-smoothness questions in complex algebraic geometry (cf. [FW10]). The first example of a relation of this second type appears in [S90] in the case of semisimple complex Lie algebras. The paper [F06] contains the generalization to the symmetrizable, non-critical Kac-Moody case. In the same spirit, the paper [F07] links restricted representations of a modular Lie algebra as well as representations of the small quantum group to parity sheaves on affine flag varieties (one of the main ingredients for this is the combinatorial description in [AJS94]). In our research project we hope to establish a similar result for the critical level representation theory of an affine Kac-Moody algebra. Note that both approaches outlined above potentially are sufficient to determine representation theoretic data such as characters, but it is only after one takes them together that they release their full potential. For example, the celebrated Koszul duality for the category O of a semisimple Lie algebra is constructed by combining the Beilinson-Bernstein localization as well as the Soergel approach (cf. [BGS96]). In the following we review our approach and state the main results of the articles [AF08,AF09]. Moreover, we complement these articles by a new and simplified treatment of projective covers in deformed versions of the affine category O. Affine Kac-Moody algebras We fix a finite dimensional simple complex Lie algebra g and we denote by k : g × g → C its Killing form. First, we explain the main steps of the construction of the affine Kac-Moody algebra associated with g (for more details, see [K90]). The loop algebra associated with g is the Lie algebra with underlying vector space g ⊗ C C[t, t −1 ] that is endowed with the C[t, t −1 ]-bilinear extension of the bracket of g. So we have [x ⊗ t n , y ⊗ t m ] = [x, y] ⊗ t m+n for x, y ∈ g and m, n ∈ Z. The loop algebra has an up to isomorphism unique non-split central extension g of rank one. Its underlying vector space is g ⊗ C C[t, t −1 ] ⊕ CK and the bracket is given by [x ⊗ t n , y ⊗ t m ] = [x, y] ⊗ t m+n + nδ m,−n k(x, y)K, [K, g] = {0} (here δ a,b denote the Kronecker symbol). In order to obtain the affine Kac-Moody algebra g associated with g we add the outer derivation operator D = t ∂ ∂t to g. So we obtain the vector space g ⊗ C C[t, t −1 ] ⊕ CK ⊕ CD with bracket [x ⊗ t n , y ⊗ t m ] = [x, y] ⊗ t m+n + nδ m,−n k(x, y)K, [K, g] = {0}, [D, x ⊗ t n ] = nx ⊗ t n for x, y ∈ g, m, n ∈ Z. Note that g naturally appears as a subalgebra in g via the map x → x ⊗ 1. 2.1. Roots and coroots. We fix a Cartan subalgebra h in g and a Borel subalgebra b ⊂ g that contains h. Then the corresponding Cartan and Borel subalgebras in g are h := h ⊕ CK ⊕ CD, b := g ⊗ tC[t] ⊕ b ⊕ CK ⊕ CD. Let us denote by R ⊂ h ⋆ = Hom C (h, C) the root system of g with respect to h and by R + ⊂ R the subset of positive roots, i.e. the set of roots of b. Let g = h ⊕ α∈R g α be the root space decomposition. The coroot associated with α ∈ R is the unique element α ∨ ∈ [g α , g −α ] with the property α, α ∨ = 2. Note that the dual of the projection h → h along the decomposition h = h ⊕ CK ⊕ CD allows us to view h as a subset of h ⋆ . We denote by δ ∈ h ⋆ the unique element with δ(h ⊕ CK) = {0}, δ(D) = 1. Then the set of roots of g with respect to h is R = R re ∪ R im , where R re = {α + nδ \| α ∈ R, n ∈ Z}, R im = {nδ \| n ∈ Z, n = 0}. The first set is called the set of real roots, the second set is called the set of imaginary roots. The corresponding root spaces are g α+nδ = g α ⊗ t n , g nδ = h ⊗ t n . Let Π ⊂ R + be the set of simple roots. The set of simple affine roots is then Π = Π ∪ {−γ + δ}, where γ ∈ R + is the highest root. The set of positive affine roots (i.e., the set of roots of b) is R + = R + ∪ {α + nδ \| α ∈ R, n > 0} ∪ {nδ \| n > 0}. For a real root α + nδ the space [ g α+nδ , g −(α+nδ) ] is a one-dimensional subspace of h. The coroot associated to α + nδ is the unique element (α + nδ) ∨ ∈ [ g −(α+nδ) , g α+nδ ] with the property α + nδ, (α + nδ) ∨ = 2. Explicitely, this is (α + nδ) ∨ = α ∨ + 2n k(α, α) K. Here we denote by k : h ⋆ × h ⋆ → C the bilinear form induced by the Killing form. 2.2. The affine Weyl group. To a real affine root α+nδ we associate the following reflection on h ⋆ : s α+nδ (λ) = λ − λ, (α + nδ) ∨ (α + nδ). The affine Weyl group is the subgroup W ⊂ GL( h ⋆ ) that is generated by all reflections s α+nδ with α + nδ ∈ R re . We need the following shifted, non-linear action of W on h ⋆ . Let us choose an element ρ ∈ h ⋆ with the property that ρ(α ∨ ) = 1 for each simple affine coroot α ∨ . Note that ρ is not uniquely defined, as the simple coroots do not generate h. Instead, ρ + xδ would do equally well for each x ∈ C. However, as δ is stabilized by the action of W, the dot-action w.λ = w(λ + ρ) − ρ is independent of the choice. It fixes the line −ρ + Cδ. It is well-known that there is an up to isomorphism unique simple gmodule L(λ) with highest weight λ, i.e. that is generated by its λ-weight space and has the property that L(λ) µ = 0 implies λ ≥ µ. Then the complex dimension of L(λ) µ is finite, hence we can consider the formal sum ch L(λ) = µ≤λ dim C L(λ) µ e µ as an element in the formal completion (with respect to ) of the group algebra of the additive group h ⋆ . Now, if λ is integral and dominant (i.e., if λ, α ∨ ∈ Z ≥0 for all simple affine roots α), then ch L(λ) is given by the Weyl-Kac character formula (cf. [K90]). More generally, if λ is non-critical (i.e., if λ + ρ, K = 0), then ch L(λ) is given by an appropriate version of the formula conjectured by Kazhdan and Lusztig (cf. [KT00]). In the case that λ is critical, Feigin and Frenkel conjectured a formula for ch L(λ) (see [AF08]). This conjecture, however, is yet unproven in general. In the integral dominant critical case, the conjecture follows from the results in [FG09]. The objective of this paper is to supplement the papers [AF08, AF09] that provide a first step in a program that aims to solve the character problem at the critical level. 2.4. The category O. It is most convenient to introduce now a categorical framework for the above mentioned problem. Definition 2.1. (1) M is called a weight module (with respect to h), if h acts semisimply, i.e. if M = λ∈ h ⋆ M λ . (2) M is called locally b-finite, if each element of M is contained in a finite dimensional b-submodule. We denote by O the full subcategory of all g-modules which are weight modules and on which b acts locally finitely. This is an abelian subcategory of g-mod. Each simple object L(λ) is contained in O, as is, more generally, each module with highest weight. For λ ∈ h ⋆ the Verma module with highest weight λ is defined as ∆(λ) = U( g) ⊗ U ( b) C λ , where C λ is the simple b-module corresponding to the character b → h λ → C, where the map on the left is the homomorphism of Lie algebras that is left invers to the inclusion h ⊂ b. The dual Verma module ∇(λ) is the restricted dual of ∆(λ), i.e. it is the set of h-finite vectors in the representation of g that is dual to ∆(λ). Each ∇(λ) is contained in O as well. 2.5. The level. For a g-module M and a complex number k we define M k := {m ∈ M \| K.m = km}, the eigenspace of the action of the central element K in g with eigen- value k. Clearly, each M k is a submodule in M. A g-module M is said to be of level k if M = M k . If M is a weight module, then K acts semisimply on M, so M = k∈C M k . In fact, in this case we have M k = λ∈ h ⋆ ,λ(K)=k M λ . If we denote by O k the full subcategory of O that contains all modules of level k, then the functor k∈C O k → O {M k } k∈C → k∈C M k is an equivalence of categories. 2.6. A graded structure. In the following we construct a grading functor T on O (i.e., an autoequivalence T : O → O). Let us consider the simple g-module L(δ) with highest weight δ. It is one-dimensional. In fact, the algebra g = [ g, g] = g ⊗ C[t, t −1 ] ⊕ CK acts trivially on L(δ), while D ∈ g acts as the identity operator. Recall the usual tensor structure on the category of g-modules: If M and N are gmodules, then M ⊗ C N becomes a g-module with the action determined by X.(m ⊗ n) = (X.m) ⊗ n + m ⊗ X.n for X ∈ g and m ∈ M, n ∈ N. We define the functor T : g-mod → g-mod, M → M ⊗ C L(δ) with the obvious action on morphisms. It is an equivalence with inverse T −1 : M → M ⊗ C L(−δ), and it preserves weight modules, as (T M) λ = M λ−δ ⊗ C L(δ) for each g-module M and λ ∈ h ⋆ . Moreover, if N ⊂ M is a b-submodule, then N ⊗ C L(δ) ⊂ M ⊗ C L(δ) is a b-submodule. Hence T also preserves the b-local finiteness condition. So T preserves the category O, hence it makes O into a graded category. As δ, K = 0, the functor T in addition preserves the level, i.e. it induces a grading on the subcategories O k for all k ∈ C. 2.7. The graded center. With any graded category (C, T ) we can associate the following: Definition 2.2. The graded center of (C, T ) is the graded vector space A = A(C, T ) = n∈Z A n , where A n is the space of natural transfor- mations τ : id C → T n with the property that for all objects M of C and all m ∈ Z, the morphism T m (τ M ) : T m M → T m T n M = T n T m M equals the morphism τ T m M . Note that A carries a natural multiplication that makes it into a commutative, associative, unital algebra (cf. [AF08]). In particular, we have the graded centers A of ( O, T ) and A k of ( O k , T ) for all k ∈ C. Clearly, A = k∈C A k . Now there is only one value k = crit := −ρ, K for which (A k ) n is non-zero for all n ∈ Z. In fact, if k = crit, then (A k ) n is the trivial vector space for all n = 0 (for more information about (A k ) 0 for k = crit, see [F03]). However, the spaces (A crit ) n are huge for any n = 0 (cf. [AF08]). We define the restricted Verma module corresponding to λ ∈ h ⋆ as ∆(λ) := ∆(λ) and the restricted dual Verma module as ∇(λ) := ∇(λ). Deformed category O One of the main methods in our approach to the representation theory of Kac-Moody algebras is the following deformation idea. Let us denote by S = S(h) and S := S( h) the symmetric algebras associated with the vector spaces h and h. The projection h → h along the decomposition h = h ⊕ CK ⊕ CD yields a homomorphism S → S of algebras. Let A be a commutative, Noetherian, unital S-algebra. Then we can consider A as an S-algebra via the above homomorphism. We call such an algebra in the following a deformation algebra. As A contains a unit, we have a canonical map τ : h → A, f → f.1 A . Note that, as we start with an S-algebra, we will always have τ (K) = τ (D) = 0. By a g A -module we mean in the following an A-module together with an action of g that is A-linear, i.e. a g-A-bimodule. Let M be a g A -module and λ ∈ h ⋆ . We define the λ-weight space of M as M λ := {m ∈ M \| H.m = (λ(H) + τ (H))m for all H ∈ h}. (Note that it would be more appropriate to call this the λ + τ -weight space.) Let M be a g A -module. Definition 3.1. ( 1) We say that M is a weight module if M = λ∈ h ⋆ M λ . (2) We say that M is locally b A -finite, if every element in M is contained in a b A -submodule that is finitely generated as an A-module. We denote by O A the full subcategory of the category of g A -modules that contains all locally b A -finite weight modules. Note that if A = K is a field, then O K is a direct summand of the usual category O defined for the Lie algebra g ⊗ C K. It contains all modules M whose weights have the special form λ + τ with λ ∈ h ⋆ (note that the later element can be considered as a K-linear form on the Cartan subalgebra h ⊗ C K). 3.1. Verma modules in O A . Let λ ∈ h ⋆ and denote by A λ the b Amodule that is free of rank one as an A-module and on which h acts by the character λ + τ and [ b, b] acts trivially. The deformed Verma module with highest weight λ is ∆ A (λ) := U( g) ⊗ U ( b) A λ . Then ∆ A (λ) is a weight module and each weight space ∆ A (λ) µ is a free A-module of finite rank. Moreover, ∆ A (λ) is b A -locally finite, so it appears as an object in As before there is a functor · : O A → O A that is left adjoint to the inclusion functor (it is constructed as in the non-restricted case). The deformed restricted Verma module with highest weight is ∆ A (λ) := ∆ A (λ). O A . If A → A ′ is a homomorphism of deformation algebras, then ∆ A (λ) ⊗ A A ′ ∼ = ∆ A ′ (λ). Projective objects in O A Let A be a local deformation algebra. Now we want to study projective objects in the deformed category O A . In particular, we want to study projective covers of simple objects L A (λ), i.e. projective objects P together with a surjective map c : P → L A (λ) with the following property: If g : M → P is a homomorphism such that c•g : M → L A (λ) is surjective, then g is surjective. Such projective covers do not always exists in O A . But when we restrict ourselves to truncated categories, then the situation improves. Theorem 4.2. Suppose that A is a local deformation algebra. Let µ ∈ J . Then there exists a projective cover P J A (µ) → L A (µ) in O J A . In order to prove the above theorem, we first consider the universal enveloping algebra U( b) under the adjoint action of h, so we obtain a weight space decomposition U( b) = γ∈Z ≥0 R + U( b) γ . For µ ∈ J we define J ′ = J − µ = {ν ∈ h ⋆ \| ν = γ − µ for some γ ∈ J } and I ′ = h ⋆ \ J ′ . Then the vector space U( b) I ′ = γ∈I ′ U( b) γ is a (two-sided) ideal in U( b), and hence U( b) J ′ = U( b)/U( b) I ′ is a U( b)-module. As S = U( h), we get a (right) action of S on U( b) J ′ , and hence we can form the tensor product Q J A (µ) := U( g) ⊗ U ( b) U( b) J ′ ⊗ S A µ . This is a g A -module. As in [RCW82] (see also [F03]) one shows that this object represents the functor O J A → A-mod, M → M µ , so by the definition of O A it is projective in O A . Moreover, it admits a Verma flag with multiplicities (Q J A (µ) : ∆ A (ν)) = dim C U( n) ν−µ , if ν ∈ J , 0, if ν ∈ J , where n = α∈ R + g α . In particular, we have (Q J A (µ) : ∆ A (ν)) = 0 only if ν ≥ µ and (Q J A (µ) : ∆ A (µ)) = 1. Every direct summand of a module with a Verma flag admits a Verma flag as well. As ∆ A (µ) occurs with multiplicity one, there is a direct summand P J A (µ) of Q J A (µ) with (P J A (µ) : ∆ A (µ)) = 1. Note that we do not yet claim that P J A (µ) is unique up to isomorphism, yet this will be a consequence once we proved Theorem 4.2. For now, it suffices to choose a direct summand with the above properties. As all other Verma subquotients of P J A (µ) have highest weights µ ′ with µ ′ > µ, there is a surjection P J A (µ) → ∆ A (µ), hence a surjection P J A (µ) → L A (µ) and this surjection is unique up to non-zero scalars in C. We can now prove the above theorem. Proof of Theorem 4.2. We prove the statement by induction on the number of elements in the set J ≥µ . If it contains only the element µ, then P J A (µ) = Q J A (µ) ∼ = ∆ A (µ) and the locality of A implies that ∆ A (µ) → L A (µ) is a projective cover. So let us fix µ ∈ J and let us assume that the statement is proven for all pairs µ ′ ∈ J ′ such that J ′ ≥µ ′ contains strictly less elements then J ≥µ . As a next step we prove that L A (µ) is then the only simple quotient of P J A (µ). Suppose that this is not the case, hence that there exists a surjection P J A (µ) → L A (ν) for some ν = µ. As P J A (µ) is generated by its weight spaces corresponding to weights in J ≥µ , this implies ν ∈ J ≥µ . By induction assumption, P J A (ν) → L A (ν) is a projective cover. Now by the projectivity of P J I I I I I I I I I P J A (ν) z z u u u u u u u u u L A (ν) commutes. As P J A (ν) → L A (ν) is a projective cover, the homomorphism P J A (µ) → P J A (ν) is surjective, and from the projectivity of P J A (ν) and the indecomposability of P J A (µ) we deduce P J A (µ) ∼ = P J A (ν), which contradicts what we already know about the Verma subquotients of both objects. Hence we have proven that L A (µ) is the only simple quotient of P J A (µ). A (µ) there is a homomorphism P J A (µ) → P J A (ν) such that the diagram P J A (µ) / / $ $Now let J ′ ⊂ J be another open subset of h ⋆ . Then P J A (µ) J ′ is a quotient of P J A (µ) and it is projective in O J ′ A , as the functor M → M J ′ , O J A → O J ′ A , is left adjoint to the inclusion functor. As P J A (µ) has a unique simple quotient, we have P J A (µ) J ′ ∼ = P J ′ A (µ). Now we can prove that c : P J A (µ) → L A (µ) is a projective cover. So let g : M → P J A (µ) be a homomorphism such that c • g : M → L A (µ) is surjective. Then the projectivity implies that there is a homomorphism h : P J A (µ) → M such that the diagram P J A (µ) h / / $ $ I I I I I I I I I M g / / P J A (µ) z z u u u u u u u u u L A (µ) is commutative. We will now prove that the composition f = g • h is surjective, from which the surjectivity of g readily follows. Let ν ∈ J be a maximal element and consider J ′ = J \ {ν}. Then f J ′ is an endomorphism of P J A (µ) J ′ ∼ = P J ′ A (µ). By induction we know that P J ′ A (µ) → L A (µ) is a projective cover (in O J ′ A ), hence f J ′ is an automorphism. Hence the quotient P J A (µ)/imf is generated by its νweight space. But this quotient then has to be trivial, as P J A (µ) has no simple quotient of highest weight ν. Hence f is surjective, which is what we wanted to show. 4.3. Restricted projective covers. Again we suppose that A is a local deformation algebra. Each simple object L A (λ) is restricted, and {L A (λ)} λ∈ h ⋆ is a full set of representatives of the simple objects in O A as well. We will now show that projective covers also exist in the truncated restricted categories O J A = O A ∩ O J A . Theorem 4.3. Suppose that A is a local deformation algebra and let J ⊂ h ⋆ be an open, locally bounded subset. Then there exists for each λ ∈ J a projective cover P J A (λ) → L A (λ) in O J A . Proof. Consider the projective cover P J A (λ) → L A (λ) in O J A and consider its restriction P J A (λ) → L A (λ) = L A (λ). As the functor M → M is left adjoint to the (exact) inclusion functor O A ⊂ O A , P J A (λ) is projective in O A . We have seen in the proof of Theorem 4.2 that L A (λ) is the only simple subquotient of P J A (λ). Hence P J A (λ) is indecomposable. Now we show that P J A (λ) := P J A (λ) → L A (λ) is a projective cover. As we have seen in the proof of Theorem 4.2, for this it is enough to show that if f is an endomorphism of P J A (λ) that has the property that the composition P J A (λ) f → P J A (λ) → L A (λ) is surjective, then f is surjective. By projectivity of P J A (λ) we can find an endomorphism f ′ of P J A (λ) such that the diagram P J A (λ) f ′ / / P J A (λ) P J A (λ) f / / P J A (λ) is commutative (here the vertical maps are the quotient maps). But now f ′ is surjective, as we have seen in the proof of Theorem 4.2, so f has to be surjective as well. Block decomposition of O A We denote by ∼ A the equivalence relation on h ⋆ that is generated by λ ∼ A µ if there exists some open, locally bounded subset J of h ⋆ and a non-zero homomorphism P J A (λ) → P J A (µ). As P J A (λ) → L A (λ) is a projective cover in O J A , this condition is equivalent to the fact that L A (λ) occurs as a subquotient of P J A (µ) (in the following we write [P J A (µ) : L A (λ)] = 0 if this is the case). For an equivalence class Λ ⊂ h ⋆ with respect to ∼ A we define the full subcategory O A,Λ of O A that contains all objects M with the property that if L A (λ) is a subquotient of M, then λ ∈ Λ. Theorem 5.1 (Block decomposition). The functor Λ∈ h ⋆ /∼ A O A,Λ → O A , {M Λ } → M Λ is an equivalence of categories. Proof. Theorem 5.2 (Restricted block decomposition). The functor Λ∈ h ⋆ /∼ A O A,Λ → O A , {M Λ } → M Λ is an equivalence of categories. BGGH-reciprocity. The linkage principle and the restricted linkage principle now describe the equivalence classes under ∼ A and ∼ A explicitely. The first step towards these results are the respective BGGHreciprocity statements. Theorem 5.3 (Deformed BGGH-reciprocity). Let A be a local deformation algebra with residue field K and let J be an open locally bounded subset of h ⋆ and µ ∈ J . Then P J A (µ) admits a Verma flag and we have (P J A (µ) : ∆ A (λ)) = [∇ K (λ) : L K (µ)], if λ ∈ J , 0, if λ ∈ J for all λ ∈ h ⋆ . Note that the right hand side refers to the K-linear versions of the objects. Proof. By construction, P J A (µ) is a direct summand of an object that admits a Verma flag, hence also admits a Verma flag. In the proof of Theorem 4.2 we have shown that P J A (µ) has a unique simple quotient, hence P J A (µ) ⊗ A K must be indecomposable. As it is a direct summand of the projective object Q J K (µ) in O J K , it is projective. Hence it has to be isomorphic to P J K (µ). Hence the Verma multiplicities of P J A (µ) and P J K (µ) coincide. So it suffices to prove the above statement in the case that A = K is a field, in which case it reduces to the well-known non-deformed BGGH-reciprocity. The following is the restricted analogue of the above theorem. Theorem 5.4 (Restricted BGGH-reciprocity). Let A be a local deformation algebra and J an open locally bounded subset of h ⋆ and µ ∈ J . Then P J A (µ) admits a restricted Verma flag and for the multiplicities we have (P J A (µ) : ∆ A (λ)) = [∇ K (λ) : L K (µ)], if λ ∈ J , 0, otherwise for all λ ∈ h ⋆ . The proof can be found in [AF09]. 5.3. The equivalence relation. Let us define the equivalence rela- tion ∼ ′ A on h ⋆ as generated by λ ∼ ′ A µ if [∆ K (λ) : L K (µ)] = 0. Lemma 5.5. We have ∼ ′ A =∼ A . Proof. Since the characters of ∆ K (λ) and ∇ K (λ) coincide, the BGGHreciprocity implies that we have ∼ ′ A =∼ ′′ A , where ∼ ′′ A is generated by λ ∼ ′′ A µ if there exists some J with (P J A (λ) : ∆ A (µ)) = 0. Hence we have to show that ∼ A =∼ ′′ A . It is clear that λ ∼ ′′ A µ implies λ ∼ A µ. So let us suppose that [P J A (λ) : L A (µ)] = 0. Then there is a non-zero homomorphism P J A (µ) → P J A (λ). So there must be a Verma subquotient of P J A (λ) that admits a non-zero homomorphism from P J A (µ), so if its highest weight is ν, then ν ∼ ′′ A λ and there is a non-zero homomorphism P J A (µ) → ∆ A (ν). This implies that [∆ K (ν) : L K (µ)] = [∇ K (ν) : L K (µ)] = 0, hence µ ∼ ′′ A ν, again by BGGH-reciprocity. So λ ∼ ′′ A µ. Moreover, the restricted version of the above statement holds as well: using analogous arguments (in particular, using the restricted BGGH-reciprocity) one can prove that ∼ A is also generated by λ∼ A µ if [∆ K (λ) : L K (µ)] = 0. Note, however, that ∼ A is a finer relation than ∼ A , i.e. λ∼ A µ implies λ ∼ A µ. The linkage principle In some sense, the results of the previous section are quite abstract and do not give us enough information about the structure of category O. The next step is to prove the linkage principles, i.e. to determine the equivalence classes with respect to ∼ A and ∼ A . In the non-restricted case, the linkage principle follows from our results above together with a theorem of Kac and Kazhdan. 6.1. The theorem of Kac and Kazhdan. Let A be a local deformation algebra with residue field K. As before, we consider τ as an element in h ⋆ A = Hom A ( h ⊗ C A, A) = h ⋆ ⊗ C A. Let (·, ·) A : h ⋆ A × h ⋆ A → A be the A-bilinear extension of the bilinear form (·, ·) : h ⋆ × h ⋆ → C that is induced by the usual non-degenerate, invariant bilinear form on g (cf. [K90]). Now let us consider the partial order ↑ A on h ⋆ that is generated by µ ↑ A λ if there exists a positive root β ∈ R + and some n ∈ N such that 2(λ + τ + ρ, β) K = n(β, β) K and µ = λ − nβ. In particular, the equivalence relation ∼ A is generated by the partial order ↑ A . The following lemma is immediate from the above theorem. Lemma 6.2. If λ ∼ A µ, then {α ∈ R \| 2(λ + τ + ρ, α) K ∈ Z(α, α) K } = {α ∈ R \| 2(µ + τ + ρ, α) K ∈ Z(α, α) K }. Hence any equivalence class Λ ∈ h ⋆ defines R A (Λ) := {α ∈ R \| 2(λ + τ + ρ, α) K ∈ Z(α, α) K for some λ ∈ Λ}, = {α ∈ R \| 2(λ + τ + ρ, α) K ∈ Z(α, α) K for all λ ∈ Λ}. We also define W A (Λ) := s α+nδ \| α + nδ ∈ R re ∩ R A (Λ) . Clearly, the elements in a fixed equivalence class Λ have the same level, so we can talk about the level of an equivalence class. Note that an equivalence class if of critical level if and only if δ ∈ R A (Λ), i.e. if (λ + ρ, δ) = (δ, δ) = 0 for all λ ∈ Λ. This is the case if and only if nδ ∈ R A (Λ) for all n = 0. In this case, W A (Λ) is an affine Weyl group isomorphic to the affinization of its finite analogue W A (Λ) that is generated by the reflections s α for all finite roots α in R A (Λ). The Kac-Kazhdan theorem now immediately implies the following. Theorem 6.3 (The non-restricted linkage principle). Let Λ ⊂ h ⋆ be an equivalence class with respect to ∼ A . (1) If Λ is non-critical, then Λ is a W A (Λ)-orbit in h ⋆ . (2) If Λ is critical, then Λ is an orbit under W A (Λ) × Zδ. 6.2. Base change. Now we explain one of the main reasons for the use of the deformation theory in our approach. Let us look at the special case A = S = S(h) (0) , the localization of the symmetric algebra S(h) at the maximal ideal generated by h ⊂ S(h). For any prime ideal p of S we denote by S p the localization of S at p and by K p the residue field of S p . Then S is the intersection of S p inside the quotient field Q of S for all prime ideals of height one. Proposition 6.4. The equivalence relation ∼ S is the finest relation on h ⋆ that is coarser than ∼ Sp for all prime ideals p of S of height one. Proof. Recall that the equivalence relation ∼ A is generated by λ ∼ A µ if (P J A (λ) : ∆ A (µ)) = 0 for some open, locally bounded set J . If A → A ′ is a homomorphism of deformation algebras, then P J A (λ) ⊗ A A ′ is projective in O J A ′ , hence splits into a direct sum of restricted projective covers. Hence λ∼ Sp µ implies λ∼ S µ. Let ∼ ′ S be the finest relation on h ⋆ that is coarser than ∼ Sp for all prime ideals p of S of height one. Let Λ be an equivalence class with respect to ∼ S . Then Λ is a union of equivalence classes with respect to ∼ ′ S . Let us write this decompositon as Λ = i∈I Λ i . Now P J S (µ) ⊗ S Q splits into a direct sum of Verma modules in O Q , hence we have a canonical decomposition P J S (µ) ⊗ S Q = i∈I P i , where P i is the direct summand that contains all Verma modules with highest weight belonging to Λ i . By our assumption on ∼ ′ S , this direct sum decomposition induces a direct sum decomposition of P J S (µ) ⊗ S S p for each prime ideal p of height one, i.e. P J S (µ) ⊗ S S p = i∈I P J S (µ) ⊗ S S p ∩ P i . After taking the intersection we get a direct sum decomposition P J S (µ) = i∈I P J S (µ) ∩ P i and we deduce that only one direct summand on the right hand side appears, i.e. that Λ is already an equivalence class with respec to ∼ ′ S . So we have ∼ S = ∼ ′ S . The advantage now is that the equivalence relations ∼ Sp can be described explicitely. 6.3. The restricted linkage principle. In the restricted case, we do not yet have such an explicit description of the highest weights of simple subquotients of a given Verma module as we have, by the Kac-Kazhdan theorem, in the non-restricted case. Nevertheless, we can explicitely determine the equivalence relations ∼ Sp for each prime ideal p of height one and then use Proposition 6.4. Let λ ∈ h ⋆ be a weight at critical level and define R(λ) = {α ∈ R \| λ, α ∨ ∈ Z} (note that this definition only refers to finite roots!). For any α ∈ R(λ) we denote by α ↓ λ the maximal element in {s α+nδ .λ \| n ∈ Z} that is smaller or equal to λ. Here is our first result: Theorem 6.5 ([AF08]). Let λ ∈ h ⋆ be of critical level and let p ⊂ S be a prime ideal of height one. (1) If α ∨ ∈ p for all α ∈ R(λ), then ∆ Kp (λ) is simple, i.e. [∆ Kp (λ) : L Kp (µ)] = 1, if λ = µ, 0, otherwise. In particular, λ ∼ Sp µ implies λ = µ. (2) If α ∨ ∈ p for some α ∈ R(λ), then p = α ∨ S and we have [∆ Kp (λ) : L Kp (µ)] = 1, if µ ∈ {λ, α ↓ λ}, 0, otherwise. In particular, the equivalence class of λ with respect to ∼ Sp is the orbit of λ under the subgroup of W that is generated by the reflections s α+nδ with n ∈ Z. We can now deduce the restricted linkage principle. Let Λ ⊂ h ⋆ be a critical, restricted equivalence class and define W(Λ) := s α+nδ \| α ∈ R(λ) for some λ ∈ Λ and n ∈ Z . Theorem 6.6 ([AF09]). Suppose that Λ is a restricted critical equivalence class. Then Λ is a W(Λ)-orbit in h ⋆ . Proof. Let λ ∈ Λ. Let p ⊂ S be a prime ideal of height one. If α ∨ ∈ p for some α ∈ R(λ), then p = α ∨ S and Theorem 6.5 implies that the ∼ Sp -equivalence class of λ is its s α+nδ \| n ∈ Z -orbit. If α ∨ ∈ p for all α ∈ R(λ), then λ forms an ∼ Sp -equivalence class by itself, again by Theorem 6.5. As by Proposition 6.4 the relation ∼ S is generated by the relations ∼ Sp , the equivalence class of λ is its W(Λ)-orbit. Department Mathematik, Universität Erlangen-Nürnberg, 91054 Erlangen, Germany E-mail address: [email protected] 2. 3 . 3Simple highest weight characters. Let M be a g-module. For any linear form λ ∈ h ⋆ we denote by M λ = {m ∈ M \| H.m = λ(H)m for all H ∈ h} the eigenspace of the h-action on M with eigenvalue λ. The set h ⋆ carries a natural partial order (with respect to our choice of b): we set λ ≥ µ if and only if λ − µ can be written as a sum of positive roots, i.e. if and only if λ − µ ∈ Z ≥0 R + . 2. 8 . 8The restricted category O. In the following we abbreviate A = A( O, T ). We now define the restricted subcategory O of O as a special fiber for the action of A. Definition 2.3. An object M of O is called restricted if for all n = 0 and τ ∈ A n we have that τ M : M → T n M is the zero homomorphism. We denote by O ⊂ O the full subcategory that contains all restricted objects and by O k ⊂ O the full subcategory of restricted modules of level k. Note that if k = crit, then each M ∈ O k is restricted, i.e. O k = O k , as (A k ) n = 0 for n = 0. A simple highest weight module L(µ) is always restricted. The inclusion functor O → O has adjoints on both sides. For M ∈ O we denote by M ′ the submodule of M that is generated by the images of all homomorphisms T −n τ M : T −n M → M with τ ∈ A n and n = 0. Set M := M/M ′ . Dually, denote by M the submodule of M that contains all elements m with τ M (m) = 0 for all τ ∈ A n , n = 0. Then these definitions extend to functors ·, · : O → O. The next result follows easily from the definitions. Lemma 2.4. The functor M → M is left adjoint to the inclusion functor O ⊂ O, and the functor M → M is right adjoint to the inclusion functor. Definition 3. 2 . 2We say that an object M of O A admits a Verma flag if there is a finite filtration0 = M 0 ⊂ M 1 ⊂ · · · ⊂ M n = M such that for each i = 1, . . . , n the quotient M i /M i−1 is isomorphic to a deformed Verma module ∆ A (µ i ) for some µ i ∈ h ⋆ .It turns out that the multiset {µ 1 , . . . , µ n } is independent of the chosen filtration and hence for each M that admits a Verma flag the multiplicity (M : ∆ A (ν)) := #{i \| µ i = ν} is well defined for all ν ∈ h ⋆ .3.2. Simple objects in O A .Let us now assume that A is a local deformation algebra with maximal ideal m and residue field K = A/m. Note that we can consider each g K -module as a g A -module on which A acts via the quotient map A → K. This even extends to a functor Res : O K → O A . It is clearly isomorphic to the unique simple quotient of ∆ A (λ). For λ ∈ h ⋆ we have the simple quotient L K (λ) of the Verma module ∆ K (λ) in O K and we define L A (λ) := Res(L K (λ)). Proposition 3.3 ([F03]). The set {L A (λ)} λ∈ h ⋆ is a full set of representatives for the simple objects in O A . 3.3. The restricted deformed category O A . As in the non-deformed situation we can define the shift functor T on O A that maps an object M to M ⊗ C L(δ) and has the obvious impact on morphisms. Then T : O A → O A is an equivalence and we obtain the graded Aalgebra A A = A( O A , T ). An object M of O A is called restricted if τ M : M → T n M is the zero morphism for each τ ∈ (A A ) n , n = 0. We define O A as the full subcategory of O A that contains all restricted objects. 4. 1 . 1Truncated subcategories. The truncations that we are going to consider are associated to open, locally bounded subsets of h ⋆ . Definition 4.1. A subset J of h ⋆ is called open, if for all λ ∈ J and all µ ∈ h ⋆ with µ ≤ λ we have µ ∈ J . A subset J is locally bounded, if for all µ ∈ h ⋆ the set J ≥µ := {γ ∈ J \| γ ≥ µ} is finite.Let us fix an open, locally bounded subset J of h ⋆ . We define the full subcategory O J A of O A that contains all objects M with the property that M λ = 0 implies λ ∈ J . Let M ∈ O A and let M ′ be the submodule of M that is generated by the weight spaces M ν with ν ∈ J and set M J := M/M ′ . Then M → M J is a functor from O A to O J A that is left adjoint to the inclusion functor O J A ⊂ O A . 4.2. Existence of projective covers. The main objective of this section is to give a new proof of the following result. For an equivalence class Λ and an object M of O A let M Λ be the submodule of M that is generated by the images of all homomorphisms P J A (λ) → M with λ ∈ Λ and arbitrary open, locally bounded J . By definition of ∼ A the sum of all M Λ is direct. Moreover, we have M = Λ M Λ , as M is isomorphic to a quotient of a direct sum of various P J A (λ)'s. 5.1. Restricted block decomposition. The block decomposition above has an immediate analogue in the restricted case: We define the equivalence relation ∼ A on h ⋆ as generated by λ∼ A µ if [P J A (λ) : L A (µ)] = 0 for some open, locally bounded subset J . As before one proves the following result. Theorem 6.1 ([KK79]). We have [∆ K (λ) : L K (µ)] = 0 if and only if µ ↑ A λ. Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p. Jens Carsten Henning Haahr Andersen, Wolfgang Jantzen, Soergel, Astérisque. 220Henning Haahr Andersen, Jens Carsten Jantzen, Wolfgang Soergel, Rep- resentations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque (1994), no. 220. On the restricted Verma modules at the critical level, to appear in Trans. Tomoyuki Arakawa, Peter Fiebig, arXiv:0812.3334Amer. Math. 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Roman Bezrukavnikov, Ivan Mirković, Dmitriy Rumynin, Ann. of Math. 1672Roman Bezrukavnikov, Ivan Mirković, Dmitriy Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2), 167, no. 3, 2008, 945-991. Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. Peter Fiebig, Geordie Williamson, arXiv:1008.0719Peter Fiebig, Geordie Williamson, Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties, preprint 2010, arXiv:1008.0719. Centers and translation functors for category O over symmetrizable Kac-Moody algebras. Peter Fiebig, Math. Z. 2434Peter Fiebig, Centers and translation functors for category O over sym- metrizable Kac-Moody algebras, Math. Z. 243 (2003), No. 4, 689-717. The combinatorics of category O for symmetrizable Kac-Moody algebras. Transform. Groups. 111, The combinatorics of category O for symmetrizable Kac-Moody algebras, Transform. Groups 11 (2006), No. 1, 29-49. 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[ "Crowding of Brownian spheres", "Crowding of Brownian spheres" ]	[ "Krzysztof Burdzy [email protected] ", "Soumik Pal [email protected] ", "Jason Swanson [email protected] ", "\nDepartment of Mathematics\nDepartment of Mathematics\nUniversity of Washington\nBox 354350, Box 35435098195SeattleWA\n", "\nDepartment of Mathematics\nUniversity of Washington\n98195SeattleWA\n", "\nUniversity of Central Florida\n4000 Central Florida BlvdP.O. Box 16136432816-1364OrlandoFL\n" ]	[ "Department of Mathematics\nDepartment of Mathematics\nUniversity of Washington\nBox 354350, Box 35435098195SeattleWA", "Department of Mathematics\nUniversity of Washington\n98195SeattleWA", "University of Central Florida\n4000 Central Florida BlvdP.O. Box 16136432816-1364OrlandoFL" ]	[ "Alea" ]	We study two models consisting of reflecting one-dimensional Brownian "particles" of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small.	null	[ "https://arxiv.org/pdf/1002.1057v2.pdf" ]	7,210,377	1002.1057	efbf594c0f2bf455ac79825a2e39773b0ac46918	Crowding of Brownian spheres 2010 Krzysztof Burdzy [email protected] Soumik Pal [email protected] Jason Swanson [email protected] Department of Mathematics Department of Mathematics University of Washington Box 354350, Box 35435098195SeattleWA Department of Mathematics University of Washington 98195SeattleWA University of Central Florida 4000 Central Florida BlvdP.O. Box 16136432816-1364OrlandoFL Crowding of Brownian spheres Alea 72010The original article is published by the Latin American Journal of Probability and Mathematical Statistics We study two models consisting of reflecting one-dimensional Brownian "particles" of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small. Introduction In this article we consider the dynamics of a collection of hard Brownian spheres with drifts or boundary conditions that includes instantaneous reflections upon collisions. The models are similar to existing ones in the literature that consider point masses instead of spheres of a positive radius. We will show that the (empirical) distribution of a family of Brownian spheres behaves differently from the (empirical) distribution of the point Brownian particles in some natural models. In particular, the distribution of Brownian spheres fails to satisfy the usual heat equation under circumstances that lead to the heat equation for the infinitely many infinitesimally small Brownian particles. Various models of colliding Brownian particles have been considered in the statistical physics literature. One stream, pioneered by Harris (1965), considers a countable collection of Brownian point masses on the line that collide and reflect instantaneously. Also see the follow-up work on tagged particle in the Harris model by Dürr, Goldstein, & Lebowitz, Dürr et al. (1985). A variation on the theme has been to replace the instantaneous reflection by a potential, and goes by the name of gradient systems. In these gradient systems, one studies the behavior of countably many particles under a repelling potential. Usually the potential is modeled as smooth with a singularity at zero; see the article by Cépa and Lépingle (1997). A particular example of this class includes the famous Dyson Brownian motion from Random Matrix theory; see Dyson (1962), and Cépa and Lépingle (2001). The other class of models, closer to our article, goes by the name of hard-core interactions, in which the Brownian particles are assumed to be hard balls of small radius, and consequently, there is instantaneous reflection whenever two such balls collide (plus possible additional interactions). This is the spirit taken in the articles by Dobrushin & Fritz in dimension one, , and Fritz & Dobrushin in dimension two, Fritz and Dobrushin (1977), Lang (1977a,b) (with a correction by Shiga, 1979). The main focus of these authors is the non-equilibrium dynamics of the gradient systems. Also see the articles by Osada (1996, 1998), and Tanemura (1996 all of which consider properties of a tagged particle in the infinite system. In the discrete case, the various models of symmetric and asymmetric exclusion processes have been considered. Closest in spirit to the models discussed here is the totally asymmetric exclusion process (TASEP) considered by Baik, Deift, &Johansson, Baik et al. (1999), andJohansson (2000) in connection with random matrices and the longest increasing subsequence problem. Specifically, if the initial configuration in TASEP is Z − , then the probability that a particle initially at −m moves at least n steps to the right in time t equals the probability distribution of the largest eigenvalue in a unitary Laguerre random matrix ensemble. In recent subsequent articles Widom (2008, 2009), Tracy & Widom explicitly compute transition probabilities of individual particles in the asymmetric exclusion process, extending Johansson's work. In this paper, we consider only one dimensional models, so our "spheres" are actually intervals. The title of this paper reflects our intention to study multidimensional models in future articles. We leave more detailed discussion to Section 4. That section also contains references to related research projects. We consider two models, which have the following common features. Informally speaking, both models consist of families of Brownian "particles". The k-th "particle" is represented by an interval I k t = (X k t − ε/2, X k t + ε/2), where X k t is a Brownian-like process. The intervals I k and I j are always disjoint, for k = j. The processes X k are driven by independent Brownian motions. When two intervals I k and I j collide, they reflect instantaneously. In the first model, the number of particles is constant and they are pushed by a barrier moving at a constant speed. In the second model, particles enter the interval [0, 1] at the left, they reflect at 0, and they are killed when they hit the right endpoint. The second model is our primary focus because it is related to other models considered in mathematical physics literature-see Section 4. We are grateful to Thierry Bodineau, Pablo Ferrari, Claudio Landim, Mario Primicerio and Jeremy Quastel for very helpful advice. We would like to thank the referee for the suggestions for improvement. Extreme crowding We start with an informal description of our first model, which consists of a fixed number n of "particles". The k-th leftmost "particle" is represented by an interval I k t = (X k t −ε/2, X k t +ε/2). The intervals I k and I j are always disjoint. The processes X k are driven by independent Brownian motions. When two intervals I k and I j collide, they reflect instantaneously. The intervals are pushed from the left by a barrier with a constant velocity, that is, the leftmost interval reflects on the line x = ct. Formally, we define {X 0 , X 1 , . . . , X n } to be continuous processes such that X 0 t = −ε/2 + ct, X k t − X k−1 t ≥ ε for all k ≥ 1 and all t ≥ 0, and for k ≥ 1, dX k t = dB k t + dL k t − dM k t , where {B 1 , . . . , B n } are iid Brownian motions, and L k and M k are nondecreasing processes such that ∞ 0 1 {X k t −X k−1 t >ε} dL k t = 0 and ∞ 0 1 {X k+1 t −X k t >ε} dM k t = 0. (Here, we may interpret X n+1 ≡ ∞.) The distributions of X k 0 for 1 ≤ k ≤ n will be specified later. To construct the solution to this Skorohod problem, consider first the processes Y k t = X k t − (k − 1)ε − ε/2 − ct. These processes satisfy Y 0 ≡ 0, Y k t − Y k−1 t ≥ 0 for all k ≥ 1 and all t ≥ 0, and for k ≥ 1, dY k t = dB k t − c dt + dL k t − dM k t , where ∞ 0 1 {Y k t −Y k−1 t >0} dL k t = 0 and ∞ 0 1 {Y k+1 t −Y k {I k } as n → ∞. In other words, we will keep the total length of all intervals I k constant. The stationary distribution for Z k has the density ϕ(z) = ce −cz for z ≥ 0, with c = 2c 1 , because 1 2 d 2 dz 2 ϕ(z) + c 1 d dz ϕ(z) = 0. Consider any 0 ≤ x 1 < x 2 < ∞, let λ denote the Lebesgue measure, and let d([x 1 , x 2 ]) = d t ([x 1 , x 2 ]) = λ [x 1 + ct, x 2 + ct] ∩ 1≤k≤n I k t x 2 − x 1 . (2.1) The quantity d([x 1 , x 2 ]) represents the average density of "particles" I k on the interval [x 1 , x 2 ]. We will say that the intervals {I k } have the pseudo-stationary distribution if all Z k t 's are independent and have the stationary distribution ϕ for t = 0 and, therefore, for every t ≥ 0. Theorem 2.1. Suppose that the intervals {I k } have the pseudo-stationary distri- bution. Fix arbitrary p 1 , d 1 < 1, d 2 > 0, and 0 ≤ x 1 < x 2 < b < x 3 < x 4 < ∞. There exist c 0 , n 0 < ∞ such that for c ≥ c 0 , n ≥ n 0 and any t ≥ 0, we have P (d t ([x 1 , x 2 ]) ≥ d 1 ) ≥ p 1 , (2.2) P (d t ([x 3 , x 4 ]) ≤ d 2 ) ≥ p 1 . (2.3) The theorem says that the "particles" I k clump together and there is a sharp transition in density of "mass" around x = b. This is in contrast with infinitely small "particles" Z k whose empirical distribution is close to the distribution with the density ϕ(z) = ce −cz that displays no sharp drop-off. Proof of Theorem 2.1: Without loss of generality, we let t = 0. We define y k ∈ (0, ∞) in an implicit way by the following formula, for k = 1, 2, 3, 4, x k = b y k 0 ϕ(z)dz + y k . Note that y 1 < y 2 , and that for ε > 0 sufficiently small (that is, for n = bε −1 sufficiently large), it is possible to choose y 5 , y 6 such that y 1 < y 5 < y 6 < y 2 , and b y6 y5 ϕ(z)dz − 2ε y 2 − y 1 + b y2 y1 ϕ(z)dz ≥ b y2 y1 ϕ(z)dz y 2 − y 1 + b y2 y1 ϕ(z)dz − (1 − d 1 )/2. (2.4) Since b − x 2 > 0, we can find c so large that, c(b − x 2 ) 1 + c(b − x 2 ) ≥ 1 − (1 − d 1 )/2. (2.5) Let ⌈a⌉ denote the smallest integer greater than or equal to a. By the law of large numbers, if n is sufficiently large, the number of Z k 0 's in the interval [0, y 1 ] is smaller than or equal to n y5 0 ϕ(z)dz, with probability greater than 1 − (1 − p 1 )/2. If this event holds then there are exactly n y5 0 ϕ(z)dz processes Z k 0 in some (random) interval [0, y 7 ] with y 7 ≥ y 1 . This implies that there are exactly n y5 0 ϕ(z)dz processes X k 0 in [0, ε n y5 0 ϕ(z)dz + y 7 ]. Note that ε n y5 0 ϕ(z)dz + y 7 ≥ b y5 0 ϕ(z)dz + y 7 ≥ b y1 0 ϕ(z)dz + y 1 = x 1 . Hence, the number of X k 0 's in the interval [0, x 1 ] is smaller than or equal to n y5 0 ϕ(z)dz + 1, with probability greater than 1 − (1 − p 1 )/2. A completely analogous argument shows that, if n is sufficiently large, then the number of X k 0 's in the interval [x 2 , ∞] is smaller than or equal to n ∞ y6 ϕ(z)dz + 1, with probability greater than 1 − (1 − p 1 )/2. Both events hold with probability greater than 1 − 2(1 − p 1 )/2 = p 1 , and then the number of X k 0 's in [x 1 , x 2 ] is greater than or equal to n y6 y5 ϕ(z)dz − 2. This and (2.4) imply that d([x 1 , x 2 ]) ≥ εn y6 y5 ϕ(z)dz − 2ε x 2 − x 1 = b y6 y5 ϕ(z)dz − 2ε b y2 0 ϕ(z)dz + y 2 − b y1 0 ϕ(z)dz − y 1 = b y6 y5 ϕ(z)dz − 2ε y 2 − y 1 + b y2 y1 ϕ(z)dz ≥ b y2 y1 ϕ(z)dz y 2 − y 1 + b y2 y1 ϕ(z)dz − (1 − d 1 )/2. (2.6) We have x 2 = b y2 0 ϕ(z)dz + y 2 = b y2 0 ce −cz dz + y 2 = y 2 + b − be −cy2 , so e −cy2 = (y 2 − x 2 + b)/b and, therefore, for z ≤ y 2 , ϕ(z) = ce −cz ≥ ce −cy2 = (c/b)(y 2 − x 2 + b). We combine this estimate with (2.6) and (2.5) to see that, with probability greater than p 1 , d([x 1 , x 2 ]) ≥ b y2 y1 ϕ(z)dz y 2 − y 1 + b y2 y1 ϕ(z)dz − (1 − d 1 )/2 ≥ b y2 y1 (c/b)(y 2 − x 2 + b)dz y 2 − y 1 + b y2 y1 (c/b)(y 2 − x 2 + b)dz − (1 − d 1 )/2 = c(y 2 − y 1 )(y 2 − x 2 + b) y 2 − y 1 + c(y 2 − y 1 )(y 2 − x 2 + b) − (1 − d 1 )/2 = c(y 2 − x 2 + b) 1 + c(y 2 − x 2 + b) − (1 − d 1 )/2 ≥ c(b − x 2 ) 1 + c(b − x 2 ) − (1 − d 1 )/2 ≥ 1 − (1 − d 1 )/2 − (1 − d 1 )/2 = d 1 . This completes the proof of (2.2). The proof of (2.3) is completely analogous. 2 Brownian gas under pressure In this model, "particles" I k are confined to the interval [0, 1]. More precisely, their centers are confined to this interval. The k-th leftmost "particle" is represented by an interval I k t = (X k t −ε/2, X k t +ε/2). The intervals I k and I j are always disjoint. The processes X k are driven by independent Brownian motions with the diffusion coefficient σ 2 . When two intervals I k and I j collide, they reflect instantaneously. The particles are added to the system at the left endpoint of [0, 1] at a constant rate. In other words, they are pushed in at the speed a, so that a new particle enters the interval every ε/a units of time. As soon as X k reaches 0, it starts moving as a Brownian motion reflected at 0. The k-th interval is removed from the system when X k hits the right endpoint of [0, 1]. Formally, we define {X 1 , X 2 , . . .} to be a collection of right-continuous, [0, ∞]valued processes such that X k 0 = −kε + ε/2 for all k, (3.1) If S k = inf{t > 0 : X k t− = 1 − ε/2}, then X k t is continuous on [0, S k ) and X k t = ∞ for all t > S k , (3.2) X k t − X k+1 t ≥ ε for all k ≥ 1 and all t ≥ 0, and (3.3) dX k t = a dt if t ∈ [0, kε/a), σ dB k t + dL k t − dM k t if t ∈ [kε/a, S k ),(3.4) where a and σ are positive constants, {B 1 , B 2 , . . .} are iid Brownian motions, and L k and M k are nondecreasing processes such that S k kε/a 1 {X k t −X k+1 t >ε} dL k t = 0 and S k kε/a 1 {X k−1 t −X k t >ε} dM k t = 0. (Here, we may interpret X 0 ≡ ∞.) To construct the solution to this Skorohod problem, consider first the processes Y k t = X k t + kε − ε/2 − at. These processes satisfy Y k 0 = 0 for all k, (3.5) If S k = inf{t > 0 : Y k t− = 1 − at + (k − 1)ε}, then Y k t is continuous on [0, S k ) and Y k t = ∞ for all t > S k , (3.6) Y k t − Y k+1 t ≥ 0 for all k ≥ 1 and all t ≥ 0, and (3.7) dY k t = 0 if t ∈ [0, kε/a), σ dB k t − a dt + dL k t − dM k t if t ∈ [kε/a, S k ), (3.8) where S k kε/a 1 {Y k t −Y k+1 t >0} dL k t = 0 and S k kε/a 1 {Y k−1 t −Y k t >0} dM k t = 0. Again, we shall construct the processes {Y 1 , Y 2 , . . .} using order statistics. Let Z k be a [0, ∞)-valued process, satisfying the SDE dZ k t = σdB k t − adt, and reflected at 0. The process Z k t is defined on the time interval t ∈ [kε/a, ∞), and starts at Z k kε/a = 0. At any time t ∈ [kε/a, (k+1)ε/a), only processes Z j , 1 ≤ j ≤ k, are defined. Let ⌊a⌋ denote the greatest integer less than or equal to a, and S 0 = 0, A 1 t = {j ∈ Z : 1 ≤ j ≤ ⌊ta/ε⌋}, t ≥ 0, S 1 = inf{t > 0 : sup j∈A 1 t Z j t ≥ 1 − at}, A k t = A k−1 t \ {m ∈ Z : Z m S k−1 = 1 − at + (k − 2)ε}, t ≥ S k−1 , k ≥ 2, S k = inf{t > S k−1 : sup j∈A k t Z j t ≥ 1 − at + (k − 1)ε}, k ≥ 2. Note that it is possible that A k t = ∅ for some random k and t > 0. Convention (C). For the sake of future reference, it is convenient to say that the process Z m is killed at the time S k−1 , where {m} = A k−1 S k−1 \ A k S k−1 . In other words, Z m is killed when the corresponding interval I k , defined below, hits the right endpoint of the interval [0, 1]. For every t ∈ [S k−1 , S k ), note that there are ⌊ta/ε⌋ − (k − 1) elements in A k t . Let Y k t , Y k+1 t , . . . , Y ⌊ta/ε⌋ t be reverse-ordered Z j t 's, j ∈ A k t , that is, {Y k t , . . . , Y ⌊ta/ε⌋ t } = {Z j t , j ∈ A k t } and Y k t ≥ Y k+1 t ≥ · · · ≥ Y ⌊ta/ε⌋ t . Let Y j t = ∞ for j < k and Y j t = 0 for j > ⌊ta/ε⌋. It is elementary to check that {Y 1 , Y 2 , . . .} satisfy (3.5)-(3.8). We may therefore define X k t = Y k t − kε + ε/2 + at, I k t = (X k t − ε/2, X k t + ε/2). We have to modify slightly the definition (2.1) of density to match the current model. For t ∈ [S k−1 , S k ), let d([x 1 , x 2 ]) = d t ([x 1 , x 2 ]) = λ [x 1 , x 2 ] ∩ k≤j≤⌊ta/ε⌋ I j t x 2 − x 1 . (3.9) Theorem 3.1. Fix arbitrary 0 < x 1 < x 2 < 1, p 1 < 1 and a, σ, c 0 > 0. There exist t 0 < ∞ and ε 0 > 0 such that for t ≥ t 0 and ε ∈ (0, ε 0 ), P 1 − x 2 1 − x 2 + σ 2 /(2a) − c 0 ≤ d t ([x 1 , x 2 ]) ≤ 1 − x 1 1 − x 1 + σ 2 /(2a) + c 0 ≥ p 1 . (3.10) Intuitively speaking, the theorem says that the mass density at x ∈ (0, 1) is close to (1 − x)/(1 − x + σ 2 /(2a)), for large t and small ε. Proof of Theorem 3.1: We will use the coupling technique. Recall processes Z 1 , Z 2 , . . . used in the definition of Y k 's-we will use the same Z k 's to construct auxiliary processes. Fix some v 1 > 0, let S k = inf{t ≥ 0 : Z k t = v 1 }, and let Z k t be the process Z k killed at the time S k . Let n t be the number of processes Z k alive at time t. Let Y 1 t , Y 2 t , . . . , Y nt Every process X j t is defined on the interval [jε/a, S j ) and it is continuous on this interval. Although it may not be apparent from the above formulas, the processes Y j , X j and I j are constructed from Z j 's in the same way as Y j , X j and I j were constructed from Z j 's. We leave the verification of this claim to the reader. We will find the Green function G v1 (v) of Z k , i.e., the density of its occupation measure. Consider a process V with values in [−v 1 , v 1 ], satisfying the SDE dV t = dB t − a sign(V t )dt, where B is Brownian motion, V 0 = 0, and such that V is killed when it hits −v 1 or v 1 . Note that the Green function G V v1 (v) of V is one half of G v1 (v) for v > 0. The scale function S(v) and the speed measure m(v) for V can be calculated as follows (see Karlin and Taylor, 1981, pp. 194-195), s(v) = exp v 0 −(−2a sign(x)/σ 2 )dx = exp(2av sign(v)/σ 2 ), S(v) = v 0 s(x)dx = sign(v)σ 2 2a exp(2av sign(v)/σ 2 ) − 1 , m(v) = 1/(σ 2 s(v)) = (1/σ 2 ) exp(−2av sign(v)/σ 2 ). We will use formula (3.11) on page 197 of Karlin and Taylor (1981). In that formula, we take x = 0, so u(0) = 1/2, by symmetry. We apply the formula to functions g(v) of the form g(v) = 1 [v3,v4] (v), to conclude that for v ∈ (0, v 1 ), the Green function G V v1 (v) is given by G V v1 (v) = (S(v 1 ) − S(v))m(v) = 1 2a (exp(2av 1 /σ 2 ) − exp(2av/σ 2 )) exp(−2av/σ 2 ) = 1 2a (exp(2a(v 1 − v)/σ 2 ) − 1). It follows that G v1 (v) = 2G V v1 (v) = 1 a (exp(2a(v 1 − v)/σ 2 ) − 1). Define v 0 ∈ (0, ∞) by setting ϕ(v) = aG v0 (v) = 1 a (exp(2a(v 0 − v)/σ 2 ) − 1), and the following condition, 1 = v0 0 ϕ(v)dv + v 0 = v0 0 (exp(2a(v 0 − v)/σ 2 ) − 1)dv + v 0 (3.11) = (σ 2 /2a)(exp(2av 0 /σ 2 ) − 1). We define y k ∈ (0, ∞), k = 1, 2, by the following formula, x k = y k 0 ϕ(v)dv + y k = y k 0 (exp(2a(v 0 − v)/σ 2 ) − 1)dv + y k = (−(σ 2 /2a) exp(2a(v 0 − v)/σ 2 ) − v) v=y k v=0 + y k = (σ 2 /2a)(exp(2av 0 /σ 2 ) − exp(2a(v 0 − y k )/σ 2 )). (3.12) Choose y 1 < y 3 < y 4 < y 2 and v 1 < v 0 such that, a y4 y3 G v1 (v)dv y 2 − y 1 + y2 y1 ϕ(z)dz ≥ y2 y1 ϕ(z)dz y 2 − y 1 + y2 y1 ϕ(z)dz − c 0 . (3.13) Recall that ⌈a⌉ denotes the smallest integer greater than or equal to a. Let ⌊a⌋ denote the largest integer smaller than or equal to a. Let c 1 = 1 − p 1 and p 2 = 1 − c 1 /8. By the law of large numbers, we can find a large t 0 and make ε 0 > 0 smaller, if necessary, such that if t ≥ t 0 and ε ∈ (0, ε 0 ) then with probability greater than p 2 , the number of processes Z k t in the interval [0, y 1 ] is smaller than or equal to (a/ε) y3 0 G v1 (v)dv. If this event holds then there are exactly (a/ε) y3 0 G v1 (v)dv processes Z k 0 in some (random) interval [0, y 5 ] with y 5 ≥ y 1 . This implies that there are exactly (a/ε) y3 0 G v1 (v)dv processes X k 0 in [0, ε (a/ε) y3 0 G v1 (v)dv + y 5 ]. For fixed y 1 and y 3 , we make v 1 < v 0 larger, if necessary, so that ε (a/ε) y3 0 G v1 (v)dv + y 5 ≥ a y3 0 G v1 (v)dv + y 5 ≥ a y3 0 G v1 (v)dv + y 1 ≥ a y1 0 G v0 (v)dv + y 1 = y1 0 ϕ(v)dv + y 1 = x 1 . Hence, the number of X k 0 's in the interval [0, x 1 ] is smaller than or equal to (a/ε) y3 0 G v1 (v)dv, with probability greater than p 2 . We can make t 0 larger and ε 0 > 0 smaller, if necessary, so that by the law of large numbers, if t ≥ t 0 and ε ∈ (0, ε 0 ) then with probability greater than p 2 , the number of processes Z k t3 in the interval [0, y 2 ] is greater than or equal to (a/ε) y4 0 G v1 (v)dv. If this event holds then there are exactly (a/ε) y4 0 G v1 (v)dv processes Z k 0 in some (random) interval [0, y 6 ] with y 6 ≤ y 2 . This implies that there are exactly (a/ε) y4 0 G v1 (v)dv processes X k 0 in [0, ε (a/ε) y4 0 G v1 (v)dv + y 6 ]. Note that, ε (a/ε) y4 0 G v1 (v)dv + y 6 ≤ a y4 0 G v1 (v)dv + y 2 ≤ a y2 0 G v0 (v)dv + y 2 = x 2 . Hence, the number of X k 0 's in the interval [0, x 2 ] is greater than or equal to (a/ε) y4 0 G v1 (v)dv, with probability greater than p 2 . Let d be defined as in (3.9) but relative to I k in place of I k . The two events described in the last two paragraphs hold simultaneously with probability greater than 1 − c 1 /4. Then the number of X k 0 's in [x 1 , x 2 ] is greater than or equal to (a/ε) y4 y3 G v1 (v)dv. This and (3.13) imply that d t ([x 1 , x 2 ]) ≥ ε (a/ε) y6 y5 G v1 (v)dv x 2 − x 1 = a y6 y5 G v1 (v)dv y2 0 ϕ(z)dz + y 2 − y1 0 ϕ(z)dz − y 1 = a y6 y5 G v1 (v)dv y 2 − y 1 + y2 y1 ϕ(z)dz ≥ y2 y1 ϕ(z)dz y 2 − y 1 + y2 y1 ϕ(z)dz − c 0 . (3.14) It follows from (3.12) that exp(2a(v 0 − y 2 )/σ 2 ) = exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 and, therefore, for v ≤ y 2 , ϕ(v) = exp(2a(v 0 − v)/σ 2 ) − 1 ≥ exp(2a(v 0 − y 2 )/σ 2 ) − 1 = exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − 1. We combine this estimate with (3.14) to see that, with probability greater than 1 − c 1 /4, d t ([x 1 , x 2 ]) ≥ y2 y1 ϕ(v)dv y 2 − y 1 + y2 y1 ϕ(v)dv − c 0 ≥ y2 y1 (exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − 1)dv y 2 − y 1 + y2 y1 (exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − 1)dv − c 0 = (y 2 − y 1 )(exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − 1) y 2 − y 1 + (y 2 − y 1 )(exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − 1) − c 0 = exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − 1 exp(2av 0 /σ 2 ) − 2ax 2 /σ 2 − c 0 = (σ 2 /2a)(exp(2av 0 /σ 2 ) − 1) − x 2 (σ 2 /2a)(exp(2av 0 /σ 2 ) − 1) − x 2 + σ 2 /2a − c 0 = 1 − x 2 1 − x 2 + σ 2 /2a − c 0 . (3.15) stationary regime for this process has the density (1 − x)/(1 − x + σ 2 /(2a)), just like in model (C), where a, σ and b are related by the following formula, 1 0 1 − x 1 − x + σ 2 /(2a) = b. Heuristically, we expect processes X k in model (R) to jump at a more or less constant rate in the stationary regime, so this is why we believe that models (R) and (C) have the same hydrodynamic limit. We chose not to analyze model (R) in this paper as it appears to be harder from the technical point of view while it seems to illustrate the same phenomenon as model (C). Remark 4.2. Model (R) is closely related to a model studied by T. Bodineau, B. Derrida and J. Lebowitz (Bodineau et al., 2010). In their model, one considers a periodic system of L sites with N particles. The particles perform random walks but cannot cross each other-it is the symmetric simple exclusion process. At some fixed edge, the jump rates are no longer symmetric but jumps occur with rates p in one direction and 1 − p in the other direction. The case p = 1 corresponds to model (R) described in the previous remark. In the stationary state, the rescaled density varies linearly on the unit line segment, with a discontinuity located where the jump rates are biased. Hence, away from the singularity, the stationary empirical distribution is harmonic for the generator of the single particle process, i.e., Laplacian. This does not apply to the density of mass d in our models (C) and (R). Remark 4.3. Since the density d of the intervals I k has the form (1 − x)/(1 − x + σ 2 /(2a)), it is elementary to check that the typical gap size between I k 's is εσ 2 /(2a (1 − x)). In a model with infinitely small particles X k , the gap size is also c/(1 − x) but we do not have any heuristic explanation why the two functions representing the typical gap size should have the same form in both models. Remark 4.4. We conjecture that the motion of an individual tagged particle I k in model (C) converges, as ε → 0, to a deterministic motion with fluctuations having the "fractional Brownian motion" structure. In other words, we conjecture that the fluctuations are Gaussian with the local scaling of space and time given by ∆x = (∆t) 1/4 . Our conjecture is inspired by the results in Dürr et al. (1985); Harris (1965); Swanson (2007Swanson ( , 2008 on families of one dimensional Brownian motions reflecting from each other. Remark 4.5. A d-dimensional counterpart of model (C) can be represented as follows. Let I k be balls with radius ε and center X k . Our d-dimensional model consists of a constant number n of I k 's which are confined to the cube [0, 1] d . The balls I k and I j are always disjoint. The processes X k are driven by independent ddimensional Brownian motions with the diffusion coefficient σ 2 . When two balls I k and I j collide, they reflect instantaneously. Let S ℓ and S r be two opposite (d − 1)dimensional sides on the boundary of [0, 1] d . Balls I k are pushed into the cube through S ℓ at a constant rate a, i.e., ε −(d−1) balls are pushed into the cube every ε/a units of time, uniformly over S ℓ . Once inside the cube, the balls reflect from all sides except S r . When a ball hits S r , it is removed from the cube. We conjecture that in the stationary regime, when ε is small, the density of the mass analogous to d will be a function of the distance x from S ℓ , i.e., a function of depending only on one coordinate. We do not see any obvious reason why the density should have the form (1 − x)/(1 − x + σ 2 /(2a)). In relation to Remark 4.4, we conjecture that the motion of a tagged particle in the present model is diffusive, with the diffusion coefficient depending on x. If this is true, it means that the "pressure" applied to particles in one direction can have a dampening effect on the size of oscillations of an individual particle in orthogonal directions. t >0} dM k t = 0.We may therefore construct the processes {Y 1 , . . . , Y n } using order statistics. Namely, let {Z 1 , . . . , Z n } be defined by dZ k t = dB k t − cdt, and reflected at 0. For every fixed t ≥ 0, we let Y 1 t , Y 2 t , . . . , Y n t be ordered Z k t 's, that is, {Y 1 t , . . . , Y n t } = {Z 1 t , . . . , Z n t } and Y 1 t ≤ Y 2 t ≤ · · · ≤ Y n t . Finally, we let X k t = Y k t +(k−1)ε+ε/2+ct and I k t = (X k t − ε/2, X k t + ε/2). Let nε = b. We will fix b > 0 and analyze the behavior of the system of intervals t be ordered Z j t 's, that is, { Y 1 t , . . . , Y nt t } = { Z j t , S j > t} and Y 1 t ≤ Y 2 t ≤ · · · ≤ Y nt t .For t ∈ [ S k−1 , S k ) and j = k, . . . , k + n t − 1, we let X j t = Y nt+k−j t + (n t + k − j − 1)ε + ε/2 + (t − ⌊ta/ε⌋ε/a)a, I j t = ( X j t − ε/2, X j t + ε/2). The last equality follows from (3.11).Recall that n t is the number of processes Z k alive at time t. Note that for any 0 ≤ t 1 < t 2 < ∞ with t 2 − t 1 ≥ ε/a, we have, n t2 − n t1 ≤ (a/ε)(t 2 − t 1 ).(3.16) Fix arbitrary t 1 ≥ t 0 and choose δ > 0 such thatWe make ε 0 > 0 smaller, if necessary, so that, by the law of large numbers, if ε ∈ (0, ε 0 ) then with probability greater than 1 − c 1 /4, for all s k of the form s k = kδ/2, k = 0, . . . , ⌊t 1 /δ⌋ + 1, we have n s k < (a/ε) v1 0 G v1 (v)dv + δ/2 . It follows from (3.16) and (3.17) that for ε < aδ/2,Suppose that this event holds. Then, for every t ≤ t 1 , the right edge of the rightmost interval I k t is to the left ofThe second equality in the above formula follows from (3.11).Recall the definitions given before the statement of the theorem. A process Z k is killed when the right end of the rightmost interval I k hits 1. Since the processes Z k are driven by the same Brownian motions as Z k , (3.18) implies that every Z k has a longer lifetime than Z k . This implies that d t1 ([x 1 , x 2 ]) ≥ d t1 ([x 1 , x 2 ]). We combine this with (3.15) to conclude that, with probability greater than 1 − c 1 /2,A completely analogous argument shows that, with probability greater than 1−c 1 /2,This completes the proof of the theorem. 2DiscussionRemark 4.1. In the following remarks we will refer to the model analyzed in Section 3 as model (C). We will present another model, which we will call (R). Here, C represents the "constant" rate of influx of new particles, and R stands for the "random" rate of influx. Model (R) consists of a constant number n of "particles" I k which are confined to the interval [0, 1]. The k-th leftmost "particle" is represented by an interval I k t = (X k t − ε/2, X k t + ε/2). The intervals I k and I j are always disjoint. The processes X k are driven by independent Brownian motions with the diffusion coefficient σ 2 . When two intervals I k and I j collide, they reflect instantaneously. The number of particles n is such that nε = b, a constant. When X k hits 1, it jumps to 0. 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[ "QUASICONFORMAL EXTENSION FIELDS", "QUASICONFORMAL EXTENSION FIELDS" ]	[ "Pekka Pankka ", "Kai Rajala " ]	[]	[]	We consider extensions of differential fields of mappings and obtain a lower energy bound for quasiconformal extension fields in terms of the topological degree. We also consider the related minimization problem for the q-harmonic energy, and show that the energy minimizers admit higher integrability.2000 Mathematics Subject Classification. Primary 30C65, 35J60; Secondary 46E35, 49J52, 57M12.	10.1007/s00526-010-0380-9	[ "https://arxiv.org/pdf/1006.2259v1.pdf" ]	119,296,818	1006.2259	6932eda2431bb3e18937c1f5e79c4a7c931d58e1	QUASICONFORMAL EXTENSION FIELDS 11 Jun 2010 Pekka Pankka Kai Rajala QUASICONFORMAL EXTENSION FIELDS 11 Jun 2010 We consider extensions of differential fields of mappings and obtain a lower energy bound for quasiconformal extension fields in terms of the topological degree. We also consider the related minimization problem for the q-harmonic energy, and show that the energy minimizers admit higher integrability.2000 Mathematics Subject Classification. Primary 30C65, 35J60; Secondary 46E35, 49J52, 57M12. Introduction A continuous mapping f : R n \ A → R n , where A is an open annulus in R n , can be extended, by classical methods, to a continuous mappinḡ f : R n → R n . Although orientation preserving mappings do not need to admit orientation preserving extension in general, for homeomorphisms this extension problem has a solution in the form of the annulus theorem; see e.g. [5] and [6] for detailed discussions. In the quasiconformal category the annulus theorem is due to Sullivan [11] and it yields that given a quasiconformal embedding f : R n \ A → R n , where A is an annulus, there exists a quasiconformal mappingf : R n → R n so thatf \|R n \ A ′ = f , whereĀ ⊂ intA ′ ; the distortion of the extension is quantitatively controlled. We refer to Tukia-Väisälä [12] for a detailed discussion on the annulus theorem in the quasiconformal category. A simple consequence of the annulus theorem is that a mapping f : R n \A → R n , that is quasiconformal embedding in the components of R n \ A, can be extended to a quasiconformal mapping R n → R n if we are allowed to precompose f with a Euclidean similarity in one of the components of R n \ A. For more general non-injective mappings of quasiconformal type, i.e., for quasiregular mappings, extension results of this type are not known. We refer to [9] and [10] for the theory of quasiregular mappings. In this article, we discuss quantitative estimates, in terms of the degree, for the non-existence of extensions. If we focus on matrix fields instead of the differential fields of mappings, it is easy to see that the extension problem admits an orientation preserving solution in the sense that the differential of an orientation preserving C 1mapping f : R n \A → R n admits an extension to a continuous matrix field M on R n having non-negative determinant. This matrix field is not, in general, a differential field of a mapping, but it can be integrated to obtain a mapping f : R n → R n so that the differencef − f is bounded. The difference M − Df can be viewed to measure the non-exactness of the extension field M . We estimate the non-exactness of extensions for differential fields in the context of quasiconformal geometry, i.e. we consider matrix fields satisfying the quasiconformality condition (1.1) \|M (x)\| n ≤ K det M (x) a.e. in A, where \|M (x)\| is the operator norm of the matrix M (x). Our main theorem gives a quantitative estimate for the non-exactness of the extension in terms of the degree information on the underlying mappings. For the statement, we introduce some notation. Let B n (r) be a Euclidean ball of radius r > 0 about the origin. We denote A(r, R) = B n (R) \B n (r) for 0 < r < R. For notational convenience, we consider 1-forms and 1-(co)frames instead of vectors and matrix fields. We say that an n-tuple ρ = (ρ 1 , . . . , ρ n ) of measurable 1-forms on a domain Ω is a measurable frame. Moreover, for p ≥ 1 and q ≥ 1, we say that ρ is a W p,q -frame if ρ i ∈ L p ( 1 Ω) and dρ i ∈ L q ( 2 Ω) for every i = 1, . . . , n. The local space W loc p,q of frames is defined similarly. A frame ρ is said to be K-quasiconformal in A(r, R) if (QC) \|ρ\| n ≤ K ⋆ (ρ 1 ∧ · · · ∧ ρ n ) a.e. in A(r, R), where \|ρ\| is the operator norm of ρ, see Section 2. After the natural identification of frames and matrix fields, the two conditions (1.1) and (QC) coincide. Let 0 < r < R, and let ρ 0 and ρ 1 be frames defined on B n (r) and R n \B n (R), respectively. We say that a frame ρ K-quasiconformally connects ρ 0 and ρ 1 in A(r, R) if ρ is K-quasiconformal in A(r, R) and satisfies ρ\|B n (r) = ρ 0 \|B n (r) and ρ\|(R n \B n (R)) = ρ 1 . In our main theorem we assume that ρ 0 and ρ 1 are (the restrictions of) df 0 and dx, respectively, where f 0 : R n → R n is a continuous W 1,n loc -mapping and dx is the standard frame dx = (dx 1 , . . . , dx n ). Theorem 1.1. Let f 0 ∈ W 1,n loc (R n , R n ) be a continuous mapping, 0 < r < ∞, p > n, and n ≥ 3. Suppose that a W p,n/2 -frame ρ K-quasiconformally connects df 0 and dx in A(r/2, r). Then (1.2) R n max{deg(y, f 0 , B n (r/2)) − 1, 0} dy ≤ C dρ n n/2 , where C = C(n, K) > 0. Similar results also hold in the plane, but with the L n/2 -norm replaced by other norms. The estimate (1.2) can be interpreted as a lower bound for the minimal energy of the extension frame. For the statement of our next result, let A be an open annulus in R n . Given W loc n,q -frames ρ 0 and ρ 1 in R n , we denote by E q,K (ρ 0 , ρ 1 ; A) the set of W loc n,q -frames ρ K-quasiconformally connecting ρ 0 and ρ 1 in A. Theorem 1.2. Let q > n/2, n ≥ 2, A an annulus in R n , and let ρ 0 and ρ 1 be K-quasiconformal W loc n,q -frames in R n that can be K-quasiconformally connected in A. Then there exists a W loc n,q -frame ρ ∈ E q,K (ρ 0 , ρ 1 ; A) so that (1.3) A \|dρ\| q 2 = inf ρ ′ A \|dρ ′ \| q 2 , where the infimum is taken over ρ ′ ∈ E q,K (ρ 0 , ρ 1 ; A), and the norm \| · \| 2 is the Hilbert-Schmidt norm in 2 R n . Moreover, there exists p = p(n, K) > n so that ρ ∈ L p loc ( 1 A). This paper is organized as follows. In Section 3, we discuss the L p -Poincaré homotopy operator T of Iwaniec and Lutoborski. This operator plays a crucial role in both of our theorems by providing a Sobolev-Poincaré inequality for W p,q -frames. The interplay between degree of the potential T ρ and the energy of ρ is then discussed in Section 4. A continuity estimate for T ρ is proven in Section 5, and the proof of Theorem 1.1 is given in Section 6. In Section 7 we consider the variational problem for the energy and prove Theorem 1.2. Preliminaries The open ball in R n about x 0 with radius r > 0 is denoted by B n (x 0 , r). For x 0 = 0 we abbreviate B n (r) = B n (0, r) and B n = B n (1). The corresponding closed balls are denoted byB n (x 0 , r),B n (r), andB n . The sphere of radius r about the origin is denoted by S n−1 (r) and the unit sphere in R n by S n−1 . Given a ball B = B n (x, r) we commonly use also notation λB to denote the ball B n (x, λr) for λ > 0. Given a frame ρ, we denote by \|ρ\| the operator norm \|ρ\| = sup (v 1 ,...,vn) \|(ρ 1 (v 1 , . . . , v n ), . . . , ρ n (v 1 , . . . , v n ))\|, where the supremum is taken over n-tuples (v 1 , . . . , v n ) satisfying i \|v i \| 2 = 1. We abuse the common terminology slightly and call J ρ = ⋆(ρ 1 ∧ · · · ∧ ρ n ) the Jacobian of ρ, although we also write J f = det(Df ) when f is mapping. Let Ω be a domain in R n . The weak exterior differential of an ℓ-form ω ∈ L 1 loc ( ℓ Ω) is the unique form dω ∈ L 1 loc ( ℓ+1 Ω), if exists, that satisfies Ω dω ∧ ϕ = (−1) ℓ+1 Ω ω ∧ dϕ for every ϕ ∈ C ∞ 0 ( n−ℓ−1 Ω). Given 1 ≤ p < ∞ and 1 ≤ q < ∞, we denote by W p,q ( ℓ Ω) the (p,q)-partial Sobolev space of the ℓ-forms ω ∈ L p ( ℓ Ω) having dω ∈ L q ( ℓ+1 Ω). We will also say that a measurable ℓ-form ω in Ω belongs to the Sobolev space W 1,p ( ℓ Ω) if ω I ∈ W 1,p (Ω), where ω = I ω I dx I . Here dx I = dx i 1 ∧ · · · ∧ dx i ℓ for I = (i 1 , . . . , i ℓ ). We call an n-tuple ρ = (ρ 1 , . . . , ρ n ) of (Borel) measurable 1-forms on Ω a measurable frame. We say that a measurable frame is a W p,q -frame if the forms ρ i , i = 1, . . . , n, belong to W p,q . We then denote dρ = (dρ 1 , . . . , dρ n ). 2.1. Topological degree. Let f :B n (r) → R n be a continuous mapping and y ∈ R n \ f S n−1 (r). Then the local degree deg(y, f, B n (r)) of f at y with respect to B n (r) is the mapping degree of g : S n−1 → S n−1 , g(x) = f (rx) − y \|f (rx) − y\| . If f : Ω → R n is C ∞ , and if G ⊂ Ω is a domain compactly contained in Ω, then the local degree satisfies the change of variables formula (2.1) G η(f (x))J f (x) dx = R n η(y)deg(y, f, G) dy for every non-negative η ∈ L 1 (G). In fact, the degree can be defined by using (2.1) and the property that deg(y, f, G) = deg(z, f, G) whenever y and z lie in the same component of R n \ f (∂Ω). We will use the fact that (2.1) remains valid for mappings f ∈ W 1,p (Ω, R n ) when p > n; see e.g. [7]. We will use the following properties of the local degree; see e.g. [10, I.4.2]. Suppose that f i : Ω → R n , i = 0, 1, are continuous, G ⊂⊂ Ω is a domain, and y ∈ R n . If there exists a homotopy H : [0, 1] × G → R n so that y / ∈ H([0, 1] × ∂G), H(0, x) = f 0 (x), and H(1, x) = f 1 (x) for every x ∈ G, then (2.2) deg(y, f 0 , G) = deg(y, f 1 , G). Also, if U ⊂ G is open, and if y / ∈ f 0 (∂U ∪ ∂G), then (2.3) deg(y, f 0 , G) = deg(y, f 0 , U ) + deg(y, f 0 , G \ U ). Averaged Poincaré homotopy operator Iwaniec and Lutoborski introduced the L p -averaged Poincaré homotopy operator in [3]. Given y ∈ R n , we denote by K y : C ∞ ( ℓ R n ) → C ∞ ( ℓ−1 R n ), ℓ = 1, . . . , n − 1, the Poincaré homotopy operator (at y) K y ω(x; v 1 , . . . , v ℓ−1 ) = 1 0 t ℓ−1 ω(y + t(x − y); x − y, v 1 , . . . , v ℓ−1 ) dt. As in [3] we define an averaged Poincaré homotopy operator T as follows. Let ϕ ∈ C ∞ 0 (B n (1/4)) be non-negative with integral one. From now on we consider ϕ to be fixed. We set T : L 1 loc ( ℓ R n ) → L 1 loc ( ℓ−1 R n ) by (3.1) T ω(x; v 1 , . . . , v ℓ−1 ) = R n ϕ(y)K y ω(x; v 1 , . . . , v ℓ−1 ) dy; T is well-defined by [3, (4.15)]. Both operators K and T satisfy a chain homotopy condition, which for T reads as (3.2) id = dT + T d. For all p > 1, we can consider T as a bounded operator T : L p ( ℓ B n ) → W 1,p ( ℓ−1 B n ); see [3, Proposition 4.1]. The chain homotopy condition together with the Sobolev embedding theorem then give the Sobolev-Poincaré inequality (3.3) T dω p * ,B n ≤ C(n, p) dω p,B n , where ω ∈ W 1,p ( ℓ B n ), 1 < p < n, ℓ ≥ 1, and p * = np/(n − p) is the Sobolev exponent; see [3,Corollary 4.2]. The formula (3.1) naturally defines an averaged homotopy operator on L p ( ℓ B n (r)) for all r > 0 if the function ϕ is scaled properly. To avoid such technicalities, we define for all r > 0 the Poincaré homotopy operator by T r = λ * 1/r • T • λ * r , where λ r : R n → R n is the mapping λ r (x) = rx. By scale invariance of (3.3), operators T r satisfy the same Sobolev-Poincaré inequality as T . Moreover, due to condition spt ϕ ⊂ B n (1/4), they satisfy T r df = f + c on B n (r/4) for functions f ∈ W 1,1 (B n (r)). Given a frame ρ, we extend the notation T ρ to denote the mapping T ρ = (T ρ 1 , . . . , T ρ n ) : Ω → R n . We end this section with an application of the isoperimetric inequality (3.4) B \|J f (x)\| dx ≤ C(n) ∂B \|Df (y)\| n−1 dH n−1 (y) n/(n−1) in balls B compactly contained in Ω; see e.g. [9, Chapter II, p.81] or [8, Section 6]. Lemma 3.1. Let r > 0, f 0 : R n → R n be a mapping in W 1,n loc , n ≥ 3, and let ρ be a W loc n,n/2 -frame in R n . Then there exists C = C(n) > 0 so that B n (r) \|J Tr ρ \| ≤ C ρ n,A(r,2r) + dρ n/2,B n (r) n . Proof. By the isoperimetric inequality (3.4), B n (t) \|J Tr ρ \| ≤ C ∂B n (t) \|dT r ρ\| n−1 n n−1 for almost every r ≤ t ≤ 2r. Thus the Sobolev-Poincaré inequality (3.3) gives 2r r B n (t) \|J Tr ρ \| n−1 n dt ≤ C 2r r ∂B n (t) \|dT r ρ\| n−1 dt ≤ C A(r,2r) \|dT r ρ\| n−1 ≤ C A(r,2r) (\|ρ\| + \|T r dρ\|) n−1 ≤ C ρ n−1,A(r,2r) + T r dρ n−1,A(r,2r) n−1 ≤ C r 1 n−1 ρ n,A(r,2r) + r 1 n−1 dρ n/2,B n (2r) n−1 ≤ Cr ρ n,A(r,2r) + dρ n/2,B n (2r) n−1 . Therefore, B n (r) \|J Tr ρ \| n−1 n ≤ 1 r 2r r B n (t) \|J Trρ \| n−1 n dt ≤ C ρ n,A(r,2r) + dρ n/2,B n (2r) n−1 . The claim follows. Energy and local degree In this section we prove an integral estimate which relates the degree of the mapping T ρ and the energy of the frame ρ. Given a continuous mapping f : A(r, R) → R n we denote by Ω − (f ) the set Ω − (f ) = {y ∈ R n : deg(y, f, A(r, R)) < 0}. The main result of this section reads as follows. Proposition 4.1. Let 0 < r < R < ∞, p > n, and n ≥ 3. Suppose that ρ is a W loc p,n/2 -frame on R n such that ρ is K-quasiconformal in A(r, R). Then (4.1) − Ω − (T R ρ) deg(y, T R ρ, A(r, R)) dy ≤ C dρ n n/2 , where C = C(n, K, ϕ) > 0. Proof. SetG = (T R ρ) −1 (Ω − (T R ρ)) ∩ A(r, R) and I = − Ω − (T R ρ) deg(y, T R ρ, A(r, R)) dy. Since T R ρ ∈ W 1,p loc with p > n, we have, by the change of variables (2.1) and by (3.2), that −I ≥ G dT R ρ 1 ∧ . . . ∧ dT R ρ n = G (ρ 1 − T R dρ 1 ) ∧ . . . ∧ (ρ n − T R dρ n ). The last integrand can be estimated from below by J ρ (x) − C n k=1 \|T R dρ(x)\| k \|ρ(x)\| n−k almost everywhere. Here C = C(n). Then, by Hölder's inequality and the Sobolev-Poincaré inequality (3.3), we obtain −I ≥ G J ρ (x) dx − C 0 n k=1 T R dρ k n,B n (R) ρ n−k n,G ≥ G J ρ (x) dx − C 0 n k=1 dρ k n/2,B n (R) ρ n−k n,G ,(4.2) where C 0 = C 0 (n) ≥ 1. Since ρ is K-quasiconformal, K G J ρ (x) dx ≥ ρ n n,G . Thus (4.3) 1 K ρ n n,G + I ≤ C 0 n k=1 dρ k n/2,B n (R) ρ n−k n,G . We show that (4.4) ρ n,G ≤ KC 0 n dρ n/2,B n (R) . Suppose towards contradiction that (4.4) does not hold. Then, by (4.3), ρ n n,G ≤ C 0 K n k=1 dρ k n/2,B n (R) ρ n−k n,G < C 0 K n k=1 1 C 0 Kn k ρ k n,G ρ n−k n,G ≤ ρ n n,G . This is a contradiction. Thus (4.4) holds. We may now estimate I using (4.3) and (4.4) to obtain I ≤ (2KC 0 n) n dρ n n/2,B n (R) . This concludes the proof. A continuity estimate The second main ingredient in the proof of Theorem 1.1 is the following continuity estimate for T ρ. Lemma 5.1. Let u ∈ W 1,1 loc (R n ), n ≥ 2, and let ρ be a W loc 1,1 -form in R n so that ρ = du in R n \B n . Then there exists C = C(n) so that (5.1) \|T ρ(x) − T ρ(y) − (u(x) − u(y))\| ≤ C dρ 1 for almost every x and y ∈ R n \ B n (2). For the proof of this lemma, we introduce some notation. Given points x and y in R n we denote the (oriented) line segment from x to y by [x, y]. Given points x, y, and z in R n we denote by [x, y, z] the (oriented) 2-simplex that is the convex hull of {x, y, z}. Similarly, for affinely independent points, we define L(x, y) and P (x, y, z) to be the (unique) line and plane containing {x, y} and {x, y, z}, respectively. Proof. By the density of smooth frames in W 1,1 , we may assume that ρ is smooth. We assume that n ≥ 3, the simpler planar case is left for the reader. Let a and b ∈ R n \ B n (2). We may assume that \|a\| ≤ \|b\| and that L(a, b) ∩ B n = ∅. Otherwise, we consider an additional point c ∈ S n−1 (2) so that ([a, c] ∪ [c, b]) ∩ B n = ∅. Indeed, we can take c ∈ S n−1 (2) so that [0, c] bisects the angle between [0, a] and [0, b] in the plane P (0, a, b). Then, since ρ = du outside B n , K y ρ(b) − K y ρ(a) = [y,b] ρ − [y,a] ρ − [a,b] ρ + u(b) − u(a) = [y,b,a] dρ + u(b) − u(a) for all y ∈ R n . Thus y,a,b) \|dρ\| dy. \|T ρ(b) − T ρ(a) − (u(b) − u(a))\| ≤ B n ϕ(y) P (y,a,b) \|dρ\| dy ≤ \|\|ϕ\|\| ∞ B n P ( Let ψ be a Euclidean isometry so that ψ(0) = a and ψL(0, e n ) = L(a, b). Then B = ψ −1 (B n ) is a ball in A(\|a\| − 1, \|a\| + 1). Since ψ is an isometry, we have (5.2) B n P (y,a,b) \|dρ\| dy = B P (x,0,en) \|dψ * ρ\| dx = B Ψ(x) dx, where Ψ : R n → R is defined by Ψ(x) = P (x,0,en) \|dψ * ρ\|. To estimate the integral in (5.2), we observe first that, given x ∈ R n \ L(0, e n ), we have Ψ(x) = Ψ(y) for y ∈ P (x, 0, e n ). Then, writing x = (s, φ, x n ) in cylindrical coordinates, we see that Ψ(x) = Ψ(φ). We denote p = (p 1 , . . . , p n ) = ψ −1 (0). Then B Ψ(x) dx ≤ C pn+1 pn−1 S n−2 \|a\|+1 \|a\|−1 s n−2 Ψ(φ) ds dφ dx n ≤ C\|a\| n−2 S n−2 Ψ(φ) dφ. We write P (x, 0, e n ) = P (φ). Then, as Ψ(φ) = P (φ) \|dψ * ρ\| = ∞ −∞ ∞ −∞ \|dψ * ρ(s, φ, x n )\| ds dx n , another application of cylindrical coordinates yields \|a\| n−2 S n−2 Ψ(φ) dφ ≤ C\|a\| n−2 B \|dψ * ρ(x)\| dist (x, L(0, e n )) n−2 dx ≤ C dρ 1 . The claim follows by combining the estimates. 6. Degree estimate and Proof of Theorem 1.1 Lemma 6.1. There exists ε = ε(n) > 0 so that if ρ is a W p,n/2 -frame, p > n, n ≥ 3, satisfying dρ 1 ≤ ε, and if ρ = dx in R n \ B n , then deg(y, T ρ, B n (2)) ≤ 1 for all y ∈ R n \ T ρS n−1 (2). Proof. By Lemma 5.1, (2). Suppose from now on that C dρ 1 < 1/8. \|T ρ(x) − T ρ(2e 1 ) − (x − 2e 1 )\| ≤ C dρ 1 for x ∈ S n−1 Let f : R n → R n be the mapping f (x) = T ρ(x) + (2e 1 − T ρ(2e 1 )). We denote v = 2e 1 − T ρ(2e 1 ). Since deg(y + v, f, B n (2)) = deg(y, T ρ, B n (2)) for all y ∈ T ρS n−1 (2), the claim of the lemma holds if and only if deg(z, f, B n (2)) ≤ 1 for all z ∈ f S n−1 (2). Since (6.1) \|f (x) − x\| = \|T ρ(x) − T ρ(2e 1 ) − (x − 2e 1 )\| < C dρ 1 < 1/8 for x ∈ S n−1 (2), we have, by a homotopy argument, (6.2) deg(y, f, B n (2)) = deg(y, id, B n (2)) for y ∈ A(15/8, 17/8). Moreover, (6.3) \|\|f (x)\| − 2\| < C dρ 1 < 1/8 on S n−1 (2). Suppose now that there exists y ∈ A(15/8, 17/8) so that (6.4) deg(y, f, B n (2)) ≥ 2. By continuity, we can fix r > 0 so that deg(y ′ , f, B n (2)) ≥ 2 for every y ′ ∈ B n (y, r). By density of smooth frames in W p,q and continuity of T : L p ( 1 B n (2)) → W 1,p (B n (2)), we may fix a smooth frameρ so thatf = Tρ satisfies (6.1), and hence also (6.3), in place of f and deg(y ′ ,f , B n (2)) ≥ 2 for y ′ ∈ B n (y, r/2). We may also assume thatρ = ρ = dx on R n \ B n . By (6.1), the mapping g : S n−1 → S n−1 , g(x) =f (2x) \|f (2x)\| , is well-defined and smooth. We show that J g ≥ 0 almost everywhere on S n−1 . This contradicts deg(y, f, B(2)) ≥ 2 and the claim follows. Indeed, since g is homotopic to id : S n−1 → S n−1 , we have, by the degree theory, S n−1 J g = deg(g)\|S n−1 \| = \|S n−1 \|. Since deg(y ′ ,f , B n (2)) ≥ 2 for y ′ ∈ B n (y, r/2), there exists a set E ⊂ S n−1 of positive H n−1 -measure so that # g −1 (z) ≥ 2 for z ∈ E. Hence, by the change of variables, S n−1 \|J g \| = S n−1 N (z, g) dH n−1 (z) = S n−1 \E N (z, g) dH n−1 (z) + E N (z, g) dH n−1 (z) > \|S n−1 \| = S n−1 J g . This contradicts the non-negativity of J g . It remains to show the non-negativity of J g . We denote ω 0 = n j=1 (−1) j+1 x j \|x\| n dx 1 ∧ . . . ∧ dx j ∧ . . . ∧ dx n = n j=1 x j \|x\| n (⋆dx j ). Here ⋆ is the Hodge star operator. Then we have \|1 − J g (x)\| = \|(id * − g * )ω 0 \| = n j=1 x j (⋆dx j ) − g * y j \|y\| n (⋆dy j ) = n j=1 x j (⋆dx j ) − (f j /\|f \| n )f * (⋆dy j ) ≤ n j=1 (x j −f j /\|f \| n ) ⋆ dx j + (f j /\|f \| n )(⋆dx j −f * (⋆dy j )) ≤ n (M 1 + M 2 ) , where M 1 = max j x j −f j \|f \| n and M 2 = max jf j \|f \| n ⋆dx j −f * (⋆dy j ) . To estimate M 1 we observe that, by (6.3) and (6.1), x j −f j \|f \| n ≤ 1 − 1 \|f \| n \|x j \| + \|x j −f j \| \|f \| n ≤ C dρ 1 on S n−1 (2), where C = C(n). To estimate M 2 , we observe first that, on R n \ B n , we have ⋆dx j −f * (⋆dy j ) ≤ \|dx − df \| = \|dx − dTρ\| = \|dx −ρ + T dρ\| = \|T dρ\|. Since \|T dρ(x)\| = B n ϕ(y)K y dρ(x) dy ≤ C S n−1 (x,4) 1 0 \|dρ(y + t(x − y))\| dt dH n−1 (y) ≤ C R n \|dρ(y)\| \|x − y\| n−1 = C B n \|dρ(y)\| \|x − y\| n−1 dy ≤ C dρ 1 for x ∈ S n−1 (2), we have that M 2 ≤ C dρ 1 , where C = C(n). We choose ε = ε(n) > 0 so that M 1 + M 2 < 1/(2n) for dρ 1 < ε. Then J g > 1/2. The claim follows. Proof of Theorem 1.1. Suppose first that r = 1. For brevity, we denote f = T ρ : R n → R n . Since f = T ρ = f 0 +c, where c ∈ R n , on B n (1/2), we have deg(y, f, B n (1/2)) = deg(y − c, f 0 , B n (1/2)) for y ∈ f S n−1 (1/2). Then, by Lemma 3.1, R n max{deg(y, f 0 , B n (1/2)) − 1, 0} dy = R n max{deg(y, f, B n (1/2)) − 1, 0} dy ≤ B n (1/2) \|J f \| ≤ C 1 + dρ n/2 n , (6.5) where C = C(n) > 0. If dρ n/2 ≥ ε, where ε = ε(n, K) is the constant in Lemma 6.1, the claim follows. Thus we may assume that dρ n/2 < ε. Since, for every ε > 0, the n-measure \|f S n−1 (2(1 + t))\| = \|f S n−1 ((1 + t)/2)\| = 0 for almost every t ∈ (−ε, ε), we may assume that \|f S n−1 (2)\| = \|f S n−1 (1/2)\| = 0 by applying a rescaling to ρ if necessary. Since deg(y, f, A(1/2, 2)) = deg(y, f, B n (2)) − deg(y, f, B n (1/2)) for y ∈ f S n−1 (1/2)∪f S n−1 (2), we have, by Lemma 6.1, deg(y, f, B n (2)) ≤ 1 for all y ∈ R n \ f S n−1 (2). Thus, by Proposition 4.1, where C = C(n) > 0 and Ω − = {y ∈ R n : deg(y, f, A(1/2, 2)) < 0}, as in Section 4. For general r > 0 the argument above can be applied to ρ ′ = (λ * r ρ) /r and f ′ 0 = (f 0 • λ r )/r. The proof is complete. Quasiconformal energy minimizers In this section we consider the minimization problem for the q-energy of extension frames. We obtain Theorem 1.2 in two parts. The existence of minimizers is shown in Theorem 7.2. For the higher integrability of minimizers, we derive an Euler-Lagrange equation (Lemma 7.4) and a Caccioppoli type inequality (Corollary 7.6) for this variational problem. We then establish a reverse Hölder inequality (Theorem 7.7) which yields the higher integrability by Gehring's lemma. We denote by ·, · the inner product X, Y = 1 m tr X t Y for (m × k)-matrices and by \| · \| 2 the (normalized) Hilbert-Schmidt norm \|X\| 2 2 = X, X = 1 n m i=1 k j=1 X 2 ij . In this section, we identify frames with matrix fields, and use the inner product ·, · and the norm \| · \| 2 also for frames. Let q > n/2 and 0 < r < R < ∞. Let ρ 0 and ρ 1 be W loc n,q -frames in B n (r) and R n \ B n (R), respectively. The following lemma shows that ρ 0 and ρ 1 can be quasiconformally connected if they have quasiconformal extensions to the neighborhoods of S n−1 (r) and S n−1 (R), respectively. Lemma 7.1. Let 0 < r < r ′ < R ′ < R < ∞ and let ρ 0 and ρ 1 be W loc n,qframes in B n (r ′ ) and R n \B n (R ′ ), respectively, so that ρ 0 is K-quasiconformal in A(r, r ′ ) and ρ 1 is K-quasiconformal in A(R ′ , R). Then there exists a W n,q -frame ρ so that ρ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)), whereK =K(n, K, r, r ′ , R, R ′ ). Proof. Let r 0 = (r ′ + R ′ )/2. We define mappings λ 0 : A(r, r 0 ) → A(r, r ′ ) and λ 1 : A(r 0 , R) → A(R ′ , R) by λ 0 (x) = r ′ − r r 0 − r (\|x\| − r) + r x \|x\| and λ 1 (x) = R − R ′ R − r 0 (\|x\| − r 0 ) + R ′ x \|x\| . Let also θ : [0, ∞) → [0, 1] be a smooth function so that θ(t) = 1 for t < r ′ and t > R ′ , θ(t) = 0 in a neighborhood of r 0 , and that \|dθ\| ≤ 3/(R ′ − r ′ ). We set ρ to be the frame ρ(x) =        ρ 0 , x ∈ B n (r) θ(\|x\|)(λ * 0 ρ 0 )(x), x ∈ A(r, r 0 ) θ(\|x\|)(λ * 1 ρ 1 )(x), x ∈ A(r 0 , R) ρ 1 , x ∈ R n \ B n (R). Since ρ 0 and ρ 1 are K-quasiconformal in A(r, r ′ ) and A(R ′ , R), respectively, frames λ * i ρ i areK-quasiconformal forK =K(n, K, r, r ′ , R, R ′ ) for i = 0, 1. Since \|dρ(x)\| ≤ \|dθ(\|x\|)\|\|λ * i ρ i (x)\| + \|θ(x)\|\|λ * i dρ i (x)\| for i = 0, 1 in A(r, r 0 ) and A(r 0 , R), respectively, we have that dρ ∈ L q ( 2 R n ). Thus ρ is a W n,qframe. In what follows, we assume that E q,K (ρ 0 , ρ 1 ; A(r, R)) is non-empty and we consider the minimization problem (7.1) I q,K (ρ 0 , ρ 1 , A(r, R)) = inf ρ∈E q,K (ρ 0 ,ρ 1 ;A(r,R)) A(r,R) \|dρ\| q 2 , q > n/2. In the forthcoming discussion, we use the observation that for every frame ρ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)) there exists an affine subspace of frames conformally equivalent to ρ; more precisely, (1 + h)ρ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)) for all ρ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)) and all h ∈ C ∞ 0 (A(r, R)) satisfying h ≥ −1. Theorem 7.2. The minimization problem (7.1) admits a minimizer ρ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)), i.e., there exists ρ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)) so that A(r,R) \|dρ\| q 2 = I q,K (ρ 0 , ρ 1 , A(r, R)). To this end, we would like to note that, since the minimization problem is considered in E q,K (ρ 0 , ρ 1 ; A(r, R)), standard convexity arguments are not at our disposal and the uniqueness of the minimizer is not guaranteed. We begin the proof of Theorem 7.2 with the following lemma. We assume in what follows that 0 < r < R < ∞. Lemma 7.3. Let q > n/2, n ≥ 2, and let ρ be a W loc n,q -frame in R n so that ρ is K-quasiconformal in A(r, R). Then ρ n,A(r,R) ≤ C ρ n,A(R,2R) + dρ q,B n (2R) , where C = C(n, K, q, R) > 0. Proof. Let A = A(R, 2R) and B = B n (2R). We set ω = n j=1 (−1) j T 2R ρ j dT 2R ρ 1 ∧ · · · ∧ dT 2R ρ j ∧ · · · ∧ dT 2R ρ n . As in the proof of Proposition 4.1, we obtain B n (t) dT 2R ρ 1 ∧ · · · ∧ dT 2R ρ n ≥ 1 K ρ n n,A(r,R) − C n k=1 T 2R dρ k n,B ρ n−k n,B for R ≤ t ≤ 2R, where C = C(n). On the other hand, 2R R S n−1 (t) ω dt = 2R R B n (t) dω dt = 2R R B n (t) dT 2R ρ 1 ∧ · · · ∧ dT 2R ρ n dt and 2R R S n−1 (t) ω dt ≤ A \|ω\| ≤ n A \|T 2R ρ\|\|dT 2R ρ\| n−1 ≤ n T 2R ρ n,A dT 2R ρ n−1 n,A . Thus R ρ n n,A(r,R) 1 − C n k=1 T 2R dρ k n,B ρ k n,B ≤ Kn T 2R ρ n,A dT 2R ρ n−1 n,A . There exists ε = ε(n) > 0 so that either (7.2) ε ρ n,B ≤ T 2R dρ n,B or (7.3) R ρ n n,A(r,R) ≤ C T 2R ρ n,A dT 2R ρ n−1 n,A , where C = C(n, K). Suppose first that (7.2) holds. Then ρ n,A(r,R) ≤ ρ n,B ≤ 1 ε T 2R dρ n,B ≤ C T 2R dρ q * ,B ≤ C dρ q,B , where C = C(n, q, R). Here we used the Sobolev-Poincaré inequality (3.3). Suppose now that (7.3) holds. The Sobolev-Poincaré inequality applies to the mapping T 2R ρ in A, so T 2R ρ n,A ≤ C dT 2R ρ n,A . Therefore, another application of (3.3) gives ρ n,A(r,R) ≤ C dT 2R ρ n,A ≤ C ( ρ n,A + T 2R dρ n,A ) ≤ C ( ρ n,A + dρ q,B ) , where C = C(n, K, q, R). Having Lemma 7.3 at our disposal, the standard methods in non-linear potential theory can be used to prove Theorem 7.2; see [1,Chapter 5]. Proof of Theorem 7.2. Suppose (ρ k ) is a minimizing sequence for (7.1). Then (dρ k ) is a bounded sequence in L q ( 2 R n ). Sinceρ k coincides with ρ 0 in B n (r) and with ρ 1 in R n \ B n (R) for every k, we have, by Lemma 7.3, that (ρ k ) is a bounded sequence in L n ( 1 B n (R)). By passing to a subsequence if necessary, we may assume thatρ k →ρ ∞ weakly and dρ k → dρ ∞ weakly as k → ∞, whereρ ∞ ∈ L n ( 1 B n (R)) with dρ ∞ ∈ L q ( 2 B n (R)). By the weak lower semi-continuity of norms, we obtain dρ ∞ q,A(r,R) = I q,K (ρ 0 , ρ 1 , A(r, R)). Since q > n/2, the K-quasiconformality ofρ ∞ is a consequence of compensated compactness [3, Theorem 5.1]; see also [2,Proposition 4.5]. Finally, the boundary conditionsρ ∞ \|B n (r) = ρ 0 andρ ∞ \|R n \ B n (R) = ρ 1 follow from weak convergence of the sequence (ρ k ) toρ ∞ . Thusρ ∞ ∈ E q,K (ρ 0 , ρ 1 ; A(r, R)). Following the standard arguments in the elliptic theory we can show that minimizers satisfy an Euler-Lagrange equation; we refer to [1, 5.13] for details. Lemma 7.4. A minimizer ρ of the problem (7.1) satisfies the equation (7.4) A(r,R) \|dρ\| q−2 2 dρ, d(hρ) = 0 for every h ∈ C ∞ 0 (A(r, R)). Having the Euler-Lagrange equation at our disposal, we find an Euler-Lagrange equation for the minimizers of (7.1). Lemma 7.5. Let ρ be a minimizer of the problem (7.1). Then (7.5) A(r,R) \|dρ\| q 2 h q 1/q ≤ q A(r,R) \|ρ\| q 2 \|dh\| q 1/q for every non-negative h ∈ C ∞ 0 (A(r, R)). Proof. By the Euler-Lagrange equation (7.4), 0 = A(r,R) \|dρ\| q−2 2 dρ, d(h q ρ) = A(r,R) \|dρ\| q−2 2 dρ, qh q−1 dh ∧ ρ + h q dρ . Thus A(r,R) h q \|dρ\| q 2 ≤ q A(r,R) \|dρ\| q−1 2 h q−1 \|dh\|\|ρ\| ≤ q A(r,R) \|dρ\| q 2 h q (q−1)/q A(r,R) \|dh\| q \|ρ\| q 2 1/q . The claim follows. Caccioppoli's inequality (7.5) readily yields the following corollary. Corollary 7.6. Let B = B n (x 0 , s) be a ball so that 2B ⊂ A(r, R). Then ∦ dρ q,B ≤ 2q s ∦ ρ q,2B . Here, and in what follows, we denote the integral average ∦ ω p,Ω = − Ω \|ω\| p 2 1/p whenever Ω is a bounded domain in R n and ω is an n-tuple of forms in Ω. The main result in this section is the following reverse Hölder's inequality. Theorem 7.7. Let ρ 0 be a minimizer of the problem (7.1). Then there exists C = C(n) > 0 so that Corollary 7.8. Let ρ 0 be a minimizer of the problem (7.1). Then there exists p > n and C 0 = C 0 (p, n) > 0 so that ∦ ρ 0 p,B ≤ C 0 ∦ ρ 0 n,2B whenever B = B n (x 0 , s) is a ball satisfying 2B ⊂ A(r, R). Proof of Theorem 7.7. For the purpose of this proof, we define T x 0 to be the averaged Poincaré homotopy operator centered at x 0 , that is, T x 0 = (τ −1 ) * • T 2s • τ * , where τ is the translation x → x + x 0 . Naturally, the properties of T discussed in Section 3 hold also for T x 0 . Let ρ = (ρ 1 , . . . , ρ n ). By quasiconformality of ρ in A(r, R), we obtain, as in the proof of Proposition 4.1, that where C = C(n, K). We estimate the integral I 1 first. Since T x 0 ρ ∈ W 1,n loc (2B), R n ), we have by the isoperimetric inequality (3.4), B n (x 0 ,t) \|dT x 0 ρ 1 ∧ · · · ∧ dT x 0 ρ n \| ≤ C for almost every r ≤ t ≤ 2r. Thus B \|dT x 0 ρ 1 ∧ · · · ∧ dT x 0 ρ n \| (n−1)/n ≤ C − 2r r S n−1 (x 0 ,t) \|dT x 0 ρ\| n−1 ≤ C r 2B \|dT x 0 ρ\| n−1 . Since dT x 0 ρ n−1,2B ≤ ρ n−1,2B + T x 0 dρ n−1,2B , we have, by the Sobolev-Poincaré and Caccioppoli's inequality, I 1/n 1 ≤ Cr −1/(n−1) ρ n−1,2B + Cr ∦ T x 0 dρ n−1,2B ≤ Cr −1/(n−1) ρ n−1,2B + Cr 2 ∦ dρ q,2B ≤ Cr ∦ ρ n−1,2B + Cr ∦ ρ q,2B ≤ Cr ∦ ρ max{n−1,q},2B . To estimate I 2 we use first the Sobolev-Poincaré inequality and then Caccioppoli's inequality to obtain I 2 ≤ C n k=1 r k ∦ T x 0 dρ k n,B ρ n−k n,B ≤ C n k=1 r 2k ∦ dρ k q,B ρ n−k n,B ≤ C n k=1 r k ∦ ρ k q,2B ρ n−k n,B ≤ Cr n n k=1 ∦ ρ k max{n−1,q},2B ∦ ρ n−k n,B . Combining estimates for I 1 and I 2 we have ∦ ρ n n,B ≤ Cr −n (I 1 + I 2 ) ≤ C ∦ ρ n n−1,2B + C n k=1 ∦ ρ k max{n−1,q},2B ∦ ρ n−k n,B , where C = C(n, K, ϕ). Suppose that (7.6) does not hold with C 0 = 1/(2 + 2nC). Then, by (7.7), ∦ ρ n n,B ≤ C ∦ ρ n n−1,2B + (1/2) ∦ ρ n n,B . and (7.7) holds with C 0 = 2C. The proof is complete. balls B = B n (x 0 , s) satisfying 2B ⊂ A(r, R). 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[ "Generalized Linear Bandits with Local Differential Privacy", "Generalized Linear Bandits with Local Differential Privacy" ]	[ "Yuxuan Han \nDepartment of Industrial Engineering and Decision Analytics\n\n", "Zhipeng Liang \nThe Hong Kong University of Science and Technology\n\n", "Yang Wang \nDepartment of Industrial Engineering and Decision Analytics\n\n\nThe Hong Kong University of Science and Technology\n\n", "Jiheng Zhang \nDepartment of Industrial Engineering and Decision Analytics\n\n\nThe Hong Kong University of Science and Technology\n\n", "\nDepartment of Mathematics\n\n" ]	[ "Department of Industrial Engineering and Decision Analytics\n", "The Hong Kong University of Science and Technology\n", "Department of Industrial Engineering and Decision Analytics\n", "The Hong Kong University of Science and Technology\n", "Department of Industrial Engineering and Decision Analytics\n", "The Hong Kong University of Science and Technology\n", "Department of Mathematics\n" ]	[]	Contextual bandit algorithms are useful in personalized online decision-making. However, many applications such as personalized medicine and online advertising require the utilization of individual-specific information for effective learning, while user's data should remain private from the server due to privacy concerns. This motivates the introduction of local differential privacy (LDP), a stringent notion in privacy, to contextual bandits. In this paper, we design LDP algorithms for stochastic generalized linear bandits to achieve the same regret bound as in non-privacy settings. Our main idea is to develop a stochastic gradient-based estimator and update mechanism to ensure LDP. We then exploit the flexibility of stochastic gradient descent (SGD), whose theoretical guarantee for bandit problems is rarely explored, in dealing with generalized linear bandits. We also develop an estimator and update mechanism based on Ordinary Least Square (OLS) for linear bandits. Finally, we conduct experiments with both simulation and real-world datasets to demonstrate the consistently superb performance of our algorithms under LDP constraints with reasonably small parameters (ε, δ) to ensure strong privacy protection.*Equal contributions.	null	[ "https://arxiv.org/pdf/2106.03365v1.pdf" ]	235,358,954	2106.03365	18a868e40e991c67b86d619d4e108db594d0ff37	Generalized Linear Bandits with Local Differential Privacy 7 Jun 2021 June 8, 2021 Yuxuan Han Department of Industrial Engineering and Decision Analytics Zhipeng Liang The Hong Kong University of Science and Technology Yang Wang Department of Industrial Engineering and Decision Analytics The Hong Kong University of Science and Technology Jiheng Zhang Department of Industrial Engineering and Decision Analytics The Hong Kong University of Science and Technology Department of Mathematics Generalized Linear Bandits with Local Differential Privacy 7 Jun 2021 June 8, 2021 Contextual bandit algorithms are useful in personalized online decision-making. However, many applications such as personalized medicine and online advertising require the utilization of individual-specific information for effective learning, while user's data should remain private from the server due to privacy concerns. This motivates the introduction of local differential privacy (LDP), a stringent notion in privacy, to contextual bandits. In this paper, we design LDP algorithms for stochastic generalized linear bandits to achieve the same regret bound as in non-privacy settings. Our main idea is to develop a stochastic gradient-based estimator and update mechanism to ensure LDP. We then exploit the flexibility of stochastic gradient descent (SGD), whose theoretical guarantee for bandit problems is rarely explored, in dealing with generalized linear bandits. We also develop an estimator and update mechanism based on Ordinary Least Square (OLS) for linear bandits. Finally, we conduct experiments with both simulation and real-world datasets to demonstrate the consistently superb performance of our algorithms under LDP constraints with reasonably small parameters (ε, δ) to ensure strong privacy protection.Equal contributions. Introduction Contextual bandit algorithms have received extensive attention for their efficacy for online decision making in many applications such as recommendation system, clinic trials, and online advertisement Bietti et al. (2018); Slivkins (2019); Lattimore and Szepesvári (2020). Despite their success in many applications, intensive utilization of user-specific information, especially in privacy-sensitive domains such as clinical trials and e-commerce promotions, raises concerns about data privacy protection. Differential privacy, as a provable protection against identification from attackers Dwork et al. (2006); Dwork and Roth (2013), has been put forth as a competitive candidate for a formal definition of privacy and has received considerable attention from both academic research Rubinstein et al. (2009) Tang et al. (2017). While increasing attention has been paid to bandit algorithms with jointly differential privacy Shariff and Sheffet (2018); Chen et al. (2020), we introduce in this paper a more stringent notion, locally differential privacy (LDP), in which users even distrust the server collecting the data, to contextual bandits. In contextual bandit, at each time round t with individual-specific context X t , the decision maker can take an action a t from a finite set (arms) to receive a reward randomly generated from the distribution depending on the context X t and the chosen arm through its parameter θ ⋆ at which is not unknown to the decision maker. We use the standard notion of expected regret to measure the difference between expected rewards obtained by the action a t and the best achievable expected reward in this round. While several papers consider the adversarial setting (i.e., X t can be arbitrary determined in each round), this paper considers the stochastic contextual case where X t is generated i.i.d. from a distribution P X . The goal is to maximize the rewards accumulated over the time horizon. An algorithm achieves LDP guarantee if every user involved in this algorithm is guaranteed that anyone else can only access her context (and related information such as the arm chosen and the reward) with limited advantage over a random guess. Recently there is an emerging steam of works combining LDP and bandit. Basu et al. (2019); Ren et al. (2020); Chen et al. (2020) consider the LDP contextual-free bandit and design algorithms to achieve the same regret as in the non-privacy setting. For contextual bandits, Zheng et al. (2020) considers the adversarial setting. Despite their pioneering work, their regret bounds O(T 3/4 ) leave a gap from the corresponding non-privacy results O(T 1/2 ), which is conjectured to be inevitable. A natural question arises: can we close this gap for stochastic contextual bandits? In this paper, we design several algorithms and show that they can achieve the same regret rate in terms of T as in the non-private settings. If we don't assume any structure on the arms' parameters, the above formulation is referred to as multi-parameter contextual bandits. If we impose structural assumptions such as all arms share the same parameter (see Section 2.2 for details), then the formulation is referred to as singleparameter contextual bandits. Although multi-parameter and single-parameter settings can be shown to be equivalent, they need independent analysis and design of algorithms because of their distinct properties based on different modeling assumptions (e.g., Raghavan et al. (2018)). In this paper, we consider the privacy guarantee in both settings. In fact, multi-parameter setting is more difficult since we need to estimate the parameters for all K arms with sufficient accuracy to make good decisions. However, privacy protection also requires protecting the information about which arm is pulled in each round. Such a requirement hinders the identification of optimal arm and may incur considerable regret in the decision process. A proper balance between privacy protection and estimation accuracy is the key to design algorithms with desired performance guarantee in this setting. Contributions. We organize our results for various settings in Table 1.1. Our main contributions can be summarized as follows: 1. We develop a framework for implementing LDP algorithms by integrating greedy algorithms with a private OLS estimator for linear bandits and a private SGD estimator for generalized linear bandits. We prove that our algorithms achieve regret bound matching the corresponding non-privacy results. 2. In the multi-parameter setting, to ensure the privacy of the arm pulled in each round, we design a novel LDP strategy by simultaneously updating all the arms with synthetic information instead of releasing the pulled arm. By conducting such synthetic updates for unselected arms, we protect the information of the pulled arm from being identified by the server or other users. This is at the cost of corrupting the estimation of the un-selected arms. To deal with this issue, we design an elimination method that is only based on data collected during a short warm up period. We show that such a mechanism can be combined with the OLS and SGD estimators to achieve the desired performance guarantees. O(log T /ε 2 ) Stochastic Single β = 1 Theorem 3.3Õ(T 1−β 2 /ε 1+β ) Stochastic Single 0 ≤ β < 1 Theorem 4.1 O((log T /ε) 2 ) Stochastic Multiple β = 1 Theorem 4.1Õ(T 1−β 2 /ε 1+β ) Stochastic Multiple 0 < β < 1 3. We introduce the SGD estimator to bandit algorithms to tackle generalized linear reward structure. To the best of our knowledge, few papers have ever considered SGD-based bandit algorithms. (2018). We establish such theoretical regret bounds for SGD-based bandit algorithms. Our private SGD estimator for bandits is highly computationally efficient, and more importantly, greatly simplifies the data processing mechanism for LDP guarantee. Preliminaries Notations. We start by fixing some notations that will be used throughout this paper. For a positive integer n, [n] denotes the set {1, · · · , n}. \|A\| denotes the cardinality of the set A. · 2 is Euclidean norm. W (i, j) denotes the element in the i-th row and j-th column of matrix W . We write W > 0 if the matrix W is symmetric and positive definite. We denote I d as the d-dimensional identity matrix. Let ⊗ denote the Kronecker product. Let B d r denote the ddimensional ball with radius r and S d−1 r denotes the (d−1)-dimensional sphere for the ball. Given a set A, Unif(A) denote the uniform distribution over A. For a tuple (Z i,j ) i≤N,j≤M and 1 ≤ k 1 < k 2 ≤ M , we denote Z i,k 1 :k 2 = (Z i,k 1 , · · · , Z i,k 2 ). We adopt the standard asymptotic notations: for two non-negative sequences {a n } and {b n }, {a n } = O({b n }) iff lim sup n→∞ a n /b n < ∞, a n = Ω(b n ) iff b n = O(a n ), a n = Θ(b n ) iff a n = O(b n ) and b n = O(a n ). We also writeÕ(·),Ω(·) andΘ(·) to denote the respective meanings within multiplicative logarithmic factors in n. Local Differential Privacy Definition 2.1 (Local differential privacy). We say a (randomized) mechanism M : X → Z is (ε, δ)-LDP, if for every x = x ′ ∈ X and any measurable set C ⊂ Z we have P (M (x) ∈ C) ≤ e ε P (M (x ′ ) ∈ C) + δ. When δ = 0, we simply denote ε-LDP. We now present some tools that will be useful for our analysis. (2013)). For any f : X → R n , let σ ε,δ = 1 ε sup x,x ′ ∈X f (x) − f (x ′ ) 2 2 ln(1.25/δ). The Gaussian mechanism, which adds random noise independently drawn from distribution N (0, σ 2 ε,δ I n ) to each output of f , ensures (ε, δ)-LDP. Although all our results can be extended in parallel to ε-LDP if using Laplacian noise instead of Gaussian noise, we focus on (ε, δ)-LDP in this paper. Besides the Gaussian mechanism, we also use the following privacy mechanism for bounded vectors. R > 0, let r ε,d = R √ π 2 e ε + 1 e ε − 1 dΓ( d+1 2 ) Γ( d 2 + 1) where and Γ is the Gamma function. For any x ∈ B d R , consider the mechanism Ψ ε,R : B d R → S d−1 r ε,d of generating Z x as the follows. First, generate a random vector X = (2b − 1)x where b is a Bernoulli random variable with success probability 1 2 + x 2 2R . Next, generate random vector Z x via Z x ∼ Unif{z ∈ R d : z TX > 0, z 2 = r ε,d } with probability e ε /(1 + e ε ) Unif{z ∈ R d : z TX ≤ 0, z 2 = r ε,d } with probability 1/(1 + e ε ). Then Ψ ε,R is ε-LDP and E[Ψ ε,R (x)] = x. Lemma 2.3 (Post-Processing property Dwork and Roth (2013)). If M : X → Y is (ε, δ)-LDP and f : Y → Z is a fixed map, then f • M : X → Z is (ε, δ)-LDP. Lemma 2.4 (Composition property Dwork and Roth (2013)). If M 1 : X → Z 1 is (ε 1 , δ 1 )-LDP and M 2 : X → Z 2 is (ε 2 , δ 2 )-LDP, then M = (M 1 , M 2 ) : X → Z 1 × Z 2 is (ε 1 + ε 2 , δ 1 + δ 2 )-LDP. Local Differential Privacy in Bandit We consider contextual bandits with LDP guarantee in the context of the user-server communication protocol described in Figure 2.1. The user in round t with context X t ∈ R d receives (processed) historical information S t−1 from the server, and chooses an action a t ∈ [K] to obtain a random reward r t = v(X t , a t ) + ǫ t . Define F t as the filtration of all historical information up to time t, i.e., F t = σ(X 1 , · · · , X t , ǫ 1 , · · · , ǫ t−1 ), and we require E[ǫ t \|F t ] = 0, E[exp(λǫ t )\|F t ] ≤ exp( σ 2 ǫ λ 2 2 ), ∀λ ∈ R. Then the user processes the tuple (X t , r t ) by some mechanism ϕ with LDP guarantee and send the processed information Z t = ϕ(X t , r t ) to the server. After receiving Z t , the server updates the historical information S t to get S t+1 . We consider the generalized linear bandits by allowing v(X t , a t ) = µ(X T t θ ⋆ at ), where µ : R → R is a link function and θ ⋆ i ∈ R d is the underlying parameter of the i-th arm. For a fix time t, we denote a t = arg max i∈ [K] µ(X T t θ ⋆ i ). The regret over time horizon T is Reg (T ) = T t=1 µ(X T t θ ⋆ a * t ) − µ(X T t θ ⋆ at ) . If we don't assume any structure on {θ ⋆ i } i∈ [K] , we refer it as the multi-parameter setting. We also consider d-dimensional single-param setting by assuming In the rest of paper, we always assume that θ ⋆ i 2 ≤ 1, ∀i ∈ [K], the reward is bounded by c r and the context is bounded by C B , our analysis can be easily generalized to the case where ǫ t and the context follow sub-gaussian distributions. We also impose regularize assumptions on the link function, which are common in previous work Zheng et al. θ ⋆ i = e i ⊗ θ ⋆ for some θ ⋆ ∈ R d where {e i } i∈[K] is canonical basis of R K . In this case, x t,i ∈ R d is the i-th segment of X t ∈ R dK and X T t θ ⋆ i = x T t,i θ ⋆ , so choosing arm i becomes choosing the i-th segment x t,i of the context. User Side Server Side · · · · · · · · · · · · X t r t Z t X t+1 r t+1 Z t+1 S t S t+1 S t+2 a t a t+1> 0 such that inf x∈[−C B ,C B ] µ ′ (x) = ζ > 0. Single-Parameter Setting In this section, we develop a LDP contextual bandit framework (Algorithm 1) by combining statistical estimation and privacy mechanisms in the single-param bandit setting to achieve optimal regret bound in various cases. We use an abstract privacy mechanism ψ in (3.1) and estimator ϕ in (3.2) to allow the plug-in of various components. Privacy Guarantee For the linear case where the link function µ(x) = x, we can use the following ordinary least square (OLS) estimator. Let with σ ε,δ = 2 2 ln(1. 25/δ)/ε. Define M t = x t,at x T t,at +W t where W t is a random matrix with W t (i, j) ∼ N (0, 4C 2 B σ 2 ε,δ ) and W t (j, i) = W t (i, j), and u t = r t x t,at + ξ t where ξ t is a random vector following distribution N (0, C 2 B c 2 r σ 2 ε,δ I d ). The OLS privacy mecha- Receiveθ t−1 from the server. 5 Pull arm a t = argmax a∈ [K] x T t,aθt−1 and receive r t . 6 Generate Z t by Z t = ψ t (x t,at , r t ;θ t−1 ). (3.1) 7 Server side: 8 Receive Z t from the user. Update the estimation viaθ t = ϕ t (Z 1 , . . . , Z t ;θ t−1 ). (3.2) 10 end nism and the corresponding estimator are ψ OLS t (x t,at , r t ;θ t−1 ) = (M t , u t ), (3.3) ϕ OLS t (Z 1 , . . . , Z t ;θ t−1 ) = t i=1 M i +c √ tI −1 t i=1 u i , (3.4) wherec > 0 is to be determined. We have the following LDP guarantee using the Gaussian mechanism (Lemma 2.1) and post-processing (Lemma 2.3). Proposition 3.1. Algorithm 1 with the private OLS update mechanism ψ OLS t and estimator ϕ OLS t is (ε, δ)-LDP. For the general link function µ, its non-linearity adds to the difficulty in terms of both privacypreserving and bandits. To estimate parameters in generalized linear bandits, one common approach to use a maximum likelihood estimator (MLE) at each step. In contrast to OLS solution, MLE does not have a close form solution with simple sufficient statistics in general. Thus, solving an MLE optimization procedure requires using all the previous data points and conducting costly operations at each round, resulting in time complexity and memory usage increasing with time. Instead, we use a one-step stochastic gradient approximation to incrementally update the estimator with the new observation at each round. To obtain a LDP version of this approximation, we use the LDP l 2 -ball mechanism in Lemma 2.2. ψ SGD t (x t,at , r t ;θ t−1 ) = Ψ ε,R µ(x T t,atθt−1 ) − r t x t,at , (3.5) ϕ SGD t (Z 1 , . . . , Z t ;θ t−1 ) =θ t−1 − η t ψ SGD t . (3.6) where η t > 0 is the stepsize to be determined and R = 2c r C B . Similarly, we can prove the following LDP guarrantee using the l 2 -ball mechanism Lemma 2.2 and post-processing Lemma 2.3. Proposition 3.2. Algorithm 1 with the private SGD update mechanism ψ SGD t and estimator ϕ SGD t is ε-LDP. Regret Analysis To derive the regret bound of our framework, we need the following assumptions on the marginal distribution P X of the stochastic contexts {x t,a } a∈ [K] . Assumption 2. There exists some κ u > 0 such that λ max (Σ a ) ≤ κu d where Σ a is the covariance matrix of P X and λ max (Σ a ) is the largest eigenvalues of Σ a . Assumption 3. For every u 2 = 1, denote a * = arg max a∈[K] x T t,a u, there exist some κ l > 0, p * > 0 such that P u ((x T v) 2 > κ l /d) ≥ p * holds for any u ∈ S d−1 1 , where P u (·) is the distribu- tion of x t,a * . Similar assumptions are common in the analysis of single-parameter contextual bandits, e.g. Ding et al. (2021); Han et al. (2020), and our conditions contain a wide range of distributions, including sub-gaussian with bounded density. See appendix A for discussion. Now we can show that our framework indeed achieves optimal regret bound. Theorem 3.1. Under Assumptions 2 and 3, with the choice ofc = 2σ ε,δ (4 √ d + 2 log(2T /α)) in (3.4), Algorithm 1 with OLS mechanism ψ OLS t and estimator ϕ OLS t achieve the following regret with probability at least 1 − α for some constant C, Reg(T ) ≤ C √ T (C B (σ ε,δ + σ ǫ )d (d + log(T /α)) log(KT /α) κ l p * + o(1)) Under Assumptions 1-3, with the choice of η t = c ′ d/(κ l ζp * t) for some c ′ > 1 in (3.6), Algorithm 1 with SGD mechanism ψ SGD t and estimator ϕ SGD t achieves the following regret with probability at least 1 − α for some constant C, (1)). Reg(T ) ≤ C √ T ( r ε,d √ d ζκ l p * log log(T /α) + o with o(1) means some factor that turns to 0 as T → ∞. In the algorithm we shift the sample covariance matrix byc √ t to ensure the positive-definiteness of the noise matrix as in Shariff and Sheffet (2018). Such a shift guarantee the estimation accuracy in the early stage. Note that the optimal worst-case regret bound in the non-privacy case isÕ(T 1/2 ), our results show that we can achieve the same regret bound as in the non-privacy case in terms of time T . In fact, we can show a Ω( √ T /ε) lower bound in this setting even when K = 2, which verified our optimal dependence on both T and ε. Theorem 3.2. For θ ∈ R d and an algorithm π, we denote E[Reg π (T ; θ)] the expectation regret of π when the underlying parameter is θ. When K = 2 and x t,a ∼ N (0, I d /d) are independent over a ∈ [K], we have for any possible ε-LDP algorithm π, sup θ ⋆ : θ ⋆ 2 ≤1 E[Reg π (T ; θ ⋆ )] = Ω( √ T /ε). Given the best known O(T 3/4 ) regret bound of adversarial contextual LDP bandit in Zheng et al. (2020), our O( √ T /ε) result points out a possible gap between stochastic contextual bandits and adversarial contextual bandits under the LDP constraint. The bounds given above are problemindependent, which do not dependent on the underlying parameters. If we consider an additional assumption that there is a gap between the optimal arm and the rest, which is usually the case when the number of contexts is small, then we can obtain sharper bounds than the problemindependent ones in Theorems 3.1. Assumption 4 ((γ, β)-margin condition). We say P X satisfies the (γ, β)-strong margin condi- tion with γ > 0, 0 < β ≤ 1, if for △ t := µ(x T t,a * t θ ⋆ ) − max j =a * t µ(x T t,j θ ⋆ ) and h ∈ [0, b] with some positive constant b, we have P[△ t ≤ h] ≤ γh β . Theorem 3.3. Under Assumptions 2-4 with the same choice ofc in Theorems 3.1, Algorithm 1 with OLS mechanism ψ OLS t and estimator ϕ OLS t achieves the following regret with probability at least 1 − α for some constant C, Reg(T ) ≤ C ·        γC B log T [( C B d(C B σ ǫ + σ ε,δ ) d + log(T /α) κ l p * ) 2 + o β,γ (1)], β = 1, γC B 1 − β T 1−β 2 [( C B d(C B σ ǫ + σ ε,δ ) d + log(T /α) κ l p * ) 1+β + o β,γ (1)], 0 ≤ β < 1. Under Assumptions 1-4 and with the same choice of η t in Theorems 3.1, Algorithm 1 with SGD mechanism ψ SGD t and estimator ϕ SGD t achieves the following regret with probability at least 1− α for some constant C, Reg(T ) ≤ C ·        γLC B log T [( r ε,d LdC B log(log(T )/α) ζκ l p * ) 2 + o β,γ (1)], β = 1, γLC B 1 − β T 1−β 2 [( r ε,d LdC B log(log(T )/α) ζκ l p * ) 1+β + o β,γ (1)], 0 ≤ β < 1. with o β,γ (1) being a factor depending on β, γ that converges to 0 as T → ∞. Multi-parameter Setting In this section, we present our LDP framework for the multiple parameter setting. Compared with the single parameter setting, this framework introduces three non-trivial components to match classical regret bounds while still guarantee LDP: warm up, synthetic update and elimination. Warm up. In the warm up stage, all arms are given equal opportunities to be explored for a preliminary estimation of their parameters. Such estimation does not aim for the accuracy to select the optimal arm with high probability. Instead, we only need accuracy at the level of ruling out the substantially inferior arms. Thus, this stage only needs O(log T ) steps. Since the actions in this stage are independent of the contexts, there is no need to protect the pulled arm. However, we still need to protect the contexts by using a privacy mechanism similar in the single-parameter setting. Synthetic update. After the warm up, we need to make decisions based on the contexts to achieve vanishing regret. In order to obtain the privacy guarantee, we introduce our synthetic update mechanism. Although in each time only one arm is pulled, we create synthetic data for all unselected arms. In this way, the server receives synthetic feedback about all arms, regardless of whether it is selected or not, and thus cannot figure out which one is selected. Another method to provide LDP protection for the selected arm is to ensure the action a t satisfies LDP. However, the regret will grow linearly, as shown in Shariff and Sheffet (2018). Elimination. We use the information obtained during warm up to exclude obviously inferior arms. Such a method has been applied in Bastani et al. (2017) to guarantee a certain kind of Receivingθ t−1,1:K from the server. 5 Pulling arm a t := (t mod K) + 1 and receive r t . 6 Generate and update Z t,i = 1{a t = i}ψ t (X t , r t ;θ t−1,i ), i ∈ [K] to the server. 7 Server side: 8 Receive the update Z t,1:K from the user. 9 Re-estimate parameters viaθ t,i := ϕ t (Z 1,i , . . . , Z t,i ), ∀i ∈ [K]. 10 end 11 for t ← Ks 0 + 1 to T do 12 User side: 13 Receiveθ t−1,1:K from the server. 14 Determine a subsetK t of [K] by settinĝ K t := {a ∈ [K] : X T tθKs0,a > max a∈[K] X T tθKs0,a − h 2 } (4.1) 15 Pulling arm a t := argmax a∈Kt µ(X T tθt−1,a ) and receive r t . 16 Generating information for all arms {Z i,t } i∈[K] by setting Z i,t = ψ t (X t , r t ;θ t−1,i ) if a t = i, ψ t (0, 0;θ t−1,i ) otherwise. 17 Server side: 18 Receive the update {Z i,t } i∈[K] from the user. 19 Re-estimate parameters viaθ t,i := ϕ t (Z 1,i , . . . , Z t,i ). 20 end independence of the information in each round. However, we use this method for a different purpose. The necessity of such an elimination strategy comes from protecting privacy in the multi-parameter setting. Although we have obtained an estimation to a certain level of accuracy in the warm up stage, our knowledge on un-selected arms will be gradually corrupted by the noise incurred in the synthetic update in each round. Such corruption will make us fail to distinguish arms that are possibly optimal from the surely sub-optimal ones. To avoid corruption, we may need to pick the sub-optimal arms frequently but this will result in large regret. That is why we use the warm up information to eliminate the arms with extremely poor performance as in (4.1). Privacy Guarantee The OLS/SGD mechanisms and estimators are the same as (3.3)-(3.6) in the single-parameter setting. To prevent the server from distinguishing the selected arm from the other K − 1 arms, a straightforward idea is to use (ε/K, δ/K)-LDP mechanism for the synthetic update by composition property in lemma 2.4. However, we can prove that our algorithm can still achieve the same LDP guarantee with a much less stringent privacy mechanism, say (ε/2, δ/2)-LDP, in Propositions 4.1 and 4.2. Proposition 4.1. Algorithm 2 with the private OLS update mechanism ψ OLS t and estimator ϕ OLS t is (ε, δ)-LDP. Proposition 4.2. Algorithm 2 with the private SGD update mechanism ψ SGD t and estimator ϕ SGD t is ε-LDP. Regret Analysis Assumption 5 (Diversity condition). Let K opt and K sub be a partition of [K] such that for any i ∈ K sub , µ(X T θ i ) < max j =i µ(X T θ j ) − h sub for some h sub > 0 and every X ∈ X . For any i ∈ K opt define the set U i := {X : µ(X T θ i ) > max j =i µ(X T θ j )}. There exists κ l > 0, p ′ > 0 such that for all i ∈ K opt and unit vector v,P((v T X) 2 1{X ∈ U i } ≥ κ l /K opt ) > p ′ . Assumption 6 ((γ, β)-margin condition). This is almost identical to Assumption 4 except that we replace △ t with △ t := µ(X T t θ a * t ) − max j =a * t µ(X T t θ j ). In our algorithm, diversity condition guarantees that conditioning on the arm i is pulled, the distribution of X t still can provide enough information about θ i . We would remark here that we need no longer any deterministic gap in the definition of U i , which weakens the assumption made in Bastani and Bayati (2020), Bastani et al. (2017). Now we are in the suited position to present our theoretical guarantee of the algorithm. Theorem 4.1. Under Assumptions 1, 5 and 6, with the choice ofc = 2σ ε/2,δ/2 (4 √ d+2 log(2T K/α)) in (3.4), s 0 = C · K( C B σ ǫ + σ ε,δ min{λ 0 , h}p ′ κ l ) 2 (d + log(T K/α)) and h = h sub , λ 0 = (2γLC B ) −1 ( p ′ 2 ) 1/β , Algorithm 2 with OLS mechanism ψ OLS t and estimator ϕ OLS t achieve the following regret with probability at least 1 − α for some constant C, Reg(T ) ≤ γCC B KC B (C B σ ǫ + σ ε,δ ) d + log((T K)/α) κ l p ′ 1+β + o h sub ,β,γ (1) ·      log T, β = 1, T 1−β 2 1 − β , 0 < β < 1. Under Assumptions 1, 5 and 6, with the choice of step-size η t := (1{t ≤ Ks 0 }((t mod K) + 1) + 1{t > Ks 0 }(t − (K − 1)s 0 )) −1 K −1 opt ζκ l p ′ c ′ for any c ′ ≥ 1 and h = h sub , Algorithm 2 with SGD mechanism ψ SGD t and estimator ϕ SGD t achieve the following regret with probability at least 1 − α for some constant C, Reg(T ) ≤ C · γLC B Kr ε,d LC B log((T K log T )/α) ζκ l p ′ 1+β + o h sub ,β,γ (1) ·      log T, β = 1, T 1−β 2 1 − β , 0 < β < 1. Experiment To the best of our knowledge, the contextual bandit algorithms with LDP guarantee has only been studied by Zheng et al. (2020), who propose a variant of LinUCB algorithm for linear bandits and a variant of Generalized Linear Online-to-confidence-set Conversion (GLOC) framework Jun et al. (2017) for generalized linear bandits. We refer their methods as LDP-UCB and LDP-GLOC. We call our method LDP-OLS if we plug in the OLS mechanism and estimator into Algorithms 1 and 2, and LDP-SGD if we plug in the SGD ones. We evaluate all the four methods on two different privacy levels ε = 1, 5 in synthetic datasets, which are industry standards. For example, Apple uses ε = 4 in their projects on Emojis and Safari usage Team (2017). Similar choices of the privacy parameter ε can be found in Bassily et al. (2017); Erlingsson et al. (2014). We also demonstrate the efficacy of our algorithms with real data on auto lending in Appendix F. For the sake of comparison, the learning step parameter for LDP-GLOC and LDP-SGD are tuned in the same way. 1 . The first and second columns in Figure F.1 are for single-param and multi-param settings, respectively, which are simulation studies on linear bandits. The context is generated from Unif(S d−1 1 ) at each round. In conclusion, our methods significantly outperform existing ones in all settings consistently. In particular, LDP-SGD achieves better performance under more strigent privacy requirements. Conclusion In this paper, we propose LDP contextual bandit frameworks in both single-parameter and multi-parameter settings with flexibility to deal generalized linear reward structure, and establish theorectical guarrentee of our algorithms based on the frameworks. Our algorithms are highly efficient and have superior empirical performance. There are still some open questions to be explored. Whether our regret bounds are optimal in terms of ε in the multi-parameter setting is still unknown. It will be interesting to explore estimators and mechanisms beyond the private OLS and SGD ones to study the optimality in terms of ε. Moreover, whether there is a fundamental limit in adversarial contextual bandit under LDP constraints is still an open question. It also remains an open question to analyze the regret bound in the multi-parameter setting when β = 0 in the margin condition. Smith, A. (2011). Privacy-preserving statistical estimation with optimal convergence rates. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pp. 813-822. Tang, J., A. Korolova, X. Bai, X. Wang, and X. Wang (2017). Privacy loss in apple's implementation of differential privacy on macos 10.12. arXiv preprint arXiv:1709.02753 . A Randomness Condition In this section, we show that a sub-gaussian random vector with bounded density satisfies Assumption 3: We say a random vector x is σ 2 -sub-gaussian vector with bounded density, if for every v ∈ S d−1 1 , v T x is σ 2 -sub-gaussian and its density function exists and is bounded by γ for some γ > 0. For such kind of random vector, Ren and Zhou (2020) Proof of Proposition 3.2. Since we assume that the features and rewards are bounded, x t,a ≤ C B , r t ≤ c r for all t ∈ [T ] and a ∈ [K], we have (µ(x T t,atθt−1 ) − r t )x t,at bounded by 2c r C B . Lemma 2.2 implies that ψ SGD t is ε-LDP. B.2 Proof of Results in Section 4.1 Proof of Proposition 4.1. We simply denote ψ OLS t by ψ t in this proof. At time t, for any two x = x ′ , without loss of generality assuming the action corresponding x and x ′ are a t = 1 and a t = 2, then the output corresponding x, x ′ is given by (ψ t (x, x T θ 1 + ǫ t ), ψ t (0, 0), . . . , ψ t (0, 0)) and (ψ t (0, 0), ψ t (x ′ , x ′T θ 2 + ǫ t ), . . . , ψ t (0, 0)). Since ψ t (0, 0) has the same distribution, we have for any subset A 1 × A 2 × · · · × A K ⊂ R Kd with A i a Borel set in R d , P(ψ t (x, x T θ 1 + ǫ t ) ∈ A 1 , ψ t (0, 0) ∈ A 2 , . . . , ψ t (0, 0) ∈ A K ) P(ψ t (0, 0) ∈ A 1 , ψ t (x ′ , x ′T θ 2 + ǫ t ) ∈ A 2 , . . . , ψ t (0, 0) ∈ A K ) = P(ψ t (x, x T θ 1 + ǫ t ) ∈ A 1 , ψ t (0, 0) ∈ A 2 ) P(ψ t (x ′ , x ′T θ 2 + ǫ t ) ∈ A 2 , ψ t (0, 0) ∈ A 1 ) . (B.1) Setψ(v 1 , v 2 ) := (ψ t (v 1 ), ψ t (v 2 )), and (v 1 , v 2 ) := (x, 0), (v ′ 1 , v ′ 2 ) := (0, x ′ ), then we have (B.1) equals toψ(v 1 , v 2 )/ψ(v ′ 1 , v ′ 2 ) , thus applying Lemma 2.4 to it implies that (B.1) is upper bounded by e ε + δP(ψ t (x ′ , x ′T θ 2 + ǫ t ) ∈ A 2 , ψ t (0, 0) ∈ A 1 ) −1 , leading to the desired result. Proof of Proposition 4.2. That is nearly the same as the proof of Proposition 4.1, but replacing e ε + δP(ψ t (x ′ , x ′T θ 2 + ǫ t ) ∈ A 2 , ψ t (0, 0) ∈ A 1 ) −1 by e ε in the last step. C Proof of Results in Section 3.2 In the following analysis, without special explaination, all the c and C denote absolute constants. Sometimes we state the inequality of type A 1 ≤ C log(A 2 /α)A 3 holds with probability at least 1 − α while in proof we derive the results hold with 1 − cα for some constant c. In fact, they are equivalent by re-scaling α and changing C to some larger constant. C.1 Proof of Worst-Case Bounds Proof of Theorem 3.1. Since x t,at is the greedy selection, we have x T t,atθt−1 ≥ x T t,aθt−1 for any time t ∈ [T ] and a ∈ [K]. Consequently we have the following upper bound for the instantaneous regret at time t, max a∈[K] (x t,a − x t,at ) T θ ⋆ ≤ max a∈[K] (x t,a − x t,at ) T θ ⋆ −θ t−1 ≤ max a,a ′ ∈[K] x t,a − x t,a ′ T θ ⋆ −θ t−1 ≤ 2 max a∈[K] x T t,a θ ⋆ −θ t−1 . For any fixed a ∈ [K], x t,a is independent ofθ t−1 . By Assumption 3, conditioning on the historical information up to time t, x T t,a (θ ⋆ −θ t−1 ) is a κu d θ ⋆ −θ t−1 2 -sub-gaussian random variable. Now by the maximal concentration inequality for a sub-gaussian sequence, we have with probability at least 1 − α T , max a∈[K] \|x T t,a (θ ⋆ −θ t−1 )\| = O κ u log(KT /α) d θ ⋆ −θ t−1 . To control the regret bound, we bound the estimation error θ ⋆ −θ t−1 in each time in the following lemma. Lemma C.1 (Estimation Error for OLS). Using the private OLS update mechanism ψ OLS t and estimator ϕ OLS t , for any 8 d log 9+log(T /α) p 2 * < t ≤ T , we have with probability at least 1 − α T , θ t − θ ⋆ 2 ≤ C(C B σ ǫ σ ε,δ d) 2 d + log(T /α) κ 2 l p 2 * t , (C.1) for some C independent of d, K and T. Lemma C.2 (Estimation Error for SGD). Using the private OLS update mechanism ψ SGD t and estimator ϕ SGD t , for any 3 ≤ t ≤ T , we have with probability at least 1 − α T , θ t − θ ⋆ 2 ≤ (624 log(log T /α) + 1)r 2 ε,d d 2 4κ 2 l ζ 2 p 2 * t . (C.2) Plugging OLS estimation error (C.1) into the regret bound, denote t 1 := 8 d log 9+log(T /α) p 2 * , the following holds with probability at least 1 − α, T t=1 max a∈[K] (x t,a − x t,at ) T θ ⋆ ≤t 1 c r + T t=t 1 +1 CC B σ ǫ σ ε,δ d κ u log(KT /α) d d + log(T /α) κ l p * √ t (C.3) ≤8 d log 9 + log(T /α) p 2 * + CC B σ ε,δ σ ǫ √ d d + log(T /α) κ l p * κ u log(KT /α) √ T . Plugging the SGD estimation error (C.2) into the regret bound, we have T t=1 max a∈[K] (x t,a − x t,at ) T θ ⋆ ≤2c r + T t=3 κ u log(KT /α) (624 log(log T /α) + 1)r ε,d √ d 2κ l ζp * √ t ≤2c r + (624 log(log T /α) + 1)r ε,d √ d 2ζκ l p * κ u log(KT /α) √ T . (C.4) So now it suffices to prove the Lemmas C.1 and C.2. C.2 Proof of lemma C.1 Lemma C.3. As long as t > 8 d log 9+log(T /α) p 2 * , the following lower bound λ min ( t i=1 x i,a i x T i,a i ) ≥ C · tκ l p * d , holds with probability at least 1 − α T , for some C independent of d and T . Proof. Define F − t as the filtration generated by {x i,a i } i∈[t−1] , {ǫ i } i∈[t−1] and the randomness from {ψ OLS i } i∈ [t−1] . By greedy algorithm, in each time i, x i,a i is selected as a i = argmax a∈ [K] x T i,aθ i−1 . Thus by the Assumption 3, we have for any 0 < s < p * , P( t i=1 (x T i,a i v) 2 < tκ l (p * − s)/d) ≤ P( t i=1 1{(x T i,a i v) 2 > κ l /d} < t(p * − s)) ≤ P( 1 t t i=1 (1{(x T i,a i v) 2 > κ l /d} − E[1{(x T i,a i v) 2 > κ l /d}\|F − i ])) < −s) ≤ exp(− s 2 t 2 ), where in the last inequality we use the Azuma-Hoeffding's inequality for bounded martingaledifference sequence (see Corollary 2.20 in Wainwright (2019)). For every d × d positive-definite matrix A, with an abuse of notation, we denote N ε as the ε-net of S d−1 1 for some ε > 0 to be determined, λ max (A) ≤ 1 1 − 2ε sup x∈Nε x T Ax, which then implies λ min (A) = −λ max (−A) ≥ −1 1 − 2ε sup x∈Nε x T (−A)x = 1 1 − 2ε inf x∈Nε x T Ax. By choosing ε = 1/4, we can find an ε-net N ε with cardinality \|N ε \| ≤ 9 d . Therefore λ min (A) ≥ 2 inf x∈Nε x T Ax. Note that P( min v =1 t i=1 (x T i,a i v) 2 < 2tκ l (p * − s)/d) ≤ P( t i=1 (x T i,a i v) 2 < tκ l (p * − s)/d, ∃v ∈ N ε ) ≤ 9 d exp(− s 2 t 2 ) . By setting s = 2d log 9+2 log(T /α) t , we have when t > 8 d log 9+log(T /α) p 2 * with probability at least 1 − α T , λ min ( t i=1 x i,a i x T i,a i ) = min v =1 t i=1 x i,a i , v 2 ≥ κ l p * t d . Proof of Lemma C.1. By lemma C.3 we know that with probability at least 1 − α T , λ min ( t i=1 x i,a i x T i,a i ) ≥ C 1 κ l p * t/d, for some C 1 independent of d, K and T . Since {W i } i∈ [t] are independent, therefore by concentration bounds for Wigner matrix we have with probability at least 1 − α T , t i=1 W i 2 ≤ C 2 tσ 2 ε,δ (d + log(T /α)), for some C 2 independent of d, K and T. However, it is important to note that the perturbation of privacy noise matrix t i=1 W i may destroy the positive definite property of the Gram matrix t i=1 x i,a i x T i,a i when t is still small. Therefore, we shift t i=1 W i by addingc √ tI d wherec := C 2 σ ε,δ ( √ d + log(T /α)). We denote A t := t i=1 (x i,a i x T i,a i + W i ) +c √ tI. Therefore, by Weyl's inequality we have with probability at least 1 − α T , λ min (A t ) = λ min t i=1 (x i,at i x T i,a i + W i ) +c √ tI d ≥ λ min t i=1 x i,a i x T i,a i ≥ C 1 κ l p * t/d. So now we we study the OLS estimator with x i,a i , ǫ i given above and r i = x T i,a i θ ⋆ + ǫ i . In that case, the estimation error of the OLS estimator under LDP constraints at time t is given bŷ θ t − θ ⋆ = A −1 t t i=1 (x i,a i r i + ξ i ) − θ ⋆ = A −1 t t i=1 (x i,a i x T i,a i θ ⋆ + x i,a i ǫ i + ξ i ) − θ ⋆ = A −1 t ( t i=1 x i,a i ǫ i ) − A −1 t t i=1 W i θ ⋆ + A −1 t t i=1 ξ i −c √ tA −1 t θ ⋆ . Define F t as the filtration generated by {x i,a i } i∈ [t] , {ǫ i } i∈[t−1] and the randomness from {ψ i } i∈ [t−1] . Notice that for every unit vector u, E[exp(λ t i=1 u T x i,a i ǫ i )] = E[E[exp(λ t i=1 u T x i,a i ǫ i )\|F t ]] = E[ t−1 i=1 exp(λu T x i,a i ǫ i )E[exp(λu T X i ǫ i )\|F t ]] (1) ≤ exp( λ 2 C 2 B σ 2 ǫ 2 )E[ t−1 i=1 exp(λu T x i,a i ǫ i )](2) ≤ exp( λ 2 C 2 B σ 2 ǫ t 2 ). Inequality (2) is due to the mathematical induction using the same technique in the equality (1). Thus t i=1 x i,a i ǫ i is σ 2 C 2 B t-sub-gaussian vector, and by the concentration of norm for subgaussian vectors, we have then with probability at least 1 − α T , t i=1 x i,a i ǫ i 2 ≤ C 3 σ 2 ǫ C 2 B t(d + log(T /α)), where C 3 is a positive constant independent of d, K and T . Therefore, A −1 t ( t i=1 x i,a i ǫ i ) 2 ≤ A −1 t 2 ( t i=1 x i,a i ǫ i ) 2 ≤ C 3 σ 2 C 2 B d 2 t(d + log(T /α)) (C 1 κ l p * t) 2 . (C.5) Moreover, A −1 t t i=1 W i θ ⋆ 2 ≤ A −1 t 2 t i=1 W i 2 θ ⋆ 2 ≤ A −1 t 2 t i=1 W i 2 ≤ C 2 tσ 2 ε,δ (d + log(T /α)) (C 1 κ l p * t) 2 , (C.6) where the second inequality is from the assumption that θ ⋆ ≤ 1. Third, Since ξ i are random vector with independent, sub-gaussian coordinates that satisfy Eξ 2 i,j = σ 2 ε,δ , t i=1 ξ i is a random vactor with independent sub-gaussian coordinates that satisfy E t i=1 ξ 2 i,j = tσ 2 ε,δ . Therefore for all t ∈ [T ], with probability at least 1 − α T , t i=1 ξ i 2 ≤ C 4 tσ 2 ε,δ (d + log(T /α)), for some positive constant C 4 independent of d, K and T . Therefore, A −1 t t i=1 ξ i 2 ≤ C 4 tσ 2 ε,δ d 2 (d + log(T /α)) (C 2 κ l p * t) 2 . (C.7) Lastly, c √ tA −1 t θ ⋆ 2 ≤c 2 t (C 2 κ l p * t) 2 , (C.8) holds with probability at least 1 − α T . Plugging all bounds (C.5) (C.6) (C.7) and (C.8) together we get then with probability at least 1 − α T , θ t − θ ⋆ 2 ≤ C 5 σ 2 ǫ C 2 B σ 2 ε,δ d 2 d + log(T /α) κ 2 l p 2 * t , for some positive constant C 5 independent of d, K and T. D Proof of Results in Section 4.2 To lighten the notation, in this section we denote θ i the underlying parameter of arm i. In the following analysis, without special explaination, all the c and C denote absolute constants. Sometimes we state the inequality of type A 1 ≤ C log(A 2 /α)A 3 holds with probability at least we define h 0 = h sub 8LC B ,λ 0 = (2LC B ) −1 ( p ′ 2γ ) 1/β , A t := {sup i∈Kopt θ t,i − θ i ≤ λ 0 }, H 0 := {sup i∈[K] θ Ks 0 ,i − θ i ≤ h 0 }. Lemma D.5. Define F t the filtration generated by {X i } i∈[t] ,{ǫ i } i∈[t] together with all randomness from {ψ i } i∈ [t] . Then we have: λ min (E[X t X t 1{a t = i}\|F t−1 ]) ≥ p ′ κ l 2K 1 A t−1 1 H 0 , ∀i ∈ K opt . Proof. We have for every unit vector v E[v T X t X T t v1{a t = i}\|F t−1 ] ≥1 H 0 κ l K E[1{\|v T X1{X t ∈ U i }\| 2 ≥ κ l /K, a t = i, A t−1 }\|F t−1 ] ≥1 H 0 1 A t−1 κ l K E[1{\|v T X1{X t ∈ U i }\| 2 ≥ κ l /K} − 1{a t = i, X t ∈ U i , A t−1 }\|F t−1 ] ≥1 H 0 1 A t−1 κ l K [p ′ − P({a t = i, X t ∈ U i } ∩ H 0 ∩ A t−1 \|F t−1 )]. P({a t = i, X t ∈ U i } ∩ H 0 ∩ A t−1 \|F t−1 ) = 1 H 0 1 A t−1 P({a t = i, X t ∈ U i } ∩ E t ∩ A t−1 \|F t−1 ) ≤ 1 A t−1 1 H 0 P(△ t < 2LC B λ 0 ) ≤ 1 A t−1 1 H 0 γ(2LC B λ 0 ) β ≤ 1 A t−1 1 H 0 p ′ 2 , where the last inequality is by the choice of λ 0 . Then the proof is finished. D.2 Proof of Lemma D.3 We first establish the lower bound of the sample-covaraince matrix sampled by the greedy action based on the following matrix-martingale concentration result: Lemma D.6 (Theorem 3.1 in Tropp (2011)). Let z 1 , . . . , z t be a sequence of random, positivesemidefinite d×d matrices adapted to a filtration F ′ t , let Z t := t i=1 z i andZ t := t i=1 E[z i \|F ′ i−1 ] . Suppose that λ max (z i ) ≤ R 2 almost surely for all i, then for any µ and α ∈ (0, 1), P[λ min (Z t ) ≤ (1 − α)µ, λ min (Z t ) ≥ µ] ≤ d( 1 e α (1 − α) 1−α ) µ/R 2 . Now we can show the following result: Lemma D.7. For t 1 < t 2 ∈ N such that (t 2 − t 1 ) · κ l p ′ 8K > 10C 2 B log(d/α ′ ), for a fixed i ∈ [K] we have P(λ min ( t 2 t=t 1 X t X t 1{a t = i}) ≤ t 2 − t 1 8K κ l p ′ , sup t 1 ≤t≤t 2 ,i∈Kopt θ t,i − θ i ≤ λ 0 , H 0 ) ≤ α ′ . Now we can claim our first result about the private OLS-estimator in the warm up stage: Lemma D.9. Selecting s 0 as in Lemma D.8 . For the warm up stage with private-OLS-estimator and length Ks 0 , we have for any α > 0, with probability at least 1 − α, sup i∈ [K] θ t,i − θ i ≤ (4C B σ ǫ + σ ε,δ ) t(log( T K α ) + d) s 0 p ′ κ l holds for all Ks 0 ≤ t ≤ T . Proof. Denote U t = t s=1 (1{a s = i}X s X T s + (1{a s = i, s ≤ Ks 0 } + 1{s > Ks 0 })W s ) +c √ tI d , we havê θ t,i = U −1 t ( t s=1 1{a s = i}X s y s + (1{a s = i, s ≤ Ks 0 } + 1{s > Ks 0 })ξ s ) = U −1 t ( t s=1 1{a s = i}[X s X T s θ i + X s ǫ s ] + (1{a s = i, s ≤ Ks 0 } + 1{s > Ks 0 })ξ s ) = θ i + U −1 t ( t s=1 (1{a s = i}X s ǫ s + (1{a s = i, s ≤ Ks 0 } + 1{s > Ks 0 })(ξ s − W s θ i )) −c √ tI d θ i ). By Ks 0 s=1 1{a s = i}W s + t s=Ks 0 +1 W s ≤c √ t, ∀Ks 0 ≤ t ≤ T, i ∈ [K] with probability at least 1 − α, we have with probability at least 1 − 2α, [λ min (U )] −1 ≤ λ min ( Ks 0 s=1 1{a s = i}X s X T s ) −1 ≤ 2 s 0 p ′ κ l , ∀Ks 0 ≤ t ≤ T. On the other hand, we have by the concentration of sub-gaussian random vector, the following bounds hold with probability at least 1 − α/(T 2 K): t s=1 1{a s = i}X s ǫ s ≤ CC B σ ǫ t(d + log(T K/α)), (D.1) t s=1 (1{a s = i, s ≤ Ks 0 } + 1{s > Ks 0 })ξ s ≤ Cσ ε,δ t(d + log(T K/α)), (D.2) Ks 0 s=1 1{a s = i}W s θ i + t s=Ks 0 +1 W s θ i ≤c √ t θ i ≤ Cσ ε,δ t(d + log(T K/α)). (D.3) Gathering all bounds together, we have with probability at least 1 − (2 + 1 T 2 )α, sup i∈ [K] θ t,i − θ i ≤ 2C s 0 p ′ κ l (C B σ ǫ + σ ε,δ ) t(log(T K/α) + d). That finishes the proof. Lemma D.10. As long as s 0 ≥ CK( C B σ ǫ + σ ε,δ min{λ 0 , h 0 }p ′ κ l ) 2 (d + log(T K/α)), and the competitor's rate as the feature vector for each customer. Note that a description of the data set (with descriptive statistics on the demand and available features) is available in Ban and Keskin (2020). The objective is to offer a personalized lending price (from a range of choices) based on personal information such as FICO score to a customer who will either accept or reject it. In contrast to linear bandits, the binary reward is non-linear. Therefore we leave LDP-UCB and LDP-OLS out of considerations. To formulate a bandit environment, first we need to recover the underlying true parameter. Since the lender's decision, i.e., the price for each customer, is not presented in the dataset, we follow Ban and Keskin (2020) (1 + Rate ) −τ − Loan Amount . After imputing the loan prices, to represent customers' binary loan choices, we employ the logit demand model. To be specific, given a price p and a context x ∈ R d , the binary variable apply takes value of 1 with probability exp(v) 1+exp(v) and takes value of 0 with probability 1 1+exp(v) where the linear predictor v = (x, px) T θ ⋆ . We conduct one-hot encoding for categorical features in the dataset and use the python package sklearn Pedregosa et al. (2011) for the estimation of the underlying parameter θ ⋆ . We use the interval [0, 25000] as the feasible region of the prices, which covers the lending prices computed from the dataset, and we discrete the feasible region uniformly into 25 options {p i } i∈ [25] . We use LDP-SGD and LDP-GLOC to sequentially compute the loan prices for the 10 5 with randomly selected customers in the dataset, and compute the company's expected regret based on the population model mentioned above. ; Dwork and Lei (2009); Wasserman and Zhou (2010); Smith (2011); Chaudhuri et al. (2011) and industry adoption Erlingsson et al. (2014); Ding et al. (2017); Lemma 2 . 1 ( 21Gaussian Mechanism Dwork et al. (2006); Dwork and Roth Lemma 2 . 2 ( 22Privacy Mechanism for l 2 -ball Duchi et al. (2018)). For any Figure 2 . 1 : 21User-server communication protocol (2020); Ren et al. (2020); Toulis et al. (2014) and the corresponding family contains a lot of commonly-use model, e.g., linear model, logistic model. Assumption 1. The link function µ is continuously differentiable, Lipschitz and there exists some ζ the non-privacy bound inBastani et al. (2017) under similar condition up to a logarithmic factor. Notice that unlike Theorem 3.3 in the single-parameter case, we cannot establish the regret when β = 0. The reason is that in our analysis, we need the probability of △ t > h vanish as h → 0 to guarantee the estimation error for θ i , i ∈ K opt converges. The corresponding theoretical result in this setting when β = 0 is left as an open question. Figure 5 . 1 : 51We perform 10 replications for each case and plot the mean and 0.5 standard deviation of their regrets. particular, Han et al. (2020) shows that when x follows N (0, Σ), with λ min (Σ) ≥ κ d , we can have κ l = c 1 κ d and p * = c 2 for constants c 1 and c 2 . B Proof of Privacy Guarantee B.1 Proof of Results in Section 3.1Proof of Proposition 3.1. Since we assume that the features and rewards are bounded, x t,a ≤ C B , r t ≤ c r for all t ∈ [T ] and a ∈ [K], by Lemma 2.1, M t is (ε/2, δ/2)-LDP and u t is (ε/2.δ/2)-LDP. Thus Lemma 2.4 implies that ψ OLS t is (ε, δ)-LDP. ;Cheung et al. (2018) and impute it by using the net-present value of futher payment minus the loan amount, i.e., Figure F. 1 : 1We perform 10 replications for each case and plot the mean and 0.5 standard deviation of their regrets. Table 1 . 11: Summary of our main results in (ε, δ)-LDP, whereÕ(·) omits poly-logarithmic factors. Theoretical regret bounds are established in Ding et al. (2021) by combining SGD and Thompson Sampling, while most of the others are limited to empirical studies Bietti et al. (2018); Riquelme et al. Algorithm 1: LDP Single-parameter Contextual BanditInput: Time horizon T ; Privacy Level ε, δ.1 Initialization: Settingθ 0 = 0. 2 for t ← 1 to T do 3 User side: 4 Algorithm 2 : 2LDP Multi-parameter Contextual Bandit Input: Time horizon T ; Warm up period length s 0 ; Privacy Level ε, δ.1 Initialization: Settingθ 0,i = 0, i ∈ [K]. 2 for t ← 1 to Ks 0 do 3 User side: 4 Zheng, K., T. Cai, W. Huang, Z. Li, and L. Wang (2020). Locally Differentially Private (Contextual) Bandits Learning. arXiv (NeurIPS), 1-20.Team, D. P. (2017). Learning with privacy at scale. Toulis, P., E. Airoldi, and J. Rennie (2014). Statistical analysis of stochastic gradient methods for generalized linear models. In International Conference on Machine Learning, pp. 667-675. PMLR. Tropp, J. A. (2011). User-friendly tail bounds for matrix martingales. Tsybakov, A. B. (2008). Introduction to nonparametric estimation. Springer Science & Business Media. Wainwright, M. J. (2019). High-dimensional statistics: A non-asymptotic viewpoint, Volume 48. Cambridge University Press. Wasserman, L. and S. Zhou (2010). A statistical framework for differential privacy. Journal of the American Statistical Association 105 (489), 375-389. Then Theorem 4.1 follows from combining Lemma D.2, D.3 and D.4 . The proof of Lemma D.3 and Lemma D.4 needs the following result: For a fixed β ∈ (0, 1], The source code to reproduce all the results is available at the GitHub repo liangzp/LDP-Bandit. − α while in proof we derive the results hold with 1 − cα for some constant c. In fact, they are equivalent by re-scaling α and changing C to some larger constant. Proof of Lemma D.3. Lemma D.3 is implied directly by Lemma D.11 and Lemma D.10. Proof of Lemma C.2. Denote g t as the gradient at time t,ĝ t := Ψ ε [(µ(x T t,atθ t ) − r t )x t,at ] is the LDP private estimator of g t andẑ t = g t −ĝ t . By the unbiasedness of Ψ ε in Lemma 2.2 we havewhere the last inequality is from Lemma C.3 and Markov's inequality λ min (E xa t [x at x T at \|F t−1 ]) ≥ κ l p * /d. Moreover, notice that ĝ t = r ε,δ . Let λ := 2κ l ζp * /d and η t = 1 λt ,It follows from the same proof as in Proposition 1 inRakhlin et al. (2011), we can obtain for any 0 < α ≤ 1 eT , T ≥ 4 and for all 3 ≤ t ≤ T , with probability at least 1 − α, θ t − θ ⋆ 2 ≤ (624 log(log(T )/α) + 1)r 2 ε,d d 2 4κ 2 l ζ 2 p 2 * t .C.3 Proof of Problem-dependent BoundTo prove the problem-dependent bound, we need only combine Lemma C.1 and Lemma C.2 together with the following lemma.Lemma C.4. Under the (β, γ)-margin condition, if we have θ t − θ ⋆ ≤ U 0 √ t holds uniformly for all t 0 ≤ t ≤ T 0 for some t 0 and U 0 with probability at least 1 − α, we have then with probability at least 1 − 2α,Proof. We have, with probability at least 1 − α,by Hoeffding's inequality we have with probability at leastfor 0 ≤ β < 1. Then the claim holds.D.1 Proof of Theorem 4.1Lemma D.1. If after the warm up stage of length Ks 0 , the estimatorθ Ks 0 ,i achieves the following error bound with probability at least 1 − α,Proof. Firstly, to show a * t ∈K t , without loss of generality we assume that a * t = 1, and argmax i∈[K]µ(X T tθKs0,i ) = 1. Then by the optimality of θ a * t , condition on sup i∈Now for any j ∈ K sub , we have condition on sup i∈[K]θ Ks 0 ,i − θ i ≤ h 0 ,where the final equation is due to the sub-optimality gap assumed in Assumption 5.Proof of Theorem 4.1. We first show the following lemma, which converts the regret bound under margin condition to the estimation error bound:Lemma D.2. Under the (β, γ)-margin condition, given h 0 defined in Lemma D.1, suppose there exists some s 0 such that with a warm up stage of length Ks 0 , sup i∈[K] θ t,i − θ i ≤ h 0 , and there exists some t 0 , U 0 (α) such that with probability at least 1 − α,Then, we have with probability at least 1 − 2α, for some constant C,when 0 < β < 1. This completes the proof.Given Lemma D.2, we need only show that for both the private OLS estimator and the private SGD estimator, we can find the corresponding s 0 , t 0 and U 0 (α).satisfy the requirements in Lemma D.2.satisfy the requirements in Lemma D.2.Proof. Denote S t 1 ,t 2 := ∩ t 1 ≤t≤t 2 A t , by Lemma D.5 we haveThat implies.That leads to the claim.In warm up stage, we have the following lemma.Lemma D.8. As long as s 0 ≥ C(κ l p ′ ) −2 max{log 1 α , d} for some absolute constant C, we have with probability at least 1 − α,Proof. Since X t are i.i.d. for (i − 1)s 0 + 1 ≤ t ≤ is 0 , using classical concentration results for i.i.d. sub-gaussian covariance matrix result (e.g. Theorem 6.5 in Wainwright(2019)), we have when s 0 > C(κ l p ′ ) −2 max{log 1 α , d}, with probability at least 1 − α,On the other hand, we have by Markov's inequality. Thus we have with probability at least 1 − α,we have with probability at least 1 − α,with probability at least 1 − t ′ j=1 2 j 2 α, thenApplying the inequalities (D.1),(D.2) (D.3), we haveBy the selection of s 0 , we haveŨ s (α) ≤ λ 0 fort 0 ≤ s ≤ t + 1, and as a result, P(H 0 , St 0 ,t+1 ) = P(H 0 , St 0,t ,Ã t+1 ). Thus the claim holds.D.3 Proof of Lemma D.4Proof. For the estimatorθ Ks 0 ,i at the end of warm up stage, since the action is independent of the contexts, everyθ Ks 0 ,i can be seen as an output of performing private gradient descent over s 0 i.i.d. samples. Without loss of generality, we perform the analysis for the parameter of the first armθ Ks 0 ,1 (notice that by the sampling strategy in the warm up stage, we haveθ Ks 0 ,1 =θ s 0 ,1 ). The result for otherθ Ks 0 ,i can be established using the same argument. For 2 ≤ t ≤ s 0 ,We getNotice ĝ t 2 2 is upper bounded by r 2 ε,d . Now using the same argument as in the proof of Proposition 1 ofRakhlin et al. (2011)leads to the following result:Lemma D.12. If we pick η t = 1/(ζκ l p ′ t) in the warm up stage, then with probability at least 1 − α,Notice that in our algorithm, when t > Ks 0 , for any i ∈ K opt , the private gradient descent formula is given byθ. Again without loss of generality we assume that 1 ∈ K opt , and we provide the analysis for i = 1, the argument is same for other i ∈ K opt :If we denote S t := ∩ t s=Ks 0 A s , then using the above inequality recursively until t = Ks 0 + 1(i.e. until t ′ = s 0 + 1) , we haveThen it follows from the same proof as in Proposition 1 inRakhlin et al. (2011)that for any fixed Ks 0 < t ≤ T , we have with probability at least 1 − α/T ,, so that the second term in (D.8) is less or equal to λ 0 /2, we have P(S Ks 0 +1 , H 0 ) ≥ 1 − 2α by (D.7). And by calling (D.8) recursivelyThe above inequality is because the termwhich can be absorbed into the constant C.E Proof of Theorem 3.2In this section, we would give a proof on the Theorem 3.2 by combining the argument in Han et al.(2020) and the divergence contraction inequality inDuchi et al. (2018).Proof of Theorem 3.2. Consider the two-arm stochastic contextual bandit environment: for each d-dimensional context i = 1 or 2, x t,i ∼ N (0, 1 d I d ) independently. If choosing action a t at time t, the reward y t is generated via y t = x T t,at θ + ǫ t with ǫ t ∼ i.i.d. N (0, 1). Given any fixed ε-LDP bandit algorithm π with ε ≤ 1, we denote its decision at t-th step by a t , by definition a t can be seen as a function of current contextual x t,1 , x t,2 and all history outputs (x 1,a 1 , y 1 , x 2,a 2 , y 2 , . . . , x t−1,a t−1 , y t−1 ). Since the algorithm is under the ε-LDP constraint, each a t can only access S t := (M 1 (x 1,a 1 , y 1 ), M 2 (x 2,a 2 , y 2 ), . . . , M t−1 (x t−1,a t−1 , y t−1 )) with M 1 , . . . , M t−1 a sequence of ε-LDP mechanisms. We denote the distribution of S t by Q t θ , and we havewhere (x) + denote max{x, 0} and Q 0 denote the uniform distribution over △S d−1 1 with △ > 0 some positive number to be determined, we define Q 1 , Q 2 as dQ 1 dQ 0 := ((x t,1 − x t,2 ) T θ) + Z 0 , dQ 2 dQ 0 := ((x t,2 − x t,1 ) T θ) + Z 0 ,By the argument of in Han et al. (2020), we have (F.1) is lower bounded byx t,at x T t,at )u t ).Now taking expectation over x t,1 , x t,2 , and using the convexity of function f (x) = exp(−x) we getSelecting △ ≍ d (e ε − 1) √ t and taking summation over 1 ≤ t ≤ T leads to Ω( √ T d/(e ε − 1)) lower bound, finally noticing e ε − 1 ≤ Cε for ε ≤ 1 leads to the desired lower bound when ε ≤ 1.F Auto Loan Experiment DetailsWe use On-Line Auto Lending dataset CRPM-12-001 in our real data case study 2 . We use the same features selection as in Ban and Keskin(2020);Cheung et al. (2018)in the dataset and select FICO score, the term of contract, the loan amount approved, prime rate, the type of car, 2 On-Line Auto Lending dataset CRPM-12-001 provided by Columbia University https://www8.gsb.columbia.edu/cprm/research/datasets.	[]
[ "Prepared for submission to JHEP", "Prepared for submission to JHEP" ]	[ "Thiago R Araujo [email protected] \nInstituto de Física Teórica\nUNESP-Universidade Estadual Paulista R. Dr\nBento T. Ferraz 271, Bl. II01140-070Sao PauloSPBrazil\n", "Horatiu Nastase [email protected] \nInstituto de Física Teórica\nUNESP-Universidade Estadual Paulista R. Dr\nBento T. Ferraz 271, Bl. II01140-070Sao PauloSPBrazil\n" ]	[ "Instituto de Física Teórica\nUNESP-Universidade Estadual Paulista R. Dr\nBento T. Ferraz 271, Bl. II01140-070Sao PauloSPBrazil", "Instituto de Física Teórica\nUNESP-Universidade Estadual Paulista R. Dr\nBento T. Ferraz 271, Bl. II01140-070Sao PauloSPBrazil" ]	[]	We consider the action of a non-abelian T-duality on backgrounds with an AdS 5 factor in type IIA supergravity, finding a new type IIB background, as well as the non-abelian T-dual of a domain wall solution which has as limits the non-abelian T-dual of AdS 5 × T 1,1 and the non-abelian T-dual of AdS 3 × R 2 × S 2 × S 3 . We explore some consequences of non-abelian T-duality for the dual conformal field theories.	10.1103/physrevd.91.126015	[ "https://arxiv.org/pdf/1503.00553v3.pdf" ]	118,856,709	1503.00553	70d1b075021370d2a363dc44bac28776482d21cf	Prepared for submission to JHEP 23 Apr 2015 Thiago R Araujo [email protected] Instituto de Física Teórica UNESP-Universidade Estadual Paulista R. Dr Bento T. Ferraz 271, Bl. II01140-070Sao PauloSPBrazil Horatiu Nastase [email protected] Instituto de Física Teórica UNESP-Universidade Estadual Paulista R. Dr Bento T. Ferraz 271, Bl. II01140-070Sao PauloSPBrazil Prepared for submission to JHEP 23 Apr 2015arXiv:1503.00553v3 [hep-th] ON N = 1 SUSY BACKGROUNDS WITH AdS FACTOR FROM NONABELIAN T-DUALITYT-dualityAdS/CFT correspondencenon-abelian We consider the action of a non-abelian T-duality on backgrounds with an AdS 5 factor in type IIA supergravity, finding a new type IIB background, as well as the non-abelian T-dual of a domain wall solution which has as limits the non-abelian T-dual of AdS 5 × T 1,1 and the non-abelian T-dual of AdS 3 × R 2 × S 2 × S 3 . We explore some consequences of non-abelian T-duality for the dual conformal field theories. Introduction Since the begining of string theory, the notion of duality symmetries has played an important role. In the early days of string theory, when it was a model for strong interactions, the observation that the amplitude of a scattering process could be written equally well in terms of the s-or t-channel Mandelstam variables led to the name of "dual models" [1]. Nowadays, many different dualities exist in string theory, for instance Gauge/Gravity duality [2,3], S-duality, T-duality [4], Mirror Symmetry [5,6], Langlands duality [7,8]. In this paper we are interested in the non-abelian generalization of T-duality (started by the paper [9]), which is the case where the isometry group of the background is non-abelian. Differently than its abelian cousin, the non-abelian T-duality has been poorly understood, and just recently the action of the transformation on the RR fields was found [10,11]. The general procedure for T-duality follows the original idea of Buscher [12], that is, we start with a σ-model which supports an isometry such as U (N ). Then we gauge the isometry, but we need to impose a constraint by means of Lagrange multipliers which guarantees that the connection field strength remains equal to zero. This constraint enforces the condition that after gauging the isometry, the initial degrees of freedom remain unchanged. The duality works as follows. On one hand, by solving the equation of motion for the Lagrange multipliers and replacing the solution into the action, we recover the original model. If instead we solve the equation of motion for the connection and we gauge fix, we find the dual σ-model. Non-abelian T-duality can be used as a solution generating technique, that is, starting from a solution of supergravity, we can find another solution by a simple set of transformations rules. These solutions can be understood better through another type of duality, the gauge/gravity correspondence. Particularly interesting solutions in light of this fact are those with AdS 5 factors. In fact, one of the first examples of the application of a non-abelian T-duality transformation in a background supporting a non-trivial RR field was in the Klebanov-Witten solution which consists of a space of the form AdS 5 × T 1,1 , where T 1,1 is the homogenous space (SU (2) × SU (2))/U (1). Recently, [13] reported a large class of new solutions with AdS 5 factors and made the analysis of the field theory 1 , following [17], which performed a non-abelian T-duality in a type IIB solution of the type AdS 5 × X 5 , where this solution was obtained in [18] after a dimensional reduction of D = 11 supergravity, followed by an abelian T-duality. The study of non-abelian T-duality of AdS backgrounds was initiated in [11,19] In this article we explore the non-abelian T-duality on the type IIA supergravity solution (that is, before the abelian T-duality which gives AdS 5 ×X 5 ) of the form AdS 5 × w M 5 , where the internal manifold is obtained after a dimensional reduction of a space that consists of a 2-sphere bundle over S 2 × T 2 [18]. Another application considered relates to the background found in [20]. It consists of a domain wall with non trivial fluxes in the NS-NS and RR sectors. This domain wall solution flows to the background AdS 3 × R 2 × S 2 × S 3 in the IR limit, and in the UV to AdS 5 × T 1,1 . We study the T-dual of this domain wall and see that it has as limits the T-dual of AdS 5 × T 1,1 and AdS 3 × R 2 × S 2 × S 3 . We then study the implication of non-abelian T-duality for the dual conformal field theories, through a calculation of central charges. The paper is organized as follows. In section 2 we review non-abelian T-duality and in section 3 we apply it to the warped AdS 5 solution. In section 4 we consider the T-dual of the domain wall solution. In section 5 we consider dual conformal field theory aspects of the T-dual solution and calculate central charges, and in section 6 we conclude. Non-Abelian T-Duality in a nutshell Since the present work deals with the uses of the Non-Abelian T-duality as a solution generating technique, we start with a review of this procedure, following mostly [11]. Consider a background that supports an SU (2)-structure, so that we write the metric in the form ds 2 = G µν (x)dx µ dx ν + 2G µi (x)dx µ L i + g ij (x)L i L j (2.1) where µ, ν = 1, . . . , 7, and L i are the Maurer-Cartan forms for SU (2). In general we also have non-trivial Kalb-Ramond two-forms B = B µν dx µ ∧ dx ν + B µi dx µ ∧ L i + 1 2 b ij L i ∧ L j ,(2. 2) 1 Nonabelian T-duality on solutions with AdS factors was considered also in [14][15][16]. and a dilaton Φ = Φ(x). The important point here is that all dependence on the SU (2) Euler angles θ, ψ, φ is contained in the one-forms L i . Next, define the vielbeins e A = e A µ dx µ e a = κ a j L j + λ a µ dx µ ,(2.3) with A = 1, . . . , 7 and a = 1, 2, 3. Imposing ds 2 = η AB e A e B + e a e a ,(2.G µν = η AB e A e B + K µν , κ a i κ a j = g ij , κ a i λ a µ = G µi ,(2.5) where we defined λ a µ λ a ν = K µν . If we combine the metric and B field into Q and E by Q µν = G µν + B µν , Q µi = G µi + B µi Q iµ = G iµ + B iµ , E ij = g ij + b ij ,(2.6) one can show that the non-abelian T-dual background is Q µν = Q µν − Q µi M −1 ij Q jν , E ij = M −1 ij Q µi = Q µj M −1 ji , Q iµ = −M −1 ij Q jµ , (2.7) where the matrix M is defined by M ij = E ij + α ′ f k ij v k . (2.8) Here f k ij = √ 2ǫ ijk are the structure constants of the group SU (2) and v i are originally Lagrange multipliers, now dual coordinates. We can make the scaling v i → 1 √ 2 v i , so that the dual fields are written as dŝ 2 = G µν (x)dx µ dx ν + 2 √ 2 G µi (x)dx µ dv i + 1 2ĝ ij (x)dv i dv j (2.9) and B = B µν dx µ ∧ dx ν + 1 √ 2 B µi dx µ ∧ dv i + 1 4b ij dv i ∧ dv j . (2.10) and dilaton (transformed at the quantum level as usual) φ = φ − 1 2 ln det M α ′3 . (2.11) Besides the spectator fields x µ , the dual theory depends on θ, ψ, φ, v i , so we have too many degrees of freedom. We need to impose a gauge fixing in order to remove three of these variables, usually taken to be θ = ψ = φ = 0. Then one finds (M −1 ) ij = 1 det M det gg ij + y i y j − ǫ ijk g kl y l (2.12) where we have defined b ij = ǫ ijk b k and y i = b i + α ′ v i . For a gauge fixing different than θ = ψ = φ = 0, one definesv i = D ji v j , where D ij = 1 2 Tr (τ i gτ j g −1 ), g = e i 2 φτ 3 e i 2 θτ 2 e i 2 ψτ 3 (2.13) (τ i are the Pauli matrices) and replaces everywhere v i byv i . The dualization acts differently on the left-and the right-movers which produces two different sets of framesê i + andê i − that are related by a Lorentz transformationê a + = Λ a bê b − . The action on the spinor representation of the Lorentz group is given by Ω −1 Γ a Ω = Λ a b Γ b . (2.14) Considering the RR sector in the democratic formalism (we consider the fluxes and their Hodges dual as well), we define the polyforms in type II supergravity IIB: P = e φ 2 4 n=0 / F 2n+1 , IIA: P = eφ 2 5 n=0 / F 2n (2.15) Then the non-abelian T-dual forms are obtained by the transformation (applied to the non-abelian case by [17], following work in the abelian case by [21]) P = P · Ω −1 . (2.16) Warped AdS 5 solution Supersymmetric solutions of D = 11 supergravity of the form AdS 5 × w M 6 , with nontrivial four form flux living in the internal Riemann manifold were considered in [18]. The authors found that the six dimensional Riemannian manifold always admits a Killing vector, and that locally, the five-dimensional space orthogonal to the Killing vector is a warped product of a one dimensional space parametrized by the coordinate y and a fourdimensional complex space M 4 . Also, the authors found a large class of regular solutions. One of this solutions, namely M 4 = S 2 × T 2 is peculiar. Firstly we can reduce on an S 1 direction in the torus T 2 so that we can obtain a regular solution of type IIA solution of the form, AdS 5 × X ′ 5 . Moreover, after a T-duality on the other S 1 we get a type IIB solution of the form AdS 5 × X 5 , where X 5 is a family of Sasaki-Einstein manifolds, and the global aspects of these spaces was studied in [22,23]. The type IIA solution of [18] is of the form 1 R 2 ds 2 = ds 2 (AdS 5 ) + α 1 (y)dy 2 + α 2 (y)dx 2 Gµν (x)dx µ dx ν + β 1 (y)(L 2 1 + L 2 2 ) + β 2 (y)L 2 3 g ij (x)L i L j , (3.1a) 1 R 2 B = γ(y)dx ∧ L 3 B µi dx µ ∧L i (3.1b) φ = φ(y) (3.1c) 1 R 3 F (RR) 4 = η(y)dy ∧ V ol(S 2 ) ∧ L 3 (3.1d) where L i = σ i / √ 2, with i = 1, 2, 3 are the Maurer-Cartan forms of the group SU (2), satisfying dL i = − 1 2 √ 2ǫ ijk L j ∧ L k , (3.2) with the left invariant forms σ 1 = cos ψdθ + sin ψ sin θdφ σ 2 = − sin ψdθ + cos ψ sin θdφ (3.3) σ 3 = dψ + cos θdφ. The coefficients of this solution are given by α 1 (y) = e −6λ sec 2 ζ, α 2 (y) = e −6λ , β 1 (y) = 1 − cy 3 , β 2 (y) = 2 cos 2 ζ 9 , γ(y) = − √ 2(ca + cy 2 − 2y) 6(a − y 2 ) and η(y) = − 2 √ 2(1 − cy) 9 = − 2 √ 2 3 β 1 ,(3.4) so that the metric is ds 2 = R 2 ds 2 (AdS 5 ) + R 2 e −6λ sec 2 ζdy 2 + R 2 e −6λ dx 2 + R 2 1 − cy 6 (dθ 2 + sin 2 θdφ 2 ) + R 2 9 cos 2 ζ(dψ + cos θdφ) 2 , (3.5a) where x parametrizes the circle S 1 of length 2πα ′ /(lR 2 ), with 2 l = q 3q 2 − 2p 2 + p 4p 2 − 3q 2 , (3.5b) (θ, φ) are the polar and azimuthal angles in S 2 , y ∈ (y 1 , y 2 ) and 0 ≤ ψ ≤ 2π (note that in our conventions, x and y are dimensionless, i.e. are written in units of R). The angle ζ is defined by sin ζ = 2ye −3λ and e 6λ = 2(a − y 2 )/(1 − cy) and a, c are constants such that, if c = 0 then 0 < a < 1, and if c = 0 then a = 0, and if c = 0 one can set it to 1 and find a = 1 2 + 3q 2 − p 2 4p 3 4p 2 − 3q 2 , (3.5c) where p, q ∈ Z. The dilaton is φ = −3λ (3.5d) and the Kalb-Ramond field is B = R 2 (ca + cy 2 − 2y) 6(a − y 2 ) (dψ + cos θdφ) ∧ dx. (3.5e) In the RR sector, we have only a nonzero four-form field F 4 = −R 3 2(1 − cy) 9 dy ∧ (dψ + cos θdφ) ∧ V ol(S 2 ). (3.6) In what follows, it is convenient to use the frame fields i a = e â a dxâ AdS 5 directions e x = Rα e 1 = Rβ 1/2 1 L 1 , e 2 = Rβ 1/2 1 L 2 , e 3 = Rβ 1/2 2 L 3 , so that we have the matrix κ a j given by κ =    Rβ 1/2 1 0 0 0 Rβ 1/2 1 0 0 0 Rβ 1/2 2    . (3.8) Nonabelian T-dual model We want to T-dualize [11] (see also [26] for the complete list of dual transformations) with respect to the SU (2). As in section 2, we form the matrix M ij , given by M ij = g ij + b ij + α ′ ǫ ijkvk , so (b ij = 0, g ij = κ a i κ a j ), M =    R 2 β 1 αv 3 −α ′v 2 −α ′v 3 R 2 β 1 α ′v 1 α ′v 2 −α ′v 1 R 2 β 2    . (3.9a) We pick a gauge where θ = φ = v 2 = 0, so thatv = (cos ψv 1 , sin ψv 1 , v 3 ). This gauge is useful when the vector ∂ ψ is a Killing vector as the present case (see [11], for further possible choices). Therefore, the matrix M in this gauge is M =    R 2 β 1 α ′ v 3 −α ′ sin ψv 1 −α ′ v 3 R 2 β 1 α ′ cos ψv 1 α ′ sin ψv 1 −α ′ cos ψv 1 R 2 β 2    . (3.9b) The dilaton in the dual theory is given by φ = φ − 1 2 ln ∆ α ′3 , (3.10) where ∆ ≡ det M = R 2 [(R 4 β 2 1 + α ′2 v 2 3 )β 2 + α ′2 v 2 1 β 1 ]. To simplify the notation, from now on we absorb R 2 in β 1 , β 2 , α ′ in v 1 , v 3 , as well as R 2 in α 1 , α 2 , γ. The inverse of the matrix M is then (M −1 ) T = 1 ∆    β 1 β 2 + v 2 1 cos 2 ψ v 3 β 2 + v 2 1 cos ψ sin ψ v 1 v 3 cos ψ − v 1 β 1 sin ψ −v 3 β 2 + v 2 1 cos ψ sin ψ β 1 β 2 + v 2 1 sin 2 ψ v 1 β 1 cos ψ + v 1 v 3 sin ψ v 1 v 3 cos ψ + v 1 β 1 sin ψ −v 1 β 1 cos ψ + v 1 v 3 sin ψ v 2 3 + β 2 1    . (3.11) Finally, taking the symmetric and skew-symmetric part of (2.7), we get the following T-dual fields G µν = G µν − 1 2 Q µi M −1 ij Q jν + Q νi M −1 ij Q jµ G µi = 1 2 Q µj M −1 ji − Q jµ M −1 ij ĝ ij = 1 2 M −1 ij + M −1 ji B µν = B µν − 1 2 Q µi M −1 ij Q jν − Q νi M −1 ij Q jµ B µi = 1 2 Q µj M −1 ji + Q jµ M −1 ij b ij = 1 2 M −1 ij − M −1 ji For the solution (3.1a -3.1d), where x µ = {x, y, AdS 5 coordinates} and i = 1, 2, 3, we consider just the terms which will be affected by the non-abelian T-duality, namely, Q xx , Q xi and Q ij , giving Q xx = G xx = α 2 (y) Q x3 = B x3 = γ(y) Q 11 = Q 22 = g 11 = β 1 (y) Q 33 = g 33 = β 2 (y) For the metric, we obtain G µν = G µν , G µi = 0 ∀ µ, ν = x. Moreover, we have the diagonal component G xx = α 2 (y) + 1 ∆ (v 2 3 + β 2 1 )γ 2 , (3.12) the crossed terms G x1 = 1 ∆ γv 1 v 3 cos ψ G x2 = 1 ∆ γv 1 v 3 sin ψ (3.13) G x3 = 1 ∆ γ(v 2 3 + β 2 1 ) , and the g ij componentŝ g 11 = 1 ∆ (β 1 β 2 + v 2 1 cos 2 ψ),ĝ 12 = 1 ∆ v 2 1 cos ψ sin ψ,ĝ 13 = 1 ∆ v 1 v 3 cos ψ g 21 = 1 ∆ v 2 1 cos ψ sin ψ,ĝ 22 = 1 ∆ (β 1 β 2 + v 2 1 sin 2 ψ),ĝ 23 = 1 ∆ v 1 v 3 sin ψ (3.14) g 31 = 1 ∆ v 1 v 3 cos ψ,ĝ 32 = 1 ∆ v 1 v 3 sin ψ,ĝ 33 = 1 ∆ (v 2 3 + β 2 1 ). All in all, we have the type IIB metric dŝ 2 = ds 2 + 1 ∆ dΣ 2 ,(3.15) where ds 2 = ds 2 AdS + α 1 (y)dy 2 + α 2 (y)dx 2 (3.16a) and dΣ 2 = γ 2 (v 2 3 + β 2 1 )dx 2 + 2γ √ 2 dx v 1 v 3 (cos ψdv 1 + sin ψdv 2 ) + (v 2 3 + β 2 1 )dv 3 + 1 2 (β 1 β 2 + v 2 1 cos 2 ψ)dv 2 1 + (β 1 β 2 + v 2 1 sin 2 ψ)dv 2 2 + 2v 2 1 cos ψ sin ψdv 1 dv 2 (3.16b) + 2v 1 v 3 cos ψdv 1 dv 3 + 2v 1 v 3 sin ψdv 2 dv 3 + (v 2 3 + β 2 1 )dv 2 3 . Remembering thatv = (v 1 cos ψ, v 1 sin ψ, v 3 ), we rewrite it as dΣ 2 =γ 2 (v 2 3 + β 2 1 )dx 2 + 2γ √ 2 dx v 1 v 3 dv 1 + (v 2 3 + β 2 1 )dv 3 + 1 2 β 1 β 2 v 2 1 dψ 2 + + 1 2 (β 1 β 2 + v 2 1 )dv 2 1 + v 1 v 3 dv 1 dv 3 + 1 2 (v 2 3 + β 2 1 )dv 2 3 . (3.16c) For later use, we calculate √ det g int for this metric, where g int refers to the internal, non-AdS, part of the metric. Writing explicitly the factors of R and α ′ , we obtain √ g int = 1 ∆ 2 R 3 α ′3 √ α 1 β 1 β 2 v 1 √ 2 detM , (3.17) whereM is the matrix M =     ∆R 2 α 2 + γ 2 R 4 (α ′2 v 2 3 + β 2 1 R 4 ) γ √ 2 R 2 α ′2 v 1 v 3 γ √ 2 R 2 (α ′2 v 2 3 + β 2 1 R 4 ) γ √ 2 R 2 α ′2 v 1 v 3 β 1 β 2 R 4 +α ′2 v 2 1 2 α ′2 v 1 v 3 2 γ √ 2 R 2 (α ′2 v 2 3 + β 2 1 R 4 ) α ′2 v 1 v 3 2 α ′2 v 2 3 +β 2 1 R 4 2     (3.18) and we find detM = α 2 β 1 R 4 4 ∆ 2 ⇒ det g int = R 5 α ′3 ∆ √ α 1 α 2 β 1 β 2 v 1 2 √ 2 . (3.19) Finally, the T-dual Kalb-Ramond field is given by B = γv 1 β 1 √ 2∆ dx ∧ (− sin ψdv 1 + cos ψdv 2 ) + 1 2∆ (−v 3 β 2 dv 1 ∧ dv 2 + v 1 β 1 sin ψdv 1 ∧ dv 3 − v 1 β 1 cos ψdv 2 ∧ dv 3 ) = 1 ∆ v 2 1 β 1 √ 2 γdx + 1 √ 2 dv 3 − 1 2 v 1 v 3 β 2 dv 1 ∧ dψ. (3.20) The T-dual vielbeins are 3 e ′ 1 = − √ β 1 √ 2∆ v 1 v 3 β 2 dψ + (v 2 1 + β 1 β 2 )dv 1 + v 1 v 3 dv 3 − γ √ β 1 ∆ v 1 v 3 dx (3.21a) e ′ 2 = − √ β 1 √ 2∆ (v 1 β 1 β 2 dψ − β 2 v 3 dv 1 + v 1 β 1 dv 3 ) − γ √ β 1 ∆ v 1 β 1 dx (3.21b) e 3 = − √ β 2 √ 2∆ −v 2 1 β 1 dψ + v 1 v 3 dv 1 + (v 2 3 + β 2 1 )dv 3 − γ √ β 2 ∆ (v 2 3 + β 2 1 )dx, (3.21c) where we have defined the rotated vielbeins ê ′ 1 e ′ 2 = cos ψ sin ψ − sin ψ cos ψ ê 1 e 2 . (3.21d) In term of this basis we write the Kalb-Ramond field (3.20) as − B 2 = − v 3 β 1ê ′ 1 ∧ê ′ 2 + v 1 (β 1 β 2 ) 1/2ê 3 ∧ê ′ 2 . (3.22) Using these results,we are able to find the RR forms in this type IIB background. We write the four-form (3.1d) as ( e 1 ∧ e 2 ∧ e 3 = β 1 √ β 2 2 vol(S 2 ) ∧ L 3 , remembering that β i contain R 2 ) F 4 = Ξ 0 dy ∧ e 1 ∧ e 2 ∧ e 3 ≡ G(3)1 ∧ e 1 ∧ e 2 ∧ e 3 ,(3.23) where G 1 = Ξ 0 dy with Ξ 0 = −4 √ 2R/(3β 1/2 2 ) = 4 √ 2/ 3(1 − cy).(3) In this way we have written the RR 4-form in the way suited to apply the nonabelian T-duality as described in the Appendix. Using these rules, we findF 4 =F 2 = 0 and (reintroducing all factors of R and α ′ ) F 1 = −e φ−φ A 0 G (3) 1 = d C 0 = R 3 α ′3/2 4 √ 2 3 β 1 dy (3.24) F 3 = d C 2 − C 0 d B = 1 2 e φ−φ G (3) 1 ∧ ǫ abc A cêa ∧ê b = Ξ 0 1 2 ǫ abc A a dy ∧ê b ∧ê c = R 5 √ α ′ 4 √ 2 3∆ β 1 dy ∧ v 2 1 β 1 √ 2 1 √ 2 dv 3 + R 2 γdx − v 1 v 3 β 2 2 dv 1 ∧ dψ = − 1 α 3/2 4 √ 2 3 1 β 1/2 2 dy ∧ β 1/2 2 v 3 e ′ 1 ∧ e ′ 2 + β 1/2 1 v 1 e ′ 2 ∧ e 3 =B ∧F 1 , (3.25) where the coefficients from the appendix are A a = 1 ∆ 1/2 A a ,(3.26) and A a = κ a iv i = Rα ′ (β 1/2 1 v 1 cos ψ, β 1/2 1 v 1 sin ψ, β 1/2 2 v 3 ) . This background is supplemented by the forms F 9 = ⋆ F 1 and F 7 = − ⋆ F 3 . Using these expressions it is straightforward to verify that the Bianchi identities dF 1 = 0 and dF 3 = H ∧ F 1 are satisfied. Moreover, B ∧ F 3 = 0. For later use, we also compute the Page charges in this geometry. The quantized Page charges in this background are given by 4 Q P age D3 = 1 2κ 2 10 T D3 Σ 5 (F 5 −B ∧F 3 ) = 0 4 Note that 2κ 2 10 = (2π) 7 α ′4 and TDp = (2π) −9 α ′− p+1 2 , so 2κ 2 10 TDp = (2πls) 7−p . Q P age D5 = 1 2κ 2 10 T D5 Σ 3 (F 3 −B ∧F 1 ) = 0 Q P age D7 = 1 2κ 2 10 T D7 y 2 y 1 F 1 = R 3 α ′3/2 4 √ 2 9 (y 2 − y 1 ) 1 − c(y 1 + y 2 ) 2 = N D7 (3.27) where, since after an abelian T-duality along the x-direction on the solution (3.1a-3.1d) we get a the Sasaki-Einstein manifold, we have [22,25] y 1 = 1 4p (2p − 3q − 4p 2 − 3q 2 ) y 2 = 1 4p (2p + 3q − 4p 2 − 3q 2 ) ,(3.28) the solutions to cos 2 ζ = 0, and p, q ∈ N with (p, q) = 1 for p > q. One may verify that this new background has N = 1 supersymmetry, under the criteria of [11]. In fact, in [27] the authors have proved that the vanishing of the Kosmann derivative in the dualized directions of the Killing spinors means supersymmetry is preserved. 5 In the present case, the derivative trivially vanishes, because the Killing spinors are independent of the dualized directions. Moreover, in [27] a proof was given for the formula (2.16), with closed expressions for the dual p-form potentials, that can be applied more easily to specific cases. Note that we could have considered the same calculation with a different gauge fixing for the Lagrange multipliers. Consider that the matrix M is instead M =    β 1 v 3 −v 2 −v 3 β 1 v 1 v 2 −v 1 β 2    ,(3.29) with v = (ρ cos ζ sin χ, ρ cos ζ sin χ, ρ cos χ). In this coordinate system, we have that ∆ = β 2 (β 2 1 + ρ 2 cos 2 χ) + β 2 1 ρ 2 sin 2 χ. The inverse of the matrix M gives equation (3.11), but with the replacements ψ ζ, v 1 ρ sin χ, v 2 ρ cos χ. In a recent paper [20], the authors considered the construction of a supersymmetric domain wall that approaches AdS 5 × T 1,1 in the UV limit, and AdS 3 × R 2 × S 2 × S 3 in the IR limit. In this section we consider the non-abelian T-dual solution of the domain wall ansatz and see that it has as its limit the non-abelian T-dual of the AdS 5 ×T 1,1 and AdS 3 ×R 2 ×S 2 ×S 3 in the UV and IR respectively. In fact, the non-abelian T-dual solution of AdS 5 × T 1,1 is already known from [11]. We therefore start with a short review of this solution. We consider the conventions of [20]. Then the type IIB solution is 1 R 2 ds 2 AdS 5 ×T 1,1 = ds 2 AdS 5 + 1 6 (ds 2 1 + ds 2 2 ) + 1b) and B = 0, φ =constant, where ds 2 i = dθ 2 i + sin 2 θ i dφ 2 i and P = cos θ 1 dφ 1 + cos θ 2 dφ 2 and we make the replacements v 1 2y 1 and v 3 2y 2 . The NS-NS sector of the T-dual background is given by 1 9 (dψ + P ) 2 (4.1a) 1 R 4 F 5 = 4(vol AdS 5 + vol T 1,1 ),(4.dŝ 2 T (AdS 5 ×T 1,1 ) = ds 2 AdS 5 + λ 2 0 ds 2 1 + λ 2 0 λ 2 ∆ y 2 1 σ 2 3 + 1 ∆ (y 2 1 + λ 2 λ 2 0 )dy 2 1 + (y 2 2 + λ 4 0 )dy 2 2 + 2y 1 y 2 dy 1 dy 2 (4.2a) B = − λ 2 ∆ y 1 y 2 dy 1 + (y 2 2 + λ 4 0 )dy 2 ∧ σ3, (4.2b) e −2φ = 8∆α ′−3/2 , (4.2c) where λ 2 0 = 1/6, λ 2 = 1/9, σ3 = dψ + cos θ 1 dφ 1 , and ∆ ≡ det M = 8∆ = 8[λ 2 0 y 2 1 + λ 2 (y 2 2 + λ 4 0 )] = β 1 v 2 1 + β 2 (v 2 3 + β 2 1 ). (4.3) Here β 1 = 2λ 2 0 , β 2 = 2λ 2 , v 1 = 2y 1 and v 3 = 2y 2 , and as in section 2, we have absorbed a factor of R 2 in β 1 , β 2 , and a factor of α ′ in v 1 , v 3 . The RR-sector is given by α ′3/2 R F 2 = 8 √ 2λ 4 0 λ sin θ 1 dφ 1 ∧ dθ 1 α ′3/2 R F 4 = −8 √ 2λ 4 0 λ y 1 ∆ sin θ 1 dφ 1 ∧ dθ 1 ∧ σ3 ∧ (λ 2 0 y 1 dy 2 − λ 2 y 2 dy 1 ). (4.4) For completeness, the T-dual vielbeins are given bŷ This completes the type IIA background T-dual to AdS 5 × T (1,1) in type IIB supergravity. e ′ 1 = − λ 0 ∆ (y 2 1 + λ 2 λ 2 0 )dy 1 + y 1 y 2 (dy 2 + λ 2 σ3) (4.5a) e ′ 2 = λ 0 ∆ λ 2 y 2 dy 1 − λ 2 0 y 1 (dy 2 + λ 2 σ3) (4.5b) e 3 = − λ ∆ y 1 y 2 dy 1 + (y 2 2 + λ 4 0 )dy 2 − λ 2 0 y 2 1 σ3 ,(4. AdS 3 solution and its non-abelian T-dual The solution with metric AdS 3 × R 2 × S 2 × S 3 is given by 1 R 2 ds 2 AdS 3 ×R 2 ×S 2 ×S 3 = 1 3 √ 3 2ds 2 AdS 3 + dz 2 1 + dz 2 2 + ds 2 1 + ds 2 2 + 1 2 (dψ + P ) 2 (4.7a) 1 R 2 B = −τ 6 √ 6 z 1 (vol 1 − vol 2 ) ≡ −τβ 2 2 √ 2R 2 z 1 (vol 1 − vol 2 ) (4.7b) 1 R 2 F 3 = τ 6 √ 6 dz 2 ∧ (vol 1 − vol 2 ) (4.7c) 1 R 4 F 5 = 1 27 vol AdS 3 ∧ 4dz 1 ∧ dz 2 + τ 2 2 (vol 1 + vol 2 ) +(dψ + P ) ∧ vol 1 ∧ vol 2 + τ 2 8 dz 1 ∧ dz 2 ∧ (vol 1 + vol 2 ) , (4.7d) where τ is a constant. In order to find its T-dual, we consider the Maurer-Cartan forms L 1 = 1 √ 2 (cos ψdθ 2 + sin ψ sin θ 2 dφ 2 ) L 2 = 1 √ 2 (− sin ψdθ 2 + cos ψ sin θ 2 dφ 2 ) (4.8) L 3 = 1 √ 2 (dψ + cos θ 2 dφ 2 ), such that vol 2 = 2L 1 ∧ L 2 . Using the set-up of section 2, the vielbeins related to the directions to be T-dualized are With these definitions, we may write the metric as ds 2 =β 2 (2ds 2 AdS 3 + ds 2 1 + ds 2 2 + dz 2 1 + dz 2 2 ) + (e 1 ) 2 + (e 2 ) 2 + (e 3 ) 2 (4.10) and the RR-forms as (vol 2 = 2 β 1 e 1 ∧ e 2 , dψ + P = √ 2 √β 2 e 3 ) 1 R 2 F 3 = τ 6 √ 6 dz 2 ∧ vol 1 − τ √ 2 dz 2 ∧ e 1 ∧ e 2 (4.11a) 1 R 4 F 5 = 1 27 vol AdS 3 ∧ 4dz 1 ∧ dz 2 + τ 2 2 vol 1 + 2 β 1 e 1 ∧ e 2 + √ 2 β 2 e 3 ∧ vol 1 ∧ 2 β 1 e 1 ∧ e 2 + τ 2 8 dz 1 ∧ dz 2 ∧ vol 1 + 2 β 1 e 1 ∧ e 2    , (4.11b) or as F 3 =G (0) 3 + G 12 1 ∧ e 1 ∧ e 2 (4.11c) F 5 =G (0) 5 + G 3 4 ∧ e 3 + G 12 3 ∧ e 1 ∧ e 2 + G (3) 2 ∧ e 1 ∧ e 2 ∧ e 3 , (4.11d) where 1 R 4 G (0) 5 = 1 27 vol AdS 3 ∧ 4dz 1 ∧ dz 2 + τ 2 2 vol 1 1 R 4 G 3 4 = √ 2τ 2 216β 1/2 2 dz 1 ∧ dz 2 ∧ vol 1 , 1 R 2 G (0) 3 = τ 6 √ 6 dz 2 ∧ vol 1 , 1 R 4 G 12 3 = τ 2 27β 1 vol AdS 3 1 R 4 G (3) 2 = 4 27 √ 2 1 β 1/2 2β 1 vol 1 + τ 2 8 dz 1 ∧ dz 2 1 R 2 G 12 1 = − τ √ 2 dz 2 . (4.12) The matrix M is given by M ij = g ij + b ij + α ′ ǫ ijkvk , so (after absorbing α ′ factors inv i ) M =   β 1 τ z 1 √ 2β 2 +v 3 −v 2 − τ z 1 √ 2β 2 −v 3β1v1 v 2 −v 1β2    ,(4.13) As before, we consider the gauge fixing θ = φ = v 2 = 0, so thatv = (cos ψv 1 , sin ψv 1 , v 3 ), and for simplicity we defineṽ 3 = τ z 1 √ 2β 2 +v 3 , in such a way that the inverse of M is (3.11), with the replacement v 3 ṽ 3 , that is, (M −1 ) T = 1 ∆   β 1β2 + v 2 1 cos 2 ψṽ 3β2 + v 2 1 cos ψ sin ψ v 1ṽ3 cos ψ − v 1β1 sin ψ −ṽ 3β2 + v 2 1 cos ψ sin ψβ 1β2 + v 2 1 sin 2 ψ v 1β1 cos ψ + v 1ṽ3 sin ψ v 1ṽ3 cos ψ + v 1β1 sin ψ −v 1β1 cos ψ + v 1ṽ3 sin ψṽ 2 3 +β 2 1    ,(4. 14) where the determinant det M is ∆ ≡ det M = (β 2 1 +ṽ 2 3 )β 2 + v 2 1β 1 . Under these definitions, we must apply the duality on the following fields 6 Q φφ = G φφ =β 2 sin 2 θ 1 + 1 2 cos 2 θ 1 Q φ3 = G φ3 = Q 3φ = √ 2 2β 2 cos θ 1 Q θθ = G θθ =β 2 Q θφ = B θφ = − τ 2 √ 2 z 1β2 sin θ 1 E 12 = b 12 = 2τ √ 2β 2 z 1 E 11 = E 22 = g 11 =β 1 E 33 = g 33 =β 2 Using these results and the same procedure as in section 3, we find that the dual metric, dilaton and B field are dŝ 2 AdS 3 ×R 2 ×S 2 ×S 3 =β 2 2ds 2 AdS 3 + dz 2 1 + dz 2 2 + ds 2 1 + 1 2 ∆β 1β2 v 2 1 (dψ + cos θ 1 dφ 1 ) 2 6 Note that since the dependence on the angular coordinates (φ2, θ2) is encapsulated into the Maurer-Cartan forms Li, in what follows the subscript (φ, θ) refers logically to (φ1, θ1). + 1 2 ∆ (β 1β2 + v 2 1 )dv 2 1 + (ṽ 2 3 +β 2 1 )dv 2 3 + 2v 1ṽ3 dv 1 dv 3 (4.15) φ = φ − 1 2 ln∆ α ′3 B = − v 1 2∆ (ṽ 3β2 dv 1 − v 1β1 dv 3 ) ∧ dψ −β 2 τ z 1 2 √ 2 sin θ 1 dθ 1 ∧ dφ 1 +β 2 2∆ v 1ṽ3 cos θ 1 dφ 1 ∧ dv 1 +β 2 2∆ cos θ 1 (ṽ 2 3 +β 2 1 )dφ 1 ∧ dv 3 = − τ Rz 1 6 √ 6 vol 1 +β 2 2∆ σ3 ∧ (v 1ṽ3 dv 1 + (ṽ 2 3 +β 2 1 )dv 3 ). (4.16) For later use, the √ det g int for this metric (g int is as before the internal, i.e. non-AdS, part of the metric) is det g int = α ′3 sin θ 1 2 √ 2β 1β 5/2 2 ∆ v 1 . (4.17) With F 3 and F 5 written as in (4.11c) and (4.11d), we can apply the formulas in the appendix, reintroduce the factors of α ′ in (4.12), (4.15) and (4.16) and obtain the RR-sector T-dual forms F 1 = F 3 = F 5 = 0 and (F 6 andF 8 would be redundant, as we consider their Poincaré dualsF 4 andF 2 ) F 2 = e φ−φ −A 0 G (3) 2 + G 12 1 ∧ (A 2ê 1 − A 1ê 2 − A 0ê 3 ) F 4 = e φ−φ A 3 G 3 4 + G 12 3 ∧ (A 2ê 1 − A 1ê 2 − A 0ê 3 ) + G (0) 3 (A 1ê 1 + A 2ê 2 + A 3ê 3 ) +G (3) 2 ∧ (A 3ê 1 ∧ê 2 + A 1ê 2 ∧ê 3 + A 2ê 3 ∧ê 1 ) + A 3 G 12 1ê 1 ∧ê 2 ∧ê 3 , (4.18) where as before, e φ−φ = ∆ α ′−3/2 , α ′3/2 e φ−φ A 0 =β 1 β 2 and α ′3/2 e φ−φ A a = A a , and the dual vielbeins arê e ′1 AdS 3 = −β 1/2 1 √ 2∆ (β 1β2 + v 2 1 )dv 1 + v 1ṽ3 dv 3 + v 1ṽ3β2 (dψ + cos θ 1 dφ 1 ) (4.19a) e ′2 AdS 3 =β 1/2 1 √ 2∆ β 2ṽ3 dv 1 − v 1β1 dv 3 − v 1β1β2 (dψ + cos θ 1 dφ 1 ) (4.19b) e 3 AdS 3 = −β 1/2 2 √ 2∆ v 1ṽ3 dv 1 + (ṽ 2 3 +β 2 1 )dv 3 − v 2 1β1 (dψ + cos θ 1 dφ 1 ) . (4.19c) Domain Wall and its non-abelian T-dual The Domain Wall solution which has as limits the above AdS 3 and AdS 5 solution is given by 1 R 2 ds 2 DW =e 2A (−dt 2 + dx 2 ) + e 2B (dx 2 1 + dx 2 2 ) + dρ 2 + 1 6 e 2U (ds 2 1 + ds 2 2 ) + 1 9 e 2V ( √ 2L 3 + cos θ 1 dφ 1 ) 2 (4.20a) 1 R 2 B = −τ 6 x 1 (vol 1 − vol 2 ) (4.20b) 1 R 2 F 3 = τ 6 dx 2 ∧ (vol 1 − vol 2 ) (4.20c) 1 R 4 F 5 =4e 2A+2B−V −4U dt ∧ dx ∧ dx 1 ∧ dx 2 ∧ dρ + 1 27 ( √ 2L 3 + cos θ 1 dφ 1 ) ∧ vol 1 ∧ vol 2 + τ 2 36 dx 1 ∧ dx 2 ∧ ( √ 2L 3 + cos θ 1 dφ 1 ) ∧ (vol 1 + vol 2 ) (4.20d) + τ 2 12 e 2A−2B−V dt ∧ dx ∧ dρ ∧ (vol 1 + vol 2 ). Here τ is a constant and A, B, U, V are functions of the radial coordinate ρ. From this solution, we see that we can recover AdS 5 × T (1,1) by setting the constant τ = 0 and A = B = ρ and U = V = 0. On the other hand, to recover the AdS 3 × R 2 × S 2 × S 3 solution, we set A = 3 3/4 √ 2 ρ, B = U = −V = 1 4 ln 4 3 , (4.21) and change variables by x i z i / √ 6 . As before, the T-dual model is given by dŝ 2 DW = R 2 e 2A (−dt 2 + dx 2 ) + R 2 e 2B (dx 2 1 + dx 2 2 ) + R 2 dρ 2 + R 2 6 e 2U ds 2 1 + 1 2∆β 1β2 v 2 1 (dψ + cos θ 1 dφ 1 ) 2 + 1 2∆ (β 1β2 + v 2 1 )dv 2 1 + (β 2 1 +v 2 3 )dv 2 3 + 2v 1v3 dv 1 dv 3 B = − τ Rx 1 6 vol 1 +β 2 2∆ σ3 ∧ (v 1v3 dv 1 + (v 2 3 +β 2 1 )dv 3 ) φ = φ − 1 2 ln∆ α ′3 , (4.22) where we have defined β 1 = 1 3 e 2U ,β 2 = 2 9 e 2V ,v 3 = τ 3 x 1 +v 3 ,∆ = (β 2 1 +v 2 3 )β 2 + v 2 1β1 ,(4.23) and as before we absorbed R 2 factors inβ i and α ′ in v i . We can easily see that we can obtain the correct limits in the NS-NS sector. The UV and IR limits of the T-dual solution to the domain wall are the non-abelian T-duals of the AdS 5 × T (1,1) and the AdS 3 × R 2 × S 2 × S 3 solutions, respectively. In the RR sector, we could verify term by term that the equality holds, but alternatively, one can find the RR-forms components in the same way as in (4.12). In the present case, we obtain 1 R 4 G (0) 5 = dt ∧ dx ∧ dρ ∧ 4e 2A+2B−V −4U dx 1 ∧ dx 2 + τ 2 12 e 2A−2B−V vol 1 (4.24) 1 R 4 G 3 4 = √ 2τ 2 36β 1/2 2 dx 1 ∧ dx 2 ∧ vol 1 (4.25) 1 R 2 G (0) 3 = τ 6 dx 2 ∧ vol 1 , 1 R 2 G 12 3 = τ 2 6β 1 e 2A−2B−V dt ∧ dx ∧ dρ (4.26) 1 R 4 G (3) 2 = 2 √ 2 27β 1β 1/2 2 vol 1 + 2 √ 2τ 2 36β 1β 1/2 2 dx 1 ∧ dx 2 (4.27) 1 R 2 G 12 1 = − τ 3β 1 dx 2 ,(4.28) Then the T-dual RR-forms are as in (4.18), i.e., F 2 = e −φ −A 0 G (3) 2 + G 12 1 ∧ (A 2ê 1 − A 1ê 2 − A 0ê 3 ) F 4 = e −φ A 3 G 3 4 + G 12 3 ∧ (A 2ê 1 − A 1ê 2 − A 0ê 3 ) + G (0) 3 ∧ (A 1ê 1 + A 2ê 2 + A 3ê 3 ) +G (3) 2 ∧ (A 3ê 1 ∧ê 2 + A 2ê 3 ∧ê 1 + A 1ê 2 ∧ê 3 ) + A 3 G 12 1ê 1 ∧ê 2 ∧ê 3 . (4.29) Finally, we can also compute the vielbeins and see that they have the correct limits, therefore the RR-sector also has the correct limits. For instance, the frame field e 3 of the Domain Wall is e 3 AdS(DW ) = −β 1/2 2 √ 2∆ v 1v3 dv 1 + (v 2 3 +β 2 1 )dv 3 − v 2 1β1 (dψ − cos θ 1 dφ 1 ) , (4.30a) and we can easily verify that the UV and IR limits are the frame field e 3 in the AdS 5 , AdS 3 e 3 AdS 5 = − β 1/2 2 √ 2∆ v 1 v 3 dv 1 + (v 2 3 + β 2 1 )dv 3 − v 2 1 β 1 (dψ + cos θ 1 dφ 1 ) (4.30b) e 3 AdS 3 = −β 1/2 2 √ 2∆ v 1ṽ3 dv 1 + (ṽ 2 3 +β 2 1 )dv 3 − v 2 1β 1 (dψ + cos θ 1 dφ 1 ) (4.30c) respectively. Dual conformal field theories, central charges and RG flow An interesting question is, what happens to the conformal field theories dual to the gravity backgrounds with AdS factor under nonabelian T-duality on the extra dimensional space? The answer is not obvious. Abelian T-duality on a direction transverse to a Dp-brane turns it into a D(p + 1)-brane, but if the original direction is infinite in extent, the T-dual direction is infinitesimal in extent. However, this discussion makes sense only in the region far from the region where AdS/CFT is relevant, the core of the D-brane. Naively, abelian T-duality on the transverse part of a gravity dual should increase the dimensionality of the brane, therefore of the field theory dual to the background. But if we perform a nonabelian T-duality on a space with an AdS factor, in such a way that the AdS factor is not affected, and moreover the T-duality does not introduce a new AdS direction, then it seems that the dimensionality of the dual conformal field theory is unaffected. And yet since the gravity dual is modified, it is logical to assume that the conformal field theory is modified as well. To understand the effect of nonabelian T-duality on the conformal field theory, we need some probes of the transverse space in AdS/CFT. Such probes are for instance wrapped branes, dual to solitonic states in the field theory, like the example of the 5-brane wrapped on S 5 in AdS 5 × S 5 , giving the baryon vertex operator [24]. 7 But a more relevant probe was considered in [13], namely the central charge of the dual field theory as a function of the number of branes. One can calculate Page charges in a gravitational background, and identify those with the number of branes that generate the geometry. For the central charge of the dual conformal field theory, a slight generalization of the usual formula was provided in [13]. For a metric on M D = AdS d+2 × X n , of the type ds 2 D = A d z 2 (1,d) + AB dr 2 + g ij dθ i dθ j ,(5.1) with a dilaton φ, define the modified internal volume aŝ V int = d θ e −4φ det[g int ]A d (5.2) and thenĤ =V 2 int . Then the central charge is given by C = d d B d/2Ĥ 2d+1 2 G N (Ĥ ′ ) d (5.3) where G N = (α ′ ) D 2 −1 is the Newton constant in D dimensions and prime denotes the derivative with respect to r. The expectation of increase in dimensionality through T-duality affects the D-brane charges of the gravity background. For a geometry with an AdS 5 factor in type IIB, generated only by D3-branes (with only D3-brane Page charges), after T-duality we expect the geometry to be generated by D4-and D6-branes only, i.e. to have only D4-and D6brane Page charges Q P age D4 = 1 2κ 2 10 T D4 Σ 4 (F 4 −B ∧F 2 ) Q P age D6 = 1 2κ 2 10 T D6 Σ 2F 2 . (5.4) For an abelian T-duality, we would expect only D4-brane charge, but for nonabelian Tduality (in some sense a T-duality on 3 coordinates), the expectation, confirmed by a calculation, is that only D6-brane charges appear. One can calculate the central charges and express them as a function of the Page charges. In the AdS 5 × S 5 case, we find that C = 32π 3 R 8 α ′−4 = 2π 5 N 2 D3 before, and C = (8π 5 /3)R 8 α ′−4 = (2π 5 /24)N 2 D6 after the nonabelian T-duality, leading to the relation 8 C bef ore C af ter = 24N 2 D3 N 2 D6 , (5.5) which is found to be satisfied also in other cases of non-abelian T-duality on type IIB geometries generated by D3-branes. An interesting question which we will try to answer in this section is whether a similar formula is valid in more general contexts in the case of geometries with an AdS factor. Page charges • In the case of section 3, the starting geometry is in type IIA, the reverse of the situation considered in [13]. Since F 2 = 0 in the background before T-duality, Q P age D6 = 0, and we only have a nonzero result for N D4 = \|Q P age D4 \| = R 3 2κ 2 10 T D4 y 2 y 1 η(y)dy X 3 vol(S 2 ) ∧ L 3 = R 2π √ α ′ 3 2 √ 2 9 (y 2 − y 1 ) 1 − c y 1 + y 2 2 4π 2 √ 2 ≡ R l s 3 2 9π K. (5.6) After the nonabelian T-duality, we have calculated in section 3 that Q P age D3 = Q P age D5 = 0 and N D7 = \|Q P age D7 \| = R 3 α ′3/2 4 √ 2 9 (y 2 − y 1 ) 1 − c y 1 + y 2 2 = R l s 3 4 √ 2 9 K. (5.7) • In the case of section 4, the we have a Domain Wall solution that interpolates between an AdS 5 × T 1,1 and an AdS 3 × R 2 × S 2 × S 3 . This can be also found in the N = 4 D=5 gauged supergravity arising as a consistent KK truncation of type IIB on T 1,1 [20], and as such it can be interpreted as an RG flow between two fixed points in the dual field theory. A relevant question is then, is the ratio of the central charges before and after the nonabelian T-duality modified by the RG flow? For AdS 5 × T 1,1 , the Page charges before and after the nonabelian T-duality were found in [13], Q P age D5 = Q P age D7 = 0 and \|Q P age D3 \| = N D3 before, and \|Q P age D6 \| = N D6 , Q P age D4 = 0 after the T-duality, with (in our conventions) N D3 = 4R 4 27πα ′2 , N D6 = 4 √ 2 27α ′2 R 4 . (5.8) 8 The formula in [13] is actually with a factor of 3 instead of 24, since different conventions for T-duality were considered, with Li = σi instead of Li = σi/ √ 2, giving an extra 2 √ 2 in the quantization of the Page charges after T-duality. For AdS 3 × R 2 × S 2 × S 3 , the Page charges before the T-duality were found in [20]. Assuming that R 2 is compactified to a T 2 = S 1 (1) × S 1 (2) with period 2πRd i √ 6, and defining s(S) as a homology 2-cycle generator in S 2 × S 3 , one has the integers Q N 5 = 1 (2πl s ) 2 S 1 (1) ×s(S) H Q D5 = 1 (2πl s ) 2 S 1 (2) ×s(S) dC 2 (5.9) and the (D3-brane) Page charge quantization condition is 1 (2πl s ) 4 Σ 5 (F 5 − B ∧ dC 2 ) ∈ Z.9 = − 1 2 Q N 5 Q D5 . (5.12) Moreover, the above flux quantization is actually valid over the whole domain wall solution. After the T-duality, we have F 2 and F 4 , so we need to consider the quantization of D4-brane Page charges 1 (2πl s ) 3 Σ 4 (F 4 − B ∧ F 2 ) ∈ Z (5.13) and 1 (2πl s ) Σ 2 F 2 ∈ Z. (5.14) For Σ 2 = T 2 , we obtain N D6 = − τ 2 2 √ 2 216 4π 2 6d 1 d 2 2πl s R 4 l 3 s ,(5.15) and for Σ 2 = S 2 , we obtainN D6 = − 2 √ 2 27 4π 2πl s R 4 l 3 s . (5.16) Central charges • For the case in section 3, the central charge before the T-duality is obtained using A = R 2 r 2 , B = r −4 and d = 3, leading to ( L 1 ∧ L 2 ∧ L 3 = 2π 2 √ 2) V int = α ′ r 3 R 6 l (2π) 4π 2 9 (y 2 − y 1 ) 1 − c(y 1 + y 2 ) 2 ≡ α ′ r 3 R 6 l 8π 3 9 K ,(5.17) and therefore C bef ore = R 6 8α ′3 8π 3 9 K l ,(5.18) where the Page charge quantization condition (5.6) means that we can write R 3 /α ′3/2 as a function of N D4 , giving C bef ore = 9π 5 4 N 2 D4 Kl . (5.19) After the T-duality, the central charge is found using the same A = R 2 r 2 , B = r −4 and d = 3, leading to (also using the √ det g int calculated in (3.19)) V int = α ′ R 6 r 3 2l (2π) 2 K 9 dv 1 α ′ v 1 α ′ dv 3 α ′ . (5.20) To calculate the integral over the v i , we can use as another gauge fixing, related to the previous coordinates by v 1 /α ′ ρ cos χ and v 3 /α ′ ρ sin χ with ρ, χ ∈ [0, π], leading to a value of 2π 3 /3 for the integral. 9 We then obtain C af ter = π 5 K 54l R l s 6 ,(5.21) and from the Page charge quantization condition (5.7) we can write R 3 /α ′3/2 as a function of N D7 , giving C af ter = 3π 5 64Kl N 2 D7 (5.22) We see that the ratio is C bef ore C af ter = 48N 2 D4 N 2 D7 ,(5.23) which is basically the same as in (5.5), with the obvious generalization to N 2 Dp /N 2 Dp+3 , and an extra factor of 2 which is probably the effect of a different normalization. • For the case in section 4, on the AdS 5 × T 1,1 side, the central charge before the T-duality was found to be [13] C (1) bef ore = π 3 R 8 27α ′4 = 27 8 π 5 N 2 D3 , (5.24) 9 The rangle of ρ was defined in [15], and was then used for the calculation of central charges in [13]. and after the T-duality (5.25) leading to the ratio in (5.5). On the AdS 3 × T 2 × S 2 × S 3 side, the central charge before the T-duality is [20] C (2) bef ore = C (1) af ter = 2R 8 π 5 λλ 4 0 3α ′4 = 9 64 π 5 N 2 D6 ,3R AdS 3 2G 3 = 3 2 R l s 8 8d 1 d 2 9 vol(T 1,1 ) 4π 4 = 3 2 \|N Q N 5 Q D5 \| = 3\|NN \| = 3N D3ND3 . (5.26) Here G 3 is the effective Newton's constant, obtained from the dimensional reduction of the action in string frame, thus proportional to (R/l s ) 7 vol(T 1,1 )(2πd 1 )(2πd 2 ). After the T-duality, using the √ det g int calculated in (4.17), and doing the integration over v i in the same way as in the case in section 3, with result 2π 3 /3, we obtain V int = R 8 r 12(2π) 4 d 1 d 2 √ 2 1 3 √ 3 7/2 2π 3 3 ,(5.27) leading to C af ter = 32 √ 2π 7 3 21/4 d 1 d 2 R 8 l 8 s = 4 τ 2 3 −1/4 √ 2π 6 N D6ND6 . (5.28) The ratio of central charges before and after the T-duality can therefore be expressed as C (2) bef ore C (2) af ter = 3 5/4 √ 2τ 2 8π 6 N D3ND3 N D6ND6 . (5.29) Note that now we can fix τ such that the prefactor equals 24, obtaining C (2) bef ore C (2) af ter = 24N D3ND3 N D6ND6 ,(5.30) which is essentially the same formula (5.5) that was valid on the AdS 5 side of the domain wall. The factor τ is related to a redefinition of the fields, coupled to a rescaling of the x i (or z i ) coordinates [20], which are the two coordinates that change from the AdS 5 on one side of the domain wall to a AdS 3 × T 2 on the other. It is therefore not surprising that changing τ allows us to change the normalization of the central charge dual to AdS 3 , with respect to the one dual to AdS 5 . Conclusions In this paper we have studied the nonabelian T-duals of some backgrounds with N = 1 supersymmetry and an AdS factor, that can have an AdS/CFT interpretation. We have considered the nonabelian T-dual of a type IIA solution with an AdS 5 factor, giving a type IIB solution with an AdS 5 factor, and the nonabelian T-dual of a type IIB domain wall solution that interpolates between AdS 5 × T 1,1 and AdS 3 × R 2 × S 2 × S 3 . We have probed the interpretation of nonabelian T-duality of these solutions from the point of view of the dual conformal field theory through a calculation of the central charges. We have found that the simple law (5.5) found in [13] for the ratio of central charges before and after the T-duality holds in all cases, with the obvious generalization of N 2 D3 /N 2 D6 to N 2 Dp /N 2 Dp+3 or to N D3ND3 /N D6ND6 . In the case of the type IIB dowain wall solution, we obtained the usual ∝ N 2 behaviour, and on the AdS 3 side we could fix the normalization of the central charge by using a rescaling parameter τ , in order to obtain the same law (5.5) valid on the AdS 5 side of the domain wall. In order to understand better the effect of nonabelian T-duality on gravity duals with AdS factors, one needs to study also other probes of the geometry, but we leave this for future work. A Nonabelian T-duality action on RR fields To act with nonabelian T-duality on the RR fields, one first writes the p-form field strengths in the form we have the transformation ruleŝ F p = G (0) p + G a p−1 ∧ e a + 1 2 G ab p−2 ∧ e a ∧ e b + G(3)G (0) p = e φ−φ (−A 0 G (3) p ) + A a G a p ) G a p−1 = e φ−φ − A 0 2 ǫ abc G bc p−1 + A b G ab p−1 + A a G (0) p−1 Ĝ ab p−2 = e φ−φ ǫ abc (A c G (3) p−2 + A 0 G c p−2 ) − (A a G b p−2 − A b G a p−2 ) Ĝ (3) p−3 = e φ−φ A a 2 ǫ abc G bc p−3 + A 0 G (0) p−3 . (A.3) Here, defining y i = b i + α ′ v i as before and z i = y i √ det g ζ a = κ a i z i = κ a i y i √ det g (A.4) the coefficients of the transformation rules are A.1 Particular cases for the coefficients In the case of section 3, we have b i = 0, so y i = α ′vi , leading to κ a i = α ′ Rdiag( β 1v1 , β 1v2 , β 2 v 3 ) , (A. 6) and det g = R 6 β 2 1 β 2 and det g + (κ a i y i ) 2 = R β 2 (R 4 β 2 1 + α ′2 v 2 3 ) + α ′2 β 1 v 2 1 = √ ∆. (A.7) We also have e φ−φ = √ ∆/α ′3/2 , so α ′3/2 e φ−φ A a ≡ A a = Rα ′ ( β 1 v 1 cos ψ, β 1 v 1 sin ψ, β 2 v 3 ) α ′3/2 e φ−φ A 0 = R 3 β 1 β 2 . (A.8) In the case of section 4, the same formulas apply, with the replacement of v 3 withṽ 3 . from AdS 5 to AdS 3 3 , absorbing the factors of R 2 in them for simplicity. Σ 5 = T 2 × M 3 , where M 3 is a homology 3-cycle generator in S 2 × S 3 p− 3 3∧ e 1 ∧ e 2 ∧ e 3 . 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[ "Bayesian adjustment for preferential testing in estimating the COVID-19 infection fatality rate: Theory and methods", "Bayesian adjustment for preferential testing in estimating the COVID-19 infection fatality rate: Theory and methods" ]	[ "Harlan Campbell [email protected] ", "Perry De Valpine \nDepartment of Environmental Science, Policy, and Management\nUniversity of California\nBerkeleyCAUSA\n", "Lauren Maxwell \nHeidelberg Institute for Global Health\nHeidelberg University Hospital\nHeidelbergGermany, USA\n", "Valentijn Mt De Jong \nJulius Center for Health Sciences and Primary Care\nJulius Center for Health Sciences and Primary Care\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 5. Cochrane Netherlands\n\nSection Clinical Tropical Medicine\nDepartment of Infectious Diseases\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 6\n\nHeidelberg University Hospital\nHeidelbergGermany\n", "Thomas Debray \nJulius Center for Health Sciences and Primary Care\nJulius Center for Health Sciences and Primary Care\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 5. Cochrane Netherlands\n\nSection Clinical Tropical Medicine\nDepartment of Infectious Diseases\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 6\n\nHeidelberg University Hospital\nHeidelbergGermany\n", "Thomas Jänisch ", "Paul Gustafson ", "\nARTICLE HISTORY\nDepartment of Statistics\nUniversity of British Columbia\nBCCanada\n" ]	[ "Department of Environmental Science, Policy, and Management\nUniversity of California\nBerkeleyCAUSA", "Heidelberg Institute for Global Health\nHeidelberg University Hospital\nHeidelbergGermany, USA", "Julius Center for Health Sciences and Primary Care\nJulius Center for Health Sciences and Primary Care\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 5. Cochrane Netherlands", "Section Clinical Tropical Medicine\nDepartment of Infectious Diseases\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 6", "Heidelberg University Hospital\nHeidelbergGermany", "Julius Center for Health Sciences and Primary Care\nJulius Center for Health Sciences and Primary Care\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 5. Cochrane Netherlands", "Section Clinical Tropical Medicine\nDepartment of Infectious Diseases\nUniversity Medical Center Utrecht\nUtrecht University\nUtrecht the Netherlands; 6", "Heidelberg University Hospital\nHeidelbergGermany", "ARTICLE HISTORY\nDepartment of Statistics\nUniversity of British Columbia\nBCCanada" ]	[]	A key challenge in estimating the infection fatality rate (IFR) of COVID-19 is determining the total number of cases. The total number of cases is not known because not everyone is tested but also, more importantly, because tested individuals are not representative of the population at large. We refer to the phenomenon whereby infected individuals are more likely to be tested than non-infected individuals, as "preferential testing." An open question is whether or not it is possible to reliably estimate the IFR without any specific knowledge about the degree to which the data are biased by preferential testing. In this paper we take a partial identifiability approach, formulating clearly where deliberate prior assumptions can be made and presenting a Bayesian model, which pools information from different samples.Results suggest that when limited knowledge is available about the magnitude of preferential testing, reliable estimation of the IFR is still possible so long as there is sufficient "heterogeneity of bias" across samples.	null	[ "https://arxiv.org/pdf/2005.08459v1.pdf" ]	218,674,079	2005.08459	5073c29dac4746d544bcb18d45f989db731fc7c8	Bayesian adjustment for preferential testing in estimating the COVID-19 infection fatality rate: Theory and methods Compiled May 19, 2020 18 May 2020 Harlan Campbell [email protected] Perry De Valpine Department of Environmental Science, Policy, and Management University of California BerkeleyCAUSA Lauren Maxwell Heidelberg Institute for Global Health Heidelberg University Hospital HeidelbergGermany, USA Valentijn Mt De Jong Julius Center for Health Sciences and Primary Care Julius Center for Health Sciences and Primary Care University Medical Center Utrecht Utrecht University Utrecht the Netherlands; 5. Cochrane Netherlands Section Clinical Tropical Medicine Department of Infectious Diseases University Medical Center Utrecht Utrecht University Utrecht the Netherlands; 6 Heidelberg University Hospital HeidelbergGermany Thomas Debray Julius Center for Health Sciences and Primary Care Julius Center for Health Sciences and Primary Care University Medical Center Utrecht Utrecht University Utrecht the Netherlands; 5. Cochrane Netherlands Section Clinical Tropical Medicine Department of Infectious Diseases University Medical Center Utrecht Utrecht University Utrecht the Netherlands; 6 Heidelberg University Hospital HeidelbergGermany Thomas Jänisch Paul Gustafson ARTICLE HISTORY Department of Statistics University of British Columbia BCCanada Bayesian adjustment for preferential testing in estimating the COVID-19 infection fatality rate: Theory and methods Compiled May 19, 2020 18 May 2020Bayesian inferenceCOVID-19selection biaspartial identificationIFR A key challenge in estimating the infection fatality rate (IFR) of COVID-19 is determining the total number of cases. The total number of cases is not known because not everyone is tested but also, more importantly, because tested individuals are not representative of the population at large. We refer to the phenomenon whereby infected individuals are more likely to be tested than non-infected individuals, as "preferential testing." An open question is whether or not it is possible to reliably estimate the IFR without any specific knowledge about the degree to which the data are biased by preferential testing. In this paper we take a partial identifiability approach, formulating clearly where deliberate prior assumptions can be made and presenting a Bayesian model, which pools information from different samples.Results suggest that when limited knowledge is available about the magnitude of preferential testing, reliable estimation of the IFR is still possible so long as there is sufficient "heterogeneity of bias" across samples. Introduction If someone is infected with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the pathogen that causes COVID-19, how likely is that person to die of COVID-19? This simple question is surprisingly difficult to answer. The "case fatality rate" (CFR) is a common measure that quantifies the mortality risk in a certain population, and is given by the ratio of deaths (D) over confirmed cases (CC) during a specific time period. However, because many COVID-19 cases are never diagnosed, the CFR almost certainly overestimates the true lethality of the virus. Instead, the better answer is captured by the infection fatality rate (IFR) (Wong et al., 2013;Kobayashi et al., 2020). The IFR, also a simple ratio, differentiates itself from the CFR by considering all cases, including the asymptomatic, undetected and misdiagnosed infections, in the denominator. For instance, if 20 individuals die of the disease in a population with 1000 infections, then the IFR is 20 / 1000 = 0.02 = 2%. Evidently, a key challenge in calculating the IFR is determining the true total number of cases. The total number of cases (C) is not known because not everyone is tested in the population (P ). A naïve estimate of the IFR might take this into account by simply considering the number of tests (T ) and estimating the number of cases as: C ≈ (CC/T ) × P . However, diagnostic tests are often selectively initiated, such that tested individuals are not representative of the population at large. In most countries/jurisdictions, those with classic COVID-19 symptoms (e.g. fever, dry cough, loss of smell or taste) are much more likely to be tested than those without symptoms. Due to this severity bias, the reported number of cases likely includes mostly people whose symptoms were severe enough to be tested and excludes the vast majority of those who are mildly-or asymptomatic. Even when testing is made equally available to all individuals (e.g., Bendavid et al. (2020)), there is potential for "selection bias" if people who have reason to believe they are infected are more likely to volunteer to be tested. We refer to the phenomenon whereby infected individuals are more likely to be tested than non-infected individuals, as "preferential testing." (Hauser et al. (2020) and others use the term "preferential ascertainment.") If the degree of preferential testing in a particular sample is of known magnitude, bias adjustment can be achieved by appropriately altering the estimated rate of infection and its uncertainty interval. However, the degree of preferential testing is no doubt difficult to ascertain and likely highly variable across different jurisdictions. An open question is whether or not it is possible to reliably estimate the IFR without any specific knowledge about the degree to which the data are biased by preferential testing (Q1). If not, should we rely only on select samples for which testing is representative and ignore the vast majority of available data? (Q2) In this paper, we address these two important questions by considering a simple Bayesian hierarchical model for estimation of the IFR. Bayesian models have been previously used in similar situations. For example, Presanis et al. (2009) conduct Bayesian inference to estimate the severity of pandemic H1N1 influenza; see also Presanis et al. (2011). More recently, Rinaldi and Paradisi (2020), and Hauser et al. (2020) employ Bayesian methods for estimating the severity of COVID-19 from early data from China and Italy. In order to address preferential testing bias, Hauser et al. (2020) apply susceptible-exposed-infected-removed (SEIR) compartmental models to age-stratified data and, in order to establish parameter identifiability, assume that all cases of infected patients aged 80 years and older are confirmed cases. The Bayesian model we propose is more general and allows one to obtain appropriate point and interval estimates for the IFR with varying degrees of prior knowledge about the magnitude of preferential testing and the distribution of other explanatory factors (e.g. age, minority status). This paper is structured as follows. In Section 2, we introduce required notation, discuss distributional assumptions and review key issues of identifiability. In Section 3, we formulate our Bayesian model and present a small illustrative example. In Section 4, we describe how the model can be scaled for larger populations and present results from a simulation study. In Section 5, we go over potential model extensions as well as model limitations. Finally, we conclude in Section 6 with a discussion of how the model could be used for an analysis of COVID-19 data and return to the questions of interest, Q1 and Q2. 2. Notation, distributions, and issues of (un)identifiability Notation and distributions Let us begin by describing the data and defining some basic notation. Suppose we have data from K groups (i.e., countries or jurisdictions) from a certain fixed period of time. For group k in 1, . . . , K, let: • P k be the population size (i.e., the total number of individuals at risk of infection); • T k be the total number of people tested; • CC k be the total number of confirmed cases resulting from the tests; and • D k be the total number of observed deaths attributed to COVID-19 infection. We do not observe the following latent variables. For the k-th group, let: • C k , the total number of infected people (cases) in the population; • IR k , be the true infection rate (proportion of the population which is infected), which is the expected value of C k /P k ; • IF R k , be the true underlying infection fatality rate (IFR), which is the expected value of D k /C k . We assume that: C k ∼ Binom(P k , IR k ),(1)D k \|C k ∼ Binom(C k , IF R k ),(2) where, in the k-th group, the unknown number of infections, C k , and the known number of deaths, D k , each follow a binomial distribution. Note that there are latent variables on both the left hand side and the right hand side of (1). Suppose that, for each population, CC k is recorded, instead of C k . Even in the absence of preferential testing, CC k will be smaller than C k because not everyone is tested. In other words, the confirmed cases (CC) are a subset of the total cases (C). We assume that the distribution of the confirmed cases depends only on the actual infection rate (C/P ) and the testing rate (T /P ) and not on the infection fatality rate (D/C) or other information. In other words, we assume that the conditional distribution of (CC\|C, T, P, D) is identical to the conditional distribution of (CC\|C, T, P ). This important assumption is similar to the assumption of "non-differential" exposure misclassification in measurement error models and may or may not be realistic; see De Smedt et al. (2018). For example, COVID-19 is thought to be deadlier amongst the elderly. If this is true, the non-differential preferentiality assumption would fail if elderly individuals were just as likely to be infected as others, yet more likely to be tested. The goal is to draw inference on the relationship between the number of deaths, D, and the number of cases, C, having only data on D, CC, P , and T . This is particularly challenging since the number of confirmed cases in each group may be subject to a certain unknown degree of preferential testing. Let the degree of preferential testing correspond to the φ non-centrality parameter, where CC k follows Wallenius' noncentral hyper-geometric distribution with: CC k \|C k ∼ N CHyperGeo(C k , P k − C k , T k , φ k ).(3) The hyper-geometric distribution describes the probability of CC k confirmed cases amongst T k tests (without any individuals being tested more than once), from a finite population of size P k that contains exactly C k cases. The non-central hypergeometric is a generalization of the hyper-geometric distribution whereby, in this case, testing may be biased so that either cases or non-cases are more likely to be tested. When φ k > 1, cases (i.e., infected individuals) are more likely to be tested than noncases (i.e., non-infected individuals); when φ k < 1, cases are less likely to be tested than non-cases. When φ k = 1, we have that the probability of being tested is equal for both cases and non-cases, and the non-central hyper-geometric distribution reduces to the standard hyper-geometric distribution. In this parameterization, the φ k parameter can be interpreted as an odds ratio: the odds of a case being tested vs. the odds of a non-case being tested. P k T k D k CC k CC k CC k CC k C k IR k IF R k φ k φ k φ Issues of (un)identifiability Given the assumptions detailed above, for each of the K groups, there are three unknown parameters (latent states), IR k , IF R k and φ k , that must be estimated for every two known values (D k /P k and CC k /T k ). This suggests that a unique solution may not be attainable without additional external data. The problem at hand is sufficiently rich and complex that forming intuition about the information-content of the data is challenging. Here we present an asymptotic argument which lays bare the flow of information. Consider a situation in which an infinite amount data are available. In so-called "asymptotia," we have that populations are approaching infinite size (i.e., for k in 1,. . . , K, we have P k → ∞), and that the number of tests also approaches infinity (i.e., for k in 1,. . . , K, we have T k → ∞). Recall that a hyper-geometric distribution is asymptotically equivalent to a binomial distribution. As such, we consider the following: D k ∼ Binom(P k , a k ); and CC k ∼ Binom(T k , b k ), where a k = IF R k × IR k and b k = 1 − (1 − IR k ) φk . Note that a k simply follows from the conditional binomial distribution (see expressions (11)-(14) and explanation in Section 4.1). However, this particular parameterization of b k does not emerge from the limit of the non-central hyper-geometric distribution. As we discuss later in Section 4.1, we have simply chosen a convenient parameterization for b k with the important connotation that φ k = 1 corresponds to testing at random, but as φ k increases, the testing is more preferentially weighted to those truly infected. For example, with an infection rate of IR k = 0.01, the binomial sampling probability b k is approximately 10 times larger than the infection rate if φ k = 10, and about 18 times larger if φ k = 20. Presume that the a priori defensible information about the preferential sampling in the k-th group is expressed in the form φ k ∈ [φ k ,φ k ],(4) i.e., φ k andφ k are investigator-specified bounds on the degree of preferential sampling for that jurisdiction. If one is certain that cases are as likely, or at least as likely, to be tested as non-cases, φ k = 1 is appropriate. If testing is known to be entirely random for the k-th group, one would set φ k =φ k = 1. Note that for fixed (a k , b k ), IF R k is a function of φ k with the form IF R k (φ k ) = a k 1 − (1 − b k ) (1/φk) .(5) Examining (5), knowledge of (a k , b k ), in tandem with (4) restricts the set of possible values for IF R k . In fact it is easy to verify that (5) is monotone in φ k , hence the restricted set is an interval. We write this interval as I k (a k , b k , φ k ,φ k ), or simply as I k for brevity. This is the jurisdiction-specific identification interval for IF R k . As we approach asymptotia for the k-th group, all values inside the interval remain plausible, while all values outside are ruled out; see Manski (2003). This is the essence of the partial identification inherent to this problem. Thinking now about the meta-analytic task of combining information, we envision that both φ k and IR k could exhibit considerable variation across jurisdictions. However, the variation in IF R could be small, particularly if sufficient jurisdiction-specific covariates are included (see Section 5.1). That is, after adjustment for a jurisdiction's age-distribution, healthcare capacity, and so on, residual variation in IF R could be very modest. When modeling, we would invoke such an assumption via a prior distribution. For understanding in asymptotia, however, we simply consider the impact of an a priori bound on the variability in IF R. Let τ be the standard deviation of IF R across jurisdictions. Then we presume τ does not exceed an investigator-specified upper bound ofτ , i.e., τ ≤τ . The jurisdiction-specific prior bounds on the extent of preferential sampling, and the prior bound on IF R variation across jurisdictions, along with the limiting signal from the data in the form of (a, b), gives rise to an identification region for the average infection fatality rate, IF R = K −1 K k=1 IF R k . Formally, this interval is defined as I(a, b, φ,φ,τ ) = IF R : τ ≤τ , IF R k ∈ I k (a k , b k , φ k ,φ k ), ∀k ∈ {1, . . . , K} .(6) Again, the interpretation is direct: in the asymptotic limit, all values of IF R inside this interval are compatible with the observed data, and all values outside are not. The primary question of interest is whether this interval is narrow or wide under realistic scenarios, since this governs the extent to which we can learn about IF R from the data. In general, evaluating (6) for given inputs is an exercise in quadratic programming nested within a grid search, hence can be handled with standard numerical optimisation. Details of this formulation are given in the Appendix. However, the special case ofτ = 0 is noteworthy in terms of developing both scientific and mathematical intuition. Consequently, we explore this case in some depth in what follows. Scientifically,τ = 0 represents the extreme limit of an a priori assumption that, possibly after covariate adjustment, IF R is a 'biological constant' which does not vary across jurisdictions. If the prospects for inference are not good when this assumption holds, they will be even less good under the less strict assumption that the IF R heterogeneity is small, but not necessarily zero. Mathematically, the case is much simpler, with (6) reducing to As can be seen immediately from Figure 1 (left-hand panels), in the present example the binding constraints arise from the first and twelfth jurisdictions, which happen to have the least and most amounts of preferential testing. However, this pattern does not hold in general. One can easily construct pairs of infection rates for which the jurisdiction with more preferential testing has a smaller upper endpoint for I k and/or a larger lower endpoint. Thus the values of φ k alone do not determine which two jurisdictions will provide the binding information about IF R. I(a, b, φ,φ, 0) = ∩ k I k (a k , b k , φ k ,φ k ). A Bayesian model for small-P data and an illustrative example A Bayesian model for small-P data Bayesian models work well for dealing with partially identifiable models; see Gustafson (2010). We describe a Bayesian model for the IFR which assumes standard Gaussian random-effects allowing both the infection rate (IR) and infection fatality rate (IFR) to vary between populations with a complimentary log-log link function: clog-log(IF R k ) ∼ N (θ, τ 2 ),(7)clog-log(IR k ) ∼ N (β, σ 2 ),(8) for k in 1, . . . , K, where θ is the parameter of primary interest, τ 2 represents between group IFR heterogeneity, β represents the mean cloglog-infection rate, and σ 2 is the variance in infection rates across the K groups. A Bayesian model begins by specifying a joint probability distribution. For our unknown parameters of interest (θ, τ 2 , β, σ 2 ), latent variables (C k , IF R k , IR k , and φ k , for k in 1,. . . , K), and aggregate data from K sources (we require data = {P k , T k , CC k , and D k }, for k in 1,. . . , K), Bayes theorem states that: p((θ, τ 2 ,β, σ 2 , C, IFR, IR, φ)\|data) ∝ p(data\|θ, τ 2 , β, σ 2 , C, IFR, IR, φ) (9) × p(θ, τ 2 , β, σ 2 , C, IFR, IR, φ) = K k=1 p(D k \|IF R k , C k )p(CC k \|T k , P k , C k , φ k )p(C k \|P k , IR k )p(IF R k \|θ, τ 2 )p(IR k \|β, σ 2 ) × p(θ)p(τ 2 )p(β)p(σ 2 ) K k=1 p(φ k ). We have that p(D k \|IF R k , C k ) is defined according to a binomial distribution as stated in (2), that p(CC k \|T k , P k , C k , φ k ) is defined according to (3), and that (1). We also have that p(IF R k \|θ, τ 2 ) and p(IR k \|β, σ 2 ) are defined according to (7) and (8) respectively. We are left to define prior distributions for the unknown parameters: θ, τ 2 , β, σ 2 , and φ k , for k in 1, . . . , K. p(C k \|P k , IR k ) is defined by Defining prior distributions is often controversial, as their choice can substantially influence the posterior when few data are available; see Lambert et al. (2005); Berger (2013); Burke et al. (2018); Gelman et al. (2006). We proceed by adopting flat priors on the probability scale to induce "uninformative" priors on the θ and β parameters (i.e., uniform priors on the inverse-clog-log scale) and weakly-informative half-normal priors for the τ and σ parameters. We fully expect that the true underlying infection rate will vary importantly across populations, while the true underlying IFR should be less variable, (especially after accounting for population level sources of heterogeneity; see Section 5.1). The priors are set accordingly. We consider the following: iclog-log(θ) ∼ U nif orm(0, 1); iclog-log(β) ∼ U nif orm(0, 1); τ ∼ half-N (0, 0.01); and: σ ∼ half-N (0, 1). The only remaining component is p(φ) = k p(φ k ). Knowledge about the magnitude of preferential testing in a given population may come from a variety of sources, yet may be difficult to quantify. Let us begin by assuming that, for k = 1, . . . , K, φ k is greater than 1, but otherwise of unknown magnitude. It also seems reasonable to assume a certain degree of heterogeneity for φ k across the K populations since the degree to which testing is available and randomly allocated might vary considerably. We therefore define a uniform prior such that: φ k ∼ U nif orm(1, 1 + γ), for k in 1,...K;(10) and: γ ∼ Exp(λ = 0.5). The prior specification therefore assumes that the uniform range of possible values for φ k is itself unknown. Setting λ = 0.5 implies that, a priori, a reasonable value for the φ k odds ratio is 2, (i.e., since E(γ) = 1/λ and E(φ k ) = (1 + (γ + 1))/2). In some scenarios, it might be conceivable to have a subset of groups for which φ k is known and equal to 1 (i.e., to have data from some samples where testing is known to be random). Without loss of generality, suppose this subset is the first k studies, such that for k = 1, . . . , k , we have φ k = 1. In a situation where testing is known to be random for all groups, k = K. We emphasize that the performance of any Bayesian estimator will depend on the choice of priors. The priors described represent a scenario where there is little to no a priori knowledge about the θ, β, and φ model parameters. Inference would no doubt be improved should more informative priors be specified based on probable values for each of these parameters. We briefly consider the impact of priors in the simulation study in Section 4.3, where we look to different values for λ. Illustrative example Let us illustrate the proposed model with the simple example dataset introduced earlier in Table 1. The illustrative dataset considers K = 12 groups with average populations of 2,000 individuals (P k obtained from a N egBin(2000, 1) distribution). The data were simulated such that, across all 12 groups, the expected overall IFR is 2% (i.e., icloglog(θ) = 0.02; θ = −3.90), and the expected overall IR is 20% (i.e., icloglog(β) = 0.20; β = −1.50). A certain degree of variability between populations was allowed by selecting τ 2 = 0.005 and σ 2 = 0.25. The testing rate for each population was obtained from a U nif orm(0.01, 0.10) distribution so that the number of tests in each population ranged from 1% of individuals to 10%. Note that the number of observed deaths is relatively small ranging from 1 to 20. The number of confirmed cases differs with differing degrees of preferential testing. The φ k values were set to be equally spaced from 1 to γ + 1 and the number of confirmed cases were then simulated from Wallenius' non-central hyper-geometric distribution (see Fog (2008)) as in equation (3) with either no preferential testing (γ = 0), "mild" preferential testing (γ = 4), "modest" preferential testing (γ = 11), or "substantial" preferential testing (γ = 22). We fit the model (M 1 ) as detailed in Section 3.1, where k = 0, and K = 12, and also fit the model (M {γ=0} ) where γ = 0 is fixed, corresponding to a situation in which one assumes that none of the populations are subject to any preferential testing. Each model is fit using JAGS (just another Gibbs' sampler) (Kruschke, 2014), with 5 CC k ∼ Binom(T k , 1 − (1 − IR k ) φk ),(11)C k ∼ Binom(P k , IR k ),(12)D k \|C k ∼ Binom(C k , IF R k ),(13) for k in 1, . . . , K. Note that the φ k parameter above no longer corresponds to an odds ratio, yet the interpretation is similar. We could have considered substituting the noncentral hyper-geometric distribution in (3) with a Gaussian asymptotic approximation to the non-central hyper-geometric (Hannan and Harkness, 1963;Stevens, 1951). However, the Gaussian approximation requires solving quadratic equations and therefore, might not necessarily help reduce the the computational complexity of our model; see Sahai and Khurshid (1995). We can further simplify by marginalizing over the cases. Since we have the distribution of C k and the conditional distribution of D k given C k , and since both of these are binomials, we have that unconditionally: D k ∼ Binom(P k , IF R k × IR k ).(14) For our unknown parameters of interest (θ, τ 2 , β, σ 2 ), latent variables (IF R k , IR k , and φ k , for k in 1,. . . , K), and aggregate data from K sources (we require data = {P k , T k , CC k , and D k }, for k in 1,. . . , K), Bayes' theorem states that: p((θ, τ 2 ,β, σ 2 , IFR, IR, φ)\|data) ∝ p(data\|θ, τ 2 , β, σ 2 , IFR, IR, φ) × p(θ, τ 2 , β, σ 2 , IFR, IR, φ) = K k=1 p(D k \|IF R k , IR k , P k )p(CC k \|T k , IR k , φ k )p(IF R k \|θ, τ 2 )p(IR k \|β, σ 2 ) × p(θ)p(τ 2 )p(β)p(σ 2 ) K k=1 p(φ k ), where p(D k \|IF R k , IR k , P k ) and p(CC k \|T k , IR k , φ k ) are defined by binomial distributions detailed in (14) and (11) respectively. We also have p(IF R k \|θ, τ 2 ) and p(IR k \|β, σ 2 ) defined according to (7) and (8) respectively, and priors defined as in Section 3. MCMC details For the large-P model, MCMC mixing can be slow because different combinations of φ k , cloglog(IR k ) and cloglog(IFR k ) can yield similar model probabilities. This is related to the identifiability issues discussed in Section 2.2. To improve mixing, we wrote this model in the nimble package (de Valpine et al., 2017), which supports an extension of the modeling language used in JAGS and makes it easy to configure samplers and provide new samplers. Using nimble, we applied two sampling strategies for the trio (φ, cloglog(IR k ), cloglog(IFR k )) for each k in 1, . . . , K. The details of these are provided in the Appendix. Simulation study We conducted a simple simulation study in order to better understand the operating characteristics of the proposed model. Specifically, we wished to evaluate the frequentist coverage of the CI for θ, and investigate the impact of the chosen prior for the magnitude of preferential testing (i.e., the impact of selecting different values for λ). As emphasized in Gustafson et al. (2009), the average frequentist coverage of a Bayesian credible interval, taken with respect to the prior distribution over the parameter space, will equal the nominal coverage. This mathematical property is unaffected by the lack of identification. However, the variability of coverage across the parameter space is difficult to anticipate and could be highly affected by the choice of prior. For example, we might expect that, in the absence of preferential testing (i.e., when γ = 0), coverage will be lower than the nominal rate. However, if this is the case, coverage will need to be higher than the nominal rate when γ > 0, so that the "average" coverage (taken with respect to the prior distribution over the parameter space) is nominal overall. We First we ran the simulation study with the 12 "unknown" φ k values, for k in 9, ..., 20, set to be evenly spaced between 1 and γ + 1, (as in the illustrative example) with γ assuming one of the eight values of interest (Study A). We also repeated the entire simulation study with the 12 "unknown" φ k values, for k in 9, ..., 20, simulated from a U nif orm(1, γ + 1) distribution (Study B). We fit three models to each unique dataset: M 1 , M 2 , and M 3 . All three models follow the same large-P framework detailed in Section 4.1 but consider different subsets of the data. The M 1 model uses only data from the samples for which φ k is unknown, i.e., {P k , T k , CC k , and D k } for k in 9, ..., 20. The M 2 model considers the data from all the groups, i.e., {P k , T k , CC k , and D k } for k in 1, ..., 20. Finally, the M 3 model uses only data from the samples for which φ k is known, i.e., {P k , T k , CC k , and D k } for k in 1, ..., 8. To be clear, the M 2 and M 3 models include the assumption of (correctly) known φ k for k = 1, . . . , 8. We simulated 200 unique datasets and, for each, fit the three different models. The suggests that appropriate estimates are achievable even in the presence of a substantial and unknown amount of preferential testing. When γ = 11, the odds (on average) for a case to be tested are more than six times the odds for a non-case (the average φ k value is equal to (1 + (γ + 1))/2), and yet the model is able to appropriately adjust. Results also point to a bias-variance trade-off with regards to one's choice for λ. The smaller the value of λ, the more robust the model is to a potentially large degree of preferential testing. The price of this additional robustness is greater uncertainty, i.e., a wider credible interval. Results from the M 2 and M 3 models suggest that for a given range of γ values, the M 2 model (which makes use of the additional "non-representative" data) is preferable Finally, note that overall the interval width is much narrower for M 2 relative to M 1 (compare dashed lines in lower-right panel to those in upper-right panel of Figure 2) which confirms that the k = 8 representative samples are very valuable for reducing the uncertainty around θ. With regards to the COVID-19 pandemic, this emphasizes the importance of conducting some amount of "unbiased testing" even if the sample sizes are relatively small; see Cochran (2020). were simulated from a U nif orm(1, 1 + γ). In this study, for a given dataset, we do not necessarily have a wide range between the lowest and highest φ k values. Based on the identifiability issues discussed in Section 2.2, we might therefore anticipate that appropriate inference is more challenging. Looking at the results with regards to coverage, this intuition appears to be correct. However, note that the M 2 model is still able to provide at or above nominal coverage for most of the γ values we considered. Model extensions and model limitations clog-log(IR k ) ∼ N (β 0 + β 1 X [1]k + . . . + β p X [p]k , σ 2 ),(16)clog-log(IF R k ) ∼ N (θ 0 + θ 1 Z [1]k + . . . + θ q Z [q]k , τ 2 ).(17) With regards to the infection rate (IR), time since first reported infection and time between first reported infection to imposition of social distancing measures might be predictive (Anderson et al., 2020). Other, potentially less obvious, covariates could also be included for IR, see Stephens-Davidowitz (2020). Age is a key factor for explaining the probability of COVID-19-related death. One might therefore consider median age of each group as a predictor for the IFR, or perform analyses that are stratified by different age groups (Onder et al., 2020). The latter strategy has, for instance, been recommended to make accurate predictions for respiratory infections (Pellis et al., 2020). To illustrate, let us briefly consider the possibility that, for each population, one has age stratified data. Suppose one has {P age k , T age k , CC age k , and D age k }, for age in {0 − 30 years, 30 − 60 years, 60 − 80 years, ≥ 80 years}, and for k in 1, . . . , K. Then, including the covariates and a simple random effect can accommodate as follows: clog-log(IF R age k ) ∼ N (θ 0 + θ 1 I {age='30−60 } + θ 2 I {age='60−80 } + θ 3 I {age='≥80 } + η k , σ 2 ),(18) where: η k ∼ N (0, ω 2 ), for k in 1, . . . , K, with unknown ω 2 variance. Finally, note that including data from serology studies (Winter and Hegde, 2020) will be crucial to inform the IR. If data from multiple serology studies are available from a single jurisdiction (or from several different regions within the jurisdiction), the proposed model could incorporate all these by including appropriate covariates and random effects. Model limitations Estimation of the IFR is very challenging due to the fact that it is a ratio of numbers where both the numerator and denominator are subject to a wide range of biases. Our proposed model only seeks to address one particular type of bias pertaining to the denominator: the bias in the number of cases due to preferential testing. With this in mind, we wish to call attention to several other important sources of bias. Cause of death information, compiled from death certificates, may not list SARS-CoV-2 as a contributing factor and certain jurisdictions may not have adopted the International Form of Medical Certificate of Cause of Death or have adopted the WHO guidelines on registering COVID-19-related deaths (WHO, 2020). As such, reported statistics on the number of deaths may be very inaccurate. To overcome this issue, many suggest looking to "excess deaths," by comparing aggregate data for all-cause deaths from the time during the pandemic to the years prior (Leon et al., 2020). Using this approach and a simple Bayesian binomial model, Rinaldi and Paradisi (2020) are able to obtain IFR estimates without relying on official (possibly inaccurate) data for the number of COVID-19 deaths. Some people who are currently sick will eventually die of the disease, but have not died yet. Due to the delay between disease onset and death, the number of confirmed and reported COVID-19 deaths at a certain point in time will not reflect the total number of deaths that will occur among those already infected (right-censoring). This will result in the number of recorded deaths underestimating the true risk of death. The denominator of the IFR must be the number of cases with known outcomes. Using time-series survival data or a defensible prior on the time from infection to death, the Bayesian model could be expanded to account for this additional source of uncertainty. The model, as currently proposed, also fails to account for the (unknown) number of false positive and false negative tests. When both the test specificity and the infection rate is low, false positives can substantially inflate estimates of infection rate and as a consequence, the IFR could be biased downwards. In principle, the model could accommodate for this by specifying priors for the sensitivity and specificity. In fact, Bayesian inference is known to be an excellent tool for adjusting for unknown testing uncertainty (Srinivasan et al., 2012;Burstyn et al., 2020). Finally, because the model uses data that are aggregated at the group level, estimates are potentially subject to ecological bias. While including group-level covariates may help reduce variability in the estimates, adjustment using group-level covariates can also lead to biased, misleading results (Li and Hua, 2020; Berlin et al., 2002). While ecological analyses can be useful for developing hypotheses (Pearce, 2000) and may be needed to make rapid use of publicly available surveillance data for inference during an epidemic, they cannot be used to make reliable inferences at the participant level. Sharing de-identified participant-level data as rapidly and widely as possible, in keeping with ethical and legal standards, is central to epidemic response (The GloPID-R Data Sharing Working Group, 2018). Conclusion Thoughts on an application for COVID-19 IFR Reducing uncertainty around the severity of COVID-19 is of great importance to policy makers and the public (Ioannidis, 2020;Lipsitch, 2020). Comparisons between the COVID-19 and seasonal influenza IFRs have impacted the timing and degree of social distancing measures and highlighted the need for more accurate estimates for the severity of both viruses (Faust, 2020). The current lack of clarity means that policy makers are unsure if cross-population differences (i.e., due to large τ 2 ) are related to clinically relevant heterogeneity or to spurious heterogeneity driven by testing and reporting biases (i.e., due to large γ). Existing efforts to understand the distribution of SARS-CoV-2 infection at the population level are unfortunately met by recruiting challenges (Gudbjartsson et al., 2020;Bendavid et al., 2020), leading to an over representation of people who are concerned about their exposure and/or an under representation of individuals who are self-quarantining, isolating, or hospitalized because of the virus. As of May 2020, nonpreferential testing has only been reported for limited populations where the entire population was tested, including the Diamond Princess Cruise Ship, four US prisons where all inmates and staff were tested, and the town of Vo', Italy (Field Briefing, 2020;Lavezzo et al., 2020;So and Sminth, 2020;Aspinwall and Neff, 2020). Of these, only the Diamond Princess Cruise Ship had made testing and outcome data publicly available at the age-group level at the time of publication (Field Briefing, 2020); (participantlevel age and test result data are available for Vo', Italy, but no outcome data were listed at the time of publication). Munich, Germany has undertaken a populationrepresentative prospective cohort study where researchers randomly selected addresses and all household members will be asked to complete COVID-19-related questions and serologic testing (Radon et al., 2020). While the research protocol does not specify when, whether, or how the data would be made publicly available, the Munich data will be an important resource as perhaps the first large-scale, population-representative dataset that is not affected by preferential testing. In an ideal world (for purposes of estimating the IFR), ample data free from preferential testing bias would be available for analysis. Unfortunately, in reality, such data are in scarce supply (Goodman and Shah, 2020). Our Bayesian model suggests that we can make headway nonetheless. Data from jurisdictions that are unrepresentative can still be used to obtain informative estimates of the IFR and can help reduce uncertainty when used alongside the limited representative data available. Final thoughts In the Introduction, we identified two important questions. First (Q1), is it possible to reliably estimate the IFR without any specific knowledge about the degree to which the data are biased by preferential testing? And secondly (Q2), must we only rely on select samples for which testing is representative and ignore the vast majority of available data? The proposed Bayesian model suggests that reliable estimation of the IFR at the group level is indeed possible when existing data do not reflect a random sample from the target population, and when limited knowledge is available about the likely magnitude of preferential testing. Importantly, the key to (partial) identifiability is sufficient heterogeneity in the degree of preferential testing across groups and sufficient homogeneity in the group-specific IFR. We also saw that, one need not ignore the vast majority of available data that may be biased by preferential testing. This data, with appropriate adjustment, can supplement any available representative data in order to sharpen the inference. In a typical situation of drawing inference from a single sample, obtaining appropriate estimates if that sample is biased by preferential testing is challenging if not impossible without some sort of external validation data. Intuition suggests that one might only be able to do a sensitivity analysis with respect to the impact of bias and indeed, applying prior distributions for the degree of preferential testing and proceeding with Bayesian inference is often regarded as a probabilistic form of sensitivity analysis (see, for instance, Greenland (2005) Code for all models, data, and analysis is available at: https://github.com/harlanhappydog/COVID19IFR 7. Appendix 7.1. Issues of (un)identifiability -(continued) In section 2.2 we described how the evaluation of the identification interval (6) for IF R reduces to a simple intersection of intervals, in the special case of τ = 0. Here we describe the evaluation of (6) for τ > 0, i.e., where a limited heterogeneity in IF R is permitted. Recall that quadratic programming constitutes the minimization of a quadratic function subject to linear constraints, and these may be a mix of equality and inequality constraints. Let x be a candidate value, which we will test for membership in the identification interval. To perform this test, we use a standard quadratic programming package (quadprog, Turlach and Weingessel (2013)) to minimize the quadratic function V ar(IF R), subject to the equality constraint IF R = x and the 2K inequality constraints which restrict IF R k to the interval I k for each k. By the definition of (6) then, x belongs in the identification interval if and only if the minimized variance does not exceed τ 2 . Thus a simple grid search over values of x numerically determines the identification interval. Note that so long as a and b arise from values of φ within the prescribed bounds, the underlying value of IF R must belong to the identification interval. Thus two numerical searches can be undertaken. One starts at the underlying value and tests successively larger x until a failing value is obtained. The other starts at the underlying value and does the same, but moving downwards. In all cases the univariate sampling method was adaptive random-walk Metropolis-Hastings. nimble MCMC details Using nimble, we applied two sampling strategies for the trio (φ k , cloglog(IR k ), cloglog(IFR k )) for each k in 1,. . . , K. In all cases the univariate sampling method was adaptive random-walk Metropolis-Hastings. For notation, we drop the subscript k and define η 1 = cloglog(IR k ) and η 2 = cloglog(IFR k ). First, we included a block sampler on (φ, η 1 , η 2 ) for each k, along with the usual univariate samplers on each element of the trio. Second, we included samplers in two transformed coordinate spaces. Define transformed coordinates (z 1 , z 2 ) = (h 1 (η 1 , η 2 ), h 2 (η 1 , η 2 )) = (exp(η 1 ) + exp(η 2 ), exp(η 1 ) − exp(η 2 )). (Based on the cloglog link, the quantities exp(η 1 ) and exp(η 2 ) may be interpreted as continuous time rates.) Now z 1 represents the more strongly identified quantity, so mixing in z 2 can be slow. Hence we wish to improve mixing in the z 2 direction. To do so, a sampler can operate in the (z 1 , z 2 ) coordinates while transforming the prior such that it is equivalent in (z 1 , z 2 ) to what was specified in the original coordinates, (η 1 , η 2 ). Using P (·) for priors, we have log(P (z 1 , z 2 )) = log(P (η 1 , η 2 )) − log(\|J\|), where \|J\| is the determinant of the Jacobian of (z 1 , z 2 ) with respect to (η 1 , η 2 ). In this case, \|J\| = 2 exp(η 1 + η 2 ). The other transformed coordinates used were (z 1 , z 2 ) = (log(φ) + η 1 , log(φ) − η 1 ). Note that log(φ) + η 1 = log(− log((1 − IR) φ )). Hence z 1 represents the more strongly identified quantity, so we wish to improve mixing by sampling in the z 2 direction. We have the same formulation as above, with \|J\| = 2/φ. Figure 1 . 1Now say the investigator pre-specifies (φ k ,φ k ) = (1, 40) for all k. As such, the a priori bounds are correct, for all jurisdictions. The resulting jurisdiction-specific identification intervals, I k , are depicted in the bottom left-hand panel ofFigure 1.(The top and middle left-hand panels correspond to the identical situation but with φ k values listed in the γ = 4 and γ = 11 columns ofTable 1respectively.) Also Black lines correspond to jurisdiction-specific identification intervals and the green rectangle corresponds to the global identification interval. Left-hand panels correspond to assumption ofτ = 0 such the global identification interval is simply the intersection of the individual intervals. Right-hand panels correspond toτ = 0.002. R-code to reproduce: https://tinyurl..) Thus, depending on the range and heterogeneity in φ k values, it appears that data can contribute substantial information about the (constant) infection fatality rate. Figure 1 ( 1right-hand panels) shows how the global identification interval is wider when τ = 0.002. For reference, for the IFR values listed inTable 1, τ = SD(IF R 1:12 ) = 0.00124. For the γ = 4, γ = 11, and γ = 22 scenarios, the global identification intervals outlined by the green rectangles are [0.0139, 0.1483], [0.0137, 0.0670] and [0.0137, 0.0386], respectively. Details for these calculations are presented in the Appendix. simulated datasets with K = 20 and k = 8. For k = 1, . . . , 8, population sizes were obtained from a N egBin(20000, 1) distribution and for k = 9, . . . , 20, population sizes were obtained from a N egBin(200000, 1) distribution. Parameter values were as in the illustrative example: θ = cloglog(0.02) = −3.90, β = cloglog(0.20) = −1.50, τ 2 = 0.005 and σ 2 = 0.25. The testing rate for each population was obtained from a U nif orm(0.01, 0.10) distribution so that the proportion of tested individuals in each population ranged from 1% to 10%. We considered eight values of interest for γ: 0, 1, 4, 11, 22, 34, 52, and 80; and three different values of interest for λ: 0.25, 0.50, and 0.75. The number of confirmed cases (CC k ) were simulated from Wallenius' non-central hyper-geometric distribution as in expression (3). M 1 and M 2 models are fit 24 different times (= 3 × 8) for each unique dataset, each time with λ assuming one of the three values of interest, and with one of the eight different sets of CC k numbers (for k in 9, ..., 20), corresponding to the eight γ values. For each case, we recorded the width of the 90% HDP CI for θ and whether or not the CI contained the target value of cloglog(0.02) = −3.90. We specifically chose to conduct 200 simulation runs so as to keep computing time within a reasonable limit while also reducing the amount of Monte Carlo standard error to a reasonably small amount (for looking at coverage with 1 − α = 0.90, Monte Carlo SE will be approximately 0.90(1 − 0.90)/200 ≈ 0.02); see Morris et al. (2019). For each simulation scenario, we used nimble to obtain 100,000 MCMC draws from the posterior (20% burn-in, thinning of 50). Figure 2 2plots the simulation study results for Study A, where the φ k values are evenly spaced between 1 and γ + 1. First of all, we note that coverage for model M 3 (based on only the unbiased data) is 0.910 as expected; see dotted black line on Figure 2 lower-left panel. The average interval width is 0.197; see dotted black line on Figure 2 lower-right panel. Results from the M 1 model (upper panels, dashed lines) show that the model provides at or above nominal coverage for a wide range of γ values. This Figure 3 . 3Simulation study results for Study B. Left-hand panels correspond to frequentist coverage and right-hand panels plot average interval width. The dotted lines corresponds to the M 1 model, the solid lines correspond to the M 2 model and the dashed line corresponds to the M 3 model. R-code to reproduce: https: //tinyurl.com/yc2z776c to the M 3 model which only uses data from those samples where testing is known to be representative/random. For instance, the M 3 model provides average coverage of 0.910, and an average interval width of 0.197. In contrast, when γ = 4 and λ = 0.5, the M 2 model provides average coverage of 0.915, and an average interval width of 0.183. However, there is a limit to the "added value" that the "non-representative" data provide. For example, for γ = 80, M 3 intervals are narrower compared to M 2 intervals (for all λ). Figure 3 3plots the simulation study results for Study B, where the φ k values Figure 4 . 4A selection of diagnostic plots for the MCMC simulation of the model with γ = 0. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 5 . 5A selection of diagnostic plots for the MCMC simulation of the model with γ = 0. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 6 . 6A selection of diagnostic plots for the MCMC simulation of the model with γ = 4. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 7 . 7A selection of diagnostic plots for the MCMC simulation of the model with γ = 4. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 8 . 8A selection of diagnostic plots for the MCMC simulation of the model with γ = 11. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 9 . 9A selection of diagnostic plots for the MCMC simulation of the model with γ = 11. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 10 . 10A selection of diagnostic plots for the MCMC simulation of the model with γ = 22. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Figure 11 . 11A selection of diagnostic plots for the MCMC simulation of the model with γ = 22. The left panels report trace plots from the posterior to check convergence. The right panels report the corresponding posterior distribution estimate (black solid line) together with the prior distribution for that parameter (red solid line). The % overlap reported in red is the PPO (prior-posterior overlap). Table 1 1provides a small artificial dataset to help illustrate the type of data being described and the impact of different degrees of preferential testing. In this dataset,we have K = 12 groups and the (unknown) infection rate varies substantially from 13% to 53%. The unknown infection fatality rate only varies slightly, from 0.017% to 0.022%. Values for φ k in this dataset are evenly distributed between 1 and γ + 1, for four different values of γ = 0, 4, 11, and 22. When γ = 0, the number of true cases (i.e. actual infections) is approximately 14 times higher than the number of confirmed cases. In contrast, when γ = 22, the number of true cases is only about 5 times higher than the number of confirmed cases. Observed, γ = 0 4 11 22 Unobserved, γ = 4 11 22 k Table 1. Illustrative Example Data, with varying degrees of preferential sampling, γ = 0, γ = 4, γ = 11, and γ = 22. R-code to reproduce: https://tinyurl.com/y7gmnpobk 1 3061 190 11 24 21 32 27 430 0.140 0.018 1.00 1 1 2 482 43 2 15 11 12 24 99 0.206 0.020 1.36 2 3 3 1882 101 20 32 40 55 74 570 0.303 0.022 1.73 3 5 4 1016 67 2 14 24 33 38 193 0.190 0.017 2.09 4 7 5 1269 109 4 13 34 54 67 201 0.159 0.021 2.45 5 9 6 3670 276 9 53 70 140 162 484 0.132 0.021 2.82 6 11 7 2409 139 7 17 34 70 94 329 0.137 0.019 3.18 7 13 8 1074 81 13 42 65 68 77 565 0.526 0.019 3.55 8 15 9 3868 289 16 60 142 205 247 821 0.212 0.019 3.91 9 17 10 151 13 2 1 5 11 8 24 0.160 0.019 4.27 10 19 11 430 25 1 6 9 16 18 70 0.164 0.019 4.64 11 21 12 429 40 2 11 23 31 33 105 0.245 0.019 5.00 12 23 Table 2. Illustrative Example Data -Posterior medians and 95% HPD credible intervals. R-code to reproduce: https://tinyurl.com/yb2gf9nk independent chains, each with 500,000 draws (20% burnin, thinning of 50). The results, including posterior medians and highest posterior density (HPD) 95% credible intervals (CI), are listed inTable 2. In the Appendix, Figures 4-11 plot diagnostics for the MCMC; the prior-posterior overlap numbers suggest that the data carry substantial information content about the magnitude of the φ variables.When γ = 0, note that the 95% CI for θ is much wider with M 1 compared to with M {γ=0} . This reflects the additional uncertainty of not knowing about the absence/presence of preferential testing. Otherwise, when γ > 0, we see that ignoring preferential testing has significant consequences. With the M {γ=0} model, the 95% CI for θ fails to include the target when γ > 0. When γ = 22, the M 1 model also fails to include the target within the 95% CI for θ. This suggests that, with limited data,Truth M 1 M {γ=0} 2.5% 50% 97.5% 2.5% 50% 97.5% γ = 0 θ -3.902 -4.000 -3.511 -2.808 -4.051 -3.821 -3.577 β -1.500 -2.567 -1.745 -1.161 -1.775 -1.401 -1.034 τ 0.071 0.000 0.067 0.193 0.000 0.070 0.198 σ 0.50 0.290 0.548 0.899 0.308 0.545 0.866 γ 0 0.001 1.120 5.636 γ = 4 θ -3.902 -4.347 -3.796 -3.213 -4.538 -4.306 -4.071 β -1.500 -2.108 -1.426 -0.649 -1.213 -0.800 -0.391 τ 0.071 0.000 0.082 0.222 0.000 0.115 0.255 σ 0.50 0.385 0.639 1.037 0.397 0.648 1.014 γ 4 0.006 3.132 8.150 γ = 11 θ -3.902 -4.511 -4.092 -3.662 -4.978 -4.723 -4.482 β -1.500 -1.646 -1.065 -0.436 -0.621 -0.218 0.191 τ 0.071 0.000 0.079 0.224 0.000 0.175 0.310 σ 0.50 0.341 0.626 1.048 0.397 0.651 1.015 γ 11 1.732 5.968 12.304 γ = 22 θ -3.902 -4.797 -4.349 -3.946 -5.135 -4.894 -4.625 β -1.500 -1.426 -0.802 -0.152 -0.406 0.044 0.447 τ 0.071 0.001 0.096 0.248 0.008 0.187 0.331 σ 0.50 0.391 0.673 1.082 0.422 0.699 1.066 γ 22 1.558 7.023 15.644 there is an upper bound on the degree of preferential testing for which the model can adjust. 4. A Bayesian model for large-P data, MCMC details, and a simulation study 4.1. A Bayesian model for large-P data When populations are large, we can simplify our model (in order to reduce the compu- tational complexity), by replacing the non-central hyper-geometric distribution with a binomial distribution as follows: Simulation study results for Study A. Left-hand panels correspond to frequentist coverage and right-hand panels plot average interval width. The dotted lines corresponds to the M 1 model, the solid lines correspond to the M 2 model and the dashed line corresponds to the M 3 model. R-code to reproduce: https: //tinyurl.com/y9dfxcno0.0 0.5 0.8 0.9 1.0 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ coverage 0.2 0.4 0.6 0.8 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ width λ 0.25 0.5 0.75 0.0 0.5 0.8 0.9 1.0 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ coverage 0.125 0.150 0.175 0.200 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ width Model M1 M2 M3 Figure 2. 0.0 0.5 0.8 0.9 1.0 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ coverage 0.2 0.4 0.6 0.8 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ width λ 0.25 0.5 0.75 0.0 0.5 0.8 0.9 1.0 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 γ coverage 0.125 0.150 0.175 0.200 0.225 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 0 1 4 11 22 34 52 80 The proposed model can be expanded in several different ways. 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[ "Factor-Based Imputation of Missing Values and Covariances in Panel Data of Large Dimensions", "Factor-Based Imputation of Missing Values and Covariances in Panel Data of Large Dimensions" ]	[ "Ercument Cahan [email protected] ", "Jushan Bai \nDepartment of Economics\nColumbia University\n420 W. 118 St. MC 330810027New YorkNY\n", "Serena Ng [email protected]. \nDepartment of Economics Columbia University and NBER\n420 W. 118 St. MC 330810027New YorkNY\n" ]	[ "Department of Economics\nColumbia University\n420 W. 118 St. MC 330810027New YorkNY", "Department of Economics Columbia University and NBER\n420 W. 118 St. MC 330810027New YorkNY" ]	[]	Economists are blessed with a wealth of data for analysis, but more often than not, values in some entries of the data matrix are missing. Various methods have been proposed to handle missing observations in a few variables. We exploit the factor structure in panel data of large dimensions. Our tall-project algorithm first estimates the factors from a tall block in which data for all rows are observed, and projections of unit specific length are then used to estimate the factor loadings. A missing value is imputed by its estimated common component which we show is consistent and asymptotically normal without further iteration. Implications for using imputed data in factor augmented regressions are then discussed. To compensate for the downward bias in sample covariance matrices created by an omitted noise in each imputed value, we overlay the imputed data with re-sampled idiosyncratic residuals many times and use the average of the covariances to estimate the parameters of interest. Simulations show that the procedures have desirable finite sample properties.	10.1016/j.jeconom.2022.01.006	[ "https://arxiv.org/pdf/2103.03045v3.pdf" ]	232,110,899	2103.03045	d9a7c40f44a99809c9132959c9bc649bfa59602b	Factor-Based Imputation of Missing Values and Covariances in Panel Data of Large Dimensions 1 Feb 2022 February 2, 2022 Ercument Cahan [email protected] Jushan Bai Department of Economics Columbia University 420 W. 118 St. MC 330810027New YorkNY Serena Ng [email protected]. Department of Economics Columbia University and NBER 420 W. 118 St. MC 330810027New YorkNY Factor-Based Imputation of Missing Values and Covariances in Panel Data of Large Dimensions 1 Feb 2022 February 2, 2022We thank Bennie Chen for excellent research assistance. This paper was presented at 2021 SOFIE webinar, the 2021 Econometric Society Asia meeting, and the 2021 Statistics Canada meeting, and the econometrics and statistics seminar at Chicago Booth. We thank seminar participants, Markus Pelger, Andrew Patton, Dacheng Xiu, two anonymous referees, the associate editor and co-editor (Torben Andersen) for helpful comments. Ng acknowledges financial support from the National Science Foundation SES-1558623 and SES-2018369.risk managementcovariance structurematrix completionincomplete data JEL Classification: C1C2 * Independent researcher Economists are blessed with a wealth of data for analysis, but more often than not, values in some entries of the data matrix are missing. Various methods have been proposed to handle missing observations in a few variables. We exploit the factor structure in panel data of large dimensions. Our tall-project algorithm first estimates the factors from a tall block in which data for all rows are observed, and projections of unit specific length are then used to estimate the factor loadings. A missing value is imputed by its estimated common component which we show is consistent and asymptotically normal without further iteration. Implications for using imputed data in factor augmented regressions are then discussed. To compensate for the downward bias in sample covariance matrices created by an omitted noise in each imputed value, we overlay the imputed data with re-sampled idiosyncratic residuals many times and use the average of the covariances to estimate the parameters of interest. Simulations show that the procedures have desirable finite sample properties. Introduction Missing data is a problem that empirical researchers frequently encounter. For example, data can be missing due to attrition in longitudinal surveys such as the PSID and SIPP, to IPOs and bankruptcies in the case of stock returns, and to staggered releases of data by different institutions. Data sampled at a lower frequency will have missing values in an analysis that involves higher frequency data. Whatever is the reason, the way we handle missing data is not innocuous. Dropping rows or columns necessarily entails loss of information, and while the EM algorithm allows imputation of missing values, it is designed for low dimensional data and may not be easily scalable. In recent years, progress has been made for situations when the missing values occur in large panels of data with a low rank component. Assuming incoherence conditions to ensure that the low rank structure is non-degenerate as in Candes and Recht (2009), the machine learning literature has developed regularization based algorithms to solve matrix completion problems of the Netflix challenge type. On the econometrics front, recent work by Jin et al. (2021) Xiong and Pelger (2019), and Bai and Ng (2021) propose different implementations that permit the entire common (low rank) component to be consistently estimated if the factor structure is strong. While the machine learning literature provides worst-case error bounds, the econometrics literature provides the distribution theory that is needed for inference, and is also the approach that this paper takes. We introduce a Tall-Project estimator (or TP for short) for imputing missing values in a panel of data with T rows and N columns, where N and T are both large. The factors are estimated from a tall block consisting of complete data for N o ≤ N units, while the loadings are obtained from N time series projections with series-specific sample size. We obtain two main results. First, under certain assumptions on the factor structure in the different blocks, we show that the tp estimates are consistent and asymptotically normal for every entry of the low rank component though the convergence rate is series specific. And while iteration is not needed for consistent estimation, one re-estimation can improve the convergence rate. Because of missing data, stronger conditions are needed for the factor estimates to be treated as known in factor-augmented regressions. 1 Our second result concerns estimation of covariances from imputed data. Covariances play an important role in portfolio analysis and in structural equation (covariance structure) modeling. While the imputed data is unbiased for its mean, the variance of the imputed data is biased because the idiosyncratic noise associated with the missing observations are set to zero. To remedy this problem, we repeatedly overlay the first step estimate of missing values with resampled idiosyncractic residuals before computing sample covariances. An average of these imputed covariances is then used to estimate the objects of interest. In simulations calibrated to CRSP data from 1990-2018, resampling from the own residuals (ie. without pooling) yields risk measures that compare favorably and sometimes outperform the ones based on a factor-based covariance estimator. In what follows, let X be a T × N panel of data, X i = (X i1 , . . . X iT ) ′ be a T × 1 vector of random variables and X = (X 1 , X 2 , . . . , X N ) be a T × N matrix. We use i = 1, . . . N to index cross-section units and t = 1, . . . T to index time series observations. In practice, X i is transformed to be stationary, demeaned, and is often standardized. It is assumed that the normalized data Z = X √ N T has singular value decomposition (svd) Z = X √ N T = U DV ′ = U r D r V r ′ + U n−r D n−r V n−r ′ where D r is a diagonal matrix of r singular values ordered such that d 1 ≥ d 2 . . . ≥ d r , while U r , V ′ r are the corresponding left and right singular vectors respectively. Analogously, U n−r and V ′ n−r are n × (n − r) matrices of left and right eigenvectors associated with d r+1 , . . . , d n . The low rank component U r D r V ′ r can be defined without probabilistic assumptions.. We consider a factor model defined as X = F Λ ′ + e(1) where F is a T × r matrix of common factors, Λ is a N × r matrix of factor loadings, and e is a T × N matrix of idiosyncratic errors. We will let (F 0 , Λ 0 ) be the true values of (F, Λ). The common component C 0 = F 0 Λ 0 ′ has reduced rank r because F 0 and Λ 0 both have rank r. We will estimate the factors and loadings by the method of static asymptotic principal components (APC). 2 The normalization F ′ F T = I r gives the APC estimates that are unique up to a column sign: (F ,Λ) = ( √ T U r , √ N V r D r ). Since r can be consistently estimated, we proceed as though r is known. We consider a large dimensional approximate factor model in which T and N are large and which satisfies the following assumptions: (iii) E(e it e jt ) = τ ij,t , \|τ ij,t \| ≤ \|τ ij \| for some τ ij ∀t, and N j=1 \|τ ij \| ≤ M , ∀i; (iv) E(e it e js ) = τ ij,st and 1 N T N i=1 N j=1 T t=1 T s=1 \|τ ij,ts \| < M ; (v) E\|N −1/2 N i=1 [e is e it − E(e is e it )] 4 ≤ M for every (t, s); (vi) E( 1 √ T T t=1 F 0 t e it 2 ) ≤ M, ∀i, and E( 1 √ N T N i=1 T t=1 Λ 0 i F 0′ t e it 2 ) ≤ M . c. (Central Limit Theorems): for each i and t, 1 √ N N i=1 Λ 0 i e it d −→N (0, Γ t ) as N → ∞, and 1 √ T T t=1 F 0 t e it d −→N (0, Φ i ) as T → ∞; the two limiting distributions are independent. Assumption A is used in Bai (2003) to develop an inferential theory for large dimensional factor analysis when X is completely observed. The moment conditions in (b) ensure that the factor structure is strong and can be separated from the idiosyncratic errors which can be weakly correlated, both in the time and the cross-section dimensions. Let D 2 r and V r be the eigenvalues and eigenvectors of the r×r matrix Σ 1/2 Λ Σ F Σ 1/2 Λ , respectively. As shown in Bai (2003), plim N,T →∞ D 2 r = D 2 r and plim N,T →∞F ′ F 0 T = Q r , where Q r = D r V r Σ −1/2 Λ . The following properties of the APC estimator are given in Bai (2003): Lemma 1. Suppose that Assumption A holds and let H = (Λ 0 ′ Λ 0 /N )(F 0 ′F /T )D −2 r . If √ N /T → 0 as N, T → ∞, then √ N (F t − H ′ F 0 t ) d −→ N 0, D −2 r Q r Γ t Q r ′ D −2 r (2a) √ T (Λ i − H −1 Λ 0 i ) d −→ N 0, (Q r ′ ) −1 Φ i Q −1 r (2b) min( √ N, √ T ) C it − C 0 it ṼC it d −→ N (0, 1). (2c) whereṼC it (N, T ) is a consistent estimate of δ 2 NT N Λ 0′ i Σ −1 Λ Γ t Σ −1 Λ Λ 0 i + δ 2 NT T F 0′ t Σ −1 F Φ i Σ −1 F F 0 t . In what follows, we will obtain results for the factor estimates when X has missing values. To overcome the bias when the data are missing at random, one approach is to re-weigh the data (such as 'inverse probability weighting' used in survey design). A second approach is to rebalance the panel by amputation which can take one of three forms: (a) listwise (complete case) deletion under which the entire row with missing values is eliminated; (b) variable deletion under which a series with one or more missing values will be eliminated; and (c) pairwise deletion where only the cases with missing data involved will be deleted. Though the solution is simple and does not entail modeling assumptions, amputation leads to significant information loss. Furthermore, the resulting balanced sample may not be representative of the population and may yield biased estimates. Missing Data Another way to rebalance the panel is imputation. The simplest procedure is to replace the missing values with the sample mean or median. A more sophisticated procedure is to impute from the unconditional sample first and second moments via the EM algorithm. Schneider (2001) proposes a regularized EM algorithm that is quite popular in analysis of climate data. Imputation on the basis of unconditional moments does not require a model. A fully specified approach is to construct the likelihood using incomplete observed data and iteratively solve for the parameters of the conditional mean function. The estimates would be more efficient if the parametric assumptions were correct. These methods are designed for imputing missing values in a small number of variables/predictors. See Robins et al. (1995) Li et al. (2013) and Raghunathan (2004) for a review. Factor models provide a simple framework for imputing missing values of many variables. Banbura and Modugno (2014), consider state-space modeling of a strict factor (i.e., a diagonal idiosyncratic error variance matrix) with missing data and uses non-linear filters to compute the likelihood as F t and Λ i are both random. Giannone et al. (2008) initializes the missing values with estimates from the balanced panel and uses the Kalman filter to perform updating. Stock and Watson (2016) discusses the issues with state space estimation of factor models with missing data. For large dimensional approximate factor models, Stock and Watson (1998) suggests to fill missing values in X with the APC estimates of the common component. These are all EM algorithms that use the factor structure to evaluate the conditional mean in the E-step, and principal components estimation in the M step. Though many factor-based imputation methods have been used for some time, the theoretical properties of the imputed estimates are studied only recently in independent works by Jin et al. (2021), Xiong and Pelger (2019), and Bai and Ng (2021). All three exploit the strong factor structure in a large panel setting to estimate the factors and the loadings from incompletely observed data. Jin et al. (2021) focuses on the case of missing at random and analyzes the EM estimator considered in Stock and Watson (1998). After initializing the missing values to zero, the factors are estimated from data reweighted by the frequency of no missing values. Though these estimates are consistent, they are not asymptotically normal without further iteration. Xiong and Pelger (2019) also re-weights the data in principal components estimation of the loadings, but uses cross-sectional regressions to estimate the factors at every t. Similar to Jin et al. (2021), the estimates are consistent and asymptotically normal when iterated until convergence. but the 'all purpose' estimator proposed in Xiong and Pelger (2019) allows for mechanisms other than missing at random. Though robustness comes at the cost of larger variances, efficiency can be improved if stronger assumptions on missingness are made. Jin et al. (2021) and Xiong and Pelger (2019) reweigh the data in APC estimation, Bai and Ng (2021) implicitly re-oganizes the data into blocks. This is partly motivated by the concern that the missing at random assumption may at times be inappropriate. Examples of microeconomic applications are discussed in Athey et al. (2018). Furthermore, for macroeconomic panels such as FRED-MD where the data are collected from different sources, assuming that the missing values are due to the same missing mechanism seems restrictive. X 1 . . . X No X N o+1 . . . , X N −1 X N x x x x x x x x x X 1 . . . X No X N o+1 . . . , X N −1 X N x x x x x While Consider the left panel of Figure 1. This systematic (non-random) missing pattern can arise if, for example, weekly data for related variables (such as inventories and orders) are released in the same week of every month. The right panel shows a case of 'semi-block missing'. This can arise if, for example in a survey, respondent N dropped out early, respondent N − 1 did not respond in period T − 1, while N o + 1 did not respond in period T . The observations can be missing for different reasons likely unknown to the researcher. While the missing data mechanism can be difficult to verify, we do observe the missing data pattern, and organizing the data into blocks offers a new way of thinking about imputation. The insight in Bai and Ng (2021) is that the quantities needed for imputation can be obtained from two completely observed blocks. Their tall-wide (or tw for short) procedure uses the tall block consisting of complete data for N o < N units to estimate the factors, and a wide block in which data for all units are available over T o < T periods to estimate the factor loadings. To align the space spanned by the estimated factors and loadings, one then estimates a new rotation matrix by regressing the N o × r matrix of loadings estimated from the tall block on a sub-block of N × r matrix of loadings estimated from the wide block. The procedure produces consistent and asymptotically normal estimates without iteration. Re-estimation using the completed (imputed) data will improve efficiency of the estimates in the balance block where the tall and wide blocks intersect to the fastest rate possible, which is the rate obtained in the complete data case. x x x x x x x X1 . . . XN o XN o+1 . . . XN−1 XN x x x x x The tw algorithm takes as given that the tall and wide blocks are 'big enough' for consistent estimation. Because the size of the missing data block is defined by number of time periods and units with complete cases, the size of the missing block is the same for both examples in Figure 1. The algorithm uses information available efficiently when the missing pattern is homogeneous. But for the example in the right panel, more information could have been used. More concerning are situations when a few series significantly reduce N o and T o . As an example, A Projection Based Procedure for Imputing X In this section, we will develop a new procedure to obtain consistent estimates without iteration as in tw, but can accommodate flexible missing data patterns while making more efficient use information available. The point of departure is to partition the T × N matrix X in a different way. We assume that there are N o series with no missing values so there is a T × N o block labeled tall, and a T × (N − N o ) block of partially observed data labeled incomplete. There is no need to reorganize the data in practice, but Figure 3 shows the data with the N o variables ordered first to help visualize the idea. tall incomplete × T × N o T × (N − N o ) × × × × × × × × × × × ×                                           It will be helpful to define two locator sets as follows: J t := {j : X jt observed, i.e. units with data in period t} J i := {s : X is observed, i.e. periods with data for unit i} Let N ot be the number of units observed at time t and T o i be the number of rows observed for series i. Then J t is a N ot × 1 vector that keeps track of the units observed at t, and J i is a T o i × 1 vector that keeps track of the periods that unit i is observed. For a set A, let \|A\| be its cardinality. In this notation, N o = \| ∩ t J t \|, T o = \| ∩ i J i \|, and T o i = T for every unit in tall. Algorithm Tall-Project (TP): i. Estimate the T × r matrixF from the tall block by APC and letF t of dimension r × 1 be the t-th row ofF . ii. For each i, regress the T o i × 1 vector consisting of the observed values of X i on the corre- sponding T o i × r submatrix ofF to obtainΛ i . iii. For each (i, t), letC it =F ′ tΛi andẽ it =X it −C it wherẽ X it = X it if X it observed C it if X it missing. Step (i) of tp is the same as tw since both procedures estimate F from the tall block. If such a block does not exist, then neither tw or tp will be appropriate. Fortunately, for macroeconomic panels, a tall block is often available. Algorithm tp accommodates staggered and irregular patterns of missingness by estimating Λ i using a customized sample for each series. The two algorithms should be numerically identical if T o i = T o for all i, but since N ot ≥ N o and T o i ≥ T o , tp will utilize more information in general. The difference can be significant if T o is dictated by a few series with many missing observations. Note, however, that tp estimates the HΛ i matrix directly, while tw estimates H and Λ i separately. Furthermore, the number of factors in Algorithm tp is determined by the tall block so Algorithm tw is more flexible in this regard. We will need additional assumptions to analyze the properties of tp. Assumption B: √ N min{No,To} → 0 and √ T min{No,To} → 0 as N → ∞ and T → ∞. Assumption C: There exists M < ∞ such that for all N and t, No t (N −No t ) N No ≤ M . Assumption D: For each i, 1 T −To i s / ∈J i F 0 s F 0′ s 1 To i s∈J i F 0 s F 0′ s −1 p −→I r , 1 √ To i s∈J i F 0 s e is d −→N (0, Φ i ). For each t, 1 N −No t k / ∈J t Λ 0 k Λ 0′ k 1 No k∈∩sJ s Λ 0′ k Λ 0 k −1 p −→I r , 1 √ No t k∈J t Λ 0 k e kt d −→N (0, Γ t ). Assumption B essentially puts a lower bound on the observed number of rows and columns. Assumption C puts an upper bound on the extent of missing values at any t and can be understood as a noise-to-signal constraint. Assumption C allows N ot /N → 0, and especially N o /N → 0. For example, if N ot = N o , then the ratio is bounded by 1. For the central limit theorems in Assumption D to hold, T o i should be independent of (F 0 s , e s ), and N ot independent of (Λ 0 k , e k ). The assumption is satisfied if the missing data are unrelated to the intrinsic properties of the series. For example, missing data due to mergers are allowed if the events are unrelated to F 0 t . However, missing data due to bankruptcies would not be allowed if the bankruptcy probability depends on F 0 t . These conditions are stronger than stationarity and need to be justified on an application by application basis. All factor-based imputation procedure requires a similar assumption because without some commonality between the observed and missing units, imputation would not be possible. Lemma 2. (First Pass Estimation) Let N o be the number of units in the tall block and T o i be number of periods that unit i is observed. Let H tall be a rotation matrix based on the tall block of the data. Under Assumptions A-D, the tp estimates (F ,Λ) have the following properties : i. √ N o (F t − H ′ tall F 0 t ) d −→N 0, D −2 r Q r Γ t Q ′ r D −2 r , for t ∈ [1, T ], if √ No T → 0 ii. (a) √ T (Λ i − H −1 tall Λ 0 i ) d −→N 0, (Q ′ r ) −1 Φ i )Q −1 r , for i ≤ N o , if √ T No → 0 (b) T o i (Λ i − H −1 tall Λ 0 i ) d −→N 0, (Q ′ r ) −1 Φ i )Q −1 r , for i > N o , if √ To i No → 0 iii. LetṼ it (N o , T o ) be a consistent estimate of V it = δ 2 No,To i No Λ 0′ i Σ −1 Λ Γ t Σ −1 Λ Λ 0 i + δ 2 No,To i To F 0′ t Σ −1 F Φ i Σ −1 F F 0 t where δ 2 No,To i = min(N o , T o i ). Then min( N o , T o i ) C it − C 0 it Ṽ it (N o , T o i ) d −→N (0, 1). Part i and ii(a) are implied by Lemma 1 becauseF t andΛ i are estimated from the tall block, which is a complete matrix of T × N o dimension. Result (b) of part ii arises because the factor loadings for unit i in the incomplete block are estimated from a regression with T o i observations. Part (iii) shows that eachC it has its own convergence rate that depends on T o i , the number of observed entries in X i . The result is based on the fact for those with X it that are missing, C it − C 0 it = Λ 0′ i 1 N o k∈∩sJ s Λ 0 k Λ 0′ k −1 1 N o k∈∩sJ s Λ 0 k e kt + F 0′ t 1 T o i s∈J i F 0 s F 0′ s −1 1 T o i s∈J i F 0 s e is + r it ∆ = u it + v it + r it .(3) The error due to estimating F from the tall block appears as the first term on the right hand side and is denoted u it , while the error due to estimating Λ i by projections is summarized by the second term and is denoted v it . The term r it = O p (δ −2 To,No ) uniformly in i and t represents high order errors when estimating the factor and factor loadings. These quantities depend on N and T but the notation is suppressed for simplicity. The representation (shown in the Appendix) is useful in understanding how N and T affect factor-based imputation. Re-estimation usingX AsF is estimated using the tall block alone, re-estimation of the factors and the loadings from the completed (imputed) matrixX appears desirable. But imputation error inX must be taken into account. Using (3), we havẽ X it = Λ 0′ i F 0 t + e it , if X it observed (4a) X it = Λ 0′ i F 0 t + u it + v it + r it if X it missing. (4b) A quantity that will play a role in APC estimation fromX is B i t =      No t N I r + No t N N −No t No 1 N −No t N k / ∈J t Λ 0 k Λ 0′ k 1 No k∈∩sJ s Λ 0 k Λ 0′ k −1 i ∈ J 1 ∩ · · · ∩ J T No t N I r i ∈ J t \ (J 1 ∩ · · · ∩ J T ) Under Assumption C, B i t is bounded. For units in the tall block (i.e., i ∈ ∩ s J s ), B i t is roughly inflated from No t N I r by N ot (N − N ot )/(N N o ) which can be thought of as the noise to signal ratio.. Proposition 1. Let (F + ,Λ + ) = ( √ TŨ r , √ NṼ rDr ) whereŨ r ,i N ot (F + t − H + ′ F 0 t ) d −→N (0, D −2 r Q r Γ * t Q ′ r D −2 r ) ii T o i (Λ + i − (H + ) −1 Λ 0 i ) d −→N (0, (Q ′ r ) −1 Φ i (Q r ) −1 )) iii Suppose that e it is cross-sectionally uncorrelated. LetC + it =Λ +′ iF + t . Then, for all (i, t). min( N ot , T o i ) C + it − C 0 it Ṽ + it (N ot , T o i ) d −→N (0, 1) whereṼ + it (N ot , T o i ) is a consistent estimate of V + it = δ 2 No t To i N ot Λ 0′ i Σ −1 Λ Γ * t Σ −1 Λ Λ 0 i + δ 2 No t To i T o i F 0′ t Σ −1 F Φ i Σ −1 F F 0 t , δ 2 No t ,To i = min(N ot , T o i ) and Γ * t = plim 1 No t i∈J t B i t Λ 0 i Λ 0 ′ i B i t ′ e 2 it . The proposition establishes the convergence rate of the factors and loadings constructed from X. There are three changes due to re-estimation. First, there is only one (instead of many) rotation matrices for the factor loadings. This is a consequence of the fact that the factors are now estimated fromX in its entirety, instead of sub-blocks. Second, re-estimation generates efficiency gains. The convergence rate forF + t is improved from √ N o to N ot or to √ N . Though the rate forΛ + is unchanged, the convergence rate ofC + it is now min( N ot , T o i ) instead of min( √ N o , T o i ). If N ot = N and T o i = T for a given pair (i, t), the convergence rate forC + it is min( √ N , √ T ), the best rate possible (the same as in complete data). Third, the asymptotic variance ofC + it depends on the r × r matrix B i t . From this result, a 95% confidence interval for C 0 it ofC + it ± 1.96se(C + it ) wherese(C + it ) = Ṽ+ it (No t ,To i ) min(No t ,To i ) . The prediction interval for a missing X it isX + it ± 1.96se(X + it ) wherẽ se(X + it ) = (σ 2 ei +se(C + it ) 2 ) 1/2 ,σ 2 ei = 1 To i s∈J iẽ 2 is is an estimate of the variance of e it . Implications for Factor Augmented Regressions Consider the infeasible regression with observed covariates W t and a latent predictor F t : y t+h = α ′ F t + β ′ W t + ǫ t+h (5) = δ ′ z t + ǫ t+h where z t = (F ′ t , W ′ t ) ′ ) and δ = (α ′ , β ′ ) ′ . Suppose that F t is replaced by an estimate based on a small number of predictors. It is known that in general, the sampling uncertainty in a generated regressor will inflate standard errors in subsequent regressions. A useful result in Bai and Ng (2006) is that if F is estimated by APC from a completely observed panel X, thenF can be used in a second step regression without the need for standard error adjustments if certain conditions on the sample size are satisfied. Our foregoing analysis suggests that even if X has missing values,F + estimated by tp is still consistent for the space spanned by F albeit at a slower convergence rate. Lemma 3. Suppose Proposition 1 holds. Then (i) 1 T T t=1 F + t −H ′ N T F t 2 = O p (min[N o , T o ] −1 ). Letẑ t = (F +′ t , W ′ t ) ′ be used in place of z t in the factor-augmented regression (5), and denote δ 0 = (α ′ H ′−1 N T , β ′ ) ′ . If √ T /N o → 0 and √ T /T o → 0, then (ii) √ T (δ − δ 0 ) d −→N (0, J ′−1 Σ −1 zz Σ zz,ǫ Σ −1 zz J −1 ). (6) where J is the probability limit of J N T = diag(H ′ N T , I dim(W ) ). Part (i) is based on Lemma 3 of Bai and Ng (2021) obtained for tw which can be used as a worse case rate for tp since T o in tw cannot exceed min i T oi in tp. A consequence of (i) is that 1 √ T T t=1ẑ t (F 0′ t H N T −F +′ t ) = O p √ T min(N o , T o ) , 1 √ T T t=1 ǫ t+hF + t = 1 √ T T t=1 ǫ t+h H ′ N T F 0 t + 1 √ T T t=1 (F + t − H ′ N T F 0 t )ǫ t+h = 1 √ T T t=1 ǫ t+h H ′ N T F 0 t + O p √ T min(N o , T o ) . These results are used to obtain (ii). In particular, let Sẑẑ = 1 T T t=1ẑ tẑ ′ t . Then √ T (δ − δ 0 ) = S −1 z ′ẑ 1 √ T T t=1ẑ t ǫ t+h + S −1 z ′ẑ 1 √ T T t=1ẑ t α ′ H ′−1 N T (H ′ N T F 0 t −F + t ) ′ = S −1 z ′ẑ J N T 1 √ T T t=1 z t ǫ t+h + o p (1) provided √ T No → 0, √ T To → 0. If, in addition, 1 √ T T t=1 z t ǫ t+h d −→N (0, Σ zz,ǫ ), the asymptotic variance ofδ is of the sandwich form A −1 BA −1 with A = JΣ zz J and B = JΣ zz,ǫ J ′ , where J is the probability limit of J N T . Simplifying yields (ii). The result implies that as in Bai and Ng (2006), the estimated factors can be treated as though F were observed albeit under more stringent conditions than the complete data case which only requires that √ T /N → 0. For macroeconomic analysis when the factors are estimated from panels with large T o and N o , one can still expectδ to be precisely estimated up to a rotation. Factor Based Estimation of Covariance Matrices Consider portfolio analysis where X is a matrix of returns and Σ X is its covariance. The population weights of a minimum variance portfolio are given by w p = Σ −1 X 1 1′Σ −1 X 1 where 1 is the unit vector. If X was completely observed and x i = X i −X i , the sample moments (SM) estimator of the N × N matrix isΣ X = 1 T T t=1 x t x ′ t .(SM) It is well known that the sample covariance matrix is singular when N > T and has large sampling uncertainty when N and T are large. Nonetheless, when the data admit a factor structure so that the decomposition Σ X = ΛΣ F Λ ′ + Σ e holds, a factor-based covariance estimator can be defined aŝ Σ X =Λ ΣFΛ ′ +Σẽ whereΣẽ is the sample covariance ofẽ, which is never exactly diagonal even when Σ e is diagonal. As argued in Chamberlain and Rothschild (1983), an approximate factor model that allows for some correlation in the idiosyncratic errors is usually a better characterization of returns data. To distinguish a strict from an approximate factor structure, let Ψ e be a diagonal matrix whose i-th entry is E[e 2 it ] and which can be consistently estimated bỹ Ψẽ = diag 1 T T t=1ẽ 2 it , i = 1, 2, ..., N .(7) The strict-factor (SF) covariance estimator is defined as Σ X =ΛΣ FΛ ′ +Ψẽ (SF) Consistency of this estimator is studied by Fan et al. (2011), among others, for large N and large T , assuming that X is completely observed. We now turn to the incomplete data case. Missing data presents a challenge for risk management because the portfolio weights depend on Σ X . Variable deletion is not an option when the variances and covariances of the missing variables are the objects of interest. Practitioners often resort to listwise deletion of the sample data, deleting the entire record from the analysis if any single value is missing. The covariance estimates can be unstable when the reduction in the sample size is sufficiently large. Though it is possible to calculate the sample covariance matrix with pairwisecomplete observations, there is no guarantee that the resulting matrix will be positive definite. This is problematic because a singular covariance matrix must have at least one eigenvalue that is zero, implying that it would be possible to construct an eigenportfolio that has zero volatility (risk) by using the corresponding eigenvector as the weighs, making it possible to have an infinite ex-ante Sharpe ratio. Not only is this unrealistic, the hedges implied by the eigenportfolio weights are spurious and would fail out of sample. For these reasons, singular covariance matrices are of little use in portfolio construction in practice. Most risk applications require an estimate of a non-singular covariance matrix. The EM algorithm is one possibility, but it is likelihood based and requires parametric assumptions. Unfortunately, successful factor-based imputation of the level of the data is not enough for precise estimation of their covariances. Even if the strict factor structure is correctly imposed, both the sample-covariance estimator and factor-based covariance estimator based on the imputed data will not be consistent. The problem with both estimators is thatẽ it is set to zero when X it is not observed, so the sample variance of series i is biased whenever the series has missing values. We can replaceΨẽ in (7) ΣX − Σ X ∞ = max 1≤i,j≤N \|ΣX ,ij − Σ X,ij \| = o p (1). where for a matrix A with A ij as its (i, j) entry, A ∞ = max i max j \|A ij \|. The proof is given in the Appendix. SM Estimation and Double Imputation This subsection considers a sample covariance estimator as an alternative to the SFA covariance estimator just described. Motivated by the fact that the variance of an imputed variable is downward biased, we use a second imputation to rectify this problem. Algorithm Residual Overlay Letẽ =X −C whereX is produced by Algorithm tp. For residual resampling scheme j (j = 1, 2, 3, 4) to be discussed below, repeat for s = 1, . . . S: a For each i with missing data, replace thoseẽ it = 0 with aẽ it (s, j) randomly sampled from those non-zeroẽ it associated with observed X it . b. Defineê it (s, j) = ẽ it if X it observed e it (s, j) if X it is missing . c. LetX it (s, j) =Λ ′ iF t +ê it (s, j) andx it (s, j) =X it (s, j) −¯i X(s, j). Estimate the covariance of X(s, j) asΣX(s, j) = 1 T T t=1x t (s, j)x t (s, j) ′ . d. The N × N covariance estmator obtained fromX imputed by TP and overlaid with residuals sampled using scheme j isΣX (j) = 1 S S s=1ΣX (s, j).(SM+j) The term residual overlay is motivated by the fact that errors are added toX. The algorithm injects randomness toX it whenever X it is missing. It uses stochastic simulations to compensate for the lost variability, an idea briefly considered in Enders (2010, Chapter 5) in a fixed N or T setting. Even though eachΣX(s, j) is large in dimension and has rank min(N, T ), in our experience, the averageΣX is always full rank, making it a viable alternative estimator in the case when N > T . The method of multiple imputation typically estimates the object of interset each time an imputed set of data is obtained, and produces as output the average over the multiple estimates. In contrast, we average the multiple imputed covariances to obtainΣX and compute the object of interest (for example, portfolio weights) from it. Averaging has the advantage of reducing sampling uncertainty especially when N and T are both large. Step a: Four Sampling Schemes Methods sm+1, sm+2, sm+3, sm+4 are likewise defined withẽ it replaced byẽ + it =X it − F +′ tΛ + i , whereF + andΛ + are obtained from the principal components of the imputed dataX. In principle, we can also compute a SF estimatorΣX(j) =ΛΣ FΛ -(sm1) Let u be a N i=1 T o i vector,′ + 1 S S s=1Ψê (s, j) withΨê ii (s) = 1 T T t=1ê 2 it (s, j), but there seems no advantage over SFA which already gives a good estimate of the variance of e it when T oi is sufficiently large. This is indeed the case in simulations, and hence not considered. Though injecting noise to imputed values is not new, overlaying the noise toX imputed using SM appears to be new. The resampling involves bootstrap draws. Bootstrapping large dimensional matrices is not a trivial exercise. Our set up is different, as we only need to bootstrap the noise corresponding to the missing data, which is much smaller in dimension. However, we have to preestimate the factors and the loadings, and bootstrapping the factor estimates is also not a trivial problem as seen in Goncalves and Perron (2020). The proof of consistency of this estimator is thus quite delicate and is beyond the scope of this analysis. However, we can use simulations to evaluate if the idea holds promise to be worthy of further investigation. Simulations In this section, we consider four experiments: the first and second assess the accuracy of the asymptotic approximations forC it andC + it given in Lemma 2 and Proposition 1. The third evaluates the residual overlay procedures assuming a strict factor structure, while the fourth uses observed SP 500 returns as complete data to mimic an approximate factor structure. Finite Sample Properties for TP The first experiment compares the performance of tp with and without re-estimation. The design of the monte-carlo is identical to the one used in Bai and Ng (2021). Data are generated from F ∼ N (0, D r ) and Λ ∼ N (0, D r ) with r = 2, the diagonal entries in D r are equally spaced between 1 and 1/r, and e it ∼ N (0, 1). For each replication, the error in estimating C it is computed for four locations of (i, t). As benchmark, we consider the infeasible case of complete data labeled complete. Also reported are results for the EM algorithm in Stock and Watson (2016) that uses the estimates from the balanced panel as initial values. The algorithm repeatedly regresses X on F and then X on Λ till convergence, but the converged estimates may not be mutually orthogonal. We consider three versions of each estimator: one applied to the raw data X, one to the demeanend data, one to the standardized data. These are labled TP2, TP1, TP0 in the tables reported. The means and standard deviations are computed using the observations available for each series. Table 1 reports the root-mean-square-error for four chosen (i, t) pairs, one in each of the four blocks: In the second experiment, we generate data from a model with two factors to assess the adequacy of the asymptotic approximations. Two configurations of (T, N ) are considered: (300, 500) and (500, 300) with T o = .4T and N o = .6N . In both cases, about 15% of the observations are missing. tall is T × N o , wide is T o × N , bal is T o × N o , The factors and the loadings are drawn from the standard normal distribution once and held fixed. In each of the 5000 replications, a new batch of idiosyncratic errors are drawn from the normal distribution, so X varies across replications but the locations of the missing values do not change. We then evaluate estimates of C it at four different (i, t) chosen in the neighborhood of N o , T o so that they come from four blocks as defined above. Table 2 reports the mean estimate of C it for (i) when all data are observed, (ii) tw0 Row 1 of (no re-estimation), (ii) tw+ (re-estimation), (iv) tp0, and (v) tp+. All five estimates are close to the true values, showing that tw0 and tp0 are consistent without iteration. The second row, which gives the standard deviation of the estimates in the monte-carlo, shows that the tw+ and tp+ estimates are slightly less variable than tw and tp, showing efficiency gains. The third row gives the mean of the estimated asymptotic standard errors which are quite similar to row two, showing that the asymptotic approximation is quite accurate. The next two rows labeled q 05 and q 95 present the 5 and 95 percentage points of the empirical distribution of the standardized estimates. They are quite close to the quantiles from the normal distribution of ±1.64. The last row shows that coverage is generally satisfactory and reinforces the adequacy of the asymptotic normal approximation. In summary, the proposed tp estimates are already consistent but the convergence rate of C it depends on the position of all (i, t) as given in Proposition 1. The updated estimates make use of additional information and have improved statistical properties. Results for Covariance Estimation Having shown that the properties ofX are satisfactory, we proceed to evaluate the covariance estimates using economically meaningful benchmarks. From a T × N * panel of complete data X * treated as stock returns, we first calculate the "true" N * × N * covariance matrix Σ X * . Using the N * × 1 vector of portfolio weights defined from Σ X * , we compute r * pt = R * ′ t w * p and treat it as the "true" portfolio return at time t. We then randomly select N stocks and assume missing values in the south-west block of the returns matrix R * . Hence the data matrix for analysis is of dimension (T, N ). For each method, the (in-sample) bias and root-mean-squared error are computed for the following performance measures over B = 1000 replications. i. pvol: Portfolio volatility defined as riskp = 1 T T t=1 (r * pt −r * p ) 2 . ii pvar α : Portfolio value-at-risk at confidence level α defined as P r(r p < −V aR α ) = 1 − α. iii. call options price assuming that the current and strike price of each security are one-dollar, a risk-free rate of 2%, and time to maturity of one year. iv var: the variance of returns v covar: the covariance of returns. Of these measures, the first two are based on equally weighted portfolios while the last three are based on returns data. Portfolio volatility is often used as a risk measure when the benchmark is cash. Portfolio value-at-risk is used as a measure of downside risk. 4 For call, we price plain vanilla European call option prices written on SP500 stocks calculated using Black-Scholes formula. The call price, variance, and covariance measures are calculated only for returns with incomplete data. Since we have N m series with missing values, we compute N m call prices and variances, and N o N m + (N m )(N m − 1)/2 covariances at each iteration. The following notation is used Tables 3 to 5. • SM (sample moments-based) covariance estimators: -sm0: single imputation. -sm+0: single imputation followed by one re-estimation. smj: double imputation after sm0 using overlay method j = 1, . . . , 4. -sm+j: double imputation after sm+ using overlay method j = 1, . . . , 4. • factor based covariance estimators: sfa and sf+a Strict Factor Model: X * is simulated from a strict factor model: X * it = λ ′ i F t + e it , i = 1, ..., N * , t = 1, ..., T F t ∼ iid N (0, σ 2 F I r ) , λ i ∼ iid N (0, σ 2 λ I r ) e it ∼ iid N   0 , 1 − R 2 i R 2 i r q=1 λ 2 iq σ 2 F   where I r is the r × r identity matrix, σ 2 F and σ 2 λ are the variances of each loading and factor, respectively, and R 2 i is the percentage contribution of the systematic component to total variance (i.e. coefficient of determination) for series i. Note that setting the variance of e it in the way above guarantees that the coefficient of determination of each series is exactly equal to R 2 i . Since r is assumed known, we set r = 5 without loss of generality. We set R 2 i to 0.6, σ 2 λ to 1, and σ 2 F to 0.035, respectively. These give average volatility of 9.6%, similar to monthly SP 500 returns of 9.4%. The DGP abstracts from time varying loadings because the covariance estimators and will be affected by omitted time variation in the same way. Here, we focus on estimating the covariance ofX. Table 3 reports results over 1000 replications for (T, N ) = (339, 100) with 15% of the values missing (60% of rows and 40% of columns). Whether we use tw or tp seems to make little difference for smj estimation. This can be due to the fact that either way, the missing idiosyncratic errors are set to zero and does not make a big difference for covariance estimation. Re-estimation is, however, beneficial for sm estimation especially when the first step is based on tw. This makes sense because tw uses a more restricted sample in the first step estimation and stands to gain more from re-estimation. Of the four methods, sm+2 and sm+4 which resample from the non-zeroẽ it of series i alone have smaller bias and root-mean-squared error. Turning to the factor-based estimators, since the idiosyncratic errors are mutually uncorrelated by design, we had expected sfa and sfa+ which impose the strict factor structure to give better estimates than the sm counterparts. While sfa+ is comparable to sm2 and sm4, it is inferior to sm+2 and sm+4. This can be due to the fact that our implementation of sm is an average of S imputed covariances which has a noise reducing property. It is also possible that the sm estimators impose sparsity through univariate re-sampling, and as mentioned above, residual cross-section dependence can only come from the non-zeroẽ it associated with the non-missing data. As these are estimates of errors that are uncorrelated by design, the sm estimators should return sample covariances that are approximately diagonal. As robustness check, the top panel of Table 4 presents results for 30% missing (80% of rows and 60% of columns). The relative performance of the estimation methods are similar across levels of missingness, and as expected, estmation accuracy of a given model is higher when the missingness is lower. The results so far are presented for N < T . Results for T = 200, N = 250 are shown in the bottom panel Table 4. Note that with this design, eachΣX(s, j) is verified to be singular. However, the averaged estimatorΣX(j) is not. Similar to results in Table 3, sm+2 and sm+4 perform well and often better than sf+a. Monte Carlo Calibrated to SP500 Returns The final exercise is a Monte-Carlo that takes X * to be SP500 monthly returns between 1990 and 2018 to capture the approximate factor structure. The first factor in X * of dimension 348 × 339 explains over 26.2% of the variations, the second explains 4.1%. The next three factors explain 3.8, 2.7, and 2.1% of the variation, respectively. As in the previous monte carlo, we random select N = 100 stocks in each replication and set some values in the south-west block to missing. The results based on 1000 replications are reported in Table 5. Imputation leads to some reduction in bias and variance, but the errors are significantly smaller than no adjustment upon comparing sm1-4 with sm0. Though double imputation is generally better than single imputation, the precise gain depends on the risk measure and the estimation sample. As in the strict factor case, tw and tp give similar risk-performance errors. Also as in the strict factor case, sfa+ is comparable to sm+1 and sm+3, but has larger errors than sm+2 and sm+4. However, improvements from re-estimation both in terms of bias and RMSE are larger for this DGP that mimics the approximate factor structure. Overall, we find that single imputation is strongly preferred over no imputation but is inferior to double imputation. Ultimately, the appropriate method depends on the data generating process and we offer several alternatives worthy of consideration. Averaging the covariances of imputed data overlaid with resampled errors has two desirable effects that we had not anticipated: it reduces noise, and the averaged estimator is full rank. This interesting finding and the role of resampling in covariance estimation is an interesting problem that warrants further investigation. Conclusion This paper provides three sets of results. The first is a tp algorithm that can consistently estimate the entire low rank matrix without iteration and a distribution theory for the estimates is provided. The second result makes precise the conditions under which factors estimated from incomplete data can be treated as known in factor-augmented regressions. The third pertains to estimation of covariances from incomplete data. We consider different schemes to compensate for an omitted error in the level estimates. Implications for using imputed factors in augmented regressions are also discussed. COMPLETE TP TP+ EM 1 tall ( We observe X, but not F 0 or Λ 0 . As F and Λ are not separately identifiable. The method of asymptotic principal components (APC) uses the normalization F ′ F T = I r and Λ ′ Λ being diagonal to produce estimates (0) (1) (2) (0) (1) (2) (0) (1) (2) (0) (1) (2) case (i, t) in(F ,Λ) = ( √ T U r , √ N V r D r ). For each t ∈ [1, T ] and for each i ∈ [1, N ], (F t ,Λ i ) consistently estimate (F 0 t , Λ 0 i ) up to a rotation matrices H. This rotation matrix is not unqiue, and for the present analysis, let H = Λ 0 ′ Λ 0 N F 0 ′F T D −2 r denote the rotation matrix for complete data, and let H tall = Λ 0 ′ o Λ 0 o N o F 0 ′F tall T D −2 r,tall denote the rotation matrix for the tall block of data, where Λ 0 o is the N o × r matrix consisting of factor loadings of Λ 0 k for all k ∈ ∩ s J s , andF tall is T × r, and D 2 r,tall is r × r. Proof of Lemma 2. By construction, the first stage estimated factors are obtained from the observed T × N o data matrix, giving rise toF tall . For (i, t) in the tall block, the regression method and the principal components estimator are the same as the complete data case explained in the text. So Lemma 2i, and ii(a) follow from Lemma 1. We consider the remaining claims. For each i > N o of the reorganized data (or i ∈ ∩ s J s of the original data), we observe T o i observations. Here for notational simplicity, we assume the first consecutive T o i observations are available. The true model is X it = F 0′ t Λ 0 i + e it . To estimate the factor loadings Λ 0 i , consider the regression X it =F ′ t Λ i + a it , t = 1, 2, ..., T o i whereF ′ t is the tth row ofF tall , a it is an error term. Then, OLS gives Λ i = (F ′ o iF o i ) −1F ′ o i X o i where X o i is T o i × 1 vector such that X o i = F 0 o i Λ 0 i + e o i , and F 0 o i is T o i × r, and e o i is T o i × 1;F o i stacksF t . It follows thatΛ i = (F ′ o iF o i ) −1F ′ o i F 0 o i Λ 0 i + (F ′ o iF o i ) −1F ′ o i e o i . Let G i = (F ′ o iF o i ) −1F ′ o i F 0 o i and rewrite F o i =F o i H −1 tall − [F o i − F o i H tall ]H −1 tall , so that G i = H −1 tall − (F ′ o iF o i ) −1F ′ o i [F o i − F o i H tall ]H −1 tall . Since (F ′ o iF o i /T o i ) −1 = O p (1), and 1 To iF ′ o i [F o i − F o i H tall ] = O p (δ −2 No,To i ), we obtain G i = H −1 tall + O p (δ −2 No,To i ). Thus,Λ i − H −1 tall Λ 0 i = (F ′ o iF o i ) −1F ′ o i e o i + O p (1/δ 2 NoTo i ) = H −1 tall (F 0 ′ o i F 0 o i ) −1 F 0 ′ o i e o i + O p (1/δ 2 NoTo i ).(8) Multiply by T o i and use the limit for H tall (see Bai and Ng (2021)), we obtain part ii(b) of Lemma 2. Here we use the assumption that √ To i No → 0. Consider the estimated common components C it =F ′ tΛi =F ′ t (F ′ o iF o i ) −1F ′ o i F 0 o i Λ 0 i +F ′ t (F ′ o iF o i ) −1F ′ o i e o i . Similar to Bai and Ng (2021), we can show, for all missing entries (i, t), C it = C 0 it + Λ 0′ i (Λ 0′ o Λ 0 o /N o ) −1 1 N o No k=1 Λ 0 k e kt + F 0′ t (F 0′ o i F 0 o i /T o i ) −1 1 T o i To i s=1 F 0 s e is + O p (1/δ 2 No,To i ). where Λ 0 o is N o × r. Here and for the remainder of the proof, we assume the first N o units have no missing observations so that No k=1 is identical to k∈∩sJ s . Let u it = Λ 0′ i (Λ 0′ o Λ 0 o /N o ) −1 1 N o No k=1 Λ 0 k e kt (9) v it = F 0′ t (F 0′ o i F 0 o i /T o i ) −1 1 T o i To i s=1 F 0 s e is(10) ThenC it = C 0 it + u it + v it + O p (1/δ 2 No,To i ).(11) This implies that 1 N o A N T + 1 T o i B N T −1/2 (C it − C 0 it ) d −→N (0, 1)(12) where A N T = Λ 0′ i (Λ 0′ o Λ 0 o /N o ) −1 Γ t (Λ 0′ o Λ 0 o /N o ) −1 Λ 0 i and B N T = F 0′ t (F 0′ o i F 0 o i /T o i ) −1 Φ i (F 0′ o i F 0 o i /T o i ) −1 F 0 t . This gives part iii of Lemma 2. Proof of Proposition 1 Consider the principal components estimator of factors and factor loadings based onX (see equations (4a) and (4b)). LetF + andΛ + denote the PCA estimator so that 1 N TXX ′F + =F +D2 r withF +′F + /T = I r , andΛ = 1 TX ′F + ;D 2 r is a diagonal matrix consisting of the first r largest eigenvalues ofXX ′ /(N T ). Define H + = (Λ 0′ Λ 0 /N )(F 0′F /T )D −F + t − H + F 0 t 2 = O p (δ −2 No,To ), and (b) plimD 2 r → D 2 r andF +′ F 0 /T → Q r , where D r and Q are the same as in complete data. The proof of this lemma follows the same argument as in Bai and Ng (2021). The details are omitted. We focus on obtaining the asymptotic distributions. Asymptotic distribution for the estimated factors. To derive the limiting distribution for the estimated factor and factor loadings, definẽ e it = e it if X it is observed u it + v it if X it is missing Because of the preceding lemma, the asymptotic representation forF + t is (see Bai (2003) and Bai and Ng (2021) ),F + t − H +′ F 0 t =D −2 (F +′ F 0 /T ) 1 N N i=1 Λ 0 iẽit + O p (1/δ 2 No,To ).(13) For a given t, if all individual variables (X 1t , X 2t , ..., X N t ) are observable, thenẽ it = e it for all i, and √ N(F + t − H +′ F 0 t ) =D −2 (F +′ F 0 /T ) 1 √ N N i=1 Λ 0 i e it + O p (1/δ 2 No,To ) (14) d −→N (0, D −2 r Q r Γ t Q ′ r D −2 r ) . For a given t, suppose that (the first) N ot series are available, denoted by (X 1t , X 2t , ...X No t t ), with N ot ≥ N o . Thenẽ it = e it for i ≤ N ot andẽ it = u it + v it for i > N ot , so (13) becomes F + t − H +′ F 0 t =D −2 (F +′ F 0 /T ) 1 N No t i=1 Λ 0 i e it + 1 N N i=No t +1 Λ 0 i (u it + v it ) + O p (1/δ 2 No,To ). Note that 1 N N i=No t +1 Λ 0 i v it is negligible (dominated by 1 N N i=No t +1 Λ 0 i u it ) . This follows from the definition of v it in (10), 1 N N i=Nt+1 Λ 0 i,j v it = O p (1) 1 N T o i N i=Nt+1 To i s=1 Λ 0 i,j F 0 s e is = O p (1/ N T o i ) = O p (1/δ 2 No,To ) where Λ 0 i,j is the jth component of Λ 0 i (j = 1, 2, ..., r). Let A t = I r + N − N ot N o 1 N − N ot N i=No t +1 Λ 0 i Λ 0′ i (Λ 0′ o Λ 0 o /N o ) −1 . We can rewrite the representation as F + t − H +′ F 0 t =D −2 (F +′ F 0 /T ) 1 N No i=1 A t Λ 0 i e it + 1 N No t i=No+1 Λ 0 i e it + O p (1/δ 2 No,To ), For the special case that N ot = N o , and 1 N −No t N i=No t +1 Λ 0 i Λ 0′ i Λ 0′ o Λ 0 o /N o −1 → I r we have A t ∼ N/N o , the above representation coincides with the TW presentation in Bai and Ng (2021). Consider the more general case of N ot with N ot ≥ N o . Define B i t =    No t N A t i ≤ N o No t N I r N o < i ≤ N ot We can further rewrite the representation as F + t − H +′ F 0 t =D −2 (F +′ F 0 /T ) 1 N ot No t i=1 B i t Λ 0 i e it + O p (1/δ 2 No,To ).(15) Under Assumption C, B i t is bounded. Then the convergence rate is N ot , and N ot (F + t − H +′ F 0 t ) =D −2 (F +′ F 0 /T ) 1 N ot No t i=1 B i t Λ 0 i e it + o p (1) d −→N (0, D −2 r Q r Γ * t Q ′ r D −2 r ), where Γ * t = plim 1 N ot No t i=1 B i t Λ 0 i Λ 0′ i B i′ t e 2 it(16) (assuming cross-sectional uncorrelation for e it ). The result includes N ot = N as a special case. When N ot = N , we have B i t = I r for all i, and Γ * t = Γ t . The convergence is √ N , and the limit is in (14). Asymptotic distribution for the estimated factor loadings: We have the asymptotic rep-resentationΛ + i − (H + ) −1 Λ 0 i = H +′ 1 T T t=1 F 0 tẽit +η N T,i whereη N T,i = O p (δ −2 No,To ) is uniformly in i and t. For a given i, if all T observations are available, thenẽ it = e it for all t, so that √ T (Λ + i − (H + ) −1 Λ 0 i ) = H +′ 1 √ T T t=1 F 0 t e it + o p (1) → d N (0, Q ′−1 r Φ i Q −1 r ) where we used the fact that the limit of H + is Q −1 r . Suppose that for the given i, there are T o i observations available. Again, for notational simplicity, we assume the first T o i are observable, the rest are missing. Thenẽ it = e it for t ≤ T o i and e it = u it + v it for t > T o i . Thus Λ + i − (H + ) −1 Λ 0 i = H +′ 1 T To i t=1 F 0 t e it + 1 T T t=To i +1 F 0 t (u it + v it ) +η N T,i Note that 1 T T t=To i +1 F 0 t u it is negligible, and 1 T T t=To i +1 F 0 t v it = T − T o i T 1 T − T o i T t=To i +1 F 0 t F 0′ t F 0′ o i F 0 o i /T o i −1 1 T o i To i s=1 F 0 s e is . Under 1 T − T o i T t=To i +1 F 0 t F 0′ t F 0′ o i F 0 o i /T o i −1 p −→I r the asymptotic representation can be written as Λ + i − (H + ) −1 Λ 0 i = H +′ 1 T o i To i t=1 F 0 t e it + O p (δ −2 No,To ).(17) Then by Assumption D, T o i (Λ + i − (H + ) −1 Λ 0 i ) → d N (0, Q ′ r Φ i Q r ). Asymptotic distribution for the estimated common components: LetC + it =F +′ tΛ + i . Using the asymptotic representations in (15) and (17), we can show as in Bai and Ng (2021) C + it − C 0 it = Λ 0′ i (Λ 0′ Λ 0 /N ) −1 1 N ot No t i=1 B i t Λ 0 i e it + F 0′ t (F 0′ F 0 /T ) −1 1 T o i To i t=1 F 0 t e it + O p (δ −2 No,To ). The above representation forC + it − C 0 it implies 1 N ot V * it + 1 T o i W it −1/2 (C + it − C 0 it ) d −→N (0, 1),(18) where V * it = Λ 0′ i Σ −1 Λ Γ * t Σ −1 Λ Λ 0 i , W it = F 0′ t (Σ −1 F Φ i Σ −1 F )F 0 t and Γ * t is defined in (16). Note that we assume the two terms in the representation are asymptotically independent so that the variance is the sum of the two variances. A useful special case occurs, if for a given entry (i, t), the corresponding row and column are observable (i.e., if (X i1 , X i2 , ..., X iT ) and (X 1t , X 2t , ..., X N t ) are observable), then 1 N V it + 1 T W it −1/2 (C + it − C 0 it ) d −→N (0, 1), where V it = Λ 0′ i Σ −1 Λ Γ t Σ −1 Λ Λ 0 i . This follows from N ot = N , B i t = I r , Γ * t = Γ t , and T 0 i = T . We remark that using the convergence rate forC + it −C + it , we can also show, under the assumption that N ot ≥ N o , and T o i ≥ T o , the Frobenius norm of the matrixC + − C 0 is bounded by C + − C 0 √ N T = O p ( 1 √ N ) + O p ( 1 √ T ) + (1 − p N )(1 − p T ) O p ( 1 √ N p N ) + O p ( 1 √ T p T )(19) where p N = N 0 /N and p T = T 0 /T . Proof of Lemma 4: Omitting the subscript X, we write Σ ij for Σ X,ij , and similarly for the estimated counterpart. For i = j, Σ ij − Σ ij =Λ ′ iΣFΛj − Λ 0 ′ i Σ F Λ 0 j = (Λ i − GΛ 0 i ) ′Σ FΛj + Λ 0 ′ i G ′Σ F GG −1Λ j − Λ 0 ′ i Σ F Λ 0 j = (Λ i − GΛ 0 i ) ′Σ FΛj + Λ 0 ′ i (G ′Σ F G − Σ F )G −1Λ j + Λ 0 ′ i Σ F (G −1Λ j − Λ 0 j ), where G is either H −1 tall or (H + ) −1 depending on the choice ofΛ. Thus max ij \|Σ ij − Σ ij \| ≤ max i Λ i − GΛ 0 i · Σ F max j Λ j + (max i Λ 0 i )(max j Λ j ) G ′Σ F G − Σ F G −1 + max i Λ 0 i Σ F G −1 max j Λ j − GΛ 0 i . By Assumption A, max i Λ 0 i is bounded. Under exponential tails for the idiosyncratic errors e it (e.g, Fan et al, 2011), then it can be shown that max i Λ i − GΛ 0 i ≤ log(max{T, N }) min{N o , T o } O p (1). By adding and subtracting terms, we have max j Λ j = O p (1). Note Σ F is a fixed dimensional matrix, we have G ′Σ F G − Σ F = o p (1). In summary, for i = j, max ij \|Σ ij − Σ ij \| = o p (1). For i = j, we need to show further that the idiosyncratic variance estimator is uniformly consistent over i, max i \|Ψ 2 e,ii − Ψ 2 e,ii \| = o p (1), where Ψ 2 e,ii = var(e it ), andΨ 2 e,ii is its estimator. Fromẽ it = e it +C it − C it , we have max i \|Ψ 2 e,ii − Ψ 2 e,ii \| ≤ max i \| 1 T o i s e 2 is − Ψ 2 ii \| + 2 max i 1 T o i s \|C is − C is \|\|e is \| + max i 1 T o i s (C is − C is ) 2 where the sum is based on T o i number of entries. Using the convergence ofC it , and the exponential tails of e it , we can show that each of the right hand side term is o p (1), implying (20). Lemma 4 is obtained by combining results. Assumption A : :There exists a constant M < ∞ not depending on N, it e is ) = γ N (s, t), T t=1 \|γ N (s, t)\| ≤ M , ∀s; Figure 1 : 1Missing Figure 2 : 2Missing Data with a Less Structured Pattern Reverse Monotone Missing Heterngeneous Missing X1 . . . XN o XN o+1 . . . XN−1 XN Figure 3 : 3The Tall-Incomplete Representation of the Data obtained by stacking up allẽ it associated with the observed X it . Resample N i=1 (T −T o i ) observations from u with replacement and randomly assign them toê it wheneverẽ it = 0.-(sm2) For each i, let u i be T o i sub-vector of non-zero entries ofẽ i . Sample with replacement T − T o i errors from u i and assign them toê it wheneverẽ it = 0.-(sm3) Computeσ u from the N i=1 T o i vector of non-zero estimated errors as in Method 1. If e it = 0, replace it byê it := u itσu , where u it ∼ N (0, 1). -(sm4) Computeσ u,i from the T o i non-zero estimated errors as in Method 2. Ifẽ it = 0, replace it byê it := u itσu.i , where u it ∼ N (0, 1).All four methods assume that the returns data are covariance stationary. 3 Methods sm1 and sm2 are non-parametric, while method sm3 and sm4 calibrate the first two moments to the estimated errors associated with the observed data. Methods sm1 and sm3 are better suited for homogeneous data and sinceê it are sampled from the stacked up vector of estimated errors. Methods sm2 and sm4 are better suited for heterogeneous data as they sample from the errors of the corresponding series. Methods sm1 and sm2 do not make distributional assumptions about the errors. As many financial time series tend to have fat tails and skewed distributions, other parametric distributions can be used in place of the normal distribution in sm3 and sm4. The four sampling schemes do not account for cross-correlation in the errors, which implicitly shrinks Σ e towards zero. This can be desirable when estimating high dimensional covariance matrices. and miss is (T − T o ) × (N − N o ), With N = T = 200, results are reported for four configurations of missingness. Evidently, the error depends on observability of X it . The estimation error is largest if X it is in the miss block and smallest when X it is in the bal block. For a given block, the estimator errors are smaller when the factors are re-estimated fromX. This is consistent with the theory. Results for (N, T ) = (300, 500) and (500, 300) are similar. DGP: The T × N data matrix X is generated by X = F Λ ′ + e, F ∼ N (0, D r ), Λ ∼ (0, D r ) with r = 2, e ∼ N (0, 2.5), and diag(D r ) = [1;.5]. (N, T) is the number of columns and rows in the block. Four configurations of missing data are considered. Case 1 havs the smallest miss block and case 4 has the largest. Reported are the root-mean-squared error over 5000 replications. 1 1(T, N ) = (300, 500), (T o , N o ) = (120, 300) (T, N ) = (500, 300), (T o , N o ) = (300, 1200) C 115,290 = −generated from a model with two factors. The factors and the loadings are drawn from the normal distribution once and hence fixed. In each of the 5000 replications, new idiosyncratic errors are resampled. The location of the missing values are fixed throughout. The apc column is the infeasible case when the data are completely observed. The columns tw and tp are one step estimators. A '+' indicates one re-estimation. A panel is said to be complete if it does not have missing values. In practice most data panels have missing values. The data are said to be missing completely at random (MCAR) if missingness is independent of the data whether or not they are observed. As the observed and missing data are drawn from the same underlying distribution and hence have no systematic differences, MCAR does not create bias. This is not the case with missing at random (MAR), though the systematic differences can be explained by other observables. For example, missing values in financial statements of smaller firms may arise because smaller firms are less regulated, in which case the missing data propensity can be related to market capitalization which is observed for publicly traded firms.But the issue remains that the observed values under MAR do not form a random sample. Ṽ r are the r left and right singular vectors ofX. Let H + be a rotation matrix. Under Assumptions A − D, the factors and loadings estimated fromX have the following properties: it is consistent for Ψ e,ii if T o i is sufficiently large. Define the strict-factor adjusted (SFA) covariance matrix estimator aŝbyΨẽ = diag 1 T o i t∈J iẽ 2 it , i = 1, 2, ..., N to recognize that someẽ it are zero, and the estimateΨê ,ii = 1 To i t∈J iẽ 2 ΣX =ΛΣ FΛ ′ +Ψẽ (SFA) Lemma 4. LetΣX be the sfa estimate of Σ X . Under Assumptions A, B, exponential tail distri- bution for e it , and that Σ e is a diagonal matrix, Table 1 : 1Root Mean-Squared-Error ofC it at four (i, t) pairs Table 2 : 2Finite Sample Properties of SelectedC it Table 3 : 3Monte Carlo: Strict Factor Model, 15% Missingness, T = 339, N = 100 Bias RMSE pvol pVaR call var covar pvol pVaR call var covar TW sm0 - Table 4 : 4Additional TP Results: Strict Factor ModelBias RMSE pvol pVaR call var covar pvol pVaR call var covar 30% Missing Table 5 : 5Monte Carlo Calibrated to CRSP Data: 15% MissingBias RMSE pvol pVaR call var covar pvol pVaR call var covar TW sm0 - 2 r 2Now we assume T o i ≥ T o for some T o . We also assume N ot ≥ N o , where T o and N o satisfy Assumption B. 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[ "A tree of linear fractional transformations", "A tree of linear fractional transformations" ]	[ "Melvyn B Nathanson " ]	[]	[]	The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form a/b, where are a and b are relatively prime positive integers. It is possible to construct an analogous tree of positive linear fractional transformations of determinant 1, and to prove that this tree possesses the basic properties of the Calkin-Wilf tree of positive rational numbers.Equivalently, if z = a/b, then the left child of z is z/(z + 1) and the right child of z is z + 1. These children give birth to children, and so on. Thus, every positive reduced rational number has infinitely many descendants. If gcd(a, b) = 1, then gcd(a, a + b) = gcd(a + b, b) = 1, and so every descendant of a reduced rational number is also reduced. Equivalently, every positive reduced rational number is the root of an infinite binary tree of reduced positive rational numbers. The two children with the same parent are called siblings. The only positive rational number with no parent is 1, that is, 1 is an orphan. Calkin and Wilf[3]introduced this enumeration of the positive rationals in 2000, and it has stimulated much research (e.g.[2,4,5]).	10.1142/s1793042115500694	[ "https://arxiv.org/pdf/1401.0012v3.pdf" ]	119,146,378	1401.0012	1c1530b5b4af74888131d7ae166f62bd8a1da0d9	A tree of linear fractional transformations 3 Jan 2014 Melvyn B Nathanson A tree of linear fractional transformations 3 Jan 2014 The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form a/b, where are a and b are relatively prime positive integers. It is possible to construct an analogous tree of positive linear fractional transformations of determinant 1, and to prove that this tree possesses the basic properties of the Calkin-Wilf tree of positive rational numbers.Equivalently, if z = a/b, then the left child of z is z/(z + 1) and the right child of z is z + 1. These children give birth to children, and so on. Thus, every positive reduced rational number has infinitely many descendants. If gcd(a, b) = 1, then gcd(a, a + b) = gcd(a + b, b) = 1, and so every descendant of a reduced rational number is also reduced. Equivalently, every positive reduced rational number is the root of an infinite binary tree of reduced positive rational numbers. The two children with the same parent are called siblings. The only positive rational number with no parent is 1, that is, 1 is an orphan. Calkin and Wilf[3]introduced this enumeration of the positive rationals in 2000, and it has stimulated much research (e.g.[2,4,5]). The Calkin-Wilf tree of rational numbers A rooted infinite binary tree is a directed graph with the following properties: (i) Every vertex is the tail of exactly two edges. (ii) There is a vertex v * such that every vertex v = v * is the head of exactly one edge, but v * is not the head of any edge. We call v * the root of the tree. (iii) The graph is connected. A forest is a directed graph whose connected components are rooted infinite binary trees. We call the rational number a/b reduced if b ≥ 1 and the integers a and b are relatively prime. The Calkin-Wilf tree is a rooted infinite binary tree whose vertex set is the set of all positive reduced rational numbers, and whose root is 1. In this tree, every positive reduced rational number a/b is the tail of two edges. The heads of these edges are the positive rational numbers a/(a + b) and (a + b)/b. Note that a/(a + b) < 1 < (a + b)/b. We draw this as follows: where [x] denotes the integer part of the real number x. This result is due to Moshe Newman [1,6]. 4. Depth formula: Let a/b be a positive reduced rational number. If a b ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ 1 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄a b = q 0 + 1 q 1 + 1 q 2 + · · · + 1 q k−1 + 1 q k = [q 0 , q 1 , . . . , q k−1 , q k ] is the finite continued fraction of a/b, then the depth of a/b is q 0 + q 1 + · · · + q k−1 + q k − 1. For example, we have the continued fraction and so 11/3 is on row 3 + 1 + 2 − 1 = 5 of the Calkin-Wilf tree. Because the integer part of 11/3 is 3, the successor formula implies that the next element on row 5 is 1 2 · 3 + 1 − 11 3 = 3 10 . Indeed, c 5,24 = 11/3 and c 5,25 = 3/10. Moreover, by the symmetry formula, 3/11 = 1/c 5,24 = c 5, 9 . In this paper we describe a forest of rooted infinite binary trees of rational functions of the form (az + b)/(cz + d) in which all of these properties hold, and which specializes to the Calkin-Wilf tree when the root of the tree is the rational function z = 1. Positive linear fractional transformations Let z be a variable. A positive linear fractional transformation is a rational function F (z) = az + b cz + d where a, b, c, d are relatively prime nonnegative integers such that (a, b) = (0, 0) and (c, d) = (0, 0). Consider the positive linear fractional transformations F (z) = az + b cz + d and G(z) = ez + f gz + h . The ordered pairs (a, b), (c, d), (e, f ), and (g, h) are different from (0, 0). We have the composite function F • G(z) = a ez+f gz+h + b c ez+f gz+h + d = (ae + bg)z + (af + bh) (ce + dg)z + (cf + dh) .(2) Suppose that (ce + dg, cf + dh) = (0, 0). If c = 0, then d = 0 and so g = h = 0, which is absurd. If d = 0, then c = 0 and so e = f = 0, which is absurd. If c = 0 and d = 0, then e = f = g = h = 0, which is also absurd. Therefore, (ce + dg, cf + dh) = (0, 0). Similarly, (ae + bg, af + bh) = (0, 0), and so F • G(z) is a positive linear fractional transformation. A monoid is a semigroup with identity. If F (z) is a positive linear fractional transformation and E(z) = z, then F • E(z) = E • F (z) = F (z). With the binary operation of composition and the identity element E(z) = z, the set of positive linear fractional transformations is a monoid. We call the integer det(F (z)) = ad − bc the determinant of F (z) = (az+b)/(cz+d). For example, z, (7z+2)/(3z+1), and (z + 5)/(2z + 4) are positive linear fractional transformations with determinants 1, 1, and −6, respectively. The determinant of the composite function (2) is det(F • G(z)) = (ae + bg)(cf + dh) − (af + bh)(ce + dg) = (ad − bc)(eh − f g) = det(F (z)) det(G(z)). If det(F (z)) = 0 and det(G(z)) = 0, then det(F • G(z)) = 0. Thus, the set of positive linear fractional transformations with nonzero determinant is a submonoid of the monoid of all positive linear fractional transformations. A special positive linear fractional transformation is a positive linear fractional transformation F (z) with det(F (z)) = 1. The multiplicativity of the determinant proves that the set of special positive linear fractional transformations is also a submonoid.. Moreover, if F (z) and G(z) are positive linear fractional transformations and F (z) is special, then det(F • G(z)) = det(G(z)). F (z) + 1 = (a + c)z + (b + d) cz + d and F (z) F (z) + 1 = az + b (a + c)z + (b + d) are positive. Moreover, det(F (z) + 1) = (a + c)d − (b + d)c = ad − bc = det(F (z)) and det F (z) F (z) + 1 = a(b + d) − b(a + c) = ad − bc = det(F (z)) and so, if F (z) is special, then both F (z) + 1 and F (z)/(F (z) + 1) are special. This completes the proof. The tree of special positive transformations Associated to every positive linear fractional transformation R(z) is a rooted infinite binary tree N (R(z)) with root R(z) whose vertices are positive linear fractional transformations with determinant det(R(z)). Every vertex w in this tree will be the parent of two children: the left child w/(w + 1) and the right child w + 1. We draw this as follows: w } } ④ ④ ④ ④ ④ ④ ④ ④ 4 4 ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ w w+1 w + 1(3) with w/(w + 1) on the left and w + 1 on the right. Note that if w = a/b is a positive reduced rational number, then this is exactly the generation rule (1). A linear function az + b will be called positive if a and b are nonnegative integers and (a, b) = (0, 0). We partially order the set of positive linear functions as follows: cz + d az + b if c ≤ a and d ≤ b. We write cz + d ≺ az + b if cz + d az + b and cz + d = az + b. Distinct positive linear functions az + b and cz + d are comparable if cz + d ≺ az + b or az + b ≺ cz + d. For example, 2z + 1 and 3z + 2 are comparable, but 2z + 1 and z + 2 are not comparable. A positive linear fractional transformation (az + b)/(cz + d) such that az + b ≺ cz + d is the left child of the positive linear fractional transformation (az + b)/((c − a)z + (d − b)). A positive linear fractional transformation (az + b)/(cz + d) such that cz + d ≺ az + b is the right child of the positive linear fractional transformation ((a − c)z + (b − d))/(cz + d). If az + b and cz + d are not comparable, then the positive linear fractional transformation (az + b)/(cz + d) has no parent and is called an orphan. For example, z, (z + 2)/(2z + 1) and 1/3z are orphans with determinants 1, -3, and -3, respectively. We define the reciprocal of the positive linear fractional transformation F (z) = (az + b)/(cz + d) as the linear fractional transformation 1/F (z) = (cz + d)/(az + b). The reciprocal of a right child is a left child, and conversely. The reciprocal of an orphan is an orphan. Moreover, det(1/F (z)) = bc − ad = − det(F (z). Let N (z) be the rooted infinite binary tree whose root is the special positive linear fractional transformation z. The first four rows of N (z) are as follows: z u u • • • • • • • • • • • • • • • • • • @ @ \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| z z+1 } } ④ ④ ④ ④ ④ ④ ④ ④ 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ z + 1~⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 3 3 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ z 2z+1 Ó Ó ✞ ✞ ✞ ✞ ✞ ✞ ✞ 3 3 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ 2z+1 z+1 } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ 3 3 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ z+1 z+2~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ z + 2~⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ( ( ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ z 3z+1 3z+1 2z+1 2z+1 3z+2 3z+2 z+1 z+1 2z+3 2z+3 z+2 z+2 z+3 z + 3 Because z is a special positive linear fractional transformation, Lemma 1 implies that every vertex in this graph is a special positive linear fractional transformation. The root z is the only element in row 0 of this tree. For every positive integer n, row n of the tree consists of the 2 n elements of the nth generation descended from the root. We say that the rational function F (z) has depth n if it is on row n of the tree, or, equivalently, if it is a member of the nth generation of descendants of the root z. We denote the ordered sequence of elements of the nth row by (w n,1 (z), w n,2 (z), . . . , w n,2 n (z)). For example, w 2,3 (z) = (z + 1)/(z + 2) and w 3,6 (z) = (2z + 3)/(z + 2). Note that w 2,3 (1) = 2/3 = c 2,3 and w 3,6 (1) = 5/3 = c 3,6 . Lemma 2 A special positive linear fractional transformation F (z) is an orphan if and only if F (z) = z. A positive linear fractional transformation F (z) of determinant -1 is an orphan if and only if F (z) = 1/z. Proof If F (z) = (az + b)/(cz + d) is an orphan, then the positive linear functions az + b and cz + d are not comparable. This can happen in only two ways. In the first case, we have a > c and b < d, and so and so b = c = 0 and a = d = 1, hence F (z) = z. In the second case, we have a < c and b > d, and so 1 = ad − bc ≤ (c − 1)(b − 1) − bc = 1 − b − c ≤ 1. It follows that 0 = b = c > a ≥ 0, which is absurd. Thus, the only orphan special positive linear fractional transformation is z. If the positive linear fractional transformation F (z) is an orphan of determinant ∆, then the reciprocal 1/F (z) is an orphan of determinant −∆. If ∆ = −1, then −∆ = 1 and so 1/F (z) = z. This completes the proof. We define the height of the positive linear fractional transformation F (z) = (az + b)/(cz + d) by ht(F (z)) = max(a + b, c + d). If G(z) is the left or right child of F (z), then ht(G(z)) = a + b + c + d > ht(F (z)) . The height of a positive linear fractional transformation is a positive integer, and the height of a child is always strictly greater than the height of the parent. Proof We have already observed that every vertex of N (z) is a special positive linear fractional transformation. Conversely, every special positive linear fractional transformation F (z) = (az + b)/(cz + d) is either an orphan or has a parent. That parent is either an orphan or has a parent. Because the height of a positive linear fractional transformation is a positive integer, and because the height of a parent is always strictly less than the height of a child, it follows that every vertex in the tree N (z) has only finitely many ancestors, and so every vertex is the descendent of an orphan. By Lemma 2, the unique orphan of determinant 1 is F (z) = z. Because every positive linear fractional transformation is descended from an orphan, every special positive linear fractional transformation is a descendent of z, and must be a vertex in the tree N (z). Moreover, every vertex has a unique parent, and so every special positive linear fractional transformation occurs exactly once as a vertex in the tree N (z). This completes the proof. Properties of the tree N (z) We shall prove that, with appropriate definitions of "integer part," "reciprocal," and "continued fraction," properties (1)-(4) of the Calkin-Wilf rational number tree also hold for the tree N (z) of special linear fractional transformations. Recall that, for j = 1, . . . , 2 n , the special positive linear fractional transformation w n,j (z) is the jth vertex on the nth row of N (z). Theorem 2 (Denominator-numerator formula) For all n ≥ 1 and j = 1, . . . , 2 n − 1, the denominator of w n,j (z) is the numerator of w n,j+1 (z). Proof The proof is by induction on n. The theorem is true for n = 1 because z + 1 is both the denominator of w 1,1 (z) and the numerator of w 1,2 (z). Let n ≥ 2, and assume that the theorem holds for n − 1. If j is odd, then w n,j (z) and w n,j+1 (z) are siblings. If their parent is the linear fractional transformation (az + b)/(cz + d), then (a + c)z + (b + d) is the denominator of w n,j (z) and the numerator of w n,j+1 (z). If j = 2i is even, then i is a positive integer with i < 2 n−1 such that w n,j (z) is the right child of w n−1,i (z) and w n,j+1 (z) is the left child of w n−1,i+1 (z). If w n−1,i (z) = (az + b)/(cz + d), then the induction hypothesis implies that w n−1,i+1 (z) = (cz + d)/(ez + f ). The right child of w n−1,i (z) is w n,j (z) = ((a + c)z + (b + d))/(cz + d); the left child of w n−1,i+1 (z) is w n,j+1 (z) = (cz + d)/((c + e)z + (d + f )). We see that cz + d is both the denominator of w n,j and the numerator of w n,j+1 (z). This completes the proof. Theorem 3 (Symmetry formula) Define the function Φ on the set of nonzero rational functions in z by Φ(F (z)) = 1 F 1 z . Then Φ is an involution, that is, Φ 2 = id, and, for every nonnegative integer n and j = 1, . . . , 2 n , Φ(w n,j )(z) = w n,2 n −j+1 (z). Proof For every rational function F (z) we have Φ 2 (F (z)) = Φ (Φ(F (z))) = 1 Φ(F (1/z)) = 1 1 F ( 1 1/z ) = F (z) and so Φ 2 = id. We shall prove (4) by induction on n. We have Φ (w 0,1 (z)) = Φ (z) = 1 1 z = z = w 0,1 (z) = w 0,2 0 −1+1 and Φ (w 1,1 (z)) = Φ z z + 1 = 1 1 z 1 z +1 = z + 1 = w 1,2 = w 1,2 1 −1+1 . Because Φ is an involution, we have Φ(w 1,2 (z)) = w 1,1 (z). Thus, (4) holds for n = 0 and n = 1. Let n ≥ 2 and suppose that (4) holds for n − 1 and j = 1, . . . , 2 n−1 . If w n−1,j (z) = az + b cz + d then w n−1,2 n−1 −j+1 (z) = Φ (w n−1,j (z)) = Φ az + b cz + d = dz + c bz + a . The children of w n−1,j (z) are w n,2j−1 (z) = az + b (a + c)z + (b + d) and w n,2j (z) = (a + c)z + (b + d) cz + d . The children of w n−1,2 n−1 −j+1 (z) are w n,2 n −2j+1 (z) = dz + c (b + d)z + (a + c) and w n,2 n −2j+2 (z) = (b + d)z + (a + c) bz + a . We see immediately that Φ (w n,2j−1 (z)) = w n,2 n −2j+2 (z) and Φ (w n,2j (z)) = w n,2 n −2j+1 (z). This completes the proof. and either r < c or s < d. Moreover, rd − sc = ad − bc. Equivalently, az + b cz + d = q + rz + s cz + d with r < c or s < d, and det az + b cz + d = det rz + s cz + d . Proof Because cz + d ≺ az + b and az + b is not a multiple of cz + d, there is a largest positive integer q such that q(cz + d) If q = [a/c], then a/c < q + 1 and r = a − qc < c. If q = [b/d], then b/d < q + 1 and s = b − qd < d. Thus, either r < c or s < d. Moreover, rd − sc = (a − qc)d − (b − qd)c = ad − bc. and so (az + b)/(cz + d) and (rz + s)/(cz + d) have the same determinant. Let q and q ′ be a positive integers and let rz + s and r ′ z + s ′ be positive linear functions satisfying az + b = q(cz + d) + rz + s = q ′ (cz + d) + r ′ z + s ′ with the property that r < c or s < d, and also that r ′ < c or s ′ < d. Then qc + r = q ′ c + r ′ and qd + s = q ′ d + s ′ . If q ′ > q, then qc + r = q ′ c + r ′ ≥ (q + 1)c + r ′ = qc + c + r ′ and so r ≥ c + r ′ ≥ c. Similarly, qd + s = q ′ d + s ′ ≥ (q + 1)d + s ′ = qd + d + s ′ and s ≥ d + s ′ ≥ d. Thus, if q ′ > q, then r ≥ c and s ≥ d, which is absurd. Similarly, if q ′ < q, then r ′ ≥ c and s ′ ≥ d, which is also absurd. It follows that q = q ′ and rz + s = r ′ z + s ′ . This completes the proof. In the division algorithm (5), we call q the integer part of the linear fractional transformation (az + b)/(cz + d) and write q = az + b cz + d . We call (rz+s)/(cz+d) the fractional part of the linear fractional transformation (az + b)/(cz + d) and write rz + s cz + d = az + b cz + d . For example, if F (z) = 21z + 16 8z + 5 then the division algorithm gives 21z + 16 = 2(8z + 5) + 5z + 6 and the integer part of F (z) is 2 and the fractional part of F (z) is (5z + 6)/(8z + 5). Note that 8z + 5 ≺ 21z + 16, but 8z + 5 and 5z + 6 are not comparable, that is, {F (z)} is an orphan. If az + b = q(cz + d), then we say that the integer part of (az + b)/(cz + d) is q and the fractional part is 0. If az + b ≺ cz + d, then we say that the integer part of (az + b)/(cz + d) is 0 and the fractional part is (az + b)/(cz + d). Note that the integer and fractional parts of (az + b)/(cz + d) are undefined if az + b and cz + d are unequal and not comparable. Lemma 4 In the infinite binary tree generated by z, the descendant of z after k generations to the right is z + k, and the descendant of z after k generations to the left is z/(kz + 1). Proof For k = 1 this is simply the definition of the right and left descendants. Let k ≥ 2. If the right descendant of z after k − 1 generations is z + k − 1, then the right descendant of z after k generations is z + k. If the left descendant of z after k − 1 generations is z/((k − 1)z + 1), then the left descendant of z after k generations is z (k−1)z+1 z (k−1)z+1 + 1 = z kz + 1 . This completes the proof. Theorem 4 (Successor formula) Let n be a positive integer. In the infinite binary tree generated by z, if w n,j (z) and w n,j+1 (z) are successive terms on the nth row of the linear fractional transformation tree, then w n,j+1 (z) = 1 2[w n,j (z)] + 1 − w n,j (z) where [w n,j (z)] is the integer part of w n,j (z). Proof Let i ∈ 1, 2, . . . , 2 n−1 and j = 2i − 1. The linear fractional transformations w n,2i−1 (z) and w n,2i (z) are successive elements on the nth row, and are the left and right children of w n−1,i (z). If w n−1,i (z) = (az + b)/(cz + d), then w n,2i−1 (z) = az + b (a + c)z + (b + d) . Because az + b ≺ (a + c)z + (b + d), we have [w n,2i−1 (z)] = 0 and {w n,2i−1 (z)} = w n,2i−1 (z). Then 1 2[w n,2i−1 (z)] + 1 − w n,2i−1 (z) = 1 1 − w n,2i−1 = 1 1 − az+b (a+c)z+(b+d) = (a + c)z + (b + d) cz + d = w n,2i (z). Let i ∈ 1, 2, . . . , 2 n−1 − 1 and j = 2i. If w n,2i (z) and w n,2i+1 (z) are successive elements on the nth row, then the former is the right child and the latter is the left child of successive elements in the (n − 1)st row. If these linear fractional transformations on the (n−1)st row are not siblings, then they are the right and left children, respectively, of successive elements in row n − 2. Every element in the tree is a descendant of the root z. Retracing the family tree, we must eventually reach an element from which both w n,2i (z) and w n,2i+1 (z) are descended. Thus, there is a smallest nonnegative integer k such that this common ancestor is on row n − k − 1. Let w * = w n−k−1,t (z) be this ancestor. Its children are w n−k,2t−1 (z) and w n−k,2t (z). Then w n,2i (z) is the k-fold right child of w n−k,2t−1 (z), and w n,2i+1 (z) is the k-fold left child of w n−k,2t (z). Thus, w n−k,2t−1 (z) = w * w * + 1 and, by Lemma 4, w n,2i (z) = w n−k,2t−1 + k = w * w * + 1 + k. Because w * ≺ w * + 1, the division algorithm (Lemma 3) implies that [w n,2i (z)] = k. Similarly, w n−k,2t (z) = w * + 1 and so, by Lemma 4, w n,2i+1 (z) = w n−k,2t (z) kw n−k,2t (z) + 1 = w * + 1 k(w * + 1) + 1 = 1 k + 1 w * +1 = 1 k + 1 − w * w * +1 = 1 2k + 1 − w n,2i (z) = 1 2[w n,2i (z)] + 1 − w n,2i (z) . This completes the proof. Continued fractions and the depth formula To prove the analogue of the depth formula, we introduce finite continued fractions of linear fractional transformations. Let az +b and cz +d be comparable relatively prime positive linear functions, that is, either cz + d ≺ az + b or az + b ≺ cz + d. Note that if cz + d ≺ az + b, then 0 ≤ c + d < a + b. We define r 0 z + s 0 = az + b and r 1 z + s 1 = cz + d. If r 0 z + s 0 ≺ r 1 z + s 1 , then we have r 0 z + s 0 = q 0 (r 1 z + s 1 ) + (r 2 z + s 2 ) where q 0 = 0, r 2 z + s 2 = r 0 z + s 0 , and r 2 z + s 2 ≺ r 1 z + s 1 . If r 1 z + s 1 ≺ r 0 z + s 0 , then, by the division algorithm (Lemma 3), there exist a unique positive integer q 0 and a unique positive linear function r 2 z + s 2 such that r 0 z + s 0 = q 0 (r 1 z + s 1 ) + (r 2 z + s 2 ) and either r 2 z + s 2 ≺ r 1 z + s 1 or the linear functions r 1 z + s 1 and r 2 z + s 2 are not comparable. If r 2 z + s 2 ≺ r 1 z + s 1 , then, first, 0 ≤ r 2 + s 2 < r 1 + s 1 , and, second, there exist a unique positive integer q 1 and a unique positive linear function r 3 z + s 3 such that r 1 z + s 1 = q 1 (r 2 z + s 2 ) + (r 3 z + s 3 ) and either r 3 z + s 3 ≺ r 2 z + s 2 (and so 0 ≤ r 3 + s 3 < r 2 + s 2 ) or the linear functions r 2 z + s 2 and r 3 z + s 3 are not comparable. Continuing inductively, we obtain a finite sequence of positive linear functions r i z + s i for i = 0, 1, . . . , j + 1 such that r i z + s i ≺ r i−1 z + s i−1 for i = 2, 3, . . . , j and r 0 z + s 0 = q 0 (r 1 z + s 1 ) + (r 2 z + s 2 ) r 1 z + s 1 = q 1 (r 2 z + s 2 ) + (r 3 z + s 3 ) r 2 z + s 2 = q 2 (r 3 z + s 3 ) + (r 4 z + s 4 ) . . . r i−2 z + s i−2 = q i−2 (r i−1 z + s i−1 ) + (r i z + s i ) . . . r j−1 z + s j−1 = q j−1 (r j z + s j ) + (r j+1 z + s j+1 ). Note that r j−1 = q j−1 r j + r j+1 s j−1 = q j−1 s j + s j+1 and so r j−1 s j − s j−1 r j = (q j−1 r j + r j+1 )s j − (q j−1 s j + s j+1 )r j = −(r j s j+1 − s j r j+1 ). This implies that r j s j+1 − s j r j+1 = (−1) j (r 0 s 1 − s 0 r 1 ) = (−1) j (ad − bc).(6) Because every strictly decreasing sequence of nonnegative integers is finite and because either 0 ≤ r i+1 < r i or 0 ≤ s i+1 < s i for i = 1, . . . , j, the process of iteration of the division algorithm must terminate, and, after, say, k divisions, we obtain positive linear functions r k z + s k and r k+1 z + s k+1 that are not comparable. We call this procedure the Euclidean algorithm. We rewrite the equations in the Euclidean algorithm to obtain a finite continued fraction for the linear fractional transformation az + b cz + d = r 0 z + s 0 r 1 z + s 1 = q 0 + r 2 z + s 2 r 1 z + s 1 = q 0 + 1 r 1 z + s 1 r 2 z + s 2 = q 0 + 1 q 1 + r 2 z + s 2 r 3 z + s 3 = q 0 + 1 q 1 + 1 r 3 z + s 3 r 2 z + s 2 . . . = q 0 + 1 q 1 + 1 q 2 + · · · + 1 q k−1 + 1 r k z + s k r k+1 z + s k+1 . The linear fractional transformation w * = (r k z+s k )/(r k+1 z+s k+1 ) is an orphan; we call it the root of the linear fractional transformation (az + b)/(cz + d). Note that q 0 = 0 if (az + b)/(cz + d) is a left child and q 0 ≥ 1 if (az + b)/(cz + d) is a right child. If (az + b)/(cz + d) is a special positive linear fractional transformation, then ad − bc = 1 and formula (6) implies that r k s k+1 − s k r k+1 = (−1) k . Because w * = (r k z + s k )/(r k+1 z + s k+1 ) is an orphan, Lemma (2) implies that w * = z if k is even 1/z if k is odd. The continued fraction of (az + b)/(cz + d) is [q 0 , q 1 , . . . , q k−1 , z] if k is even, and [q 0 , q 1 , . . . , q k−1 + z] if k is odd. In Section 3 there is a drawing of the first four rows of the tree with root z. Replacing each linear fractional transformation with its continued fraction, we obtain [z] t t ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ C C ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ [0, 1, z] x x q q q q q q q q q q 8 8 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ [1 + z] v v ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ 9 9 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ [0, 2, z]~⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 8 8 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ [1, 1, z] x x q q q q q q q q q q 8 8 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ [0, 1, 1 + z] w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ @ @ [2 + z] w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ 1 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ [0, 3,v = [q 0 , q 1 , . . . , q k−1 , z] (7) with k even, or v = [q 0 , q 1 , . . . , q k−1 + z](8) with k odd. Moreover, z is in row q 0 + q 1 + · · · + q k−1 of the tree. Proof The unique element of row 0 is the root z, whose continued fraction z = [z] is of the form (8) with k = 1 and q 0 = 0. Similarly, the rational functions on row 1 are z/(z + 1) = [0, 1, z] with k = 2, and z + 1 = [1 + z] with k = 1. Let n ≥ 1, and assume that the Theorem is true for the rational functions on the nth row. Let v be on row n + 1. If v is a right child, then there exists v ′ on row n such that v = v ′ + 1. If v = [q 0 , q 1 , . . . , q k−1 , z] is of form (7), then k is even, k−1 i=0 q ′ i = n, and v = v ′ + 1 = [q ′ 0 , q ′ 1 , . . . , q ′ k−1 , z] + 1 = [q 0 , q 1 , . . . , q k−1 + z] with q 0 = q ′ 0 + 1, and q i = q ′ i for i = 1, . . . , k. Similarly, if v ′ = [q ′ 0 , q ′ 1 , . . . , q ′ k−1 + z] is of form (8), then k is odd, k−1 i=0 q ′ i = n, and v = v ′ + 1 = [q ′ 0 , q ′ 1 , . . . , q ′ k−1 + z] + 1 = [q 0 , q 1 , . . . , q k−1 + z] with q 0 = q ′ 0 + 1, and q i = q ′ i for i = 1, . . . , k − 1. In both cases, k−1 i=0 q i = 1 + k−1 i=0 q ′ i = n + 1. If v is a left child on the (n + 1)st row, then there exists v ′ on row n such that v = v ′ /(v ′ + 1). Let v ′ = [q ′ 0 , q ′ 1 , . . . , q ′ k−1 + z] be of form (8), with k is odd and k−1 . . , q ′ k−1 + z] = [q 0 , q 1 , q 2 , . . . , q k , q k+1 + z] with q 0 = 0, q 1 = 1, and q i = q ′ i−2 for i = 2, 3, . . . , k + 1. Moreover, i=0 q ′ i = n. If q ′ 0 ≥ 1, then v = v ′ v ′ + 1 = 1 1 + 1 v ′ = 1 1 + 1 [q ′ 0 , q ′ 1 , . . . , q ′ k−1 + z] = [0, 1, q ′ 0 , q ′ 1 , .k+1 i=0 q i = 1 + k−1 i=0 q ′ i = n + 1. If q ′ 0 = 0, then v = v ′ v ′ + 1 = 1 1 + 1 v ′ = 1 1 + 1 [q ′ 0 , q ′ 1 , . . . , q ′ k−1 + z] = 1 1 + [q ′ 1 , . . . , q ′ k−1 + z] = [0, 1 + q ′ 1 , . . . , q ′ k−1 + z] = [q 0 , q 1 , q 2 , . . . , q k−1 + z] with q 0 = 0, q 1 = 1 + q ′ 1 , and q i = q ′ i for i = 2, 3, . . . , k − 1. Moreover, k−1 i=0 q i = 1 + k−1 i=0 q ′ i = n + 1. The argument is the same when v ′ = [q ′ 0 , q ′ 1 , . . . , q ′ k , z] is of form (8). This completes the proof. Remarks trees are the orphans. The vertices in each tree have the same determinant, and there are exactly h(∆) trees of determinant ∆. The root of the Calkin-Wilf tree is the number z = 1. We can construct infinite rooted binary trees with other numbers as roots. For example, choosing the complex number i as the root, we obtain an infinite binary tree whose vertices are Gaussian numbers with nonnegative real part. However, not every number is the root of an infinite rooted binary tree. The left child of −1 is formally −1/0, which is undefined, and so -1 cannot be the root of an infinite tree. We can prove that, in every field K of characteristic 0, an element z ∈ K is the root of an infinite rooted binary tree if and only if z is not a negative rational number. We can also construct beautiful trees of linear fractional transformations in characteristic p. /(a + b) on the left and (a + b)/b on the right. The fraction a/b is the parent. We call a/(a + b) the left child and (a + b)/b the right child of a/b. Lemma 1 1If F (z) is a positive linear fractional transformation, then F (z) + 1 and F (z)/(F (z) + 1) are positive linear fractional transformations, anddet(F (z)) = det (F (z) + 1) = det F (z) F (z) + 1 . If F (z) isa special positive linear fractional transformation, then F (z) + 1 and F (z)/(F (z) + 1) are special positive linear fractional transformations. Proof Let F (z) = (az +b)/(cz +d) be a positive linear fractional transformation. The integers a, b, c, d are nonnegative with (a, b) = (0, 0) and (c, d) = (0, 0). It follows that the linear fractional transformations Theorem 1 1The directed graph N (z) is a rooted infinite binary tree with root z. The set of vertices of the tree N (z) is the set of all special positive linear fractional transformations, and each special positive linear fractional transformation occurs exactly once as a vertex in this tree. Lemma 3 ( 3Division algorithm) Let az + b and cz + d be relatively prime positive linear functions. If cz + d ≺ az + b, then there is a unique positive integer q and a unique positive linear function rz + s such that the polynomials rz + s and cz + d are relatively prime, az + b = q(cz + d) + (rz + s) ≺ az + b. Indeed, if cd > 0, then q = min([a/c], [b/d]) ≥ 1. If c = 0, then q = [b/d] ≥ 1, and if d = 0, then q = [a/c] ≥ 1. In all three cases, we define r = a − qc and s = b − qd. Then r and s are nonnegative integers such that rz + s = (a − qc)z + (b − qd) = (az + b) − q(cz + d). Theorem 5 (Depth formula) Every vertex v in the infinite binary tree generated by z has a unique continued fraction in exactly one of the following two forms:z] [1, 2, z] [0, 1, 1, 1, z] [2, 1, z] [0, 2, 1 + z] [1, 1, 1 + z] [0, 1, 2 + z] [3 + z] = ad − bc ≥ (c + 1)(b + 1) − bc = 1 + b + c ≥ 1 We can prove that, for every nonzero integer ∆ there are only finitely many positive linear fractional transformations of determinant ∆ that are orphans. It would be of interest to know the exact number h(∆) of orphans of determinant ∆. Here is a Maple computation of [∆, h(∆)] for ∆ = 1. . , 50By Lemma 2, the function F (z) = z is the unique special positive linear fractional transformation that is an orphan. 1, 1], [2, 4], [3, 7. 4, 13. 5, 15], [6, 26], [7, 25], [8, 39], [9, 40], [10, 54], [11, 49. 12, 79], [13, 63], [14, 88], [15, 88], [16, 112], [17, 93], [18, 140], [19, 109], [20, 159], [21, 142], [22, 170], [23, 143], [24, 224], [25, 168], [26, 216], [27, 202], [28, 255], [29, 199], [30, 304], [31, 219], [32, 308], [33, 268], [34, 316], [35, 274], [36, 404], [37, 281], [38, 370], [39, 338. 40, 438. 41, 323. 42, 484], [43, 345], [44, 481], [45, 433], [46, 484], [47, 389], [48, 611], [49, 422], [50, 566By Lemma 2, the function F (z) = z is the unique special positive linear frac- tional transformation that is an orphan. We can prove that, for every nonzero integer ∆ there are only finitely many positive linear fractional transformations of determinant ∆ that are orphans. It would be of interest to know the exact number h(∆) of orphans of determinant ∆. Here is a Maple computation of [∆, h(∆)] for ∆ = 1, . . . , 50: [1, 1], [2, 4], [3, 7], [4, 13], [5, 15], [6, 26], [7, 25], [8, 39], [9, 40], [10, 54], [11, 49], [12, 79], [13, 63], [14, 88], [15, 88], [16, 112], [17, 93], [18, 140], [19, 109], [20, 159], [21, 142], [22, 170], [23, 143], [24, 224], [25, 168], [26, 216], [27, 202], [28, 255], [29, 199], [30, 304], [31, 219], [32, 308], [33, 268], [34, 316], [35, 274], [36, 404], [37, 281], [38, 370], [39, 338], [40, 438], [41, 323], [42, 484], [43, 345], [44, 481], [45, 433], [46, 484], [47, 389], [48, 611], [49, 422], [50, 566]. We can partition the set of all positive linear fractional transformations into a forest of pairwise vertex disjoint infinite rooted binary trees. The roots of the References. We can partition the set of all positive linear fractional transformations into a forest of pairwise vertex disjoint infinite rooted binary trees. The roots of the References Martin Aigner, M Günter, Ziegler, Proofs from The Book. BerlinSpringer-Verlagthird ed.Martin Aigner and Günter M. Ziegler, Proofs from The Book, third ed., Springer-Verlag, Berlin, 2004. Linking the Calkin-Wilf and Stern-Brocot trees. Bruce Bates, Martin Bunder, Keith Tognetti, European J. Combin. 317Bruce Bates, Martin Bunder, and Keith Tognetti, Linking the Calkin-Wilf and Stern-Brocot trees, European J. Combin. 31 (2010), no. 7, 1637-1661. Recounting the rationals. Neil Calkin, Herbert S Wilf, Amer. Math. Monthly. 1074Neil Calkin and Herbert S. Wilf, Recounting the rationals, Amer. Math. Monthly 107 (2000), no. 4, 360-363. A polynomial analogue to the Stern sequence. Karl Dilcher, Kenneth B Stolarsky, Int. J. Number Theory. 31Karl Dilcher and Kenneth B. Stolarsky, A polynomial analogue to the Stern sequence, Int. J. Number Theory 3 (2007), no. 1, 85-103. Enumerating the rationals from left to right. S P Glasby, Amer. Math. Monthly. 1189S. P. Glasby, Enumerating the rationals from left to right, Amer. Math. Monthly 118 (2011), no. 9, 830-835. Recounting the rationals, continued, solution to problem 10906. Moshe Newman, Amer. Math. Monthly. 110Moshe Newman, Recounting the rationals, continued, solution to problem 10906, Amer. Math. Monthly 110 (2003), 642-643.	[]
[ "Scaling of Urban Income Inequality in the United States", "Scaling of Urban Income Inequality in the United States" ]	[ "Elisa Heinrich Mora \nMinerva Schools at KGI\n94103San FranciscoCAUSA\n\nSanta Fe institute\nSanta Fe87501NMUSA\n", "Jacob J Jackson \nSanta Fe institute\nSanta Fe87501NMUSA\n\nBrown University\n02912ProvidenceRIUSA\n", "Cate Heine \nSanta Fe institute\nSanta Fe87501NMUSA\n\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n", "Geoffrey B West \nSanta Fe institute\nSanta Fe87501NMUSA\n", "Vicky Chuqiao Yang \nSanta Fe institute\nSanta Fe87501NMUSA\n", "Christopher P Kempes \nSanta Fe institute\nSanta Fe87501NMUSA\n" ]	[ "Minerva Schools at KGI\n94103San FranciscoCAUSA", "Santa Fe institute\nSanta Fe87501NMUSA", "Santa Fe institute\nSanta Fe87501NMUSA", "Brown University\n02912ProvidenceRIUSA", "Santa Fe institute\nSanta Fe87501NMUSA", "Massachusetts Institute of Technology\n02139CambridgeMAUSA", "Santa Fe institute\nSanta Fe87501NMUSA", "Santa Fe institute\nSanta Fe87501NMUSA", "Santa Fe institute\nSanta Fe87501NMUSA" ]	[]	Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time and space. Here, we use the urban scaling framework to study an important, but under-explored feature in this community-income inequality. We propose a new method to study the scaling of income distributions by analyzing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales close to linearly with city population, while income in the most wealthy decile scale with a significantly superlinear exponent. In contrast to the superlinear scaling of total income with city population, this decile scaling illustrates that the benefits of larger cities are increasingly unequally distributed. For the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. We repeat our analysis after adjusting income by housing cost, and find similar results. We then further analyze the 1 arXiv:2102.13150v1 [physics.soc-ph] 25 Feb 2021 shapes of income distributions. First, we find that mean, variance, skewness, and kurtosis of income distributions all increase with city size. Second, the Kullback-Leibler divergence between a city's income distribution and that of the largest city decreases with city population, suggesting the overall shape of income distribution shifts with city population. As most urban scaling theories consider densifying interactions within cities as the fundamental process leading to the superlinear increase of many features, our results suggest this effect is only seen in the upper deciles of the cities. Our finding encourages future work to consider heterogeneous models of interactions to form a more coherent understanding of urban scaling.	null	[ "https://arxiv.org/pdf/2102.13150v1.pdf" ]	232,068,913	2102.13150	1ed8c2c44802e530884f5b843963215d264224a1	Scaling of Urban Income Inequality in the United States March 1, 2021 Elisa Heinrich Mora Minerva Schools at KGI 94103San FranciscoCAUSA Santa Fe institute Santa Fe87501NMUSA Jacob J Jackson Santa Fe institute Santa Fe87501NMUSA Brown University 02912ProvidenceRIUSA Cate Heine Santa Fe institute Santa Fe87501NMUSA Massachusetts Institute of Technology 02139CambridgeMAUSA Geoffrey B West Santa Fe institute Santa Fe87501NMUSA Vicky Chuqiao Yang Santa Fe institute Santa Fe87501NMUSA Christopher P Kempes Santa Fe institute Santa Fe87501NMUSA Scaling of Urban Income Inequality in the United States March 1, 2021 Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time and space. Here, we use the urban scaling framework to study an important, but under-explored feature in this community-income inequality. We propose a new method to study the scaling of income distributions by analyzing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales close to linearly with city population, while income in the most wealthy decile scale with a significantly superlinear exponent. In contrast to the superlinear scaling of total income with city population, this decile scaling illustrates that the benefits of larger cities are increasingly unequally distributed. For the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. We repeat our analysis after adjusting income by housing cost, and find similar results. We then further analyze the 1 arXiv:2102.13150v1 [physics.soc-ph] 25 Feb 2021 shapes of income distributions. First, we find that mean, variance, skewness, and kurtosis of income distributions all increase with city size. Second, the Kullback-Leibler divergence between a city's income distribution and that of the largest city decreases with city population, suggesting the overall shape of income distribution shifts with city population. As most urban scaling theories consider densifying interactions within cities as the fundamental process leading to the superlinear increase of many features, our results suggest this effect is only seen in the upper deciles of the cities. Our finding encourages future work to consider heterogeneous models of interactions to form a more coherent understanding of urban scaling. Introduction Throughout human history, the global urban population has grown continuously. More than half of the global population is currently urbanized, placing cities at the center of human development [1]. It is estimated that by 2030, the number of megacities, cities with more than 10 million inhabitants, will increase from 10 to approximately 40 [1]. Thus, there is an urgent need for a quantitative and predictive theory for how larger urban areas affect a wide variety of city features, dynamics, and outcomes [2]. Perhaps most critically, we need this theory to address how larger cities positively and negatively affect socioeconomic outcomes and the quality of life of individuals. Previous research has demonstrated power-law-like relationships between urban population (also referred to as size later in the text) and many urban features such as GDP, patents, crime, and contagious diseases that persist globally [3][4][5][6][7]. These relationships can often be described by Y = Y 0 N β ,(1) where Y is an urban feature, such as GDP or number of crime instances, N is the population of the city, Y 0 is a constant, and β is the scaling exponent. For many urban outputs, the scaling exponent β is greater than 1, suggesting greater rates of productivity (in both the positive and negative sense) in more populated cities. These observations, known as urban scaling, suggest that a small set of mechanisms significantly influence a variety of urban features across diverse cities [8,9]. Understanding these mechanisms has important implications for developing more prosperous and safer cities. In this framework, desirable aspects with β > 1 have positive returns to scale, while desirable aspects with β < 1 have a less than linear return to scale, demonstrating a diseconomy of scale. Similarly, for undesirable features β > 1 shows a diseconomy of scale since the associated per-capita costs would be increasing with city size. One important aspect of urban features that remains under-explored in the urban scaling framework is economic inequality. Inequality has fundamental implications for individuals' quality of life and the productivity and stability of societies [10]. Past research has heightened debate about economic inequality and its relationship with economic growth and general welfare [11][12][13][14][15][16][17]. Many have raised concern of its negative effects on political stability [11,18], crime [19] and corruption [20]. It has been shown that more unequal places have higher murder rates, grow more slowly, and the correlation between area-level inequality and population growth is positive [21]. Economic inequality is usually measured in terms of the dispersion in the distribution of income or wealth, such as in the Gini Coefficient. Some past research has noted larger cities are correlated with increasing Gini coefficient in income distribution [22][23][24] , but it remains unclear if there are systematic relationships between other features of the income distributions and urban area size. Furthermore, characterizing distributions by a single metric may lose important information [11] -for example, does being poor in bigger cities correspond to a higher or lower standard of living than being poor in a smaller city? A few recent studies [25,26] have investigated the scaling of total income in various income brackets in Australia. These studies find that the total income in lower income brackets scales sublinearly or linearly, while higher brackets scale superlinearly, suggesting greater income agglomeration in the higher income categories in more populated cities. While these studies are informative and provide a new measure for inequality in terms of absolute income (instead of relative income, as in the Gini Coefficient), a limitation is that this measure confounds inequality with average income, which increases with city population. In particular, the "equal" situation in this new measure of inequality is when the total income for all income brackets scales linearly. However, given that total income scales superlinearly in cities globally [3,4], this "equal" situation is unlikely to occur. For example, even if the shapes of income distributions remain identical, income bracket aggregations follow distinct scaling relationships as a result of differences in mean. Figure 1: Illustration comparing two methodologies-scaling obtained from grouping by income bracket (A and C) and that by decile (B and D). Using simulated log-normal income distributions in two scenarios-log-mean increases with city size while log-variance remains the same (A and B), and log-mean and log-variance both increases with city size (C and D). The income distributions are illustrated on a log-scale. The income-bracket grouping (A and C) leads to differences in the groups' income scaling for both scenarios, and fails to distinguish whether larger cities have more dispersion in their income distributions. The decile grouping (B and D) leads to the differences in the groups' scaling observed only when the dispersion increases with the population. The insets show how scaling exponent (β) varies with income groups (bracket or decile). A and C illustrates this behavior using simulated log-normal distributions. While the measure of inequality proposed in [25,26] can be valuable for some applications, it would be useful to untangle the increase in mean from the greater dispersion in income. In this manuscript, we address a few keys questions: (1) How does income inequality (adjusted for shifting average income) systematically change with city size? (2) How different is the income of rich and poor people (measured by percentiles of the population) in small and large cities, and how does this difference scale with city size? (3) Are poor people in a larger city better off than poor people in a small city, after adjusting by the cost of living? How about the same for rich people? Here, we propose a new method to study the scaling of inequality by analyzing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales almost linearly with city size, while that in the most wealthy decile scales with a significantly superlinear exponent. This illustrates that the benefits of larger cities are increasingly unequally distributed, and for the poorest income deciles, city growth has no positive effect on income growth over the null expectation of a linear increase. We then introduce systematic considerations of the entire distribution of income to show which income distribution features are changing with city size. We find that the mean, variance, skewness, and kurtosis of the income distribution all scale systematically with city size. We introduce a KL-divergence procedure to systematically compare all moments and find that comparisons with the largest cities also demonstrate a systematic scaling with city size, indicating that the overall shape of income distribution is radically shifting with city size. We then attempt to identify actual changes in purchasing power with city size by normalizing income by housing costs, which also grow superlinearly with city population. Finally, we discuss how these observations can be connected with the proposed mechanisms underlying urban scaling. Data and methods Scaling of income aggregated by deciles We propose a new method to investigate the scaling of income aggregated by deciles in each city (i.e., the bottom 10%, the next 10%, and so on). The number of individuals in decile n of city i is, N (n) i = N i /10, where N i is the population of city i. The total income in decile n of city i, Y (n) i is, Y (n) i = j∈D (n) y i,j ,(2) where D (n) are the individuals in income decile n, and y i,j is the income of individual j in city i. See Supplemental Materials for more details on the decile assignment in our computational implementation. Figure 1 C and D illustrate this method on simulated log-normal income distributions. Panel C represents the situation in which cities shift in logmean with city size, but do not shift in log-standard deviation, and panel D represents the situation in which cities increase both log-mean and logstandard deviations with city size. We consider the former case an example of the "equal" situation, and this method should lead to no variation in scaling exponents across deciles. Variations in scaling exponent only occur for the latter case. We also contrast the results of our method with that of the grouping by income bracket method in Figure 1 A and C, where variations in scaling exponents occur for both scenarios. Data and income distribution estimation The primary dataset used in our analysis is the 2015 American Community Survey conducted by the US Census Bureau (see Supplementary Materials for more detail). We use the income data reported on the level of census tracts, small local areas of on average 4500 people, of which on average 2300 reported income. We infer the individual-level income distribution in Metropolitan Statistical Areas (MSAs) by applying the Gaussian kernel density estimator with a widened Silverman bandwidth function on the census-tract-level data. This method assumes income in each census tract is distributed as a Gaussian. The mean equals the average income of the census tract, and the standard deviation is calculated as a function of the number of data points. Aggregating the Gaussian probability density functions (PDFs) for each census tract in the MSA produces an estimated income PDF for the MSA. Examples of the estimated individual-level income distribution for a few MSAs are shown in Figure 2. Analysis of income scaling in deciles The estimated income distributions for US cities are grouped into deciles: the 10% of the population which reports the lowest income is grouped into the first decile (decile #1), and likewise for all ten deciles up to the 10% of the population which has the highest income (decile #10). We then estimate the scaling exponent of total income for all deciles. We estimate the scaling exponent, β, and corresponding confidence intervals, by performing an ordinary least square regression of the log-transformed variables, log(Y (n) i ) = β log(N (n) i ) + c, and β and c are the fitted parameters. This methodology is consistent with previous research such as [3]. Analysis of distributions We further analyze how the shapes of the income distributions vary with city population. We first compute the first four statistical moments, mean, variance, skewness, and kurtosis, for income distributions of each city, and analyze how they vary with population. We then compute the Kullback-Leibler (KL) divergence between each city's income distribution and that of the largest city (New York-Newark-Jersey City MSA). The KL divergence measures how different one distribution is from another, while the zero value indicates the two distributions are identical, and a greater value indicates more divergence. Mathematically, the KL divergence between two discrete distributions of random variable x, P (x) and Q(x) is, KL(P \|\|Q) = x P (x) log(P (x)/Q(x)) .(3) Adjusting income by housing cost In order to normalize income by the cost of living, we calculate total housing cost in a census tract as cost = 12 (u rent r + u own o), where the average monthly rent r, the average monthly owner costs o, and the number of units of each type u rent and u own are all taken from the 2015 American Community Survey (see Supplementary Materials for more detail and access information). We then repeat the decile-grouped analysis on income adjusted for housing cost, as well as analyze how the proportion of income spent on housing varies with city size in each decile. Results Scaling of income in deciles The results for scaling of income aggregated in deciles are summarized in Figure 3. For the lowest two deciles, the scaling exponent β is linear or slightly sublinear (0.97). For upper deciles, β is consistently superlinear, as high as 1.16 as compared to the scaling exponent of total income in our dataset, β = 1.07. This shows that scaling effects are not equivalent for all segments of the population. The poorest two deciles in bigger cities make about the same income as their counterparts in smaller cities, while the wealthiest eight deciles in bigger cities make more than their counterparts in smaller cities, where the difference increases with the decile. Analysis of income distribution characteristics We further analyze how income distributions vary with urban area population by studying the statistical moments of the income distributions. We first examine the first four moments: mean, variance, skewness, and kurtosis. The scaling of the four moments of the estimated individual income distribution for all cities in our data is shown in Figure 4. The first moment, the mean, shows the well-characterized urban agglomeration effect: per-capita income increases with city size [3]. The second, and third moments both increase similarly with city population, suggesting a widening of the distribution and increasing asymmetry with greater urban population. This can also be qualitatively observed in the example distributions in Figure 2. Lastly, the kurtosis also increases with population size, showing an increasingly heavy tail with greater urban population. We find a stronger relationship for higher statistical moments, indicating that for larger American cities, there is a more evident increase in the third and fourth moments. This means that there is a stronger increase in the growing tail of the distribution, in comparison to the first two statistical moments. This gives us an interesting indication of the distribution of economic benefits. Another useful perspective on the scaling of the income distributions is to compare large and small cities using measures that consider the entire dis- tribution through the KL divergence. Figure 5 shows the KL divergence between each US city and the largest city, as a function of the log-transformed city population. The KL divergence, in general, decreases with city population, and approaches zero as the population approaches that of the largest city. This behavior suggests that as cities get smaller, their income distributions are increasingly dissimilar to that of the largest city. The Pearson correlation between the two variables in Figure 5 is −0.259, while the Spearman correlation is −0.718. The Pearson correlation measures the linear correlation between two variables, while the Spearman correlation measures the rank correlation, and assesses how well relationship between two variables can be described by a monotonic function, regardless of linearity. This finding suggests that population and the KL divergence tend to change together, but not necessarily at a constant rate. While we can identify a general scaling trend, our data also exhibit frequent outliers and deviations. Scaling of decile income adjusted by housing cost While the differences in income scaling that we have identified are important, they are not necessarily grounded in differences in the experiences of urban residents-cost of living can vary drastically across and within US cities, and if cost of living is changing in the exact same way as income, differences in income scaling between groups begin to lose meaning. In order to understand whether the differences in income scaling we see between deciles create differences in affordability and purchasing power, we look at changes in housing cost with city size. By analyzing, in combination, ag- gregate household income and aggregate housing cost for each census tract, we find that aggregate housing cost scales faster than aggregate income for every decile, implying that while income per person increases with city size, larger cities may still be overall less affordable. This difference is more dramatic for the poorer deciles-in the bottom decile, housing cost scales with β = 1.11 while income scales with β = 1.01; in the top decile, housing cost scales with β = 1.29 while income scales with β = 1.27. This is visualized in Figure 6A-income exponents begin to catch up to housing cost exponents in richer deciles, but never as high as housing cost. Perhaps more intuitively, in Figure 6B, we can see that the ratio between total housing cost and total income grows with city size for every decile, but more dramatically for poorer deciles. Together, these results imply a widening gap between richer and poorer residents in affordability of cities with city size. Discussion Here we proposed a new method to study the scaling of income distributions and income inequality in urban areas. The aggregated income in income deciles scale systematically with city size. The bottom decile scales with an exponent slightly below 1 and the top decile with an exponent of β = 1.15. This result suggests that the benefits of larger cities are increas- Figure 6: Comparing the scaling of housing cost and income. (A) The scaling of total income, total housing cost, and the difference between total income and housing cost, for each decile. Housing cost scales with greater exponents than income for all deciles. The housing-adjusted income exhibits similar variation across deciles as total income. (B) Ratio between housing cost and household income as a function of city population. In the poorest deciles (dark brown), the proportion of income spent on housing increases sharply with city size; in the wealthiest deciles (orange), this proportion remains stagnant. ingly unequally distributed, and for the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. Much has been written about the apparent increasing gains of large cities [3,4], such as greater GDP, higher wages, and more patents per capita. Our results show that the increasing benefits of city size are not evenly distributed to people within those cities. We further show systematic variations in distribution characteristics. Besides greater mean, distributions of bigger cities also exhibit greater spread, greater asymmetry, and heavier tail. These perspectives can be explicitly connected to traditional measures of income inequality, such as the Gini coefficient. Like the Gini Coefficient, our method characterizes the overall dispersion of income distributions (see Figure 7), but it also provides more detailed information that is not characterized by Gini, such as how the urban agglomeration effect alters the incomes of relatively poor or rich people differently. Figure 7: Changes in the Gini coefficient with urban population in simulated log-normal distributions. For a scenario of parallel decile scaling ( Figure 1B) and for a scenario where the deciles have divergent scaling ( Figure 1D). As expected, the divergent scaling observation corresponds to increasing Gini coefficient with population. Although our results appear to closely align with those of Sarkar et al. [25,26], which analyze Australian income data, the difference in methodology (aggregation by income brackets vs. by deciles) should lead to different interpretations of the scaling exponents derived. In particular, the baseline "equal" situation is different in the two methods-in Sarkar et al., when total income in all income brackets scale with the linear exponent, and in our methods, when total income in all deciles scale with the same exponent (either linear or nonlinear). Our paper offers new contributions to the literature. First, we develop a new method to study income inequality in the urban scaling framework, which untangles the systematic shift in mean from the study of income inequality. This method enables us to study how income agglomeration effects vary between relatively rich and poor people, after accounting for the systematically increasing mean with population size. Second, our analysis including housing cost demonstrates that despite agglomeration effects on income, bigger cities are less affordable for people of all deciles in the sense that they spend proportionally more of their income on housing; this is especially true for lower-income people. Third, our analysis extends beyond the single-parameter characterization of income inequality. We analyze more complex properties of income distributions through analyzing statistical moments and KL divergence, and reveal systematic variations with city size. Fourth, our results suggest new directions for understanding mechanisms of urban agglomeration effects-it is important to extend beyond theories considering homogeneous densifying interactions to those which account for heterogeneity. Understanding the underlying mechanisms of why inequality is systematically scaling with city size is of great future interest with many potential implications. Urban scaling theory in general proposes densifying interactions within cities as the fundamental process leading to the superlinear increase of many features [8,9,27,28]. Our analysis shows that the superlinear scaling is not seen within all subsections of the city. The superlinear scaling of total wealth is driven by the top income deciles, and is not matched proportionally by the lowest deciles. This adds another dimension to considerations of the underlying mechanisms of urban scaling theory: what processes are leading to the increasingly unequal distribution of wealth in larger cities? We explored the idea of city heterogeneity as an indirect proxy for heterogeneous interaction rates. One hypothesis of the mechanism driving superlinear scaling of income with city size is that larger cities foster more and more diverse social and economic interactions, creating opportunities for the exchange of ideas and resources. Existing literature credits superlinear growth of income in cities to more opportunities for social contacts and interactions in large cities [3,8]. Increased social contact with city size has been empirically confirmed [29], and ties between individual's exposure to diverse social connections and economic outcomes have been shown empirically as well [30]. Together, this seems to suggest that cities that are better mixed, allowing diverse parts of the population to be exposed to one another, should be overperforming with respect to urban scaling. We hypothesize that cities with high levels of economic segregation, inhibiting mixing between diverse populations, will underperform with respect to income scaling. Our finding encourages future work to consider heterogeneous models of interactions, as those clustered in space or social/work circles, to form a more coherent understanding of urban scaling. In our income measure in the decile scaling and distribution analysis (Sections 3.1 and 3.2), we multiply mean earnings by the total number of workers aged 16 years and older from census table S2001: Earnings in the Past 12 Months. As our income measure in the housing cost analysis (Section 3.3), we multiply the mean household income from census table S1901 by the total number of households for which we have rent and owner cost information. We calculate total housing cost in each census tract as cost = 12 (u rent r + u own o), where average monthly rent r, average monthly owner costs o, and the number of units of each type u rent and u own are taken from census table DP04: Selected Housing Characteristics. We provide an example of our division of census tracts into deciles in Table S1, which depicts the census tracts in deciles 1 and 2 for Aberdeen, Washington. The process is as follows: 1. For each MSA, sort census tracts by descending mean income. 2. Iterate through census tracts in order of income, adding census tracts to decile 1 until adding the next census tract would push decile 1's population over 10% of the total MSA population. 3. Add the proportion of that census tract's population that would put decile 1 at exactly 10% of the total MSA population and the same proportion of the census tract's income to decile 1; add the remaining population and income to decile 2. 4. Repeat for all deciles, until there are 10 deciles with equal population Figure 1 1 Figure 2 : 2Examples of the estimated income distributions using census tract data. Income is measured in US dollars. The three metropolitan areas shown are: New York-Newark-Jersey City, NY-NJ-PA, population 20,316,622; Minneapolis-St. Paul-Bloomington, MN-WI, population 3,670,397; Santa Fe, NM, population 204,396. Figure 3 : 3Scaling of income (in US Dollars) by population for deciles of US MSAs. (A) Scaling of total income in deciles (B) Scaling exponents (β) of each decile and corresponding 95% confidence intervals. The dashed line is β = 1 to help guide the eye. Higher-income deciles exhibit greater scaling exponents than lower income deciles, and the lowest deciles exhibit near-linear scaling. The scaling exponents for aggregated income in city, combining all deciles, is 1.07. Figure 4 : 4First four statistical moments of the estimated income distributions as a function of city size. The texts in each panel display the scaling exponent, β, and in the bracket, corresponding 95% confidence intervals. Figure 5 : 5Kullback-Leibler divergence between the estimated income distributions and that of the largest city, as a function of log population. The Spearman correlation is -0.718.. Supplementary Materials for Scaling of Urban Income Inequality in the United States 1 Data sources and methods Income and housing cost data are from the 2015 American Community Survey, openly available through the United States Census Bureau at https: //www.census.gov/programs-surveys/acs. 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[ "Critical behavior of mean-field XY and related models", "Critical behavior of mean-field XY and related models" ]	[ "Kay Kirkpatrick \nDepartment of Mathematics\nUniversity of Illinois at Urbana-Champaign\n1409 W. Green Street61801UrbanaILUSA\n", "Tayyab Nawaz \nDepartment of Mathematics\nUniversity of Illinois at Urbana-Champaign\n1409 W. Green Street61801UrbanaILUSA\n" ]	[ "Department of Mathematics\nUniversity of Illinois at Urbana-Champaign\n1409 W. Green Street61801UrbanaILUSA", "Department of Mathematics\nUniversity of Illinois at Urbana-Champaign\n1409 W. Green Street61801UrbanaILUSA" ]	[]	We discuss spin models on complete graphs in the mean-field (infinite-vertex) limit, especially the classical XY model, the Toy model of the Higgs sector, and related generalizations. We present a number of results coming from the theory of large deviations and Stein's method, in particular, Cramér and Sanov-type results, limit theorems with rates of convergence, and phase transition behavior for these models.	10.1007/978-3-319-59671-6_10	[ "https://arxiv.org/pdf/1611.02356v1.pdf" ]	119,617,177	1611.02356	cefc6ca75b7886d190f03fc60031c5f1d77d7000	Critical behavior of mean-field XY and related models November 9, 2016 Kay Kirkpatrick Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street61801UrbanaILUSA Tayyab Nawaz Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street61801UrbanaILUSA Critical behavior of mean-field XY and related models November 9, 2016 We discuss spin models on complete graphs in the mean-field (infinite-vertex) limit, especially the classical XY model, the Toy model of the Higgs sector, and related generalizations. We present a number of results coming from the theory of large deviations and Stein's method, in particular, Cramér and Sanov-type results, limit theorems with rates of convergence, and phase transition behavior for these models. Introduction We use mean-field theory to approximate a challenging problem and to study a many-body problem by converting it into a one-body problem. We survey a number of results obtained recently using the theory of large deviations along with Stein's method-type limit theorems to describe the asymptotic behavior of the O(N ) spin models such as the N = 1 Curie-Weiss model, the N = 2 model called the XY model, the N = 3 Heisenberg model, and the N = 4 Toy model of the Higgs sector [6,19,20]. We present these results mostly without proofs. In this section, we describe the mean-field XY model and the history, including the 2D XY model (which is currently intractable). In the next section we describe the asymptotic behavior of the XY model; in the last section, the behavior of its generalizations to N -dimensional spins. The XY model on a complete graph K n with n vertices in the absence of an external field is defined as follows: there is a circular spin σ i ∈ S 1 at each site i ∈ 1, 2, ..., n. The configuration space of the XY model is Ω n = (S 1 ) n where each microstate is σ = (σ 1 , σ 2 , ..., σ n ). For the higher O(N ) spin models, we simply replace S 1 by S N −1 , and in all cases the Hamiltonian energy is defined by The mean-field interaction for the XY and O(N ) models on the complete graph is defined by J i,j = 1 2n for all i, j. The simplest spin model is the Ising model, with one-dimensional ±1 spins, a model that is used extensively in statistical mechanics, invented by Ernest Ising while working with his advisor Wilhelm Lenz [9,10]. The one-dimensional Ising model has no phase transition, but there is a phase transition on an infinite two-dimensional lattice. The mean-field Ising model, or Curie-Weiss model, describes the Ising model well for higher dimensions, and the magnetization (average spin) in this model has a Gaussian law away from the critical temperature and a non-Gaussian law at the critical temperature [15]. Recently, Chatterjee and Shao [6] proved that the total spin in the Curie-Weiss model at the critical temperature satisfies a Berry-Esseen type error bound in this non-Gaussian limit. The XY model, with two-dimensional circular spins, models superconductors and is interesting but challenging to study its phase transition rigorously [5]. On a lattice of two spatial dimensions, such a continuous circular symmetry cannot be broken at any finite temperature [30]. Thus the 2D XY model cannot have an ordered phase at low temperature quite like the Ising model, and it has a phase transition that is quite different from the Ising model [31,32]. Instead, the 2D XY model exhibits the peculiar Kosterlitz-Thouless (KT) transition, a phase transition of infinite order and the subject of a Nobel prize. Above the transition temperature T KT correlations between spins decay exponentially. At low temperatures, the system does not have any long-range order as the ground state is unstable, but there is a low-temperature quasi-ordered phase with a correlation function that decreases with the distance like a power, which depends on the temperature [33]. Because the 2D XY model is so challenging, we study the mean-field classical XY model instead, which can be viewed as the large-dimensional (d → ∞) limit of the nearest-neighbor model on Z d , with spins in S 1 , and with critical inverse temperature β c = 2 [22]. Furthermore, the large-dimensional limit approximates high-dimensional models nicely since below the critical temperature, the average spin goes to zero for all d, and above the critical temperature, the total spin has a non-zero limit as d → ∞. In the next section we will examine the XY model in detail, while section 3 deals with extensions to higher spin dimensions. The mean-field XY model and asymptotic results We consider the isotropic mean-field classical XY model on a finite complete graph K n with n vertices. That is, at each site i ∈ K n of the graph is a spin living in Ω = S 1 , so the state space is Ω n = (S 1 ) n . See Fig. 1 for a picture of the XY model on 5 vertices. The corresponding mean-field Hamiltonian energy H n : Ω n → R is given by: H n (σ) := − 1 2n n i,j=1 σ i , σ j = − 1 2n i,j cos(θ i − θ j ), where θ i is the angle that the i-th spin makes with respect to some axis. The corresponding Gibbs measure is the probability measure P n,β on Ω n with density function: where Z is the normalizing constant, also known as the partition function, which encodes the statistical properties of the model such as free energy and magnetization. Note that Gibbs measure here is a normalization of the Boltzmann distribution, and that the inverse temperature β is equal to (k B T ) −1 , where k B is the Boltzmann constant and T is the temperature of the system. We can understand the structural behavior of the spin vectors' distribution by studying extreme cases for the inverse temperature β as follows: f (σ) := 1 Z(β) e −βHn(σ) .(1) • At high temperature, from equation (1) we can predict that the Gibbs measure is uniform. • At low temperature, again from equation (1) we can predict that the Gibbs measure decays quickly, and the spin vector distribution prefers the lowest-energy ground state. The most likely physical system states corresponding to the Gibbs measure are called the canonical macrostates. We will consider the random measure of the spins {σ i }, defined by µ n,σ := 1 n n i=1 δ σ i on S 1 and study the total empirical spin, defined by S n (σ) := n i=1 σ i . The relative entropy of a probability measure ν on S 1 , with respect to the uniform probability measure µ is defined by H(ν \| µ) := S 1 f log(f )dµ if f := dν dµ exists; ∞ otherwise.(2) LDPs, free energy, and macrostates for the XY model Let M 1 (S 1 ) represent the probability measures on S 1 with the weak- * topology. We are interested in analyzing the total spin, S n := n i=1 σ i , as a function of the inverse temperature β in the Gibbs measures. This leads us to consider large deviation principles (LDPs) for the µ n,σ , and then we can rewrite the free energy more explicitly to describe the phase transition at β = 2. Part of Theorem 1 (β = 0) is a special case of Sanov's theorem, while the other cases (β > 0) follow from an abstract result of Ellis, Haven, and Turkington ( [14], Theorem 2.5). Theorem 1. If P n is the n-fold product of uniform measure on S 1 and µ n,σ is the random measure as defined above. For Γ ⊂ M 1 (S 1 ), the µ n,σ satisfy an LDP with rate function I β (ν) := H(ν \| µ) − β 2 S 1 xdν(x) 2 − ϕ(β),(3) where the free energy is given by ϕ(β) = inf ν∈M 1 (S 1 ) H(ν \| µ) − β 2 S 1 xdν(x) 2 .(4) For fixed β ≥ 0, every subsequence of P n,β [µ n,σ ∈ ·] converges weakly to a probability measure on M 1 (S 1 ) concentrated on the canonical macrostates E β := {ν : I β (ν) = 0}, i.e., the zeros of the rate function. For β = 0, the relative entropy H(· \| µ) achieves its minima of 0 only for the uniform measure µ, implying that the canonical macrostate is disordered. For β > 0, canonical macrostates are defined abstractly through zeros of the rate function (3), and later Theorem 5 will describe the macrostates explicitly. The free energy given by (4) can be transformed into the following more explicit form. Theorem 2. (Kirkpatrick-Nawaz [20]) The free energy ϕ has the formula: ϕ(β) = 0, if β < 2, Φ β (g −1 (β)), if β ≥ 2, where I i below are modified Bessel functions of first kind and Φ β is the functional defined by: Φ β (r) := r I 1 (r) I 0 (r) − log [I 0 (r)] − β 2 I 1 (r) I 0 (r) 2 ,(5) and g(r) := r I 0 (r) I 1 (r) . Here the phase transition is continuous as the function ϕ and its derivative ϕ are continuous at the critical threshold β = β c = 2. The magnetization for the classical XY model can be obtained by differentiating the partition function: \|m\| = E 1 n i σ i = E 1 n S n = I 1 (r) I 0 (r) From the free energy we can precisely explain the phase transition at β = 2. For 0 ≤ β ≤ 2, we have a unique global minimum for the free energy at the origin with a zero magnetization. For β ≥ 2, we have a unique global minimum for a positive radius. Let {σ i } n i=1 be i.i.d. uniform random points on S 1 ⊆ R 2 . We have the following Cramértype LDP for the average spin. Theorem 3. (Kirkpatrick-Nawaz [20]) Let P n,β be the Gibbs measure defined above (1). Then for β ≥ 0, the average spin M n = M n (σ) := 1 n n i=1 σ i satisfies an LDP with rate function I β (x) = Φ β (r): P n,β (M n x) e −nΦ β (r) , where Φ β is given by (5) and r = \|x\|. For an explicit representation of E β , we note from (2) that the relative entropy depends only on the distribution of f . By Fubini's theorem f log(f )dµ = ∞ 0 µ f log(f ) > t dt − ∞ 0 µ f log(f ) < −t dt. This implies that for a fixed f , the quantity xdν(x) is maximized for corresponding densities which are symmetric about a fixed pole and decreasing as the distance from the pole increases. Using this reasoning, consider a density f that is symmetric about the north pole and decreasing away from the pole i.e., ν f is the measure with density f (x, y) = f (y) which is increasing in y. Then H(ν f \| µ) = 1 2π S 1 f (x, y) log[f (x, y)]dxdy = 1 π 2π 0 π 0 f (cos(θ)) log[f (cos(θ))]dθdϕ = 1 π 1 −1 f (y) log[f (y)] 1 − y 2 dy. Similarly, S 1 xdν f (x) = 1 π 0 1 1 −1 yf (y) 1 − y 2 dy. Therefore, our minimization problem is reduced to minimizing the following functional 1 π 1 − y 2 log f (y) π dy + log(π) = −S f π + log(π). Now for xdν(x) = c ∈ [0, 1], using constrained entropy maximization (see Theorem 12.1.1 from [8]), we will minimize 1 π 1 −1 yf (y) √ 1−y 2 dy, that is, maximize S(f /π), over the ν ∈ M 1 (S 1 ) corresponding to this c. i.e., weighted integral of f is 1 while first weighted moment is bounded. Then the exponential function f * (y) = πae by uniquely maximizes S(f /π) over the densities satisfying these conditions. For c ∈ [0, 1], observe that f * increasing gives all b ∈ [0, ∞). Now for b ∈ [0, ∞), our functional minimization reduces to the following one dimensional function: 1 π 1 −1 f (y) log[f (y)] 1 − y 2 dy − β 2 c 2 = b I 1 (b) I 0 (b) − log [I 0 (b)] − β 2 I 1 (b) I 0 (b) 2 =: Φ β (b).(6) The following theorem, a special case proved using the calculus of variations in [20], describes the canonical macrostates: that the minimizing function f * = 1 and therefore the only canonical macrostate is the uniform distribution µ. (b) For β > 2, inf b≥0 {Φ β (b)} = Φ β (g −1 (β)), where b = g −1 (β) is the unique strictly positive solution to g(b) = β where g(b) = b I 0 (b) I 1 (b) , a = 1 πI 0 (b) and lim β↓2 inf b≥0 {Φ β (b)} = 0. In this case, the canonical macrostates are given by E β = {ν f,x } x∈S 1 , where ν f,x is the measure that is the rotation of ν f from north pole to x-direction, which is symmetric about the north pole with density f : [−1, 1] → R given by f (y) = πae by with a and b as above. We can also visualize the Gibbs measure corresponding to subcritical or supercritical cases as shown in Fig. 2. Limit theorems for the total spin in the XY model Next we understand the asymptotics for the total spin of the mean-field XY model, in different regimes across the phase transition, describing the central and non-central limit theorems for each phase. In the high temperature regime (0 ≤ β < 2), the average spin (magnetization) of the system goes to zero with increasing number of spins n → ∞, and we have a multivariate central limit theorem with a rate of convergence in Theorem 6. The main idea is to use Stein's method [19,27,24] with the exchangeable pair (W n , W n ) from the Gibbs sampling approach: our random variable representing the rescaled total spin of the original configuration is W n := 2 − β n n i=1 σ i , while the random variable representing the rescaled total spin of the new configuration, with I ∈ {1, . . . , n} chosen uniformly at random, is W n := W n (σ ) = W n − 2 − β n σ I + 2 − β n σ I . Theorem 6. (Kirkpatrick-Nawaz [20]) In the high temperature regime 0 < β < 2, if W n is defined as above, Z is a standard normal random variable in R 2 , c β is a function depending on β only, L(g) is the modulus of uniform continuity of g, and M (g) is the maximum operator norm of the Hessian of g, then we have: sup g:L(g),M (g)≤1 \|Eg(W n ) − Eg(Z)\| ≤ c β √ n The proof of Theorem 6 proceeds in several steps, as a special case of [20]: first we use the fact that the density of the Gibbs measure is rotationally invariant to conclude that each spin has a uniform marginal distribution. We obtain the complete asymptotic behavior of the total spin using the rotational invariance of the total spin, a strategy adapted from [19]. We calculate the variance of the total spin to arrive at the proper scaling for defining the exchangable pair and use the pair to derive expressions and bounds for the linear factor Λ appearing in the conditional expectation and the remainder terms R and R [19,20,24]. The rest follows from a theorem of Meckes [24]. As the temperature decreases to zero, the spins start aligning. For smaller values of β > 2, the spins vectors are aligned weakly, while for larger β, this alignment is strong. For any β > 2, because of the large deviation principle in Theorem 3, we have that \| σ j \| is close to bn/β with high probability, if b is the minimizer in Φ β . And due to the circular symmetry, all points on the circle of radius bn/β are equally likely. With this reasoning, similar to [19], it is natural to consider the random variable representing the fluctuations of squared-length of total spin, i.e., W n := √ n   β 2 n 2 b 2 n j=1 σ j 2 − 1   .(7) Our multivariate central limit theorem in the low temperature (ordered) regime is as follows: Theorem 7. (Kirkpatrick-Nawaz [20]) If β > 2 and b is the solution of b = βf (b) := β I 1 (b) I 0 (b) , and W n is as defined above in (7), and if Z is a centered normal random variable with variance V , where V = 4β 2 (1 − βf (b)) b 2 1 − 1 b I 1 (b) I 0 (b) − I 1 (b) I 0 (b) 2 , then there exists c β , depending only on β, such that then d BL (W n , Z) ≤ c β log(n) n 1/4 . where d BL (X, Y ) is the bounded Lipschitz distance between random variables X and Y . Again the proof of Theorem 7 follows from a univariate analogue of the abstract normal approximation of Stein [27], and relies on conditional moment bounds. The fact that the variance is positive was proved by Amos [28] while deriving the improved bounds on the ratio of Bessel functions. At the critical temperature β c = 2, we will consider the random variable W n := c n 3/2 n i,j=1 σ i , σ j ,(8) and make an exchangeable pair (W n , W n ) using Glauber dynamics. Using symmetry of the total spin and Stein's method similar to [6,19], we will obtain critical limiting density function p as defined below. Theorem 8. (Kirkpatrick-Nawaz [20]) For the critical inverse temperature β = 2, if W n is as defined above in (8), and X is the random variable with the density p(t) = 1 Z e −t 2 /64 t ≥ 0, 0 t < 0, where Z is normalizing constant, then there exists a universal constant C such that sup h ∞≤1, h ∞≤1 h ∞≤1 Eh(W n ) − Eh(X) ≤ C log(n) √ n . The proof of the limit theorem for the critical temperature is essentially via the "density approach" to Stein's method introduced by Stein, Diaconis, Holmes, and Reinert [29]. Recenlty, also Chatterjee and Shao [6] have applied this approach to the total spin of the mean-field Ising model, i.e., the Curie-Weiss model. We note that these limit theorems with explicit rates of convergence can be generalized to high-dimensional spins, but we will omit those technicalities in the following section. High-dimensional spin O(N ) models We can use similar methods to extend our results for two-dimensional spin classical XY model to classical O(N ) models, or N -vector models. In this general case, with spins in S N −1 ⊂ R N , the critical inverse temperature is β c = N [22,20]. The N -vector models on a complete graph K n have the Hamiltonian: H n (σ) := − 1 2n n i,j=1 σ i , σ j .(9) We present results about the magnetization, free energy, and critical behavior in the O(N ) models. It is important to note that we divide our asymptotic analysis into two cases: if N an even positive integer, we have modified Bessel functions of first kind with order ν = N/2 and ν − 1, while for N odd, we have hyperbolic functions arising from the half-integer order Bessel functions. The magnetization in O(N ) models Similar to the classical XY model, we can calculate the magnetization of the classical Nvector unit hyperspherical model using the conditional density, from the conditional expectations, and it turns out to be a ratio of modified Bessel function of first kind: Theorem 9. (Kirkpatrick-Nawaz [20]) Consider the O(N ) model with the above Hamiltonian (9), with N representing the dimension of the spin σ i ∈ S N −1 . Then on the complete graph K n the O(N ) magnetization M N,n = n i=1 σ i has the following mean-field limit: \|M N \| = I N 2 (r) IN 2 −1 (r)I N 2 −1 (r) I N 2 (r) = β From Fig 3, we can observe that low-dimensional spin models can be magnetized easier in some sense, and as the spin gets higher dimensional, it takes more energy to magnetize the physical system. The rate function and free energy in O(N ) models Next we will present rate functions for large deviation principles similar to Theorems 1&3, the first of which is the relative entropy for the N -vector model given by an abstract formula similar to before: I β,N (ν) := H(ν \| µ) − β 2 S N −1 xdν(x) 2 − ϕ(β) where H(ν \| µ) is the relative entropy (2) and ϕ N is the free energy defined abstractly as before: ϕ N (β) = inf ν∈M 1 (S N −1 ) H(ν \| µ) − β 2 S N −1 xdν(x) 2 .(10) We can calculate the minima in the expression of this rate function and verify that in the subcritical regime (β < N ) there is a unique minimum, while in the supercritical regime there is a family of minima parametrized by S N −1 . The free energy given by (10) can be written in the following more explicit form using a method like the one in the previous section. In particular, we have a Cramér-type LDP for the average spin M n := 1 n n i=1 σ i ∈ R N , with rate function I β,N (x) = Φ β,N (r), defined below for β ≥ 0 and r = \|x\|. In particular, ϕ and ϕ are continuous at the critical threshold β = N , implying that the phase transition is second-order or continuous. The critical density function in O(N ) models The limiting density for the critical case uses the (hyper-)spherical symmetry of the total spin for O(N ) models, giving the following non-normal limit theorem. Theorem 11. (Kirkpatrick-Nawaz [20]) At the critical temperature β = N , the random variable W n = c N \|Sn\| 2 n 3/2 has as its limit as n → ∞ the random variable X with density p N (t) = Eh(W n ) − Eh(X) ≤ C log(n) √ n . The proof of this theorem is in [20] and includes methods from [34,16,19]. Figure 1 : 1Left: The classical mean-field XY Model on the complete graph K 5 with five sample spin vectors. Right: The projection of the same spin vectors from K 5 onto S 1 . 2 dy = 1. We can rewrite the first term of the last expression to see that it involves the usual entropy S(f ) = f log(f Proposition 4 . 4) Consider a set of functions f : [−1, 1] → R + , with weight function w(y) yf (y)w(y)dy = c. Theorem 5 . 5) (a) For β ≤ 2, inf b≥0 {Φ β (b)} = 0 is achieved for b = 0 and the corresponding a = 1, so Figure 2 : 2Cross-sections of two canonical macrostates: For β ≤ 2 (the disordered regime), we have the uniform distribution f (y) = 1 as the dotted line; for β = 5 > β c = 2 (the ordered regime), we have plotted the cross-section of the distribution ν f , given by f (x, y) = f (y) = e by I 0 (y) , showing that the spins point predominantly near the north-pole direction. Figure 3 : 3Graph of magnetization limits \|M N \| for N -vector models, 1 ≤ N ≤ 4. For the mean-field Ising model, M 1 = tanh(x), for the mean-field XY model \|M 2 \| = I 1 (r) I 0 (r) , for the mean-field Heisenberg model \|M 3 \| = coth(r) − 1 r , and for the mean-field Toy model of the Higgs sector, \|M 4 \| = I 2 (r) I 1 (r) . Here r and β are related by the formula g N (r) := r Theorem 10 .Figure 4 : 104) For dimension N , the free energy ϕ has the formula:ϕ N (β) = 0, if β < N, Φ β,N (g −1 (β)),if β ≥ N, Graph of the rate function I β,N (x) = Φ β,N (r) in the supercritical regime (β = N +1) for 2 ≤ N ≤ 4, which has minimum at radius g −1 N (β) = r.where g −1 (β) = r with g(r) = g N (r) Figure 5 : 2 52Mean-field critical density functions p N for 2 ≤ N ≤ 4 and t ≥ 0. For the XY model p 2 (t) = e −t , and for the Toy model of the Higgs sector, p 4 (t) = te −t 2 /384 192 . N 2 (4N +8) and Z is the normalizing constant. 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[ "A Study of Holographic Dark Energy Models in Chern-Simon Modified Gravity", "A Study of Holographic Dark Energy Models in Chern-Simon Modified Gravity" ]	[ "M Jamil Amir ", "Sarfraz Ali ", "\nDepartment of Mathematics\nDepartment of Mathematics\nUniversity of Sargodha\nPakistan\n", "\nUniversity of Education\nLahorePakistan\n" ]	[ "Department of Mathematics\nDepartment of Mathematics\nUniversity of Sargodha\nPakistan", "University of Education\nLahorePakistan" ]	[]	This paper is devoted to study some holographic dark energy models in the context of Chern-Simon modified gravity by considering FRW universe. We analyze the equation of state parameter using Granda and Oliveros infrared cut-off proposal which describes the accelerated expansion of the universe under the restrictions on the parameter α. It is shown that for the accelerated expansion phase −1 < ω Λ < − 1 3 , the parameter α varies according as 1 < α < 3 2 . Furthermore, for 0 < α < 1, the holographic energy and pressure density illustrates phantom-like theory of the evolution when ω Λ < −1. Also, we discuss the correspondence between the quintessence, K-essence, tachyon and dilaton field models and holographic dark energy models on similar fashion. To discuss the accelerated expansion of the universe, we explore the potential and the dynamics of quintessence, K-essence, tachyon and dilaton field models.	10.1007/s1077	[ "https://arxiv.org/pdf/1509.06980v1.pdf" ]	118,574,652	1509.06980	0c2e7d7d2c052ccfa9b280f7c2354a2cad92456c	A Study of Holographic Dark Energy Models in Chern-Simon Modified Gravity 18 Sep 2015 M Jamil Amir Sarfraz Ali Department of Mathematics Department of Mathematics University of Sargodha Pakistan University of Education LahorePakistan A Study of Holographic Dark Energy Models in Chern-Simon Modified Gravity 18 Sep 2015arXiv:1509.06980v1 [gr-qc] 1CS Modified GravityDark EnergyHolographic Dark Energy Models * This paper is devoted to study some holographic dark energy models in the context of Chern-Simon modified gravity by considering FRW universe. We analyze the equation of state parameter using Granda and Oliveros infrared cut-off proposal which describes the accelerated expansion of the universe under the restrictions on the parameter α. It is shown that for the accelerated expansion phase −1 < ω Λ < − 1 3 , the parameter α varies according as 1 < α < 3 2 . Furthermore, for 0 < α < 1, the holographic energy and pressure density illustrates phantom-like theory of the evolution when ω Λ < −1. Also, we discuss the correspondence between the quintessence, K-essence, tachyon and dilaton field models and holographic dark energy models on similar fashion. To discuss the accelerated expansion of the universe, we explore the potential and the dynamics of quintessence, K-essence, tachyon and dilaton field models. Introduction The physicist and cosmologist are facing two fundamental curious problems, the "dark energy (DE)" and "dark matter (DM)". Since last decade, the astronomical observational data collected from large scale structures, type Ia Supernovae and the cosmic microwave background anisotropy supported that our universe is in accelerated expansion [1]- [3]. Type Ia supernovae observational data provided the evidences that our universe is under accelerated expansion due to an exotic energy which has negative pressure and it is so-called DE. According to the astrophysical observations [4], more than 95 percent of the contents of our universe are consist of DM and DE while only about 4 percent is byronic matter with negligible amount of radiation. It is more interesting that about 70 percent of the energy density is DE which is responsible of accelerated expansion of the universe. Although, a huge number of efforts have been made to resolve these issues but there is no satisfactory answer obtained till now. A number of DE models have been discussed on the holographic principle available in literature [5], [6]. Gao et al. purposed some cosmological constraints on the holographic Ricci dark energy. Adabi et al. [7] discussed the correspondence between the ghost dark energy model and Chaplygin scalar field in the framework of general relativity (GR). They investigated FRW universe containing DE and DM. K. Karami and Fehri [8] found the evolution equation as well as equation of state (EoS) parameters using holographic dark energy (HDE) model with Granda and Oliveros cut-off. Jamil et al. [9] studied the HDE problem with a varying gravitational constant, in flat and non-flat universe. Alongwith Setare he [10] discussed the HDE issue with a varying gravitational constant, in Hörava-Lifshitz gravity. With his collaborators [11], they investigated the model of interacting DE and derive its EoS and found the correspondence between the K-essence, tachyon and dilaton scalar fields with the interacting entropy corrected new agegraphic DE in the non-flat FRW universe. Jamil et al. [12] also, using Granda-Oliveros cut-off, studied the holographic dark energy model in the framework of Brans-Dicke gravity theory. Many other DE models have been investigated in different theories, for example, DE modal with quintessence [13], quintom field [14], K-essence field [15], tachyon field [16], dilaton field [17], phantom field [18]. The cosmic baryon asymmetry is longstanding problem of cosmology which suggests a modification in the theory of GR by introducing Chern-Simons (CS) term in inflationary process [19]. The CS modified gravity is an extension to GR introduced by Jackiw and Pi [20]. In this theory, the gravitational field is coupled with a scalar field using a parity-violating CS term. Pasqua et al. [21] investigated the HDE model using Granda-Oliveros cut-off, modified holographic Ricci dark energy model as well as they investigated a model containing higher derivatives of the Hubble parameter in the context of CS modified gravity. Jamil and Sarfraz [22] found the Ricci dark energy of Amended FRW universe in the frame work of CS modified gravity. We organize this paper in following order. The brief review of CS modified gravity is presented in section 2. In section 3, we investigate the HDE model and explored the EoS parameter in the framework of CS modified gravity. The Correspondence between holographic and scalar field models is studied in section 4. The results are summarized in the last section. Brief Review of CS Modified Gravity The Einstein-Hilbert action for CS modified gravity theory is given by [20] S = d 4 x √ −g[κR + α 4 Θ * RR − β 2 (g µν ∇ µ Θ∇ ν Θ + 2V [Θ])] + S mat ,(1) where κ = 1 16πG , ∇ µ is the covariant derivative, R is the Ricci scalar, * RR is called Pontryagin term defined as * RR = * R a b cd R b acd , is topological invariant. The R b acd is the Reimann tensor and * R a b cd is the dual Reimann tensor defined as * R a b cd = 1 2 ǫ cdef R a bef . The terms α and β are defined as coupling constants and the function Θ is called CS coupling field, a function of spacetime using as a deformation function. If function Θ is taken to be a constant, CS modified theory reduces to GR identically. Now, the variation of the action with respect to metric tensor g µν and scalar field Θ yields two field equations of CS modified gravity [20] G µν + lC µν = κT µν , g µν ∇ µ ∇ ν Θ = − α 4 * RR,(2) where G µν is the Einstein tensor, the term l is 4D coupling constant, C µν is the C-tensor defined as C µν = − 1 2 √ −g [υ σ ǫ σµαβ ∇ α R ν β + 1 2 υ στ ǫ σναβ R τ µ αβ ] + (µ ←→ ν).(4) Here, υ σ ≡ ∇ σ Θ and υ στ ≡ ∇ σ ∇ τ Θ. The energy-momentum tensor T µν is consists of the matter part T m µν and the external field part T Θ µν , defined respectively as T m µν = (ρ + p)U µ U ν − pg µν ,(5)T Θ µν = β(∂ µ Θ)(∂ ν Θ) − β 2 g µν (∂ λ Θ)(∂ λ Θ),(6) where ρ is energy density, p is pressure and U is the four-vector velocity in co-moving coordinates of the spacetime. HDE Model in CS Modified Gravity Granda and Oliveros [27] proposed an infrared cut-off for the HDE which is the sum of the square of the Hubble scale parameter and its time derivative given by ρ Λ = 3M 2 P (αH 2 + βḢ),(7) where α and β are constants which satisfy the restrictions of observational data and H =˙a a is Hubble parameter. Now, we discuss the FRW universe defined by line element ds 2 = −dt 2 + a 2 (t)[ dr 2 1 − κr 2 + r 2 (dθ 2 + sin 2 θdφ 2 )].(8) where κ is the curvature of the space. Here κ = −1, 0, 1 denotes open, flat and closed universe respectively. The 00-component of the field equation (2) of FRW metric and using Eq. (8), turns out to be H 2 + κ a 2 = (αH 2 + βḢ) + 1 6Θ 2 .(9) Now we calculate the value of Θ by using the Eq.(3). As for FRW universe the Pontryagin term * RR = * R a b cd R b acd vanishes, so Eq.(3) takes the form g µν ∇ µ ∇ ν Θ = g µν [∂ µ ∂ ν Θ − Γ τ µν ∂ τ Θ] = 0.(10) The solution of this equation can be found, in terms ofΘ, aṡ Θ = Ca −3 .(11) Substituting the value ofΘ in Eq.(9) along with assumption x = ln a, we arrive at dH 2 dx + 2(α − 1) β H 2 + C 2 3β e −6x − 2κ β e −2x = 0.(12) The solution of this differential equation is obtained using direct integration technique, given as H 2 (x) = C 1 e −2(α−1) β x − C 2 6(α − 3β − 1) e −6x + κ α − β − 1 e −2x .(13) The conservation equation is given by [27] ρ Λ + 3H(ρ Λ + p Λ ) = 0.(14) The holographic energy density and pressure density are related by the barotropic equation of state (EoS) defined as p Λ = ω Λ ρ Λ , where ω Λ is the EoS parameter. Then the last equation takes the form ω Λ = −1 − 2αḢ + βḦ H 3(αH 2 + βḢ) .(15) Using Eq. (13) in Eq.(15), we have ω Λ = − 1 3 ( (3β−2α+2) β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 2(α−3β−1) e −6x C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 6(α−3β−1) e −6x )(16) It is mentioned here that the EoS parameter is time dependent that can be transit from ω Λ > −1 to ω Λ < −1 [25]. Although, the recent studies [26] of DE properties are mildly support the models with ω Λ crossing −1. For the flat case, when κ = 0, by using the assumption α = 3β, the last equation turns out to be ω Λ = α − 2 α ,(17) which describes the EoS parameter in term of constant α only. The accelerated expansion of the universe can be obtained with restrictions on α such that 1 < α < 3 2 , if the phase −1 < ω Λ < − 1 3 is under consideration. If we consider 0 < α < 1 then the holographic energy and pressure density illustrates phantom-like theory of the evolution alongwith ω Λ < −1. Correspondence Between Holographic and Scalar Field Models Here we establish a correspondence between infrared cut-off proposed by Granda and Oliveros [27] for the holographic dark energy density and some of famous scalar field models, like quintessence model, tachyon model, K-essence model and dilaton model. We compare the holographic density defined by Gronda and Oliveros with the density of corresponding scalar field model. Further, we equate the barotropic EoS parameter, given in Eq. (17), with the EoS parameter of the corresponding scalar field models to find the scalar field and the potential energy. Quintessence Model in CS Modified Gravity Quintessence is described as canonical scalar field which was purposed to investigate the late-time cosmic acceleration. The pressure density and energy density of quintessence scalar field are defined as p φ = 1 2φ 2 − V (φ),(18)ρ φ = 1 2φ 2 + V (φ),(19) where the dot denotes derivative with respect to t. The dark energy EoS parameter for the quintessence scalar field is ω φ = 1 2φ 2 − V (φ) 1 2φ 2 + V (φ) .(20) Now, we compare the new HDE modal ω Λ , given in Eq. (16), with that of quintessence DE modal ω φ , given in Eq. (20), and obtain − 1 3 ( (3β−2α+2) β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 2(α−3β−1) e −6x C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 6(α−3β−1) e −6x ) = 1 2φ 2 − V (φ) 1 2φ 2 + V (φ)(21) On comparing Eq.(7) and Eq. (19), it comes out that 1 2φ 2 + V (φ) = 3M 2 P (αH 2 + βḢ).(22) Making use of Eq. (22) in Eq.(21) yields the explicit expression forφ and potential V (φ) φ 2 = 2M 2 p [ (α − 1) β C 1 e − 2(α−1)x β + κ(α − β) (α − β − 1) e −2x + C 2 (α − 3β) 2(α − 3β − 1) e −6x ](23) and V (φ) = M 2 P [ C 1 (3β − α + 1) β e − 2(α−1) β x + κ(α − β) α − β − 1 e −2x ](24) respectively. As we assumed x = ln a, it follows thatφ = φ ′ H, where prime denotes the derivative with respect to x. On substituting this value, Eq.(23) turns out to be φ ′ = √ 2M P [ (α−1) β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 2(α−3β−1) e −6x C 1 e −2(α−1) β x + κ α−β−1 e −2x − C 2 6(α−3β−1) e −6x ] 1 2 .(25) For the flat case, i.e., κ = 0, using the assumption α = 3β and taking φ(t 0 ) = 0 at initial time t 0 = 0, the Eqs. (25) and (24) become φ(t) = 6(α − 1) α M P ln t(26) and V (φ) = 3 α C 1 M 2 P e − 6(α−1) α φ M P .(27) The potential V (φ) becomes a source of accelerated expansion of the universe if α < 3 2 . If we discuss the phase-space analysis, the potential V (φ) corresponding to scalar field φ behaves like an attractor solution which is indication of accelerated expansion for α < 3 2 , same conditions are followed in power law accelerated expansion. The detailed analysis of dynamics of an exponential potential V (φ) is given in [28]. New Holographic Tachyon Model in CS Modified Gravity The tachyon modal is considered as a good candidate for dark energy. The idea of tachyon is 40 years old and attained much attention again after the research papers by Sen [23]. The tachyon scalar field φ is studied with Born-Infeld Lagrangian V (φ) 1 − g µν ∂ µ φ∂ ν φ which have minimal coupling with gravity. In the tachyon model the energy and pressure densities are given by ρ T = V (φ) 1 −φ 2 ,(28)P T = V (φ) 1 −φ 2 .(29) The EoS parameter for tachyon scalar field is ω T =φ 2 − 1.(30) The comparison between baroscopic EoS given in Eq. (16) and Eq.(30), yields 1 −φ 2 = 1 3 ( (3β−2α+2) β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 2(α−3β−1) e −6x C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 6(α−3β−1) e −6x ),(31) which implies thaṫ φ 2 = 2 3 [ C 1 α−1 β e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 2(α−3β−1) e −6x C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 6(α−3β−1) e −6x ]. (32) Sinceφ = φ ′ H and using corresponding values of φ and H, the evolutionary form of tachyon scalar field is yield as φ(a) − φ(0) = ln a 0 1 H 2(C 1 α−1 β e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 2(α−3β−1) e −6x ) 3(C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x − C 2 (α−3β) 6(α−3β−1) e −6x ) dx. (33) The analytic solution of this integral cannot be found explicitly. For approximate solution, assume that α = 3β and φ(0) = 0, i.e., at initial time and we consider the flat universe, i,e. κ = 0, the last equation becomes φ(a) = 2(1 − 1 α ) ln a 0 dx C 1 e −6(α−1)x α + C 2 6 e −6x .(34) The integral on R.H.S can be solved in term of hypergeometric function as φ(a) = 2 α − 1 3α e 3x C 2 F 1 [ 1 2 , α 2 , 1 + α 2 , − 6e 6x α C 1 C 2 ].(35) Finally, by substituting x = ln a, we obtain φ(a) = 2 α − 1 3α a 3 C 2 F 1 [ 1 2 , α 2 , 1 + α 2 , − 6a 6 α C 1 C 2 ].(36) Clearly, for α = 1, it yields φ(a) = 0.(37) Now, the comparison between Gronda and Oliveros cut-off, given in Eq. (7), and holographic tachyon model density, given in Eq. (28), yields ρ Λ = 3M 2 P (αH 2 + βḢ) = V (φ) 1 −φ 2 .(38) Substituting the values of φ and H, the tachyon potential energy turns to be V (φ) = √ 3M 2 P [C 1 e − 2(α−1)x β + κ(α − β) (α − β − 1) e −2x − C 2 (α − 3β) 6(α − 3β − 1) e −6x ] × [C 1 3β − 2α + 1 β e − 2(α−1)x β + κ(α − β) (α − β − 1) e −2x − C 2 (α − 3β) 2(α − 3β − 1) e −6x ]. (39) Again, for κ = 0 and α = 3β, we have V (φ) = 3M 2 P C 1 2 − α α e 6( 1−α α )x(40) For particular case α = 1, the potential V (φ) is constant which corresponds to ghost condensate scenario discussed in [29]. New Holographic K-essence Modal in CS Modified Gravity The concept of k-essence scalar field model was introduced by Armendariz and Mukhanov [24] to explain the accelerated expansion of the universe. The theory of k-essence deals with dynamical attractor solutions which act as a cosmological constant. The scalar field action for K-essence modal is defined as S = d 4 x √ −gp(φ, X),(41) where p(φ, X) denotes pressure density and most of time it corresponds to Lagrangian density defined as p(φ, X) = f (φ)ψ(X). In string theory, the Lagrangian density is transformed into p(φ, X) = f (φ)(−X + X 2 ).(42) The energy density of the field φ corresponding to the Lagrangian density expression is given by ρ(φ, X) = f (φ)(−X + 3X 2 ).(43) Using Eq.(42) and Eq.(43), one can easily obtain EoS parameter, given as ω K = X − 1 3X − 1 .(44) In particular X < 3 2 , the EoS ω φ < − 1 3 indicate the accelerated expansion. The comparison between Eq.(16) and new EoS parameter Eq.(44) yields X = 1 3 [ 3β−α+1 β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x 2β−α+1 β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 6(α−3β−1) e −6x ](45) The termφ 2 = 2X defined in [15] andφ = φ ′ H, using these expressions, the evolutionary form of K-essence scalar field takes the form φ(a) − φ(0) = 2 3 × ln a 0 1 H 3β−α+1 β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x 2β−α+1 β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 6(α−3β−1) e −6x dx. (46) The compassion of Eq. (16) and Eq.(44), alongwith values of H, yields the expression for f (φ) as f (φ) = 3M 2 P (1 − 3ω Λ ) 2 2(1 − ω Λ ) × [C 1 e − 2(α−1)x β + κ(α − β) (α − β − 1) e −2x − C 2 (α − 3β) 6(α − 3β − 1) e −6x ],(47) which can be further written as f (φ) = M 2 P 2 [ 2(2β−α+1) β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 3(α−3β−1) e −6x ] 2 2α−2 β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x .(48) Solving the Eqs.(45),(46) and (48) analytically, we consider the flat case, i,e. κ = 0, use the assumption α = 3β and φ(0) = 0 for the initial time t 0 = 0, it turns out to be X = 1 3 − α ,(49)φ(a) = 2 3 − α e 3x C 2 F 1 [ 1 2 , α 2 , 1 + α 2 , −6e 6x α C 2 C 1 ],(50)f (φ) = M 2 P 3 (3 − α) 2 α(α − 1) C 1 e − 6(α−1) α x.(51) For α = 3 2 the above equations turned into X = 2 3 ,(52) φ(a) = 4 3 a 3 C 2 F 1 [ 1 2 , 3 4 , 7 4 , −6a 6 C 2 C 1 ],(53)f (φ) = M 2 P C 1 a −2 .(54) The potential V (φ) obey the power law expansion analysed in [28]. New Holographic Dilaton Field in CS Modified Gravity The dilaton model of DE is described by a 4-dimensional effective low-energy limit action in string theory. It includes the higher order kinetic energy term which may be negative in the framework of Einstein relativity. It indicates that the dilaton model works like a phantom-type scalar field. The dilaton scalar field model is defined by the pressure density P d = −X + c 1 e λφ X 2 ,(55) where c 1 and λ are positive constants. The corresponding dilaton energy density is given by ρ d = −X + 3c 1 e λφ X 2 ,(56) where 2X =φ 2 . The EoS parameter ω d = P d ρ d can be obtained from Eqs. (55) and (56). ω d = −1 + c 1 e λφ X −1 + 3c 1 e λφ X .(57) Now, we compare Eq.(57) with new holographic EoS parameter, given in Eq. (16), i.e., ω d = ω Λ to obtain C ′ e λφ 2 X = 1 3 [ 2α−2 β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x 2(2β−α+1) β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 3(α−3β−1) e −6x ]. (58) Making use of X =φ = λ √ 6C ′ × ln a 0 1 H (2−2α) β C 1 e − 2(α−1)x β + 2κ(α−β) (α−β−1) e −2x 2(2β−α+1) β C 1 e − 2(α−1)x β + κ(α−β) (α−β−1) e −2x + C 2 (α−3β) 3(α−3β−1) e −6x dx(59) To obtain the evolutionary form of dilaton field, we consider the flat universe, i.e., κ = 0, using assumption α = 3β and at initial time t 0 = 0 the φ(0) = 0 and have φ(a) = λ 6C ′ 1 − α 3 − α e 6x C 2 2 F 1 [ 1 2 , α 2 , 1 + α 2 , −6e 6x α C 2 C 1 ].(60) Re-substituting x = ln a in the last equation, we arrived at φ(a) = λ √ 6C ′ 1 − α 3 − α a 6 C 2 2 F 1 [ 1 2 , α 2 , 1 + α 2 , −6a 6 α C 2 C 1 ],(61) which is in term of hypergeometric function. Conclusion The accelerated expansion of the universe is a most discussed issue in the recent past. In this paper, we found the EoS parameter ω Λ which describe the accelerated expansion of universe under certain restrictions on the parameter α. It is shown that for the accelerated expansion phase −1 < ω Λ < − 1 3 , the parameter α varies according as 1 < α < 3 2 . Furthermore, for 0 < α < 1, the holographic energy and pressure density illustrates phantom-like theory of the evolution when ω Λ < −1. We explored the scalar field φ and potential V (φ) of different holographic dark energy models such that quintessence, techyon, K-essence and dilaton. The potential V (φ) becomes a source of accelerated expansion of the universe if α < 3 2 . When we discuss the phase-space analysis, the potential V (φ) corresponding to scalar field φ behaves like an attractor solution which is indication of accelerated expansion for α < 3 2 , same conditions are followed in power law accelerated expansion. 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[ "Flat structure on the space of isomonodromic deformations", "Flat structure on the space of isomonodromic deformations" ]	[ "Mitsuo Kato ", "Toshiyuki Mano ", "Jiro Sekiguchi " ]	[]	[ "AMS 2010 Subject Classification: 34M56, 33E17, 35N10" ]	Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as the Frobenius manifold structure. In this paper we treat Saito structure (without metric) by C. Sabbah which is one of generalizations of Frobenius manifold structure. We connect Saito structure (without metric) with isomonodromic deformations of Okubo systems which are a special kind of systems of linear differential equations. As advantages of the introduction of Okubo systems, the following results are obtained: (I) introduction of flat coordinates on the orbit spaces of well-generated finite complex reflection groups (II) establishment of a correspondence between solutions satisfying certain semisimplicity condition to the three dimensional extended WDVV equation and generic solutions to the sixth Painlevé equation (III) explicit description of potential vector fields corresponding to algebraic solutions to the sixth Painlevé equation.	10.3842/sigma.2020.110	[ "https://arxiv.org/pdf/1511.01608v5.pdf" ]	119,671,025	1511.01608	c99f60f8ea584a59a6eb8b3b254e85984aa4345f	Flat structure on the space of isomonodromic deformations 2 Oct 2017 October 3, 2017 Mitsuo Kato Toshiyuki Mano Jiro Sekiguchi Flat structure on the space of isomonodromic deformations AMS 2010 Subject Classification: 34M56, 33E17, 35N10 2 Oct 2017 October 3, 2017flat structureFrobenius manifoldWDVV equationcomplex reflection groupPainlevé equation Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as the Frobenius manifold structure. In this paper we treat Saito structure (without metric) by C. Sabbah which is one of generalizations of Frobenius manifold structure. We connect Saito structure (without metric) with isomonodromic deformations of Okubo systems which are a special kind of systems of linear differential equations. As advantages of the introduction of Okubo systems, the following results are obtained: (I) introduction of flat coordinates on the orbit spaces of well-generated finite complex reflection groups (II) establishment of a correspondence between solutions satisfying certain semisimplicity condition to the three dimensional extended WDVV equation and generic solutions to the sixth Painlevé equation (III) explicit description of potential vector fields corresponding to algebraic solutions to the sixth Painlevé equation. Introduction At the end of 1970's, K. Saito introduced the notion of flat structure in order to study the structure of universal unfolding of isolated hypersurface singularities. Independently the WDVV equation (Witten-Dijkgraaf-Verlinde-Verlinde equation) arises from the 2D topological field theory [15,62]. B. Dubrovin unified both the flat structure formulated by K. Saito and the WDVV equation as Frobenius manifold structure. Dubrovin not only formulated the notion of Frobenius manifold but also studied its relationship with isomonodromic deformations of linear differential equations with certain symmetries. Particularly he derived a one-parameter family of Painlevé VI equation from three-dimensional massive (i.e. regular semisimple) Frobenius manifolds. Since then, there are several generalizations of Frobenius manifolds such as F -manifold by C. Hertling and Y. Manin [22,21] and Saito structure (without metric) by C. Sabbah [51]. Concerning the relationship with the Painlevé equation, A. Arsie and P. Lorenzoni [1,43] showed that three-dimensional regular semisimple bi-flat F -manifolds are parameterized by generic solutions to the (fullparameter) Painlevé VI equation, which is regarded as an extension of Dubrovin's result. Furthermore Arsie-Lorenzoni [2] showed that three-dimensional regular non-semisimple bi-flat F -manifolds are parameterized by generic solutions to the Painlevé V and IV equations. (Recently it was proved in [3,41] that Arsie-Lorenzoni's bi-flat F -manifold is equivalent to Sabbah's Saito structure (without metric).) The theory of linear differential equations on a complex domain is a classical branch of mathematics. In recent years there has been a great progress in this branch. One of its turning points is the introduction of the notions of middle convolution and rigidity index by N. M. Katz [33]. With the help of his idea, T. Oshima developed a classification theory of Fuchsian differential equations in terms of their rigidity indices and spectral types [49,50,48]. ( [19] is a nice introductory text on the "Katz-Oshima theory".) Okubo systems play a central role in these developments (cf. [13,14,50,64]): a matrix system of linear differential equations with the form (z − T ) dY dz = −B ∞ Y,(1) where T, B ∞ are constant square matrices, is said to be an Okubo system if T is diagonalizable. Particularly, any Okubo system is Fuchsian. One important feature of an Okubo system is that it keeps its form under the operation of the Euler transformation (cf. Remark 2.1): Y (z) → Y (λ) (z) := 1 Γ(λ) (u − z) λ−1 Y (u)du, λ ∈ C. Okubo system was introduced with the following motivations: • A system of differential equations in Birkhoff normal form (which has an irregular singularity of Poincaré rank one at 0 and a regular singularity at ∞ on P 1 (C)) is transformed into an Okubo system by a Fourier-Laplace transformation and thus the study on Stokes matrices of a Birkhoff normal form can be reduced to that on connection matrices of the corresponding Okubo system ( [4,55]). • Okubo system is one of generalizations of Gauss hypergeometric equation and one may expect that it would provide new special functions possessing rich properties ( [45,63,64]). Let us consider a regular semisimple Saito structure (without metric). (See Section 4 for the definition and properties of Saito structure (without metric). In the sequel, we abbreviate Saito structure (without metric) to Saito structure for brevity.) It is known that there are following two types of meromorphic connections associated with a Saito structure (cf. Remarks 4.2 and 4.3): (i) One induces a universal integrable deformation of a Birkhoff normal form. (ii) The other induces a universal integrable deformation of an Okubo system. These two meromorphic connections are called in [21] (in the case of a Frobenius manifold) the first structure connection and the second structure connection respectively and mutually equivalent since they are transformed to each other by Fourier-Laplace transformations. The first connection (i) is used in many preceding literatures (e.g. [16,51]). However in this paper we use the second connection (ii) because results in the recent developments on linear differential equations mentioned above can be exploited. On this standpoint, a regular semisimple Saito structure yields a universal integrable deformation of a regular Okubo system. In this paper, we show that the opposite of this statement is almost always true. Namely, for a universal integrable deformation of a regular Okubo system satisfying some generic condition, there exists a Saito structure which yields the given universal integral deformation as the second connection (ii) (Theorem 5.4). A rough picture of the result in this paper is regular semisimple Saito structures (2) ⇐⇒ universal integrable deformations of regular Okubo systems. Here it should be remarked that a universal integrable deformation of a regular Okubo system turns out to be an isomonodromic deformation (cf. Remark 2.2). In this paper, the following results are obtained as consequences of (2): (I) introduction of flat coordinates on the orbit spaces of well-generated finite complex reflection groups (Theorem 6.2), (II) correspondence between solutions satisfying certain semisimplicity condition to the three-dimensional extended WDVV equation and generic solutions to the Painlevé VI equation (Corollary 5.5), (III) explicit descriptions of potential vector fields corresponding to algebraic solutions to the Painlevé VI equation (Section 7). K. Saito and his collaborators [53,54] defined and constructed flat coordinates on the orbit spaces of finite real reflection groups. To extend them to finite (non-real) complex reflection groups has been a long-standing problem. (I) provides an answer to this problem for well-generated complex reflection groups (see also [29]). Recently this result has been crucially used to prove the freeness of multi-reflection arrangements of complex reflection groups in [25]. (II) is essentially equivalent to A. Arsie and P. Lorenzoni's result in [1,43], however the argument based on the picture (2) makes clear the relationship between Saito structures and the theory of isomonodromic deformations. (III) provides many concrete examples of three-dimensional algebraic Saito structures that are not Frobenius manifolds, which would be the first step toward classification of three-dimensional algebraic Saito structures and/or algebraic F -manifolds (cf. [21]), see also [29,30,31]. (In this paper, the term "flat structure" stands for the same meaning as "Saito structure" because it is a natural extension of K. Saito's flat structure, which would be justified by the result (I) above.) This paper is constructed as follows. In Section 2, we start with an Okubo system. Then we introduce the notion of Okubo system in several variables as an integrable Pfaffian system extending the Okubo system (Definition 2. 3). An Okubo system in several variables is equivalent to an isomonodromic deformation of an Okubo system (Remark 2.2). We see that an Okubo system in several variables is a universal integrable deformation of an Okubo system, which is uniquely determined up to changes of independent variables by the given Okubo system (Remark 2.3). In Section 3, we study the structure of logarithmic vector fields along a divisor defined by a monic polynomial of degree n h(x) = h(x ′ , x n ) = x n n − s 1 (x ′ )x n−1 n + · · · + (−1) n s n (x ′ ) where x ′ = (x 1 , · · · , x n−1 ) and each s i (x ′ ) is holomorphic with respect to x ′ , which appears as the defining equation of the singular locus of an Okubo system in several variables. In K. Saito's construction of flat structures on the orbit spaces of real reflection groups, the fact that the discriminants of real reflection groups have the form of (3) was crucially used (cf. [53]). The results in this section are used in order to generalize Saito's construction in Section 6. In Section 4, we review the general theory of Saito structure. Particularly we introduce an extension of the WDVV equation (Proposition 4.8, Definition 4.9), which is not mentioned in [51] explicitly. A solution to the extended WDVV equation is called a potential vector field, which completely describes a Saito structure (Proposition 4.10). This is an extension to Saito structure of the fact that a Frobenius manifold is completely described by its prepotential, which is a solution to the WDVV equation. In Section 5, we give a precise statement of the picture (2). Namely, we show that an Okubo system in several variables arises from a Saito structure if and only if its flat coordinate is non-degenerate in some sense (Theorem 5.4). For uniqueness of the Saito structure corresponding to an Okubo system in several variables, see Remark 5.1. Combining this result and the argument in Appendix A, we see that there is a correspondence between solutions satisfying certain semisimplicity condition to the three-dimensional extended WDVV equation and generic solutions to the (full-parameter) Painlevé VI equation (Corollary 5.5). In Section 6, we treat a problem on the existence of a flat coordinate system on the orbit space of an irreducible complex reflection group. In the case of a real reflection group, K. Saito [53] (see also [54]) proved the existence of a flat coordinate system on the orbit space based on the existence of a flat invariant metric on the standard representation space. In this paper, instead of the flat invariant metric, we construct a special type of Okubo systems in several variables called G-quotient system, whose fundamental system of solutions consists of derivatives by logarithmic vector fields of linear coordinates on the standard representation space and its monodromy group is isomorphic to the finite complex reflection group G (Theorem 6.2). Then as a consequence of the picture (2), we have a Saito structure on the orbit space corresponding to the G-quotient system. Actually, our definition of flat coordinates coincides with that of Saito when it is restricted to real reflection groups. Here it is remarkable that our proof of Theorem 6.2 is constructive i.e. it contains an algorithm of explicit computation on the flat generator system of G-invariants and the potential vector field for a given well-generated complex reflection group G. Besides, it is proved that the potential vector field corresponding to a well-generated complex reflection group has polynomial entries (Corollary 6.8). In Section 7, we treat algebraic solutions to the Painlevé VI equation. Algebraic solutions to the Painlevé VI equation were studied and constructed by many authors including N. J. Hitchin [23,24], B. Dubrovin [16], B. Dubrovin -M. Mazzocco [18], P. Boalch [6,7,8,9,10], A. V. Kitaev [37,38], A. V. Kitaev -R. Vidūnas [39,61], K. Iwasaki [26]. The classification of algebraic solutions to the Painlevé VI equation was achieved by Lisovyy and Tykhyy [42]. We give some examples of potential vector fields corresponding to algebraic solutions to the Painlevé VI equation. Some other examples and related topics are found in [29,30,31]. In Appendix A, we give a proof of that an Okubo system in several variables is equivalent to an isomonodromic deformation of an Okubo system. Especially, we show that there is a correspondence between Okubo systems in several variables of rank three and generic solutions to the Painlevé VI equation with the help of the result of Jimbo and Miwa [28] (Proposition A.1). In Appendix B, we explain a method of constructing an Okubo system in several variables of rank n from a completely integrable Pfaffian system of rank (n − 1). This construction is used in Appendix A. We close this introduction with addressing some problems. The first one is to construct algebraic solutions to higher order Painlevé equations. As stated above, a solution to the extended WDVV equation yields an isomonodromic deformation of an Okubo system. Recently higher order Painlevé equations are obtained by several authors (e.g. [36,59]) based on the classification theory on linear differential equations due to Oshima. Starting from polynomial potential vector fields corresponding to (real and complex) reflection groups of higher rank, it may be expected that one obtain algebraic solutions to those higher order Painlevé equations. Those solutions may be parametrized by higher-dimensional algebraic varieties. The second one is related with the theory of unfolding of isolated hypersurface singularities. Originally, the view of K. Saito is that flat coordinates arise from versal deformations of isolated hypersurface singularities. Let us focus our attention to the case of ADE singularities. One can define flat coordinates of ADE type from versal deformation of ADE singularities. Then the following question naturally arises: Can we relate flat coordinates for complex reflection groups with "non-versal families" of hypersurface singularities? The third one is related with potential vector fields having polynomial entries. Classification of polynomial prepotentials were conjectured by Dubrovin [16] and proved by Hertling [21]: Any polynomial prepotential comes from the Frobenius manifold structure on the orbit space of a real reflection group. Surprisingly enough, there are some examples in Section 7 whose potential vector fields have polynomial entries but the corresponding flat structures are not isomorphic to one on the orbit space of any complex reflection group. To classify polynomial potential vector fields would be an interesting problem. Extension of Okubo systems to several variables case In this section, we start with introducing a special type of a system of ordinary linear differential equations which is called an Okubo system. Then we introduce an extension of an Okubo system to several variables case as a completely integrable Pfaffian system which is called an Okubo system in several variables (Definition 2.3). An Okubo system in several variables is equivalent to an isomonodromic deformation of an Okubo system as we will show in Appendix A (cf. Remark 2.2) and it turns out to be a universal integrable deformation of an Okubo system (Remark 2.3). Let T and B ∞ be n × n-matrices. If T is a diagonalizable matrix, the system of ordinary linear differential equations (zI n − T ) dY dz = −B ∞ Y(4) is called a system of differential equations of Okubo type or shortly an Okubo system ( [45]). The aim in this section is to extend (4) to a completely integrable Pfaffian system of several variables in the form dY = B (z) dz + n i=1 B (i) dx i Y,(5) where B (z) and B (i) are n × n matrices whose entries depend on (z, x). As usual, x = (x 1 , . . . , x n ) denotes a coordinate of C n . We assume the existence of n × n matrices T = T (x) and B ∞ so that B (z) = −(zI n − T ) −1 B ∞ and satisfy the conditions: (A1) the entries of T (x) are holomorphic functions on a domain U in C n , (A2) B ∞ = diag[λ 1 , . . . , λ n ], where λ i are constant complex numbers satisfying λ i − λ j ∈ Z \ {0} for i = j. Let H(x, z) = det(zI n − T (x)) be the characteristic polynomial of T (x), which is a monic polynomial in z of degree n and analytic in x: H(x, z) = z n −S 1 (x)z n−1 +· · ·+(−1) n S n (x). We assume the following condition on H(x, z): (A3) det ∂S j (x) ∂x i i,j=1,. ..,n = 0 at generic points of U. It follows from (A3) that the discriminant δ H (x) = i<j (z i (x) − z j (x)) 2 of H(x, z) = n i=1 (z − z i (x)) is not identically zero and we define a divisor ∆ H = {δ H (x) = 0} ∪ {det ∂S j (x) ∂x i = 0} on U. Taking a smaller domain W ⊂ U \ ∆ H appropriately, the eigenvalues z 1 (x), . . . , z n (x) of T can be considered single-valued holomorphic functions in x on W by fixing their branches and we can take an invertible matrix P = P (x) whose entries are single-valued holomorphic functions on W such that P −1 T P = diag [z 1 (x), . . . , z n (x)].(6) We also assume (A4) H(x, z)B (i) ∈ (O U ⊗ C C[z]) n×n , where O U denotes the ring of holomorphic functions on U. Decompose B (z) into partial fractions on W B (z) = n i=1 B (z) i z − z i (x) ,(7) where B (z) i (i = 1, . . . , n) are independent of z. In addition to (A1)-(A4), we assume the condition: (A5) for each of B (z) i (i = 1, . . . , n), r i := tr B (z) i = ±1 on U \ ∆ H . Lemma 2.1. Assume that B ∞ is invertible. If the Pfaffian system (5) is completely integrable, then there are n × n matricesB (i) , B (i) ǫ ∈ O n×n U (i = 1, . . . , n) such that B (i) = −(zI n − T (x)) −1B(i) B ∞ + B (i) ǫ , i = 1, . . . , n,(8) and that B (i) ǫ = ∂E ∂x i E −1 in terms of a matrix E whose entries are holomorphic functions on U. If λ i = λ j for any i = j, the matrix E turns out to be a diagonal matrix E = diag[ǫ 1 , . . . , ǫ n ]. Moreover, it holds that P −1 T P = diag [z 1 (x), . . . , z n (x)],(9)P −1B(i) P = diag − ∂z 1 (x) ∂x i , . . . , − ∂z n (x) ∂x i , i = 1, . . . , n,(10) on W . In particular, T,B (i) (i = 1, . . . , n) are mutually commutative: [T,B (i) ] = O, [B (i) ,B (j) ] = O (∀i, j).(11) Proof. In virtue of the definition of B (z) , we can write H(x, z)B (z) = n−1 i=0 (HB (z) ) i z i ,(12) where (HB (z) ) i ∈ O n×n U , (HB (z) ) n−1 = −B ∞ . In virtue of the assumption (A4), we can write H(x, z)B (i) = m i j=0 (HB (i) ) j z j , i = 1, . . . , n.(13) As the first step, we are going to show m i ≤ n. From the integrability condition of (5), we have H ∂(HB (i) ) ∂z − ∂H ∂z HB (i) +[HB (i) , HB (z) ] = H ∂(HB (z) ) ∂x i − ∂H ∂x i HB (z) , i = 1, . . . , n. (14) The equation (14) combined with (12) and (13) implies that the left hand side of the resulting equation is a polynomial in z of degree m i + n − 1 and its right hand side is of degree 2n − 2. Besides, the coefficient of the term z m i +n−1 in the left hand side reads (m i − n)(HB (i) ) m i + [(HB (i) ) m i , (HB (z) ) n−1 ]. As a consequence we find that m i ≤ n, because (HB (z) ) n−1 = −B ∞ is diagonal and λ i − λ j ∈ Z \ {0} for i = j. Then B (i) (i = 1, . . . , n) is decomposed into partial fractions as follows on W : B (i) = n j=1 B (i) j z − z j + B (i) ǫ ,(15) where B (i) ǫ = (HB (i) ) n is a matrix which is defined on U. If λ i = λ j for any i = j, we see that (HB (i) ) n is a diagonal matrix since [(HB (i) ) n , (HB (z) ) n−1 ] = O. Paying attention to the value at z = ∞ of the equation obtained by substituting (15) for the integrability condition of the Pfaffian system (5), we have ∂B (i) ǫ ∂x j − ∂B (j) ǫ ∂x i + [B (i) ǫ , B (j) ǫ ] = O (i, j = 1, . . . , n). It means that B (i) ǫ is written by a matrix E as B (i) ǫ = ∂E ∂x i E −1 . As the second step, we shall show the equalities B (i) j = − ∂z j ∂x i B (z) j , i, j = 1, . . . , n,(16) for the residue matrices B (7) and (15). We have (z) i , B (j) i in−B (i) j − B (z) j ∂z j ∂x i + [B (i) j , B (z) j ] = O, i, j = 1, . . . , n,(17) by substituting (7) and (15) for the integrability condition. Now fix j ∈ {1, . . . , n}. Since rank (B j ) 2 = r j B (z) j .(18) From (17) and (18), we have B (z) j B (i) j = −r j B (z) j ∂z j ∂x i + B (z) j B (i) j B (z) j − r j B (z) j B (i) j , B (i) j B (z) j = −r j B (z) j ∂z j ∂x i + r j B (i) j B (z) j − B (z) j B (i) j B (z) j , B (z) j B (i) j B (z) j = −r 2 j B (z) j ∂z j ∂x i . Then, since r j = ±1, it holds B (z) j B (i) j = B (i) j B (z) j , which and (17) again, imply the equalities (16). As the last step, we take a matrix P that satisfies (9). Then P −1 B (z) P = −diag z − z 1 (x), . . . , z − z n (x) −1 P −1 B ∞ P, and from (16), we have P −1 B (i) P = −diag z − z 1 , . . . , z − z n −1 diag − ∂z 1 ∂x i , . . . , − ∂z n ∂x i P −1 B ∞ P + P −1 B (i) ǫ P. Hence we obtain B (i) = −(zI n − T ) −1B(i) B ∞ + B (i) ǫ forB (i) = P diag − ∂z 1 (x) ∂x i , . . . , − ∂z n (x) ∂x i P −1 . This proves (9), (10) and thus (11) on W . Since B ∞ is invertible, we find thatB (i) ∈ O n×n U . In virtue of the identity theorem, (11) holds on U. We have thus proved the lemma completely. In virtue of Lemma 2.1, we restrict ourselves to the case B (i) ǫ = O without loss of generality by applying a gauge transformation to Y . Namely we consider the Pfaffian system dY = B (z) dz + n i=1 B (i) dx i Y(19) with B (z) = −(zI n − T ) −1 B ∞ , B (i) = −(zI n − T ) −1B(i) B ∞ (i = 1, . . . , n).∂B (i) ∂x j − ∂B (j) ∂x i = O, i, j = 1, . . . , n,(21) then the Pfaffian system (19) is completely integrable. Moreover, in case where λ i = 0 (i = 1, . . . , n), the system of equations (20), (21), (22) is necessary and sufficient condition for that (19) is completely integrable. Proof. The lemma is clear from the relations ∂B (i) ∂z − ∂B (z) ∂x i + [B (i) , B (z) ] = (zI n − T ) −1 ∂T ∂x i +B (i) + [B (i) , B ∞ ] (zI n − T ) −1 B ∞ , and (zI n − T ) ∂B (i) ∂x j − ∂B (j) ∂x i + [B (i) , B (j) ] = ∂B (j) ∂x i − ∂B (i) ∂x j (zI n − T ) + ∂T ∂x i +B (i) + [B (i) , B ∞ ] B (j) − ∂T ∂x j +B (j) + [B (j) , B ∞ ] B (i) (zI n − T ) −1 B ∞ . Definition 2.3. The completely integrable Pfaffian system (19) is called a system of differential equations of Okubo type in several variables or shortly an Okubo system in several variables. Remark 2.1. Assume that T , B ∞ ,B (i) (i = 1, . . . , n) satisfy the conditions (20), (21) and (22). Then replacing B ∞ by B ∞ − λI n for any λ ∈ C, we see that T , B ∞ − λI n , B (i) (i = 1, . . . , n) also satisfy (20), (21) and (22). The corresponding transformation from (19) into dY (λ) = −(zI n − T ) −1 dz + n i=1B (i) dx i (B ∞ − λI n )Y (λ) is realized by the Euler transformation Y → Y (λ) (z) := 1 Γ(λ) (u − z) λ−1 Y (u)du. Note that the Euler transformation changes the monodromy, namely the monodromy of Y (λ) depends on λ. Remark 2.2. The completely integrable Pfaffian system (19) can be regarded as a Lax formalism of the isomonodromic deformation of an Okubo system (4). Indeed (19) is equivalent to dY = n i=1 B (z) i d log(z − z i )Y.(23) Then the system of nonlinear differential equations (21) is equivalent to the Schlesinger system dB (z) i = j =i [B (z) j , B (z) i ]d log(z i − z j ), i = 1, . . . , n.(24) (See Appendix A for details.) Remark 2.3. By (10) and the assumption (A3), we see that an Okubo system in several variables is a universal integrable deformation of an Okubo system (4), which is uniquely determined by (4) up to changes of independent variables. (The uniqueness follows also from that of the isomonodromic deformation (23), (24) which is proved in [27].) Lemma 2.4. The variables x 1 , . . . , x n can be taken so that T = T (x) satisfies the following condition: Condition (T): T (x) = −x n +T 0 (x ′ ), where T 0 (x ′ ) depends only on x ′ = (x 1 , . . . , x n−1 ). In the following, we always assume Condition (T). Remark 2.4. Under Condition (T), it holds thatB (n) = I n ,B (i) =B (i) (x ′ ) ∈ O n×n U ′ (i = 1, . . . , n − 1) and (19) is equivalent to the Pfaffian system for Y 0 = Y 0 (x) dY 0 = n i=1 B (i) 0 dx i Y 0 ,(25) where B (i) 0 = T −1B(i) B ∞ . We call (25) the reduced form of (19). Logarithmic vector fields In this section, we study a divisor defined by det(−T (x))) which is the singular locus of an Okubo system in several variables. A main purpose of this section is to prove that the vector fields defined by (34) forms a unique standard system of generators of logarithmic vector fields along the divisor when the divisor is free (Lemma 3.9, cf. Remark 3.2). Logarithmic vector fields We employ the notations x ′ = (x 1 , . . . , x n−1 ) and x = (x ′ , x n ) = (x 1 , . . . , x n ). Let h(x) = H(x, 0) = det(−T (x)) ∈ O U or equivalently H(x, z) = h(x ′ , x n + z) under Condition (T). Let D be the divisor in U defined by h(x) i.e. D = {(x) ∈ U; h(x) = 0}. We assume that the domain U ⊂ C n (and W ⊂ U \ ∆ H resp.) has the form of U = U ′ × C ⊂ C n−1 × C (and W = W ′ × C ⊂ U ′ × C resp.). Note that h(x) is a monic polynomial in x n of degree n: h(x) = h(x ′ , x n ) = x n n − s 1 (x ′ )x n−1 n + · · · + (−1) n s n (x ′ ) = n m=1 (x n − z 0 m (x ′ )),(26)where s i (x ′ ) ∈ O U ′ . Recall that z m (x) = −x n + z 0 m (x ′ ), 1 ≤ m ≤ n. Here we note that the assumption (A3) for H(x, z) = h(x ′ , x n + z) is equivalent to det       ∂z 1 (x) ∂x 1 ∂z 2 (x) ∂x 1 . . . ∂zn(x) ∂x 1 ∂z 1 (x) ∂x 2 ∂z 2 (x) ∂x 2 . . . ∂zn(x) ∂x 2 . . . . . . . . . . . . ∂z 1 (x) ∂xn ∂z 2 (x) ∂xn . . . ∂zn(x) ∂xn       = 0 (27) on W ′ . We give the definitions of logarithmic vector field along D and free divisor following K. Saito [52]: Definition 3.1. Let M U ′ be the field of meromorphic functions on U ′ . A vector field V = n k=1 v k (x ′ , x n )∂ x k with v k (x ′ , x n ) ∈ M U ′ ⊗ C C[x n ] is called a meromorphic logarithmic vector field along D if (V h)/h ∈ M U ′ ⊗ C C[x n ], or equivalently if (V h)\| D = 0. If moreover, v k (x ′ , x n ) ∈ O U ′ ⊗ C C[x n ], V is called a logarithmic vector field along D. Let Der(− log D) be the set of logarithmic vector fields along D, which is naturally an O U ′ ⊗ C C[x n ]-module. The divisor D is said to be free if Der(− log D) is a free O U ′ ⊗ C C[x n ]- module. Remark 3.1. It is known that D is free if there are logarithmic vector fields V i = n j=1 v i,j (x)∂ x j , i = 1, . . . , n, along D such that det(v i,j (x)) = h(x) (Saito's criterion [52]). The matrix M V (x) := (v n−i+1,j (x)) is called a Saito matrix. We rewrite the assumption (A3) in terms of meromorphic logarithmic vector fields (which will be used in the proof of Lemma 6.1): Lemma 3.2. The following (i) and (ii) are equivalent. (i) The assumption (A3) holds for H(x, z) = h(x ′ , x n + z). (ii) Let V = n k=1 v k (x ′ )∂ x k , v k (x ′ ) ∈ M U ′ be any meromorphic logarithmic vector field along D (whose coefficients do not depend on x n ). Then V = 0. Proof. First note that V = n k=1 v k (x ′ )∂ x k , v k (x ′ ) ∈ M U ′ is a meromorphic logarithmic vector field along D if and only if V h = 0.(28)Let ′ denote n−1 i=1 . Since V h = − ′ v i ∂s 1 ∂x i − v n n x n−1 n + ′ ∂s 2 ∂x i − v n (n − 1)s 1 x n−2 n + ... + (−1) n ′ ∂s n ∂x i − v n s n−1 , the equality (28) is equivalent to v 1 (x ′ ) . . . v n (x ′ )       ∂s 1 (x ′ ) ∂x 1 ∂s 2 (x ′ ) ∂x 1 . . . ∂sn(x ′ ) ∂x 1 . . . ∂s 1 (x ′ ) ∂x n−1 ∂s 2 (x ′ ) ∂x n−1 . . . ∂sn(x ′ ) ∂x n−1 −n −(n − 1)s 1 (x ′ ) . . . −s n−1 (x ′ )       = O.(29) Noting (26), the equation (29) has a non-zero solution if and only if (A3) for H(x, z) does not hold. This proves the lemma. On each W ′ , we define matrices P h (x ′ ), M h (x ′ ), M V (h) (x) by P h (x ′ ) =    ∂z 1 (x) ∂x 1 · · · ∂zn(x) ∂x 1 . . . ∂z 1 (x) ∂xn · · · ∂zn(x) ∂xn    ,(30) and M h (x ′ ) = P h diag[z 0 1 , z 0 2 , . . . , z 0 n ] P −1 h , M V (h) (x) = −P h diag[z 1 , z 2 , . . . , z n ] P −1 h . (31) By definition, we have M V (h) (x) = x n I n − M h (x ′ )(32) and det M V (h) (x) = h(x).(33) We define the vector fields V (h) i (i = 1, . . . , n) by V (h) n · · · V (h) 1 t = M V (h) ∂ x 1 · · · ∂ xn t .(34)Lemma 3.3. The vector fields V (h) i , 1 ≤ i ≤ n are meromorphic logarithmic vector fields along D. Proof. First we prove that the entries of M h (x ′ ) are meromorphic functions on U ′ . Fix W ′ ⊂ U ′ , and let F =      (z 0 1 (x ′ )) n−1 . . . (z 0 n (x ′ )) n−1 ... z 0 1 (x ′ ) . . . z 0 n (x ′ ) 1 . . . 1      . The two matrices M 0 and M 1 are determined by M 0 F = F diag[z 0 1 , z 0 2 , . . . , z 0 n ] and M 1 F = P h using elementary elimination methods. By the construction, M 0 and δ h M 1 are holomor- phic on U ′ , where δ h = i<j (z 0 i (x ′ ) − z 0 j (x ′ )) 2 . Since M h = M 1 M 0 (M 1 ) −1 ,(35) we find that the entries of (30), (31), (34). M h (x ′ ) are meromorphic functions on U ′ . Then it is clear that V (h) i (i = 1, . . . , n) are meromorphic logarithmic vector fields from z i ∂ z i h = h (i = 1, . . . , n) andLemma 3.4. Assume that V i = n j=1 v i,j (x)∂ x j , 1 ≤ i ≤ n are logarithmic vector fields along D satisfying v i,j (x) − x n δ n+1−i,j ∈ O U ′ , where δ ij denotes Kronecker's delta. Then it holds that V i = V (h) i , 1 ≤ i ≤ n, and D is free. Proof. Let V (h) i = n j=1 v (h) i,j (x)∂ x j . Then we see v (h) i,j (x) − x n δ n+1−i,j ∈ M U ′ from (34). Thus we have v i,j (x) − v (h) i,j (x) ∈ M U ′ , j = 1, . . . , n, which implies V i − V (h) i = 0 in virtue of Lemma 3.2. Then the equality (33) implies that M V (h) is a Saito matrix and D is free. Global case In this subsection we assume that the function h(x ′ , x n ) is a polynomial in (x ′ , x n ) and weighted homogeneous with respect to a weight w(·) with 0 < w(x 1 ) ≤ w(x 2 ) ≤ · · · ≤ w(x n−1 ) ≤ w(x n ) = 1. In this case, we replace O U ′ and M U ′ in the previous subsection by C[x ′ ] and C(x ′ ) respectively, and assume that the integrable system (25) is weighted homogeneous, that is, all the entries of B (k) 0 ,B (k) , T , Y 0 are weighted homogeneous. Lemma 3.5. It holds that V (h) 1 = n k=1 w(x k )x k ∂ x k , namely, V (h) 1 is the Euler operator. Proof. It is clear from that the Euler operator is a logarithmic vector field along D and the proof of Lemma 3.4. Note w(z 0 m (x ′ )) = w(x n ) = 1. Hence the equality (31) implies w((M h ) i,j ) = w(z 0 m ) + w( ∂z 0 m ∂x i ) − w( ∂z 0 m ∂x j ) for 1 ≤ i, j ≤ n, that is, w ((M h ) i,j ) = 1 − w(x i ) + w(x j ), 1 ≤ i, j ≤ n.(36) Since w ((M h ) i,j ) = w ((M V (h) ) i,j ), we have a kind of duality w(x i ) + w V (h) n−i+1 = 1, 1 ≤ i ≤ n,(37) between {w(x i )} and {w(V (h) i )}. As for the reduced form (25), we give a lemma. Lemma 3.6. (i) The following holds n i=1 w(x i )x iB (i) = −T, n i=1 w(x i )x i B (i) 0 = −B ∞ ,(38) (ii) There is a weighted homogeneous matrix C(x) such that B (i) = ∂C ∂x i , T = −V (h) 1 C.(39) Proof. Proof of (i). From (9) and (10), we have n i=1 w(x i )x iB (i) = P (x ′ ) n i=1 w(x i )x i diag − ∂z 1 ∂x i , . . . , − ∂z n ∂x i P (x ′ ) −1 = −P (x ′ ) V (h) 1 diag [z 1 , . . . , z n ] P (x ′ ) −1 = −P (x ′ ) diag [z 1 , . . . , z n ] P (x ′ ) −1 = −T (x), n i=1 w(x i )x i B (i) 0 = T −1 n i=1 w(x i )x iB (i) B ∞ = −B ∞ . Proof of (ii). From (22), n i=1B (i) dx i = dC(x) has a solutionC(x). Let C ij (x) be the weighted homogeneous part ofC ij (x) of the weight w(T ij ). Then the matrix C(x) = (C ij (x)) i,j=1,...,n satisfies the first equality of (39). The second equality of (39) then coincides with the first one of (38). Let Y 0 = (y 1 , y 2 , . . . , y n ) t , B ∞ =diag[λ 1 , λ 2 , . . . , λ n ] as before. Then, from (38), we have V (h) 1 Y 0 = n i=1 w(x i )x i ∂ x i Y 0 = n i=1 w(x i )x i B (i) 0 Y 0 = −B ∞ Y 0 , which implies w(y i ) = −λ i , 1 ≤ i ≤ n.(40) From the equality ∂ ∂xn Y 0 = B (n) 0 Y 0 , we obtain w(y i ) − 1 = w((B (n) 0 ) i,j ) + w(y j ), whence w((B (n) 0 ) i,j ) = −λ i + λ j − 1, 1 ≤ i, j ≤ n.(41) Since B (n) 0 = T −1 B ∞ as in Remark 2.4, we have w(T i,j ) = −w (B (n) 0 ) j,i , that is, w(T i,j ) = 1 − λ i + λ j , 1 ≤ i, j ≤ n.(42) Lemma 3.7. Let V be a non-zero weighted homogeneous logarithmic vector field along D. Then w(V ) ≥ 0. Proof. Let V = n j=1 v j (x)∂ x j , v j = v j (x ′ , x n ) ∈ C[x]. If w(V ) < 0, then v j (x) ∈ C[x ′ ] . This proves V = 0 from Lemma 3.2. For rational logarithmic vector fields V i = n j=1 v i,j (x)∂ x j (i = 1, . . . , n) along D, let M V (x) denotes the n × n matrix whose (i, j)-entry is defined by v n−i+1, j (x). Lemma 3.8. Let V i , 1 ≤ i ≤ n be weighted homogeneous logarithmic vector fields with w(V 1 ) ≤ w(V 2 ) ≤ · · · ≤ w(V n ). Assume det M V (x)\| x ′ =0 = cx n n for a non-zero constant number c. Then the following three facts hold. (i) w(V i ) = w(V (h) i ) (= 1 − w(x n−i+1 )), 1 ≤ i ≤ n. (ii) There is a matrix G(x ′ ) ∈ GL(n, C[x ′ ]) such that w(G(x ′ ) i,j ) = w(x j ) − w(x i ), and that V (h) n V (h) n−1 . . . V (h) 1 t = G(x ′ ) V n V n−1 . . . V 1 t holds. In particular, V (h) i (x), 1 ≤ i ≤ n are also logarithmic vector fields, and hence D is free. ( iii) det M V (x) = ch(x). Proof. Proof of (i). Since w(V i ) ≥ 0 from Lemma 3.7, it holds that (M V )\| x=0 = 0. Since det(M V )\| x ′ =0 = cx n n , it holds that M V (x)\| x ′ =0 = x n R, where R =      R 1 O . . . O O R 2 . . . O ·· ·· . . . ·· O O . . . R k      for some R i ∈ GL(n i , C) with k i=1 n i = n. This proves w(V i ) = 1 − w(x n−i+1 ). Proof of (ii). Note that R −1 M V (x)\| x ′ =0 = x n I n . Since all the entries of R −1 M V (x) are linear function of x n with the coefficients being weighted homogeneous polynomials in x ′ , we have R −1 M V (x) = x nR (x ′ ) +R(x ′ ), for someR(x ′ ),R(x ′ ) ∈ C[x ′ ] n×n satisfying R(0) = I n , w(R i,j ) = −w(x i ) + w(x j ), w(R i,j ) = 1 − w(x i ) + w(x j ). Then detR(x ′ ) = 1 andR(x ′ ) −1 R −1 M V − x n I n ∈ C[x ′ ] n×n , which (together with Lemma 3.4) provesR (x ′ ) −1 R −1 V n V n−1 . . . V 1 t = V (h) n V (h) n−1 . . . V (h) 1 t . This proves (ii). (iii) is proved by (ii) and the equality (33). Lemma 3.9. Assume that D is free. Then V (h) i , 1 ≤ i ≤ n are logarithmic vector fields along D. In particular, {V (h) i } i=1,. ..,n forms a unique system of generators of logarithmic vector fields along D such that its Saito matrix M V (h) satisfies M V (h) − x n I n ∈ C[x ′ ] n×n . Proof. Let V i , 1 ≤ i ≤ n be logarithmic vector fields such that det(M V ) = h(x). Since each homogeneous part of V i are logarithmic vector field with a non-negative weight (Lemma 3.7), all entries of ( M V )\| x ′ =0 are polynomials in x n of positive degrees. Let R(x n ) = x −1 n (M V )\| x ′ =0 ∈ C[x n ] n×n . Then det R(x n ) = 1, and hence det R(0) = 1. Let V ′ i 1 ≤ i ≤ n be logarithmic vector fields such that V ′ n V ′ n−1 . . . V ′ 1 t = R(0) −1 M V ∂ x 1 ∂ x 2 . . . ∂ xn t . Put M V ′ = R(0) −1 M V . Then (M V ′ )\| x ′ =0 = x n I n + O(x 2 n ). Let V ′′ i be the homogeneous part of V ′ i with w(V ′′ i ) = 1 − w(x n−i+1 ). Then it holds det(M V ′′ )\| x ′ =0 = x n n , and hence Lemma 3.8 implies that V (h) i , 1 ≤ i ≤ n are logarithmic vector fields. The uniqueness of {V (h) i } is clear from Lemma 3.4. Remark 3.2. In the case where h(x) is the discriminant of a well-generated complex reflection group, D. Bessis [5] showed the existence of a system of generators of logarithmic vector fields along D whose Saito matrix M V has the form M V − x n I n ∈ C[x ′ ] n×n . Such a system of generators is called flat in [5] (cf. Section 6). Saito structure (without metric) In this section, we review Saito structure (without metric) introduced by Sabbah [51]. (In the sequel, we abbreviate Saito structure (without metric) to Saito structure for brevity.) Proofs for many of statements in this section can be found in the literature [51,40]. [51]). Let X be a complex analytic manifold of dimension n (in this paper, we treat the case where X is a domain U in C n ), T X be its tangent bundle, and Θ X be the sheaf of holomorphic sections of T X. A Saito structure (without a metric) on X is a data consisting of ( △ , Φ, e, E) in (i),(ii),(iii) that are subject to the conditions (a), (b): Definition 4.1 (C. Sabbah(i) △ is a flat torsion-free connection on T X, (ii) Φ is a symmetric Higgs field on T X, (iii) e and E are global sections (vector fields) of T X, respectively called unit field and Euler field. (a) A meromorphic connection ∇ on the bundle π * T X on P 1 × X defined by ∇ = π * △ + π * Φ z − Φ(E) z + △ E dz z(43) is integrable, where π is the projection π : P 1 × X → X and z is a non-homogeneous coordinate of P 1 , (b) the field e is △ -horizontal (i.e., △ (e) = 0) and satisfies Φ e = Id, where we regard Φ as an End O X (Θ X )-valued 1-form and Φ e ∈ End O X (Θ X ) denotes the contraction of the vector field e and the 1-form Φ. Remark 4.1. To the Higgs field Φ there associates a product ⋆ on Θ X defined by ξ ⋆ η = Φ ξ (η) for ξ, η ∈ Θ X . The Higgs field Φ is said to be symmetric if the product ⋆ is commutative. The condition Φ e = Id in Definition 4.1 (b) implies that the field e is the unit of the product ⋆. The integrability of the connection ∇ implies that of the Higgs field Φ, which is equivalent to the associativity of ⋆. So the product ⋆ associated to a Saito structure is commutative and associative. Since the connection △ is flat and torsion free, we can take a flat coordinate system (t 1 , . . . , t n ) such that △ (∂ t i ) = 0 (i = 1, . . . , n) at least on a simply-connected open set of X. We assume the existence of a flat coordinate system (t 1 , . . . , t n ) on X replacing X by such an open set if necessary. In addition, we assume the following conditions: (C1) e = ∂ tn , (C2) E = w 1 t 1 ∂ t 1 + · · · + w n t n ∂ tn for w i ∈ C (i = 1, . . . , n), (C3) w n = 1 and w i − w j ∈ Z \ {0} for i = j. In this paper, a function f ∈ O X is said to be weighted homogeneous with a weight w(f ) ∈ C if f is an eigenfunction of the Euler operator: Ef = w(f )f . In particular, the flat coordinates t i (i = 1, . . . , n) are weighted homogeneous with w(t i ) = w i . We write Φ ∈ End O X (Θ X ) ⊗ O X Ω 1 X as Φ = n k=1 Φ (k) dt k , where Φ (k) ∈ End O X (Θ X ) (k = 1, . . . , n). We fix the basis {∂ t 1 , . . . , ∂ tn } of Θ X and introduce the following matrices: (i)B (k) (k = 1, . . . , n) is the representation matrix of Φ (k) , namely the (i, j)-entryB (k) ij is defined by Φ (k) (∂ t i ) = n j=1B (k) ij ∂ t j (i = 1, . . . , n),(44) (ii) T and B ∞ are the representation matrices of −Φ(E) and △ E respectively, namely − Φ ∂t i (E) = n j=1 T ij ∂ t j , △ ∂t i (E) = n j=1 (B ∞ ) ij ∂ t j .(45) We assume that −Φ(E) is generically regular semisimple on X, that is the discriminant of det(z − T ) does not identically vanish on X. Lemma 4.2. B ∞ = diag [w 1 , . . . , w n ]. Proof. It is straightforward. (46) is nothing but the integrability condition of the Pfaffian system            ∂B (i) ∂t j = ∂B (j) ∂t i , i, j = 1, . . . , n, T ,B (i) = O, B (i) ,B (j) = O, i, j = 1, . . . , n, ∂T ∂t i +B (i) + [B (i) , B ∞ ] = O, i = 1, . . . , n.(46)dY = −(zI n − T ) −1 dz + n i=1B (i) dt i B ∞ Y.(47) In other words, the existence of a Saito structure yields an Okubo system in several variables (47). In the next section, we will find a criterion for that, for an Okubo system in several variables, there exists a Saito structure which yields the given Okubo system in several variables. (43) is written in the following matrix form with respect to the flat coordinate system: dY = − T z + B ∞ dz z Y − n i=1B (i) z dt i Y.(48) The system of equations (46) is equivalent to the integrability condition of (48). The system of ordinary linear differential equations dY dz = − T z 2 + B ∞ z Y has an irregular singularity of Poincaré rank one at z = 0 and a regular singularity at z = ∞, which is called a Birkhoff normal form. So (48) may be regarded as a universal integral deformation of a Birkhoff normal form. A Birkhoff normal form can be transformed into an Okubo system using a Fourier-Laplace transformation. Lemma 4.4. Define vector fields V i (i = 1, . . . , n) by V n · · · V 1 t = −T ∂ t 1 · · · ∂ tn t(49) and put h = h(t) = det(−T ). Then V i , i = 1, . . . , n, are logarithmic vector fields along D = {t ∈ X; h(t) = 0}, and D is a free divisor. Proof. Several proofs are found in [51,29]. Lemma 4.5. There is a unique matrix C such that T = −EC,B (i) = ∂C ∂t i , i = 1, . . . , n and that each matrix entry C ij of C is weighted homogeneous with w(C ij ) = 1 − w i + w j . Proof. This lemma can be proved in a way similar to Lemma 3.6. Lemma 4.6. Let (t 1 , . . . , t n ) be a flat coordinate system of the Saito structure. Then T nj = −w j t j (or equivalently C nj = t j ), j = 1, . . . , n. Proof. As in Lemma 4.4, define vector fields V i (i = 1, . . . , n) by V n . . . V 1 t = −T ∂ t 1 . . . ∂ tn t . Then Lemma 4.4 shows that each V i is logarithmic along D. Then similarly to Lemma 3.5, it holds that V 1 = E, from which we have −T nj = w j t j . It is equivalent to C nj = t j by Lemma 4.5. Proposition 4.7 (Konishi-Minabe [40]). There is a unique n-tuple of analytic functions g = (g 1 , . . . , g n ) ∈ O n X such that C ij = ∂g j ∂t i and that g j is weighted homogeneous with w(g j ) = 1 + w j . The vector g (or precisely the vector field G = n i=1 g i ∂ t i ) is called a potential vector field. Proof. By the symmetry of the Higgs field Φ, it holds that ∂C ij ∂t k = ∂C kj ∂t i , i, j, k = 1, . . . , n. Then g j is uniquely given by g j = 1 1+w j n i=1 w i t i C ij . Proposition 4.8. The potential vector field g = (g 1 , . . . , g n ) is a solution to the following system of nonlinear equations: n m=1 ∂ 2 g m ∂t k ∂t i ∂ 2 g j ∂t l ∂t m = n m=1 ∂ 2 g m ∂t l ∂t i ∂ 2 g j ∂t k ∂t m , i, j, k, l = 1, . . . , n,(50)∂ 2 g j ∂t n ∂t i = δ ij , i, j = 1, . . . , n,(51)Eg j = n k=1 w k t k ∂g j ∂t k = (1 + w j )g j , j = 1, . . . , n.(52) Proof. The associativity of ⋆ (i.e., the integrability of Φ) is equivalent toB (k)B(l) = B (l)B(k) . We obtain (50) if we rewrite this condition in terms ofB (51) follows fromB (n) = I n (i.e., Φ e = Id). (k) ij = ∂ 2 g j ∂t k ∂t i . The equation Definition 4.9. The system of non-linear differential equations (50), (51), (52) for the vector g = (g 1 , . . . , g n ) is called the extended WDVV equation. Remark 4.4. The notion of "F -manifolds with compatible flat structures" was introduced by Manin [44] as a generalization of Frobenius manifolds. This notion does not require the existence of an Euler field. "Potential vector field" in Proposition 4.7 is called "local vector potential" in Manin's framework [44]. And the associativity conditions (50), (51) are called "oriented associativity equations" in [44]. The authors were informed these facts by A. Arsie and P. Lorenzoni. Conversely, starting with a solution of (50), (51) and (52), it is possible to reconstruct a Saito structure. Proposition 4.10. Take constants w j ∈ C, j = 1, . . . , n satisfying w i − w j ∈ Z and w n = 1 and assume that g = (g 1 , . . . , g n ) is a holomorphic solution of (50), (51) and (52) on a simply connected domain U in C n . Then there is a Saito structure on U which has (t 1 , . . . , t n ) as a flat coordinate system. In addition, the Saito structure is semisimple (i.e. −Φ(E) is semisimple) if and only if (SS) the n × n-matrix −(1 + w j − w i ) ∂g j ∂t i 1≤i,j≤n is semisimple. Proof. Define E = n i=1 w i t i ∂ t i , e = ∂ tn ,B (k) ij = ∂ 2 g j ∂t k ∂t i , Φ = n k=1B (k) dt k and △ (∂ t i ) = 0, i = 1, . . . , n. Then the conditions (a), (b) of Definition 4.1 hold and g is the potential vector field associated to C. The last part of the proposition is obvious from T ij = −EC ij . Remark 4.5. By definition, any two weighted-homogeneous flat coordinate systems (t ′ 1 , . . . , t ′ n ) and (t 1 , . . . , t n ) are connected by t ′ i = n j=1 c ij t j with c ij = 0 if w i = w j and c in = 0 (i = 1, . . . , n − 1). (In particular, if w i = w j for any i = j, the weighted homogeneous flat coordinate system is unique up to multiplication by non-zero constants: (t ′ 1 , . . . , t ′ n ) = (c 1 t 1 , . . . , c n t n ).) Let (g ′ j (t ′ 1 , . . . , t ′ n )) 1≤j≤n and (g j (t 1 , . . . , t n )) 1≤j≤n be the potential vector fields with respect to (t ′ 1 , . . . , t ′ n ) and (t 1 , . . . , t n ) respectively. Then it holds that g ′ j (t ′ 1 , . . . , t ′ n ) = c nn n k=1 c jk g k (t 1 , . . . , t n ), j = 1, . . . , n. In other words, given a solution (g j (t 1 , . . . , t n )) 1≤j≤n to the extended WDVV equation (50) is also a solution to it, where (d ij ) is the inverse matrix of (c ij ). Let J be an n × n-matrix with J ij = δ i+j,n+1 , i, j = 1, . . . , n, where δ ij denotes Kronecker's delta, and, for an n × n matrix A, define A * by A * = JA t J. Proposition 4.11. Given a Saito structure on X, the following conditions are mutually equivalent: (i) For appropriate normalization of the flat coordinate system, it holds that C * = C. (ii) For appropriate normalization of the flat coordinate system, there is a holomorphic function F ∈ O X such that ∂F ∂t i = g n+1−i = ( gJ) i , i = 1, . . . , n.(53) (iii) There is r ∈ C such that w n+1−i + w i = −2r, i = 1, . . . , n,(54) and there is a metric η (in this paper, "metric" means non-degenerate symmetric C-bilinear form on T X) such that η(σ ⋆ ξ, ζ) = η(σ, ξ ⋆ ζ), (compatibility to the product) ( △ η)(ξ, ζ) := d(η(ξ, ζ)) − η( △ ξ, ζ) − η(ξ, △ ζ) = 0,(55) (Eη)(ξ, ζ) := E(η(ξ, ζ)) − η(Eξ, ζ) − η(ξ, Eζ) = −2rη(ξ, ζ), (homogeneity) (57) for any σ, ξ, ζ ∈ Θ X . Here, as stated in Remark 4.5, a flat coordinate system admits indetermination of multiplication by constants. "Normalization" in the above conditions means to fix this indetermination. Proof. (i) ⇔ (ii) By definition, C * = C is equivalent to that CJ is a symmetric matrix. From ∂g n+1−j ∂t i = (CJ) ij , we find that the symmetry of CJ is equivalent to the existence of F ∈ O X such that ∂F ∂t i = g n+1−i . (i) ⇒ (iii) First, we show w n+1−i + w i = −2r, i = 1, . . . , n for some r ∈ C. Note that, if C * = C holds, then it also holds that T * = T andB (i) * =B (i) , i = 1, . . . , n. By the integrability condition, we have ∂T ∂x i +B (i) + [B (i) , B ∞ ] = O, ∂T * ∂x i +B (i) * − [B (i) * , B * ∞ ] = O. Taking the difference of these two equalities, we have [B (i) , B ∞ + B * ∞ ] = O, i = 1, . . . , n. Since T is written as T = − n i=1 w i t iB (i) , we have [T , B ∞ + B * ∞ ] = O. Besides, we note that T nj = −w j t j = 0, j = 1, . . . , n. Then it follows that B ∞ + B * ∞ = diag[w 1 + w n , w 2 + w n−1 , . . . , w n + w 1 ] is a scalar matrix. Next, we show the existence of a metric with the desired properties. Under the assumption that CJ is a symmetric matrix, define a metric η by η(∂ t i , ∂ t j ) = J ij , i, j = 1, . . . , n for the flat coordinate system (t 1 , . . . , t n ). Then, on one hand, we have η(∂ t i ⋆ ∂ t k , ∂ t j ) = η n l=1B k il ∂ t l , ∂ t j = n l=1 ∂C il ∂t k J lj = ∂(CJ) ij ∂t k , on the other hand η(∂ t i , ∂ t k ⋆ ∂ t j ) = η ∂ t i , n l=1B k jl ∂ t l = ∂(CJ) ji ∂t k , which concludes the compatibility of η to the product. The horizontality and the homogeneity hold obviously. (iii) ⇒ (i) By the horizontality, it follows that η(∂ t i , ∂ t j ) (i, j = 1, . . . , n) are constants. From the homogeneity, it holds that (w i + w j )η(∂ t i , ∂ t j ) = −2r η(∂ t i , ∂ t j ), which implies that we can take a flat coordinate system (t 1 , . . . , t n ) such that η(∂ t i , ∂ t j ) = J ij and we do so in the sequel. Then it holds that B (k) i,n+1−j = η(∂ t i ⋆ ∂ t k , ∂ t j ) = η(∂ t i , ∂ t k ⋆ ∂ t j ) =B (k) j,n+1−i , which impliesB (k) * =B (k) , k = 1, . . . , n. Hence T * = T and C * = C also hold. The function F in Proposition 4.11 is called a prepotential in [16] and a potential in [51]. It is well known that the prepotential satisfies the WDVV equation (cf. [16]). 5 Flat structure on the space of isomonodromic deformations We start with an Okubo system in several variables: dY = −(zI n − T ) −1 dz + n i=1B (i) dx i B ∞ Y.(58) Let the assumptions on (58) be same as in Section 2. The purpose of this section is to find a necessary and sufficient condition for that the Okubo system in several variables (58) arises from a Saito structure. Here, to avoid the confusion, we state the precise definition of that an Okubo system in several variables (58) arises from a Saito structure. Definition 5.1. We say that an Okubo system in several variables (58) We consider (58) restricted on an appropriate small domain W ⊂ U \ ∆ H so that we can take an invertible matrix P = P (x) such that Then the homomorphism ϕ from T W to C n (W ) defined by ϕ(∂ x i ) = F (i) (e n ) is isomorphic if and only if n j=1 P nj = 0, which can be seen from e n = (P n1 , . . . , P nn )P −1 (e 1 , . . . , e n ) t and thus ϕ(∂ x i ) = − ∂z 1 ∂x i P n1 , . . . , − ∂z n ∂x i P nn P −1 (e 1 , . . . , e n ) t . P −1 T P = diag [z 1 (x), . . . , z n (x)], P −1B(i) P = diag − ∂z 1 (x) ∂x i , . . . , − ∂z n (x) ∂x i . Put e := ϕ −1 (e n ), E := ϕ −1 (F (e n )) and introduce a Higgs field Φ on T W by Φ := n j=1 dx j ϕ −1 • F (j) • ϕ. Besides, define a connection △ on T W by △ Lemma 5.3. We consider an Okubo system in several variables (58). The following two conditions are equivalent to each other: Proof. From (21), it holds that := ϕ −1 • △ (0) • ϕ, where △((i) ∂T n1 ∂x 1 · · · ∂Tnn ∂x 1 . . .∂T nj ∂x i = (λ n − λ j − 1)B (i) nj , from which and (10) we have    ∂T n1 ∂x 1 · · · ∂Tnn ∂x 1 . . . ∂T n1 ∂xn · · · ∂Tnn ∂xn       λ n − λ 1 − 1 . . . −1    −1 P =   B (1) n1 · · ·B (1) nn . . . B (n) n1 · · ·B (n) nn    P = −    ∂z 1 ∂x 1 · · · ∂zn ∂x 1 . . . ∂z 1 ∂xn · · · ∂zn ∂xn       P n1 . . . P nn    . Hence we obtain that at any point on U. If (58) arises from a Saito structure on U, the set of variables t j := −(λ j − λ n + 1) −1 T nj , j = 1, . . . , n provides a flat coordinate system. Proof. First, we assume that (58) arises from a Saito structure. Take a flat coordinate system (t 1 , . . . , t n ) as independent variables of (58). Then it holds that T = T and T nj = −w j t j , from which we have . . . ∂Tnn ∂tn = (−1) n w 1 · · · w n = 0. Conversely, we assume (59). In virtue of Lemmas 5.2 and 5.3, (58) arises from a Saito structure on W ⊂ U \ ∆ H . We also see that {t j = −(λ j − λ n + 1) −1 T nj } is a flat coordinate system. Then E := n k=1 (λ k − λ n + 1)t k ∂ t k , e := ∂ tn and Φ := n j=1B (j) dx j = j,k ∂x j ∂t kB (j) dt k satisfy the axiom of Saito structure on W . Due to the identity theorem, they satisfy it also on U. Hence (58) arises from a Saito structure on U. Theorem 5.4 is used to construct flat coordinates in Sections 6 and 7. Remark 5.1. Uniqueness of the Saito structure corresponding to an Okubo system in several variables follows from the following argument. An Okubo system in several variables (58) admits the following types of gauge freedom: One is similarity transformations by a constant matrix C such that CB ∞ C −1 = B ∞ , which corresponds to the freedom of flat coordinate systems mentioned in Remark 4.5. The other is permutations on matrix entries. Let σ be a permutation on the set {1, 2, . . . , n}. For an Okubo system in several variables (58), we consider the following change of dependent variables: Y = (y 1 , . . . , y n ) t → Y σ := (y σ(1) , . . . , y σ(n) ) t . Then Y σ satisfies a new Okubo system in several variables dY σ = −(zI n − T σ ) −1 dz + n i=1B (i)σ dx i B σ ∞ Y σ ,(60) where T σ denotes the matrix whose (i, j)-entry is defined by T σ ij := T σ(i),σ(j) and the similar holds forB (i)σ , B σ ∞ . If ∂T σ(n),1 ∂x 1 · · · ∂T σ(n),n ∂x 1 . . . ∂T σ(n),1 ∂xn · · · ∂T σ(n),n ∂xn = 0 holds at any point on U, the space of independent variables of (58) can be equipped with a Saito structure by applying Theorem 5.4 to the new system (60), which differs in general from that obtained from the original system (58). Therefore the space of independent variables of (58) can be equipped with at most n mutually different Saito structures up to isomorphisms. (Note that, if σ fixes n (i.e. σ(n) = n), then the Saito structure obtained from (60) is isomorphic to that obtained from (58). Indeed, σ induces only the permutation on the flat coordinates (t 1 , . . . , t n ) → (t σ 1 := t σ(1) , . . . , t σ n := t σ(n) = t n ).) We see from this argument that, if one of eigenvectors of B ∞ is designated, the corresponding Saito structure is uniquely specified up to isomorphisms (provided that the condition in Theorem 5.4 is satisfied). For example, if all of the eigenvalues are real numbers and satisfy λ 1 ≤ · · · ≤ λ n−1 < λ n , then we can distinguish λ n (and the eigenvector belonging to it) from the remaining eigenvectors and a unique Saito structure can be determined corresponding to it. (The flat structure on the orbit space of a well-generated unitary reflection group introduced in Section 6 is in this case.) The initial value problem for regular Saito structures (including the non-semisimple case) is treated in [35]. Corollary 5.5. We consider the case of n = 3. There is a correspondence between solutions satisfying the semisimplicity condition (SS) in Proposition 4.10 to the extended WDVV equation 3 m=1 ∂ 2 g m ∂t k ∂t i ∂ 2 g j ∂t l ∂t m = 3 m=1 ∂ 2 g m ∂t l ∂t i ∂ 2 g j ∂t k ∂t m , i, j, k, l = 1, 2, 3, ∂ 2 g j ∂t 3 ∂t i = δ ij , i, j = 1, 2, 3, Eg j = 3 k=1 w k t k ∂g j ∂t k = (1 + w j )g j , j = 1, 2, 3. and generic solutions to the Painlevé VI equation d 2 y dt 2 = 1 2 1 y + 1 y − 1 + 1 y − t dy dt 2 − 1 t + 1 t − 1 + 1 y − t dy dt + y(y − 1)(y − t) t 2 (t − 1) 2 α + β t y 2 + γ t − 1 (y − 1) 2 + δ t(t − 1) (y − t) 2 . Let T be the 3 × 3 matrix whose entries are given by T ij = −(1 − w i + w j ) ∂g j ∂t i , and take P such that P −1 T P is a diagonal matrix. Let r i = − P diag[w 1 − w 3 , w 2 − w 3 , 0]P −1 ii , i = 1, 2, 3 and θ ∞ = w 1 −w 2 , where we remark that r i is an explicit form of that in (A5) in Section 2. Then the correspondence between the parameters is given by α = 1 2 (θ ∞ − 1) 2 , β = − 1 2 r 2 1 , γ = 1 2 r 2 2 , δ = 1 2 (1 − r 2 3 ). Proof. Since the condition (59) [43] shows that the three-dimensional regular semisimple bi-flat F -manifolds are parameterized by generic solutions to the (fullparameter) Painlevé VI equation. Recently it is proved in [3,41] that bi-flat F -manifold is an equivalent notion to Saito structure. Therefore Corollary 5.5 provides another proof of Arsie-Lorenzoni's result. The proof here makes clear the relationship between flat structures and isomonodromic deformations. In [2], Arsie and Lorenzoni study the relationship between three-dimensional regular non-semisimple bi-flat F -manifolds and Painlevé IV and V equations. Then it is naturally expected that there is a correspondence between solutions to the extended WDVV equation not satisfying the semisimple condition (SS) and the isomonodromic deformations of generalized Okubo systems introduced in [34]. This is treated in [35]. Theorem 5.4 on the existence of a Saito structure associated with a given Okubo system in several variables can be rephrased as follows. Theorem 5.6. We consider the reduced form (25) in Remark 2.5 of an Okubo system in several variables (58) with det B ∞ = 0: dY 0 = B 0 Y 0 = n i=1 T −1B(i) B ∞ dx i Y 0 .(61) Then the following two conditions are equivalent: (i) It holds that )) −1 · det M V ∈ O U , where M V is a matrix defined by (V n , . . . , V 1 ) t = M V (∂ x 1 , . . . , ∂ xn ) t . Proof. From (61), we see that (∂y n /∂x 1 , . . . , ∂y n /∂x n ) t = MY 0 holds, where we put Proposition 5.7. Assume that the reduced form of an Okubo system in several variables (61) arises from a Saito structure on U and take the flat coordinate system (t 1 , . . . , t n ) = (C n1 , . . . , C nn ) as the independent variables of (61). Then there is a function y = y(t) such that Y 0 = (∂y/∂t 1 , . . . , ∂y/∂t n ) t . M :=   B (1) n1 · · ·B (1) nn . . . B (n) n1 · · ·B (n) nn    T −1 B ∞ .(V n y n , . . . , V 1 y n ) t = M V (∂ t 1 y n , . . . , ∂ tn y n ) t = M V MY 0 = Y 0 . (ii) ⇒ (i) Assume that there are logarithmic vector fields along D (V n , . . . , V 1 ) t = M V (∂ x 1 , . . . , ∂ xn ) t such that (det(−T )) −1 det M V ∈ O U and Y ′′ 0 = M V MY 0 = Y 0 . Then we have det(M V M) = 1, which implies that B (1) n1 · · ·B (1) nn . . . B (n) n1 · · ·B (n) nn = n j=1 (λ n − λ j − 1) ∂T n1 ∂x 1 · · · ∂Tnn ∂x 1 . . . Proof. We consider the "contiguous" equation of (61): dY (−) 0 = B (−) Y (−) 0 := T −1 dC(B ∞ − I n )Y (−) 0 .(62) For any solution Y = M (−) B (−) (M (−) ) −1 + dM (−) (M (−) ) −1 Y ′(−) 0 = T −1 (B ∞ − I n )dC − T −1 dT Y ′(−) 0 = T −1 (B ∞ − I n )dC + dC + [dC, B ∞ ] Y ′(−) 0 = T −1 dCB ∞ Y ′(−) 0 . Hence we can take y = y (−) n as the desired function. Flat generator system for invariant polynomials of a complex reflection group In this section, we treat a problem on the existence of flat basic invariants for a complex reflection group. In the case of real reflection groups, K. Saito [53] proved the existence of flat basic invariants (see also [54]). For a well-generated complex reflection group G, we construct a Saito structure on the orbit space of G using an Okubo system in several variables called G-quotient system (Theorem 6.2). (The G-quotient system is a Pfaffian system whose fundamental system of solutions consists of derivatives by logarithmic vector fields of linear coordinates on the standard representation space of G and its monodromy group is isomorphic to G, see Theorem 6.2 and Remark 6.1.) As a consequence, we find that the potential vector field of the Saito structure for a well-generated complex reflection group G has polynomial entries (Corollary 6.8). It is underlined that the following proof of Theorem 6.2 is constructive i.e. it contains an algorithm which provides explicit computation of the flat generator system of G-invariants and the potential vector field for each well-generated complex reflection group G. See [29] for explicit formulas of potential vector fields for exceptional groups. (There is a procedure to construct flat basic invariants directly from a potential vector field corresponding to a finite complex reflection group, see Remark 6.2.) Let G be a finite irreducible complex (unitary) reflection group acting on the standard representation space U n = {(u 1 , u 2 , . . . , u n ) \| u j ∈ C}, and let F i (u) of degree d i , 1 ≤ i ≤ n be a fundamental system of G-invariant homogeneous polynomials. We assume that d 1 ≤ d 2 ≤ · · · ≤ d n . We define a coordinate functions on X := U n /G by, x i = F i (u), 1 ≤ i ≤ n. Let D ⊂ X be the branch locus of π G : U n → X. Let h(x) be the (reduced) defining function of D in the coordinates x = (x 1 , x 2 , . . . , x n ). We assume that G is well generated (see e.g. [5]). Then it is known that h(x) is a monic polynomial in x n of degree n ( [5]). We define a weight w(·) by w(x i ) = d i /d n . Then h(x) is a weighted homogeneous polynomial in x. It is known that D is free ( [47,60]). Here we give a key lemma for the discriminant h(x). Proof. It is known ( [5]) that there is a generator {V 1 , . . . V n } of logarithmic vector fields along D such that V i = n j=1 v ij (x) ∂ x j , 1 ≤ i ≤ n are weighted homogeneous and satisfy M V − x n I n ∈ C[x ′ ] n×n .(63) In particular, it holds that deg xn (v ij (x)) =    1 if i + j = n + 1, 0 if i + j = n + 1. Let V = n i=1 c i (x ′ ) ∂ x i bei (x) ∈ C[x] such that V = n i=1 a i (x)V i , which is equivalent to c j (x ′ ) = n i=1 a i (x)v ij (x), 1 ≤ j ≤ n.(64) Let I = {i \| a i (x) = 0}, and assume I = ∅. Let i 0 ∈ I be such that deg xn (a i 0 ) ≥ deg xn (a i ), i ∈ I. Then deg xn (a i 0 (x)v i 0 ,n+1−i 0 (x)) = deg xn (a i 0 (x)) + 1 > deg xn (a i (x)v i,n+1−i 0 (x)), i = i 0 , i ∈ I, which implies deg xn n i=1 a i (x)v i,n+1−i 0 (x) = deg xn (a i 0 (x)v i 0 ,n+1−i 0 (x)) > 0 = deg xn (c n+1−i 0 (x ′ )), which contradicts the equality (64) with j = n + 1 − i 0 . This implies that I = ∅, that is, V = 0. Then Lemma 3.2 asserts that H(x, z) satisfies (A3). Let B ∞ = diag [w(x 1 ), w(x 2 ), . . . , w(x n )] − 1 + 1 d n I n .(65) In this section, we prove the following theorem: (34) and (31) with respect to h(t). Let C(t) be the (weighted homogeneous) matrix satisfying V Theorem 6.2. There are special G-invariant homogeneous polynomials F f l i (u) of degree d i , 1 ≤ i ≤ n,i (t), 1 ≤ i ≤ n and M V (h) (t) be defined by(h) 1 C(t) = M V (h) (t). Then, for any homogeneous linear function y(u) of u, Y ′ = −B −1 ∞ V (h) n (t) V (h) n−1 (t) . . . V (h) 1 (t) t y(u)(66) satisfies the Okubo system dY ′ = − V (h) 1 C(t) −1 dC(t) B ∞ Y ′ .(67) Note that the n-th entry of Y ′ equals y(u), that is d n V (h) 1 (t) y(u) = y(u). (ii) It holds that C n,j (t) = t j , 1 ≤ j ≤ n, and hence {t j } gives a flat coordinate system on X associated to (67) by Theorem 5.4. If d 1 < d 2 · · · < d n , then F f l i (u) are unique up to constant multiplications. Definition 6.3. We call {F f l i (u)} a flat generator system of G-invariant polynomials or flat basic invariants of G. Remark 6.1. A generating system in [20] is a differential equation with a single unknown satisfied by homogeneous linear functions y(u). In [32], the generating system is rewritten as a Pfaffian system and called a G-quotient system. The equation (67) is a G-quotient system in a form of Okubo type. Again, we let x i = F i (u), 1 ≤ i ≤ n for arbitrarily given fundamental system of G-invariant homogeneous polynomials, and h(x) be the defining function of D in the coordinates x. Let [58] h(x(u)) = c 1 H i := {ℓ i (u) = 0}, 1 ≤ i ≤ N be all the distinct reflecting hyperplanes of G. If H i is the reflecting hyperplanes of g i ∈ G with the order m i , then we have g i (H i ) = H i , g j (H i ) = H i for j = i. As stated inN i=1 ℓ i (u) m i , det ∂x ∂u = c 2 N i=1 ℓ i (u) m i −1 ,(68) for some c 1 , c 2 ∈ C × , where ∂x ∂u = ∂x j ∂u i i,j=1,2,..,n . Leth (u) = N i=1 ℓ i (u), andD = ∪ N i=1 H i . Lemma 6.4. Fix i ∈ {1, . . . , N} arbitrarily. Let u (0) be a generic point of H i , and x (0) = x(u (0) ). Put y 1 (u) = l i (u). Then y 1 (u) m i is g i -invariant, and h(x) = c 1 (x)y 1 (u) m i for some non-vanishing holomorphic function c 1 (x) in a neighborhood of x (0) . There are g iinvariant linearly independent homogeneous linear functions y k (u), 2 ≤ k ≤ n. Then there are locally holomorphic functions η k (x) at x (0) , 2 ≤ k ≤ n such that y k (u) = η k (x(u)), and that {h(x), η 2 (x), . . . , η n (x)} is a local coordinate system at x (0) . Let M V (x) = M V (h) (x), V i (x) = V (h) i (x), 1 ≤ i ≤ n, where M V (h) (x), V (h) i (x) are defined by (34) and (31) with respect to h(x). Recall that det(M V (x)) = h(x), and V 1 (x) is the Euler operator : V 1 (x) = n i=1 w(x i ) x i ∂ x i . LetṼ i be the pull-back of V i to U n , that is, MṼ = M V ∂x ∂u −1 , and Ṽ nṼn−1 . . .Ṽ 1 t = MṼ ∂ u 1 ∂ u 2 . . . ∂ un t . Then, from (68), it holds that det(MṼ ) = ch(u), for a constant number c = 0. Proof. (i) is known by Terao and others ( [47,60]). We prove (ii). Fix i arbitrarily, and we will prove (Ṽ k MṼ )M −1 V is holomorphic along H i . Let y 1 (u) = l i (u), and choose y k (u) = l i k (u), 2 ≤ k ≤ n, so that y 1 , y 2 , . . . , y n are linearly independent. Then (i) implies thatṼ k = n j=1 y j v k,j (y) ∂ y j , 1 ≤ k ≤ n,(70) for some v k,j (y) ∈ C[y]. This is equivalent to MṼ = M ′Ṽ (y) · diag[y 1 , y 2 , . . . , y n ],(71) for some M ′Ṽ (y) ∈ C[y] n×n . Then it also holds thatṼ k MṼ = M ′′ V (y) · diag[y 1 , y 2 , . . . , y n ], for some M ′′ V (y) ∈ C[y] n×n . Thus we have (Ṽ k MṼ )M −1 V = M ′′ V (y) (M ′Ṽ (y)) −1 .(72) Since det(MṼ ) = N j=1 l j (u), the equality (71) implies that (M ′Ṽ (y)) −1 is holomorphic along {y 1 = 0}. Consequently the equality (72) implies that ( Ṽ k MṼ )M −1 V is holomorphic along H i . This proves that (Ṽ k MṼ ) M −1 V ∈ C[u] n×n . We next prove that (Ṽ k MṼ ) M −1 V is G-invariant. Let g ∈ G, and γ ∈ π 1 (X \ D, * ) be a loop such that γ * u = u g. Then we have γ * ∂u ∂x = ∂u ∂x g, γ * MṼ = γ * M V ∂u ∂x = MṼ g, γ * Ṽk MṼ = γ * V k MṼ = V k MṼ g =Ṽ k MṼ g. Consequently we have γ * (Ṽ k MṼ ) M −1 V = (Ṽ k MṼ ) M −1 V , which means that (Ṽ k MṼ ) M −1 V is G-invariant. This completes the proof of (ii). Let (dMṼ )M −1 V = n k=1B (k) (x) dx k , and putŶ = ŷ 1ŷ2 . . .ŷ n t = V n V n−1 . . . V 1 t y(u), for any homogeneous linear function y(u) of u. Then we haveŷ n = w(y)y = (1/d n ) y, and w(ŷ i ) = w(V n−i+1 ) + (1/d n ) = (d n − d i + 1)/d n , 1 ≤ i ≤ n.(73) Lemma 6.6. All entries of h(x)B (k) (x) are weighted homogeneous polynomials in x, and Y satisfies the system of differential equations dŶ = n k=1B (k) dx k Ŷ . (74) Proof. Put P k (x) = (Ṽ k MṼ ) M −1 V for k = 1, 2, . . . n. Then P k (x) ∈ C[x] n×n from (iii) of Lemma 6.5. We have (dMṼ )M −1 V = n k=1 ∂MṼ ∂u k M −1 V du k = n j,k=1 (M −1 V ) k,j (Ṽ n−j+1 MṼ )M −1 V du k = n j,k=1 (M −1 V ) k,j P n−j+1 (x)dx k ConsequentlyB (k) = n j=1 (M −1 V ) k,j P n−j+1 (x), which implies h(x)B (k) (x) ∈ C[x] n×n . Since y(u) is a solution of d ∂ u 1 ∂ u 2 . . . ∂ un t y = 0,(75) and since ∂ u 1 ∂ u 2 . . . ∂ un t = ∂x ∂u ∂ x 1 ∂ x 2 . . . ∂ xn t = ∂x ∂u M −1 V V n V n−1 . . . V 1 t , y(u) satisfies d − (dMṼ )M −1 V V n V n−1 . . . V 1 t y = 0. This proves the lemma. Lemma 6.7. LetŶ ,B (k) be the same as in Lemma 6.6. Then there is an upper triangular matrix R(x ′ ) ∈ GL(n, C[x ′ ]) such that, if we put Y 0 = (y 1 , y 2 , . . . , y n ) t = R(x ′ )Ŷ , B (k) 0 = R(x ′ )B (k) R(x ′ ) −1 + ∂R(x ′ ) ∂x k R(x ′ ) −1 , 1 ≤ k ≤ n, then the system of differential equations dY 0 = n k=1 B (k) 0 dx k Y 0(76) satisfied by Y 0 is an Okubo system in several variables with (B (k) 0 ) ǫ = 0 (See Lemma 2.1 for the E-part (B (k) 0 ) ǫ of B (k) 0 ), and the residue matrix (B 0 ) ∞ of B (n) 0 dx n at x n = ∞ is equal to the diagonal matrix B ∞ in (65). The matrix R(x ′ ) can be chosen such that all the entries are weighted homogeneous with w(R(x ′ ) i,j ) = w(ŷ i ) − w(ŷ j ), and R(0) = I n . In particular, it holds that y n =ŷ n = (1/d n )y(u). Proof. LetB ∞ (x ′ ) = −(x nB (n) )\| xn=∞ be the residue matrix ofB (n) dx n at x n = ∞. From w(B (k) i,j (x ′ )) = w(ŷ i ) − w(ŷ j ) − w(x k ) = w(V n−i+1 ) − w(V n−j+1 ) − w(x k ) = −w(x i ) + w(x j ) − w(x k ),(77) we find that the degree of h(x)B (n) i,j (x) in x n is at most n − 1, which implies that −B ∞ (x ′ ) is the coefficients of x n−1 n of h(x)B (n) (x). Consequently (B ∞ ) i,j (x ′ ) is a weighted homo- geneous polynomial in x ′ with w((B ∞ ) i,j (x ′ )) = w(x j ) − w(x i ). Let n 1 , n 2 , ..., n k be the positive integers such that n 1 + n 2 + ... + n k = n, w(x 1 ) = ... = w(x n 1 ) < w(x n 1 +1 ) = ... = w(x n 1 +n 2 ) < ... ≤ w(x n ). ThenB ∞ (x ′ ) has the formB ∞ (x ′ ) =      R 1 * ... * O R 2 ... * ·· ·· ... ·· O O ... R k      , for some R i ∈ C n i ×n i , 1 ≤ i ≤ k. Since n k=1 w(x k )x kB (k) (x) Ŷ = n k=1 w(x k )x k ∂ ∂x k Ŷ = diag[w(ŷ 1 ), w(ŷ 2 ), . . . , w(ŷ n )]Ŷ , for all solutionsŶ of (74), we have n k=1 w(x k )x kB (k) (x) = diag[w(ŷ 1 ), w(ŷ 2 ), . . . , w(ŷ n )]. In particular, we have (x nB (n) (x))\| x ′ =0 = diag[w(ŷ 1 ), w(ŷ 2 ), . . . , w(ŷ n )],(78) which shows R i are diagonal, andB ∞ (x ′ ) is an upper triangular matrix with the diagonal elements −w(ŷ i ). Then, by elementary linear algebra, we find that there is an upper triangular matrix R(x ′ ) ∈ GL(n, C[x ′ ]) with the form R(x ′ ) =      I n 1 * ... * O I n 2 ... * ·· ·· ... ·· O O ... I n k      , and satisfying R(x ′ )B ∞ (x ′ )R(x ′ ) −1 = −diag[w(ŷ 1 ), w(ŷ 2 ), . . . , w(ŷ n )] = B ∞ . By construction of R(x ′ ), we find that all entries R(x ′ ) i,j are weighted homogeneous with w(R(x ′ ) i,j ) = w(ŷ i )−w(ŷ j ). Now put B (k) 0 = R(x ′ )B (k) R(x ′ ) −1 + ∂R(x ′ ) ∂x k R(x ′ ) −1 , 1 ≤ k ≤ n, B 0 = n k=1 B (k) 0 dx k . Then we find that (B 0 ) ∞ = −(x n B (n) 0 )\| xn=∞ = R(x ′ ) −(x nB (n) )\| xn=∞ R(x ′ ) −1 = R(x ′ )B ∞ R(x ′ ) −1 = B ∞ . Since the residue matrix ofB (n) dx n at zeros of h(x ′ , x n ) is of rank one and diagonalizable, so is the residue matrix of B (n) 0 dx n . Consequently, we find that there exists a n × n matrix T 0 (x ′ ) such that B (n) 0 = −(x n I n − T 0 (x ′ )) −1 B ∞(79) at least at any generic point x ′ by an argument similar to the one in Appendix B. Let h(x) = x n n − s 1 (x ′ )x n−1 n + . . . , h(x)B (n) 0 = n−1 i=0 C i (x ′ )x i n , C n−1 (x ′ ) = −B ∞ . Then, by induction, we find C n−i (x ′ ) = − T 0 (x ′ ) i−1 − s 1 (x ′ )T 0 (x ′ ) i−2 + · · · + (−1) i−1 s i−1 (x ′ ) B ∞ , 1 ≤ i ≤ n. In particular, it holds that T 0 (x ′ ) = s 1 (x ′ )I n − C n−2 (x ′ )B −1 ∞ , which implies that all entries of T 0 (x ′ ) are weighted homogeneous polynomials in x ′ . Finally we prove that the differential system (76) satisfies (B h(x)B (k) 0 = m k j=0 (hB (k) 0 ) j x j n , 1 ≤ k ≤ n, where (hB (k) 0 ) j ∈ C[x ′ ] n×n , (hB (n) 0 ) n−1 = −(B 0 ) ∞ . From the equalities w((B (k) 0 ) i,j (x ′ )) = w((B (k) ) i,j (x ′ )) = w(x j ) − w(x i ) − w(x k ), we have m k ≤ n for k ≤ n − 1 and m n = n − 1. For m k = n, from the equality [(hB (k) 0 ) n , −(B 0 ) ∞ ] = O, we have ((hB (k) 0 ) n ) i,j (x ′ ) = 0 if d i = d j . If d i = d j , then w ((hB (k) 0 ) n ) i,j (x ′ ) = −w(x k ), and hence ((hB (k) 0 ) n ) i,j (x ′ ) = 0. This conclude that (hB (k) 0 ) n = O. Consequently B (k) 0 (x) is decomposed in the form B (k) 0 (x) = n−1 j=1 (B (k) 0 ) j (x ′ ) x n − z j (x ′ ) , which proves (B (k) 0 ) ǫ = 0. DefineB (k) 0 (x ′ ) by B (k) 0 (x) = −(x n I n − T 0 (x)) −1B (k) 0 (x ′ )(B 0 ) ∞ . Then (21) impliesB (k) 0 (x ′ ) ∈ C[x ′ ] n×n . Now we prove Theorem 6.2. Proof of Theorem 6.2. From now on, to avoid confusion, we denote by h x (x) the defining function of D in the coordinates x. Let B ∞ , R(x ′ ), Y 0 = R(x ′ )Ŷ , B (i) 0 ,B (i) 0 , T 0 (x ′ ) be as before. Let λ i = w(x i ) − 1 − 1/d n so that B ∞ = diag[λ 1 , λ 2 , . . . , λ n ]. Then w((T 0 ) ij ) = 1 − λ i + λ j . Recall V i = V (hx) i and M V = M V (hx) . Define V ′ i by V ′ n V ′ n−1 . . . V ′ 1 t = −B −1 ∞ R(x ′ ) V n V n−1 . . . V 1 t . In particular V ′ 1 = d n V 1 . Then Y ′ := −B −1 ∞ Y 0 = V ′ n . . . V ′ 2 V ′ 1 t y(u) = V ′ n y(u) . . . V ′ 2 y(u) y(u) t (80) satisfies the equation d Y ′ = n i=1 (B ′ ) (i) dx k Y ′ , where (B ′ ) (i) = B −1 ∞ B (i) 0 B ∞ . Let T ′ (x) = B −1 ∞ (T 0 (x ′ ) − x n I n )B ∞ , and C ′ (x) be such that T ′ (x) = −V 1 C ′ (x). Then n i=1 (B ′ ) (i) dx k = −(V 1 C ′ (x)) −1 dC ′ (x) B ∞ , and Y ′ satisfies d Y ′ = (T ′ (x)) −1 dC ′ (x) B ∞ Y ′ = −(V 1 C ′ (x)) −1 dC ′ (x) B ∞ Y ′ .(81) From Theorem 5.6, it holds that ∂T ′ n1 ∂x 1 · · · ∂T ′ nn ∂x 1 . . . ∂T ′ n1 ∂xn · · · ∂T ′ nn ∂xn = 0, which implies that t j = C ′ nj (x), 1 ≤ j ≤ n form a coordinate system on X. Note t j ∈ C[x ′ ], 1 ≤ j ≤ n − 1, t n − x n ∈ C[x ′ ]. Put F f l j (u) = C ′ nj (F 1 (u), F 2 (u), . . . , F n (u)), 1 ≤ j ≤ n. Since w(C ′ nj ) = w(x j ) , the equalities t j = C ′ nj (x), 1 ≤ j ≤ n are solved by weighted homogeneous polynomials x j = x j (t), 1 ≤ j ≤ n, and hence {F f l j (u)} is a fundamental system of G-invariant polynomials. Let T (t) = T ′ (x(t)), C(t) = C ′ (x(t)), h t (t) = h x (x(t)). Then h t (t) is the defining function of D in the coordinates t; C nj (t) = t j , 1 ≤ j ≤ n; T (t) = −V 1 C(t); and Y ′ satisfies d Y ′ = T (t) −1 dC(t) B ∞ Y ′ = −(V 1 C(t)) −1 dC(t) B ∞ Y ′ .(82) The fact −T ′ (x) − x n I n ∈ C[x ′ ] n×n implies that −T (t) − t n I n ∈ C[t ′ ] n×n , and Lemma 4.4 implies that n j=1 (−T (t) ij ) ∂ t j , 1 ≤ i ≤ n are logarithmic vector fields along D. From these two properties and Lemma 3.9, we find M V (h t ) (t) = −T (t). From the "(i) ⇒ (ii)" part of the proof of Theorem 5.6, (82) implies Y ′ = B −1 ∞ T (t) ∂ t 1 . . . ∂ tn t y(u),(83) which is equivalent to (66). This completes the proof of the existence of a flat generator system of G-invariant polynomials satisfying (i) and (ii) in Theorem 6.2. The uniqueness of a flat generator system under the condition d 1 < d 2 < · · · < d n is clear from Remarks 5.1 and 4.5. Corollary 6.8. The potential vector field g = (g 1 , . . . , g n ) corresponding to the G-quotient system (67) is a polynomial in t 1 , . . . , t n . In other words, the potential vector field of the G-quotient system yields a polynomial solution to the extended WDVV equation. Proof. By the above construction, h(t) and T = −M V (h) (t) are polynomials in t 1 , . . . , t n . Thus, C(t) and g are also polynomials. In fact, g j are given by g j = 1 1+w(t j ) n i=1 w(t i ) t i C ij . Remark 6.2. Irreducible finite complex reflection groups are classified by Shephard-Todd [58]: Except for rank 1 groups, there are two infinite families A n , G(pq, p, n), plus 34 exceptional groups G 4 , G 5 , . . . , G 37 . Explicit forms of potential vector fields for the exceptional groups are discussed in [29]. It is possible to directly compute the explicit form of flat basic invariants of a complex reflection group G from its potential vector field by the following procedure: Let (x 1 , . . . , x n ) = (F 1 (u), . . . , F n (u)) be arbitrary basic invariants of G and write down the discriminant h x (x) of G in terms of x. On the other hand, one can write down the discriminant h t (t) of G in terms of t from the potential vector field in the manner described in Section 4 (i.e. h t (t) := det(−T ), where T is constructed from the potential vector field as in the proof of Proposition 4.10). Find a weight preserving coordinate change t = t(x) such that h t (t(x)) = h x (x). Then F f l 1 (u), . . . , F f l n (u) := t 1 (F 1 (u), . . . , F n (u)), . . . , t n (F 1 (u), . . . , F n (u)) provides flat basic invariants of G. Remark 6.3. The papers [46,17] treat Frobenius structures constructed on the orbit spaces of Shephard groups (which consist a subclass of complex reflection groups). In the case of some Shephard groups, one can find that the Saito structure constructed in Theorem 6.2 which corresponding to the G-quotient system of the Shephard group does not have a prepotential. Therefore in this case, the Saito structure in Theorem 6.2 is distinct from the Frobenius structure treated in [46,17]. This phenomenon is naturally explained in the framework of this article: It is known that to each Shephard group there is an associated Coxeter group whose discriminant is isomorphic to that of the Shephard group. Then we see that there are (at least) two Okubo systems in several variables on the orbit space of the Shephard group which have singularities along the discriminant: one is the G-quotient system of the Shephard group, the other is the G-quotient system of the associated Coxeter group. The Frobenius structure on the orbit space of a Shephard group described in [46,17] corresponds to the G-quotient system of the associated Coxeter group in this picture. Examples of potential vector fields corresponding to algebraic solutions to the Painlevé VI equation In this section we show some examples of potential vector fields in three variables which correspond to algebraic solutions to the Painlevé VI equation. Algebraic solutions to the Painlevé VI equation were studied and constructed by many authors including N. J. Hitchin [23,24], B. Dubrovin [16], B. Dubrovin -M. Mazzocco [18], P. Boalch [6,7,8,9,10], A. V. Kitaev [37,38], A. V. Kitaev -R. Vidūnas [39,61], K. Iwasaki [26]. The classification of algebraic solutions to the Painlevé VI equation was achieved by Lisovyy and Tykhyy [42]. We remark that all the algebraic solutions in the list of [42] had previously appeared in the literature (see [11] and references therein). One of the principal aims of our study is the determination of a flat coordinate system and a potential vector field for each of such algebraic solutions. In spite that this aim is still not succeeded because of complexity of computation, we show some examples of potential vector fields. Some of the results below are already given in [29]. Other examples can be found in [31]. From the construction of polynomial potential vector fields corresponding to finite complex reflection groups of rank three (Corollary 6.8) and Corollary 5.5, we obtain a class of algebraic solutions to the Painlevé VI equation. The relationship between finite complex reflection group of rank three and solutions to the Painlevé VI equation was first studied by Boalch [6]. (More precisely speaking, it was conjectured in [6] that the solutions obtained from finite complex reflection groups by his construction are algebraic and this conjecture was proved in his succeeding papers.) The construction in this paper answers the question 3) in the last part of [6]: "Is there a geometrical or physical interpretation of these solutions?" It is remarkable that there are examples in Sections 7.4,7.5,7.6 whose potential vector fields have polynomial entries but the corresponding flat structures are not isomorphic to one on the orbit space of any finite complex reflection group because the free divisors defined by F B 6 , F H 2 , F E 14 in Sections 7.4,7.5,7.6 respectively are not isomorphic to the discriminant of any finite complex reflection group. The existence of these examples suggests that an analogue of Hertling's theorem ( [21]) does not hold in the case of non Frobenius manifolds. It would be an interesting problem to classify all the polynomial potential vector fields. To avoid the confusion, we prepare the convention which will be used in the following. We treat the case n = 3. Let t = (t 1 , t 2 , t 3 ) be a flat coordinate system and g = (g 1 , g 2 , g 3 ) denotes a potential vector field. Let w(t i ) be the weight of t i and assume 0 < w(t 1 ) < w(t 2 ) < w(t 3 ) = 1. The matrix C is defined by C =    ∂ t 1 g 1 ∂ t 1 g 2 ∂ t 1 g 3 ∂ t 2 g 1 ∂ t 2 g 2 ∂ t 2 g 3 ∂ t 3 g 1 ∂ t 3 g 2 ∂ t 3 g 3    and T = − 3 j=1 w(t j )t j ∂ t j C. In this section, an algebraic solution LTn means "Solution n" in Lisovyy-Tykhyy [42], pp.156-162. Algebraic solutions related with icosahedron We treat the three algebraic solutions to Painlevé VI obtained by Dubrovin [16] and Dubrovin-Mazzocco [18]. Icosahedral solution (H 3 ) In this case, w(t 1 ) = 1 5 , w(t 2 ) = 3 5 , w(t 3 ) = 1 and there is a prepotential defined by F = t 2 2 t 3 + t 1 t 2 3 2 + t 11 1 3960 + t 5 1 t 2 2 20 + t 2 1 t 3 2 6 .(84) Then it follows from the definition that g j = ∂ t j F (j = 1, 2, 3) give the potential vector field g = (g 1 , g 2 , g 3 ). We don't enter the details on this case. See [16,18]. Great icosahedral solution (H 3 ) ′ Let (t 1 , t 2 , t 3 ) be a flat coordinate system and their weights are given by w(t 1 ) = 3 5 , w(t 2 ) = 4 5 , w(t 3 ) = 1. We introduce an algebraic function z of t 1 , t 2 defined by the relation t 2 + t 1 z + z 4 = 0. It is clear from the definition that w(z) = 1 5 . In this case, we consider the algebraic function of (t 1 , t 2 , t 3 ) defined by F = t 2 2 t 3 + t 1 t 2 3 2 − t 4 1 z 18 − 7t 3 1 z 4 72 − 17t 2 1 z 7 105 − 2t 1 z 10 9 − 64z 13 585 . Then we see that F is a solution to the WDVV equation. Indeed, we first define C =    ∂ t 1 ∂ t 3 F ∂ t 1 ∂ t 2 F ∂ 2 t 1 F ∂ t 2 ∂ t 3 F ∂ 2 t 2 F ∂ t 2 ∂ t 3 F ∂ 2 t 3 F ∂ t 3 ∂ t 2 F ∂ t 3 ∂ t 1 F    . Then ∂ t i C (i = 1, 2, 3) commute to each other. This condition is equivalent to that F is a solution to the WDVV equation. Great dodecahedron solution (H 3 ) ′′ Let (t 1 , t 2 , t 3 ) be a flat coordinate system and their weights are given by w(t 1 ) = 1 3 , w(t 2 ) = 2 3 , w(t 3 ) = 1. We introduce an algebraic function z of t 1 , t 2 defined by the relation −t 2 1 + t 2 + z 2 = 0. It is clear from the definition that w(z) = 1 3 . In this case, we consider the algebraic function of (t 1 , t 2 , t 3 ) defined by F = t 2 2 t 3 + t 1 t 2 3 2 + 4063t 7 1 1701 + 19t 5 1 z 2 135 − 73t 3 1 z 4 27 + 11t 1 z 6 9 − 16z 7 35 . Then F is also a solution of the WDVV equation. The following two cases are related with the complex reflection group ST27. Solution 38 of P. Boalch [8] (LT26) In this case, w(t 1 ) = 1 5 , w(t 2 ) = 2 5 , w(t 3 ) = 1 g 1 = (−t 6 1 − 15t 4 1 t 2 + 15t 2 1 t 2 2 + 10t 3 2 + 30t 1 t 3 )/30, g 2 = (5t 7 1 + 3t 5 1 t 2 + 15t 3 1 t 2 2 − 5t 1 t 3 2 + 6t 2 t 3 )/6, g 3 = (−105t 10 1 + 200t 8 1 t 2 + 350t 6 1 t 2 2 + 175t 2 1 t 4 2 − 14t 5 2 + 20t 2 3 )/40. The determinant det(−T ) is regarded as the discriminant of the complex reflection group ST27, if t 1 , t 2 , t 3 are taken as basic invariants. Solution 37 of P. Boalch [8] (LT27) In this case, z is an algebraic function of t 1 , t 2 defined by −t 2 − t 1 z + 2z 3 = 0. w(t 1 ) = 2 5 , w(t 2 ) = 3 5 , w(t 3 ) = 1, w(z) = 1 5 g 1 = (175t 1 t 3 − 70t 3 1 z + 70t 2 1 z 3 + 378t 1 z 5 − 540z 7 )/175, g 2 = (10t 4 1 − 120t 1 t 2 2 + 75t 2 t 3 + 30t 2 1 z 4 − 192t 1 z 6 + 324z 8 )/75, g 3 = (16t 5 1 + 80t 2 1 t 2 2 + 25t 2 3 − 80t 3 1 z 4 + 540t 2 1 z 6 − 1080t 1 z 8 + 432z 10 )/50. The determinant det(−T ) is regarded as the discriminant of the complex reflection group ST27, if z, t 1 , t 3 are taken as basic invariants. Algebraic solutions related with the polynomial F B 6 in [56] We recall the polynomial F B 6 = 9xy 4 + 6x 2 y 2 z − 4y 3 z + x 3 z 2 − 12xyz 2 + 4z 3 which is a defining equation of a free divisor in C 3 (cf. [56]). There are two algebraic solutions which are related with the polynomial F B 6 . Solution 27 of Boalch [8] (LT13) In this case, w(t 1 ) = 1 15 , w(t 2 ) = 1 3 , w(t 3 ) = 1 g 1 = − 1 33 t 1 (3t 10 1 t 2 + 11t 3 2 − 33t 3 ), g 2 = 1 76 (−5t 20 1 + 114t 10 1 t 2 2 + 19t 4 2 + 76t 2 t 3 ), g 3 = 1 870 (100t 30 1 + 1740t 20 1 t 2 2 − 5220t 10 1 t 4 2 + 116t 6 2 + 435t 2 3 ). The determinant det(−T ) coincides with F B 6 by a weight preserving coordinate change up to a non-zero constant factor. Solution obtained by A. Kitaev [38] (LT14) In this case, z is an algebraic function of t 1 , t 2 defined by t 2 1 + t 2 z 6 + z 16 = 0. w(t 1 ) = 8 15 , w(t 2 ) = 2 3 , w(t 3 ) = 1, w(z) = 1 15 g 1 = −(2093t 4 1 − 897t 1 t 3 z 9 + 3450t 2 1 z 16 + 525z 32 )/(897z 9 ), g 2 = (−238t 5 1 + 85t 2 t 3 z 15 + 1700t 3 1 z 16 − 750t 1 z 32 )/(85z 15 ), g 3 = (49t 6 1 + 2415t 4 1 z 16 + 3t 2 3 z 18 + 795t 2 1 z 32 − 35z 48 )/(6z 18 ). The determinant det(−T ) regarded as a polynomial of z 5 , t 2 , t 3 coincides with F B 6 by a weight preserving coordinate change up to a non-zero constant factor. Algebraic solutions related with the polynomial F H 2 in [56] We recall the polynomial F H 2 = 100x 3 y 4 + y 5 + 40x 4 y 2 z − 10xy 3 z + 4x 5 z 2 − 15x 2 yz 2 + z 3 which is a defining equation of a free divisor in C 3 (cf. [56]). There are two algebraic solutions which are related with the polynomial F H 2 . Solution 29 of P. Boalch [8] (LT18) In this case, 18 1 t 2 + 2550t 12 1 t 4 2 + 12750t 6 1 t 7 2 + 595t 10 2 + 9t 2 3 )/18. The determinant det(−T ) regarded as a polynomial of t 2 , t 6 1 , t 3 coincides with F H 2 by a weight preserving coordinate change up to a non-zero constant factor. w(t 1 ) = 1 10 , w(t 2 ) = 1 5 , w(t 3 ) = 1 g 1 = −t 1 (5t 6 1 t 2 2 − 14t 5 2 − 2t 3 )/2, g 2 = (5t 12 1 + 275t 6 1 t 3 2 − 55t 6 2 + 33t 2 t 3 )/33, g 3 = (−100t Solution 30 of P. Boalch [8] (LT19) In this case, z is an algebraic function of t 1 , t 2 defined by t 6 1 + t 2 z 6 + z 9 = 0. w(t 1 ) = 3 10 , w(t 2 ) = 3 5 , w(t 3 ) = 1, w(z) = 1 5 18 1 + 595t 2 3 z 17 + 7140t 2 2 z 21 − 8160t 2 z 24 − 15113z 27 )/(1190z 17 ). The determinant det(−T ) regarded as a polynomial of z, t 2 , t 3 coincides with F H 2 by a weight preserving coordinate change up to a non-zero constant factor. 7.6 Algebraic solution related with E 14 -singularity Solution 13 of P. Boalch [9] (LT30) g 1 = t 1 (−80t 2 2 + 910t 3 z + 165t 2 z 3 + 63z 6 )/(910z), g 2 = (4t 2 t 3 − 12t 2 2 z 2 − 36t 2 z 5 − 27z 8 )/4, g 3 = (−560t In this case, 16 1 − 432320t 13 1 t 2 + 780416t 10 1 t 2 2 − 58240t 7 1 t 3 2 + 1019200t 4 1 t 4 2 +203840t 1 t 5 2 + 39t 2 3 )/78. The determinant det(−T ) in this case coincides with the polynomial F E 14 = −4x 6 y 6 − 20 3 x 3 y 7 − 3y 8 + 30x 7 y 3 z + 51x 4 y 4 z + 24xy 5 z − 243 4 x 8 z 2 −108x 5 yz 2 − 56x 2 y 2 z 2 − 8z 3 by a weight preserving coordinate change. The polynomial F E 14 is regarded as a 1parameter deformation of the defining polynomial of E 14 -singularity in the sense of Arnol'd. w(t 1 ) = 1 8 , w(t 2 ) = 3 8 , w(t 3 ) = 1 g 1 = (5t 9 1 − 84t 6 1 t 2 − 210t 3 1 t 2 2 + 140t 3 2 + 9t 1 t 3 )/9, g 2 = (140t 11 1 − 165t 8 1 t 2 + 924t 5 1 t 2 2 + 770t 2 1 t 3 2 + 11t 2 t 3 )/11, g 3 = (−95680tIn fact F E 14 \| x=0 = −3y 8 − 8z 3 . For topics related to F E 14 , see [57,29]. A Okubo systems in several variables and isomonodromic deformations Let us recall that an Okubo system in several variables of rank n is defined as follows: Especially in the case of n = 3, this equivalence can be related to the Painlevé VI equation (a proof of the following Proposition A.1 is given at the end of this Appendix A): dY = B (z) dz + n i=1 B (i) dx i Y,(85) Proposition A.1. We consider (85) for n = 3. Then there is a correspondence between Okubo systems in several variables of rank 3 and generic solutions to the Painlevé VI equation with the parameters (θ 0 , θ 1 , θ t , θ ∞ ) = (r 1 + λ 3 , r 2 + λ 3 , r 3 + λ 3 , λ 1 − λ 2 ), where λ i and r i are defined in (A2) and (A5) respectively in Section 2. Here, we briefly review the theory of isomonodromic deformation (following to [27,28]). We consider a system of ordinary differential equations of rank m dY dz = n i=1 B i z − a i Y,(86)Y (∞) =Ŷ (∞) (z −1 )z −B∞ , whereŶ ∞ (z −1 ) is a convergent power series of z −1 such thatŶ (∞) (0) = I m . Fixing paths from z = ∞ to z = a i (i = 1, . . . , n), the analytic continuations of Y (∞) to z = a i (i = 1, . . . , n) along the paths are described as follows: Y (∞) = G iŶ (i) (z − a i ) (z − a i ) Λ i C i , i = 1, . . . , n, whereŶ (i) (z − a i ) is a convergent power series of z − a i such thatŶ (i) (0) = I m , Λ i = G i B i G −1 i and C i is a constant invertible matrix. Now we deform (86) moving a 1 , . . . , a n as variables, namely we suppose that B i (i = 1, . . . , n) are functions of a 1 , . . . , a n . Then Y (∞) is a function of z, a 1 , . . . , a n and r (i) j , G i , C i are functions of a 1 , . . . , a n . A deformation of (86) with variables a 1 , . . . , a n is said to be an isomonodromic deformation (or monodromy preserving deformation) if r (i) j (i = 1, . . . , n, ∞, j = 1, . . . , m) and C i (i = 1, . . . , n) are constants independent of a 1 , . . . , a n . The following fact is well known (see [27], [28] for instance): Fact 1. A deformation of (86) is isomonodromic if B i (i = 1, . . . , n) satisfy the following system of differential equations: dB i = j =i [B j , B i ]d log(a i − a j ), i = 1, . . . , n,(87) which is called the Schlesinger system. Returning to our situation, we consider an Okubo system in several variables of rank n (85). Let z 1 , . . . , z n be the roots of det(zI n −T ), and decompose B (z) into partial fractions B (z) = n i=1 B (z) i z − z i .(88) Lemma A.2. The system of equations (20), (21), (22) is equivalent to the Schlesinger system dB (z) i = j =i [B (z) j , B (z) i ]d log(z i − z j ), i = 1, . . . , n.(89) Proof. On one hand, from Lemma 2.1, (20), (21), (22) are equivalent to the integrability condition of (85). On the other hand, it is known that the Schlesinger system (89) is equivalent to the integrability condition of the Pfaffian system dY = n i=1 B (z) i d log(z − z i )Y.(90) Therefore, it is sufficient to show that the two Pfaffian systems (85) and (90) are equivalent to each other. Changing the independent variables (x 1 , . . . , x n ) of (85) to (z 1 , . . . , z n ), (85) is rewritten as dY = −(zI n − T ) −1 dz + n i,j=1B (j) ∂x j ∂z i dz i B ∞ Y.(91) Let E i be the matrix whose (j, k)-entry are defined by (E i ) jk = δ ij δ ik . Then we have n j=1B (j) ∂x j ∂z i = −P E i P −1 and thus −(zI n − T ) −1 n j=1B (j) ∂x j ∂z i B ∞ = P E i P −1 B ∞ /(z − z i ). It is straightforward to check that B (z) i = −P E i P −1 B ∞ . As a preparation to prove Proposition A.1, we quote from [28, Appendix C] (a concise explanation is found in [7]) the following fact on the relation between the Painlevé VI equation and the isomonodromic deformation of a system of linear differential equations of rank 2. (Notations in Fact 2 are valid only in this part.) Fact 2. We consider a Pfaffian system of rank 2 dZ = A(x, t)dx + B(x, t)dt Z, where A(x, t) = A 0 x + A 1 x − 1 + A t x − t , B(x, t) = − A t x − t ,(93) and A 0 , A 1 , A t are 2 × 2 matrices whose entries are functions of t (independent of x). Put A ∞ := −A 0 − A 1 − A t . We assume that A 0 , A 1 , A t are rank 1 matrices and that A ∞ is a diagonal matrix. Put θ i := trA i (i = 0, 1, t). Then A i (i = 0, 1, t) can be written as follows: A 0 = z 0 + θ 0 −uz 0 u −1 (z 0 + θ 0 ) −z 0 , A 1 = z 1 + θ 1 −vz 1 v −1 (z 1 + θ 1 ) −z 1 , A t = z t + θ t −wz t w −1 (z t + θ t ) −z t .(94) From the assumption that A ∞ is diagonal, we have the following relations: uz 0 + vz 1 + wz t = 0, u −1 (z 0 + θ 0 ) + v −1 (z 1 + θ 1 ) + w −1 (z t + θ t ) = 0.(95) We put κ 1 := −(z 0 + θ 0 + z 1 + θ 1 + z t + θ t ), κ 2 := z 0 + z 1 + z t . Then it holds that A ∞ = diag[κ 1 , κ 2 ] and κ 1 + κ 2 + θ 0 + θ 1 + θ t = 0. From the first equation of (95), we find that the (1, 2)-entry of A(x, t) is of the form p(x) x(x − 1)(x − t) for some linear polynomial p(x) in x. Explicitly, p(x) is expressible as p(x) = k(x − y) with k = (t + 1)uz 0 + tvz 1 + wz t , y = k −1 tuz 0 . We definez := z 0 + θ 0 y + z 1 + θ 1 y − 1 + z t + θ t y − t ,z :=z − θ 0 y − θ 1 y − 1 − θ t y − t .(99) It is underlined thatz andz are given by the values at x = y of the (1, 1)-entry and (2, 2)-entry of A(x, t) respectively, namelyz = A 11 (y, t) andz = −A 22 (y, t). From the Changing the variables (x 1 , x 2 , x 3 ) to (z 1 , z 2 , z 3 ) and then setting x = (z −z 1 )/(z 2 −z 1 ), t = (z 3 − z 1 )/(z 2 − z 1 ), we find that (109) is changed into the form of (92) in Fact 2. Therefore we obtain a solution y = y(t) to the Painlevé VI equation with α = 1 2 (λ 1 − λ 2 − 1) 2 , β = − 1 2 (r 1 + λ 3 ) 2 , γ = 1 2 (r 2 + λ 3 ) 2 , δ = 1 2 (1 − (r 3 + λ 3 ) 2 ). Next we explain how to construct an Okubo system in several variables of rank 3 from a solution to the Painlevé VI equation. We take a solution y = y(t) to the Painlevé VI equation with parameters {θ 0 , θ 1 , θ t , θ ∞ }. From this y, we can constructz =z(t) and k = k(t) using (105) and (107) respectively. (We may freely take the integral constant of k.) Then, {y,z, k} constructed in this way satisfy the equations (105), (106), (107), and thus we obtain a completely integrable Pfaffian system (92) of rank 2 using (100), (101), (102), (103). Changing the variables x, t to z, z 1 , z 2 , z 3 by x = (z − z 1 )/(z 2 − z 1 ), t = (z 3 −z 1 )/(z 2 −z 1 ) and applying the procedure in Appendix B, we obtain an Okubo system in several variables of rank 3. B Okubo systems of rank n and Pfaffian systems of rank n − 1 In this Appendix B, we explain a method of constructing an Okubo system in several variables of rank n from a completely integrable Pfaffian system of rank (n − 1). For this purpose we first treat an Okubo system in several variables of rank n We construct a completely integrable Pfaffian system of rank (n − 1) from (110). From Remark 2.1, we can change the eigenvalues {λ 1 , . . . , λ n } of B ∞ such that one of them is equal to 0 and thus, we suppose λ n = 0 without loss of generality. Then the Okubo system (110) turns out to be reducible and we obtain the rank n − 1 Pfaffian system satisfied by Z = (y 1 , . . . , y n−1 ) t that consists of the first (n − 1) entries of Y : dY = B (z) dz + n i=1 B (i) dx i Y,(110)dZ = Γ (z) dz + n i=1 Γ (i) dx i Z,(111) where Find vectors b n,1 · · · b n,n and a 1,n · · · a n,n t so that    b 11 · · · b 1n . . . . . . . . . b n,1 · · · b n,n       a 11 · · · a 1,n . . . . . . . . . a n,1 · · · a n,n    = I n . We put P =    b 11 · · · b 1n . . . . . . . . . b n1 · · · b nn    , P −1 =    a 11 · · · a 1n . . . . . . . . . a n1 · · · a nn    . Let B (z) = − n j=1 1 z − z j    b 1j . . . b nj    a j1 · · · a jn B ∞ , where B ∞ = diag[λ 1 + λ, . . . , λ n−1 + λ, λ] for λ ∈ C. Then we see that the system of linear differential equations dY dz = B (z) Y(113) is an Okubo system since it holds that P −1 B (z) P = −diag[z − z 1 , . . . , z − z n ] −1 P −1 B ∞ P. Finally we obtain an Okubo system in several variables by extending (113) to an integrable Pfaffian system following Lemma 2.1. j ) ≤ 1 and trace (B (z) j ) = r j , we have (B (z) Lemma 2. 2 . 2If T,B (i) (i = 1, . . . , n), B ∞ in (19) satisfy the equations [T,B (i) ] = O, [B (i) ,B (j) ] = O (∀i, j), (20) ∂T ∂x i +B (i) + [B (i) , B ∞ ] = O, i = 1, . . . , n, Proposition 4 . 3 . 43The meromorphic connection ∇ is integrable if and only if T , B ∞ and B (i) (i = 1, . . . , n) are subject to the following relations Remark 4. 3 . 3The meromorphic connection 1l t l , . . . , n l=1 d nl t l 1≤j≤n arises from a Saito structure on U if there is a Saito structure on U such that there is a change of independent variables (t 1 , . . . , t n ) = (t 1 (x), . . . , t n (x)), where (t 1 , . . . , t n ) is a flat coordinate system, and the matrices T , B ∞ ,B (i) (i = 1, . . . , n) defined (in Section 4) from the Saito structure satisfy T = T , B ∞ = B ∞ − (λ n − 1)I n ,B (i) = n j=1 ∂x j ∂t iB (j) (i = 1, . . . , n). Lemma 5. 2 . 2An Okubo system in several variables (58) arises from a Saito structure on W if and only if P nj = 0 (j = 1, . . . , n) at any point on W . Proof. The lemma is proved along the line stated in [51, 4.a p.242]. Let T W be the tangent bundle on W , C n (W ) be a trivial bundle of rank n on W and (e 1 , . . . , e n ) t be a basis of C n (W ). Define endomorphisms F, F (k) (k = 1, . . . , n) of C n (W ) respectively by F (e i ) = − n j=1T ij e j , F (k) (e i 0) denotes the connection on C n (W ) defined by △ (0) (e i ) = 0 (i = 1, . . . , n). Then it is confirmed that ( △ , Φ, e, E) satisfies the axiom of Saito structure on W . any point on W , (ii) P nj = 0 (j = 1, . . . , n) at any point on W . any point on U. ( ii ) iiFor the divisor D = {det(−T ) = 0}, there are logarithmic vector fields along D V 1 , . . . , V n such that, for any solution Y 0 = (y 1 , . . . , y n ) t to (61), y j = V n+1−j y n , j = 1, . . . , n hold and (det(−T (i) ⇒ (ii) In virtue of Theorem 5.4, we can take flat coordinates (t 1 , . . . , t n ) = (C n1 , . . . , C nn ) as independent variables of (61). Then it holds that M = T −1 B ∞ sinceB(i) nj = ∂C nj /∂t i = δ ij . For M V := M −1 and (V n , . . . , V 1 ) t := M V (∂ t 1 , . . . , ∂ tn ) t , V 1 , . . . , V n are logarithmic vector fields along D and it holds that 1 , . . . , ∂y (−) n ∂tn ) t and M (−) = T −1 (B ∞ − I n ). Then we have dY ′(−) 0 Lemma 6. 1 . 1H(x, z) := h(x 1 , . . . , x n−1 , x n + z) satisfies the assumption (A3). any logarithmic vector field along D. Then there are weighted homogeneous polynomials a generating C[u] G and satisfying the following conditions:(i) Let t i = F f l i (u), and let h(t) = t n n − s 1 (t ′ )t n−1 n + . . . be the defining function of D in this coordinates t = (t 1 , t 2 , . . . , t n ). Let V Lemma 6. 5 . 5(i) The entries of MṼ are polynomials in u,Ṽ k (1 ≤ k ≤ n) are logarithmic vector fields alongD, andD is free.(ii) For any k ∈ {1, 2, . . . , n}, all entries of (Ṽ k MṼ ) M −1 V are G-invariant polynomials in u, that is, polynomials in x. ǫ = 0 modifying the proof of Lemma 2.1. Similar to the equation(13), it holds that where B (z) = −(zI n − T ) −1 B ∞ , B (i) = −(zI n − T ) −1B(i) B ∞ , and T, B ∞ ,B (i) are n × n matrices that satisfy the assumptions (A1)-(A5) and the equations (20),(21),(22) in Section 2. In this Appendix A, we show the equivalence between Okubo systems in several variables (85) and isomonodromic deformations of Okubo systems (Lemma A.2). where a i (i = 1, . . . , n) are mutually distinct complex numbers andB i (i = 1, . . . , n) are m × m constant matrices. Put B ∞ := − n i=1 B i . We assume that B i (i = 1, . . . , n) are semisimple and B ∞ is diagonal, namely there are invertible matrices G i (i = 1, . . . , n) such that G i B i G ∈ Z \ {0} for i = 1, .. . , n, ∞, j, k = 1, . . . , m. Take a fundamental system of solutions Y (∞) = Y (∞) (z) to (86) normalized around z = ∞ as follows: where Y = (y 1 , . . . , y n ) t ,B (z) = −(zI n − T ) −1 B ∞ , B (i) = −(zI n − T ) −1B(i) B ∞ , i = 1, . . . , n. Proof. See [51, (2.6) on p.203].Remark 4.2. In virtue of Lemma 2.2, the relations From now on, we consider (58) on U not on W ⊂ U \ ∆ H .∂T n1 ∂x 1 · · · ∂Tnn ∂x 1 . . . ∂T n1 ∂xn · · · ∂Tnn ∂xn = 0 ⇐⇒ n j=1 P nj = 0. Theorem 5.4. An Okubo system in several variables (58) arises from a Saito structure on U if and only if ∂T n1 ∂x 1 · · · ∂Tnn ∂x 1 . . . ∂T n1 ∂xn · · · ∂Tnn ∂xn = 0 (59) Remark 7.1. The prepotential F for the icosahedral solution (H 3 ) was firstly obtained by B. Dubrovin[16]. The above algebraic solutions including the remaining two cases (H 3 ) ′ , (H 3 ) ′′ were treated by B. Dubrovin and M. Mazzocco[18]. The icosahedral solution (H 3 ) is constructed by the polynomial F (cf. (84)). In the remaining two cases, the prepotentials are not polynomials but algebraic functions. The authors were informed by B. Dubrovin that his student Alejo Keuroghlanian computed the algebraic Frobenius manifold for the case of the great icosahedron (H 3 ) ′ in his master thesis "Varieta di Frobenius algebraiche di dimensione 3". The authors don't know whether the potential for (H 3 ) ′′ is known or not. Topics on these solutions are treated in[30].7.2 Algebraic solution related with the complex reflection group ST24 ([7]) 7.3 Algebraic solutions related with the complex reflection group ST27 (cf. [58]) Acknowledgments. Professor Yoshishige Haraoka taught the first author (M.K.) that integrable systems in three variables are useful to derive the Painlevé VI solutions. This is the starting point of our work. The authors would like to thank Professor Haraoka for his advice. After a preprint of this article was written, the authors were informed with useful comments by Professors B. Dubrovin, Y. Konishi, C. Hertling, P. Boalch, A. Arsie and P. Lorenzoni the existence of the papers[17,12,1,2,43,44,6]. The authors express their sincere gratitude to these people. This work was supported by JSPS KAKENHI Grant Numbers 25800082, 17K05335, 26400111, 17K05269.The following case is related with the complex reflection group ST24.Klein solution of P. Boalch[7](LT8) In this case,, w(t 2 ) = 3 7 , w(t 3 ) = 1 g 1 = (−2t 3 1 t 2 + t 3 2 + 12t 1 t 3 )/12, g 2 = (2t51 + 5t 2 1 t 2 2 + 10t 2 t 3 )/10, g 3 = (−8t71 + 21t 4 1 t 2 2 + 7t 1 t 4 2 + 28t 2 3 )/56. The determinant det(−T ) is regarded as the discriminant of the complex reflection group ST24 if t 1 , t 2 , t 3 are taken as basic invariants (cf.[58]). equations (95),(96),(98),(99), we obtain the following equalities:where we put θ ∞ := κ 1 − κ 2 . These equalities imply that {u, v, w, z 0 , z 1 , z t } and thus all the entries of A 0 , A 1 , A t are expressed in terms of {y,z, k, θ 0 , θ 1 , θ t , κ 1 , κ 2 , t}. Then, the Schlesinger system (which is nothing but the integrability condition of (92))is equivalent to the following system of differential equations(We note that, eliminatingz from(105)and(106), we obtain the Painlevé VI equation.) Now we shall prove Proposition A.1.Proof of Proposition A.1. We first explain how to construct a solution of the Painlevé VI equation from an Okubo system in several variables (85) with n = 3. By the procedure explained in Appendix B, we obtain a completely integrable Pfaffian system of rank 2 replacing B ∞ = diag[λ 1 , λ 2 , λ 3 ] to diag[λ 1 − λ 3 , λ 2 − λ 3 , 0]:. . , n.The system(111)is also completely integrable. The aim of this Appendix B is to explain a method of constructing an Okubo system in several variables of rank n from a system of the form (111). We start from a completely integrable Pfaffian system of rank (n − 1)where Γ (z) and Γ (i) are (n − 1) × (n − 1) matrices depending on (z, x). We assume the following conditions:(B1) There is a monic polynomial H(x, z) in z of degree n and analytic in x such that the entries of H(z, x)Γ (z) and H(x, z)Γ (i) are polynomials in z and holomorphic in x on U ⊂ C n . The degree of H(z, x)Γ (z) with respect to z is at most n−1 and the discriminant) does not identically vanish. 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[ "ENUMERATION OF IRREDUNDANT FORESTS", "ENUMERATION OF IRREDUNDANT FORESTS", "ENUMERATION OF IRREDUNDANT FORESTS", "ENUMERATION OF IRREDUNDANT FORESTS" ]	[ "Florian Ingels [email protected] \nLaboratoire Reproduction et Développement des Plantes\nUniv Lyon\nENS de Lyon\nUCB Lyon 1\n", "Romain Azaïs [email protected] \nCNRS\nINRAE\nF-69342LyonInriaFrance\n", "Florian Ingels [email protected] \nLaboratoire Reproduction et Développement des Plantes\nUniv Lyon\nENS de Lyon\nUCB Lyon 1\n", "Romain Azaïs [email protected] \nCNRS\nINRAE\nF-69342LyonInriaFrance\n" ]	[ "Laboratoire Reproduction et Développement des Plantes\nUniv Lyon\nENS de Lyon\nUCB Lyon 1", "CNRS\nINRAE\nF-69342LyonInriaFrance", "Laboratoire Reproduction et Développement des Plantes\nUniv Lyon\nENS de Lyon\nUCB Lyon 1", "CNRS\nINRAE\nF-69342LyonInriaFrance" ]	[]	Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees. If the literature proposes solutions to enumerate singletons of trees, we study in this article a more general problem, the enumeration of sets of trees -forests. Specifically, we mainly study irredundant forests, i.e., where no tree is a subtree of another. By compressing each such forest into a Directed Acyclic Graph (DAG), we develop a reverse search like method to enumerate DAGs compressing irredundant forests. Remarkably, we prove that these DAGs are in bijection with the row-Fishburn matrices, a well-studied class of combinatorial objects. In a second step, we derive our irredundant forest enumeration to provide algorithms for tackling related problems: (i) enumeration of forests in their classical sense (where redundancy is allowed); (ii) the enumeration of "subforests" of a forest, and (iii) the frequent "subforest" mining problem. All the methods presented in this article enumerate each item uniquely, up to isomorphism.	10.1016/j.tcs.2022.04.033	[ "https://arxiv.org/pdf/2003.08144v4.pdf" ]	248,157,416	2003.08144	21ea0d536d94e22fa9011209cb850a375d1cb315	ENUMERATION OF IRREDUNDANT FORESTS Florian Ingels [email protected] Laboratoire Reproduction et Développement des Plantes Univ Lyon ENS de Lyon UCB Lyon 1 Romain Azaïs [email protected] CNRS INRAE F-69342LyonInriaFrance ENUMERATION OF IRREDUNDANT FORESTS Directed Acyclic GraphReverse SearchUnordered TreesEnumerationForest Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees. If the literature proposes solutions to enumerate singletons of trees, we study in this article a more general problem, the enumeration of sets of trees -forests. Specifically, we mainly study irredundant forests, i.e., where no tree is a subtree of another. By compressing each such forest into a Directed Acyclic Graph (DAG), we develop a reverse search like method to enumerate DAGs compressing irredundant forests. Remarkably, we prove that these DAGs are in bijection with the row-Fishburn matrices, a well-studied class of combinatorial objects. In a second step, we derive our irredundant forest enumeration to provide algorithms for tackling related problems: (i) enumeration of forests in their classical sense (where redundancy is allowed); (ii) the enumeration of "subforests" of a forest, and (iii) the frequent "subforest" mining problem. All the methods presented in this article enumerate each item uniquely, up to isomorphism. Introduction Context of the work Enumeration of trees is a long-term problem, where Cayley was the first to propose a formula for counting unordered trees in the mid-19th century [9, I.5.2]. The exhaustive enumeration of ordered and unordered trees 1 was successfully tackled in the early 00's by Nakano and Uno in [19,20]. In the unordered case, an extension of the algorithm has been proposed to solve the problem of frequent substructure mining [1]. Moreover, in the field of machine learning, we have recently demonstrated that exhaustive enumeration of the subtrees of a tree makes it possible to design classification algorithms significantly more efficient than their counterpart without such enumeration [5]. Our ambition in this article is to take these two problems of enumeration -trees and subtrees -to a higher order, i.e. to enumerate sets of trees instead of singletons. Specifically, we call an irredundant forest (shortened to forest in the sequel) a set of trees that contains no repetition -in the sense that no tree is a subtree of another (see upcoming Subsection 1.2 for a precise definition). This condition of non-repetition is in line with a parsimonious enumeration approach, where the objects considered are all different and enumerated up to isomorphism. Besides, this condition is not restrictive since one can always introduce repetition afterwards. We are therefore interested in the problem of enumerating forests of unordered trees, and then, given a tree or forest, to enumerate all its "subforests" -as forests of subtrees. The latter has already been discussed in the literature, but without consideration on isomorphism [22]. We re-emphasize that we aim to enumerate these various items -forests and subforests -up to isomorphism. Such an ambition immediately raises a number of obstacles. First of all, the trees are indeed unordered, but so are the sets of trees. For the former the literature has introduced the notion of the canonical form of a tree [20,1], which is a unique ordered representation of an unordered tree. The enumeration therefore focuses only on these canonical trees. Unfortunately, if it is possible to order a set of vertices, there is no total order on the set of trees, to the best of our knowledge. In addition, the condition of non-repetition filters the set of forests in a non-trivial way, making, a priori, the enumeration problem trickier. Enumeration problems are recurrent in many fields, notably combinatorial optimization and data mining. They involve the exhaustive listing of a subset of the elements of a search set (possibly all of them), e.g. graphs, trees or vertices of a simplex. Given the possibly high combinatorial nature of these elements, it is essential to adopt clever exploration strategies as opposed to brute-force enumeration, typically to avoid areas of the search set not belonging to the objective subset. One proven way of proceeding is to provide the search set with an enumeration tree structure; starting from the root, the branches of the tree are explored recursively, eliminating those that do not address the problem. Based on this principle, we can notably mention the well-known "branch and bound" method in combinatorial optimization [18] and the gSpan algorithm for frequent subgraph mining in data mining [26]. Another of these methods is the so-called reverse search technique, which requires that the search set has a partial order structure, and which has solved a large number of enumeration problems since its introduction [2] until very recently [25]. Actually, the algorithms previously introduced in the literature to enumerate trees are based on this technique [19,20,1]. In the present paper, we restrict ourselves to reverse search methods, for which the following formalism is adapted from the one that can be found in [21, p. 45-51], and slightly differ from the original definition by Avis and Fukuda [2]. We refer the reader to these two references for further details. Let (S, ⊆) be a partially ordered set, and g : S → { , ⊥} be a property, satisfying anti-monotonicity ∀s, t ∈ S : (s ⊆ t) ∧ g(t) =⇒ g(s). The enumeration problem for the property g is the problem of listing all elements of E S (g) = {s ∈ S : g(s) = }. An enumeration algorithm is an algorithm that returns E S (g). The reverse search technique relies on inverting a reduction rule f : S \ ∅ → S, where f satisfies the two properties of (i) covering: ∀s ∈ S \ ∅, f(s) ⊂ s and (ii) finiteness: ∀s ∈ S \ ∅, ∃k ∈ N * , f k (s) = ∅. Then, the expansion rule is defined as f −1 (t) = {s ∈ S : f(s) = t}. This defines an enumeration tree rooted in ∅, and repeated call to f −1 can therefore enumerates all the elements of S. The reverse search algorithm is shown in Algorithm 1. E S (g) can be obtained from the call of REVERSESEARCH((S, ⊆), f −1 , g, ∅). As g is anti-monotone, if g(s) = ⊥, then all elements s ⊆ t also have g(t) = ⊥, and thefore pruning the enumeration tree in s does not miss any element of E S (g). Algorithm 1: REVERSESEARCH Input: (S, ⊆), f −1 , g, s 0 ∈ S -s.t. g(s 0 ) = 1 output s 0 2 for t ∈ {s ∈ f −1 (s 0 )\|g(s) = } do 3 REVERSESEARCH((S, ⊆), f −1 , g, t) When successfully designed, a reverse search technique should yield polynomial output delay [15,2], i.e., the time between the output of one element and the next is bounded by a polynomial function in the size of the input. Remark 1.1. It would have been possible to define directly the set S as the set of elements verifying the property g. Separating the two induces that the reduction rule f formally depends only on S, and not on g. This allows, once f is constructed once and for all, to filter S according to various properties g without additional work. In particular, this is useful in the case where g depends on a tunable parameter -as in the frequent pattern mining problem introduced in Section 6. Precise formulation of the problem A rooted tree T is a connected graph with no cycle such that there exists a unique vertex called the root, which has no parent, and any vertex different from the root has exactly one parent. Rooted trees are said unordered if the order between the sibling vertices of any vertex is not significant. As such, the set of children of a vertex v is considered as a multiset and denoted by C(v). The leaves L(T ) are all the vertices without children. The height of a vertex v of a tree T can be recursively defined as H(v) = 0 if v ∈ L(T ), 1 + max u∈C(v) H(u) otherwise.(1) The height H(T ) of the tree T is defined as the height of its root. The outdegree of a vertex v ∈ T is defined as deg(v) = # C(v) 2 ; the outdegree of T is then defined as deg(T ) = max v∈T deg(v). The depth of a vertex v is the number of edges on the path from v to the root of the tree. Two trees T 1 and T 2 are isomorphic if there exists a one-to-one correspondance φ between the vertices of the trees such that (i) u ∈ C(v) in T 1 ⇐⇒ φ(u) ∈ C(φ(v)) in T 2 and (ii) the roots are mapped together. For any vertex v of T , the subtree T [v] rooted in v is the tree composed of v and all its descendants -denoted by D(v). S(T ) denotes the set of all distinct subtrees of T , which is the quotient set of {T [v] : v ∈ T } by the tree isomorphism relation. In this article, we consider only unordered rooted trees that will simply be called trees in the sequel. We denote by T the set of all trees. As mentioned before, we are interested in this paper in the enumeration of forests. The literature acknowledges two definitions for a forest [6, p. 172]: (i) an undirected graph in which any two vertices are connected by at most one path or (ii) a disjoint union of trees. We adopt a variation of the latter one, that forbids repetitions inside the forest. ∀i = j, T i / ∈ S(T j ).(2) We denote by F the set of all irredundant forests -shortened to forests in the sequel of the paper. Our goal is to provide a reverse search method that outputs F. As already stated, this goal raises two major difficulties: firstly, the twofold unordered nature of forests (the set of trees and the trees themselves), and secondly, the non-trivial condition of non-repetition. While the latter problem is intrinsic, the main idea of this paper to address the former is to resort to the reduction of a forest into a Directed Acyclic Graph (DAG). Figure 1: A forest F (left) and its DAG reduction (right). Roots of isomorphic subtrees are identically colored, as well as the corresponding vertex of the DAG. The sources of the DAG (indicated with red arrows) correspond exactly to the roots of the trees in F. For the sake of clarity, arcs of multiplicity greater than one are drawn only once and their multiplicity is written next to the arc. T 1 T 2 T 3 F = {T 1 , T 2 , T 3 } 2 3 2 R(F) DAG reduction is a method meant to eliminate internal repetitions in the structure of trees and forests of trees. Beginning with [23], DAG representations of trees are also much used in computer graphics where the process of condensing a tree into a graph is called object instancing [12]. A precise definition of DAG reduction of trees, together with algorithms to compute it, are provided in [10], whereas one technique to extend those algorithms to forests is presented in [5,Section 3.2]. DAG reduction can be interpreted as the construction of the quotient graph of a forest by the tree isomorphism relation. However, in this paper, we provide the general idea of DAG reduction as a vertex coloring procedure. Consider a forest F = {T 1 , . . . , T n } to reduce. Each vertex of each tree is given a color such that if two distinct vertices u, v belonging respectively to T i , T j (not necessarily distinct) have the same color, then T i [u] and T j [v] are isomorphic. Reciprocally, if two subtrees are isomorphic, their roots have to be identically colored. Let us denote c(·) the function that associates a color to any vertex. Then, we build a directed graph D = (V, A) with as many vertices as colors used, i.e. #V = #Im (c). For any two vertices u, v in the forest, if u ∈ C(v), then we create an arc c(v) → c(u) in D. Note that this definition implies that multiples arcs are possible in D, as if there exist u, u ∈ C(v), for v ∈ T , such that T [u] and T [u ] are isomorphic, then the arcs c(v) → c(u) and c(v) → c(u ) are identical. The graph D is a DAG [10, Proposition 1], i.e. a connected directed (multi)graph without cycles. We refer to Figure 1 for an example of DAG reduction. In this paper, R(F) denotes the DAG reduction of F. It is crucial to notice that the function R is a one-to-one correspondence [10,Proposition 4], which means that DAG reduction is a lossless compression algorithm. Since F fulfills condition (2), no tree of F is a subtree of another. If this were the case, say T i ∈ S(T j ), then R(T i ) would be a subDAG of R(T j ), and therefore the numbers of roots in R(F) would be strictly less than #F. Since such a situation can not occur, there are exactly as many roots in R(F) as there are elements in F: no information is lost. In other words, F can be reconstructed from R(F) and R −1 stands for the inverse function. The DAG structure inherits of some properties of trees. For a vertex v in a DAG D, we will denote by C ( In the sequel, DAGs compressing forests are called FDAGs, to distinguish them from general directed acyclic graphs. Since DAG compression is lossless, and since a forest can be reconstructed from its DAG reduction, it should be clear that enumerating all forests is equivalent to enumerating all FDAGs. Yet, the latter approach has the merit of transforming set of trees into unique objects, which makes it possible, if able to design a canonical representation -like the trees in [20,1], to get rid of the twofold unordered nature of forests, as claimed earlier. Indeed, any ordering of the vertices of the DAG induces an order on the roots of the DAG, and therefore on the elements of the forest, as well on the vertices of the trees themselves. Aim of the paper To the best of our knowledge, the enumeration of DAGs has never been considered in the literature. The aim of this article is twofold, i.e (i) to open the way by presenting a reverse search algorithm enumerating FDAGs, in Section 2, and (ii) to derive from it an algorithm for enumerating substructures in Section 5. The frequent pattern mining problem is a classical data mining problem -see [11] for a survey on that question -and we provide in Section 6 a slight variation of the algorithm of Section 5 to tackle this issue. In addition, Section 3 analyses the growth of the enumeration tree defined in Section 2, while Section 4 proposes two variations of it. In more detail, our outline is as follows: • The first step is to introduce a canonical form for FDAG. For trees [20,Section 3], this consisted in associating an integer (its depth) to each vertex, and maximizing the sequence by choosing an appropriate ordering over the vertices. The notion of depth does not apply to FDAGs, which forces us to find another strategy. DAGs are characterized by the existence of a topological ordering [17], and we introduce in Subsection 2.2 a topological ordering that is unique if and only if a DAG compresses a forest. This canonical ordering is defined so that the sequence of children of the vertices is strictly increasing, where the multisets of children are ordered by the lexicographical order. In fact, these ordered multisets of children are considered as formal words, which brings us to a detour through the theory of formal languages in Subsection 2.1 to introduce useful results for the rest of the article. Compared to trees, we have here a first gain in complexity insofar as we maximize a sequence of words instead of a sequence of integers. • The expansion rule used for trees [20,Section 4] is to add a new vertex in the tree as a child of some other vertex, so that the depth-sequence remains maximal. Consequently, a single arc is also added. On the other hand, for a FDAG, we want to be able to add either vertices or arcs independently. In Subsection 2.3, we define three expansion rules, reflecting the full spectrum of possible operations, so that the DAG obtained afterward is still a FDAG. Specifically, the branching rule allows to add an arc, where the elongation and widening rules add vertices at different height. We show in Proposition 2.10 that the rules preserve the canonicalness and in Proposition 2.11 that they are "bijective": any FDAG can be reached by applying the expansion rules to a unique FDAG. In Subsection 2.5 we derive from them an enumeration tree covering the set of FDAGs. • Notably, a bijection between FDAGs and row-Fishburn matrices, a class of combinatorial objects much exploited in the literature [13,Section 2], is shown in Theorem 3.1 -which proof lies in Appendix A. The asymptotic behavior of these matrices being well known [14,7], this allows us to derive from it the behavior of the enumeration tree. In return, since our bijection is constructive, the enumeration tree can be used to enumerate row-Fishburn matrices -and all the objects they are in bijection with -via the reverse search method. Remarkably, this bijection operates between two objects that, at first sight, have little in common. • For an enumeration algorithm to have any practical interest, it is necessary that the associated enumeration tree has a "reasonable" growth -with regard to the size of the explored space. This is the case for our algorithm since we prove, in Subsection 3.2, that a FDAG with n vertices has a number of successors in the enumeration tree in the order of Θ(n) -and that those successors can be computed in quasi-quadratic time. We also show, in Theorem 3.6, that our algorithm runs with polynomial delay [15]. • Subsection 4.1 introduces a way of enumerating forests in their classical definition, i.e., with redundancy, where some trees may be equal to or subtrees of others. The proposed method takes a redundancy-free forest, as enumerated by our algorithm, and adds repetition in an extra enumeration step. Finally, Subsection 4.2 concludes on enumeration by proposing sets of constraints that make the enumeration tree finite. Indeed, since the rules only allow to increase the height, degree or number of vertices, it is sufficient to set maximum values for some of these parameters to achieve this goal; however the combination of parameters has to be wisely chosen, as we show it. • Since the structures we enumerate are forests, it is natural that the substructures we are interested in are "subforests". A precise definition of the latter is given in Section 5, i.e. forests of subtrees, and are referred to as subFDAGs. An algorithm to enumerate all subFDAGs appearing in a FDAG is also provided. The frequent subFDAG mining problem is finally addressed in Section 6. Concluding remarks concerning the implementation of our results in the Python library treex [3] are briefly mentioned at the end of the article. In Appendix B, the interested reader will find an index of frequent notations used throughout the paper. Exhaustive enumeration of FDAGs In this section, we introduce our main result, that is, a reverse search algorithm for the enumeration of FDAGs. As we will consider the multisets of children of vertices as formal words built on the alphabet formed by the set of vertices, we introduce in Subsection 2.1 some definitions and results on formal languages that will be useful for the sequel. We characterize unambiguously in Subsection 2.2 our objects of study, through the lens of topological orderings, defining a canonical topological ordering for DAGs, that is unique if and only if a DAG compresses a forest of unordered trees, i.e. it is a FDAG -see Theorem 2.4. We then define three expansion rules that are meant to extend the structures of FDAGs in Subsection 2.3, and we study their properties in Subsection 2.4. In Subsection 2.5, we show with Theorem 2.12 that these expansion rules define an enumeration tree on the set of FDAGs. Preliminary: a detour through formal languages We present in this subsection some definitions and results on formal languages that will be useful for the sequel of Section 2. Let A be a totally ordered finite set, called alphabet, whose elements are called letters. A word is a finite sequence of letters of A. The length of a word w is equal to its number of letters and is denoted by #w. There is a unique word with no letter called the empty word and denoted by . The set of all words is denoted by A * . Words can be concatenated to create a new word whose length is the sum of the lengths of the original words; is the neutral element of this concatenation operation. The lexicographical order over A * , denoted by < lex. is defined as follows. Let w 1 = a 0 · · · a p and w 2 = b 0 · · · b q be two words, with a i , b j ∈ A. If #w 1 = #w 2 , then w 1 < lex. w 2 if and only if ∃k ∈ [[0, p]], a i = b i ∀i < k and a k < b k . Otherwise, let m = min(p, q); w 1 < lex. w 2 if and only if either (i) a 0 · · · a m < lex. b 0 · · · b m or (ii) a 0 · · · a m = lex. b 0 · · · b m and m < q -that is, p < q. Note that, by convention, < lex. w for any word w. Let w ∈ A * . We define the suffix-cut operator SC(w), which removes the last letter of w: SC(w) = w if w = w a with a ∈ A and w ∈ A * , otherwise.(3) A language is a set of words satisfying some construction rules. We introduce hereafter two languages that will be useful in the sequel of the paper. Definition 2.1. The language of decreasing words is defined as Λ = w = a 0 · · · a m ∈ A * : a i ≥ lex. a i+1 ∀i ∈ [[0, m − 1]] . Definition 2.2. Let w ∈ Λ. The language of decreasing words bounded by w is defined as Λ w = {w ∈ Λ : w > lex. w} . Any word w ∈ Λ w is said to be minimal if and only if w ∈ Λ w but SC(w) / ∈ Λ w . As an example, if A = {0, 1, 2, 3}, then w = 211 ∈ Λ, whereas 121 / ∈ Λ. In addition, Λ w contains words such as 31, 22, 21110, etc. 22 is a minimal word of Λ w as 22 > lex. 211 but SC(22) = 2 < lex. 211. Our focus is now on the construction of the minimal words of Λ w . Let w = a 0 · · · a p and w = b 0 · · · b q ∈ Λ w . Taking into account that w > lex. w and that they both are decreasing words, there are only two possibles cases: (i) w and w share a common prefix a 0 · · · a m . Then w = a 0 · · · a m b m+1 · · · b q , and the word a 0 · · · a m b m+1 is minimal by applying successive suffix-cut operations. (ii) w and w do not share a common prefix. Necessarily b 0 > lex. a 0 , and then the word b 0 is minimal by applying several suffix-cut operations. From the above, we deduce a method for constructing all minimal word of Λ w . First, we partition A into disjoint -potentially empty -subsets: A 0 = {a ∈ A : a > lex. a 0 }, A i = {a ∈ A : a i−1 ≥ lex. a > lex. a i } 1 ≤ i ≤ p, A p+1 = {a ∈ A : a p ≥ lex. a}. It then follows that -empty A i 's not being considered, • ∀b ∈ A 0 , the word b is minimal, • ∀b ∈ A i with i ∈ {1, . . . , p}, the word a 0 · · · a i−1 b is minimal, • ∀b ∈ A p+1 , the word wb is minimal. As we partitioned A, we have proved the following proposition. Proposition 2.3. The number of minimal words of Λ w is exactly #A. As a follow-up of the example some lines ago, with A = {0, 1, 2, 3} and w = 211, we apply the proposed method to find the minimal elements of Λ w . We partition A into: A 0 = {3}, A 1 = {2}, A 2 = ∅, A 3 = {0, 1}. The four minimal words are therefore 3, 22, 2111 and 2110. Although the previous result is completely general, if we require that A = {0, . . . , n}, then the partition method described above can be rewritten into Algorithm 2. While this is not included in the pseudocode provided, note that the algorithm should return an empty list if a 0 > n, as in this case there would be no minimal word to look for. Algorithm 2: MINIMALWORDS Input: w = a 0 · · · a p , A = {0, . . . , n} Output: All minimal words of Λ w 1 Set L to the empty list 2 if a 0 < n then 3 for i ∈ {a 0 + 1, . . . , n} do 4 Add the word i to L 5 for k ∈ {1, . . . , p} do 6 if a k < a k−1 then 7 for i ∈ {a k + 1, . . . , a k−1 } do 8 Add the word a 0 · · · a k−1 i to L 9 for i ∈ {0, . . . , a p } do 10 Add the word a 0 · · · a p i to L 11 return L Canonical FDAGs FDAGs are unordered objects, like the trees they compress, and therefore their enumeration requires to reflect this nature. In practice, finding a systematic way to order them makes it possible to design a simpler reduction rule, as done for trees [20], ignoring the combinatorics of permutations. The purpose of this subsection is to provide a unique way to order FDAGs. We show that such an order exists in Theorem 2.4, unambiguously characterizing FDAGs. The approach chosen is based on the notion of topological order. Topological ordering Let D be a directed graph, where multiple arcs are allowed. A topological ordering on D is an ordering of the vertices of D such that for every arc uv from vertex u to vertex v, u comes after v in the ordering. ψ 1 ( ) 3 2 1 0 ψ 2 ( ) 3 2 0 1 ψ 3 ( ) 2 3 0 1 ψ 4 ( ) 2 3 1 0 ψ 5 ( ) 1 3 0 2Formally, ψ : D → [[0, #D − 1] ] is a topological ordering if and only if ψ is bijective and ψ(u) > ψ(v) for all u, v ∈ D such that there exists at least one arc uv in D. A well known result establishes that D is a DAG if and only if it admits a topological ordering [17]. Nonetheless, when a topological ordering exists, it is in general not unique -see Figure 2. A reverse search enumeration of topological orderings of a given DAG can actually be found in [2, Section 3.5]. Constrained topological ordering We aim to reduce the number of possible topological orderings of a DAG by constraining them. Let D be a DAG and ψ a topological ordering. Taking advantage of the vertical hierarchy of DAG, our first constraint is ∀(u, v) ∈ D 2 , H(u) > H(v) =⇒ ψ(u) > ψ(v).(4) Applying (4) to the topological orderings presented in Figure 2, ψ 5 must be removed, as ψ 5 ( ) > ψ 5 ( ) and H( ) > H( ). For any vertex v, and any u ∈ C(v), by definition, H(v) > H(u). Therefore, there can be no arcs between vertices at same height. Any arbitrary order on them leads to a different topological ordering. The next constraint we propose relies on the lexicographical order: ∀(u, v) ∈ D 2 , H(u) = H(v) and C ψ (u) > lex. C ψ (v) =⇒ ψ(u) > ψ(v),(5) where Table 1 illustrates the behavior of (5) on the followed example of Figure 2. C ψ (v) is the list [ψ(v i ) : v i ∈ C(v)] sorted (5) C ψ 1 ( ) 11 10 C ψ 2 ( ) 00 10 C ψ 3 ( ) 00 10 C ψ 4 ( ) 11 10 Table 1: Application of (5) to the remaining topological orderings of Figure 2 that satisfy (4). As C ψ ( ) = C ψ ( ), we only need to consider vertices and . As ψ i ( ) > ψ i ( ) ⇐⇒ i ∈ {1, 2}, the only orderings that are kept are ψ 1 and ψ 3 . The combination of those two constraints imposes uniqueness in all cases except when there exists (u, v) ∈ D 2 such that C ψ (u) = C ψ (v) and u = v. It should be clear that if we impose the upcoming condition (6), such a pathological case can not occur. ∀(u, v) ∈ D 2 , u = v =⇒ C(u) = C(v)(6) Upcoming Theorem 2.4 establishes that a DAG compresses a forest if and only if the topological order constrained by (4) and (5) is unique. In other words, an unambiguous characterization of FDAGs is exhibited. Theorem 2.4. The following statements are equivalent: (i) D fulfills (6), (ii) there exists a unique topological ordering ψ of D that satisfies both constraints (4) and (5), (iii) there exists a unique forest F ∈ F -cf. (2) -such that D = R(F), where R is the DAG reduction operation defined in Subsection 1.2. Proof. (i) ⇐⇒ (ii) follows from the above discussion. (iii) =⇒ (i) follows from the definition of R. Indeed, if there was two distinct vertices (u, v) ∈ D 2 with the same multiset of children, they would have been compressed as a unique vertex in the reduction. We now prove that (i) =⇒ (iii). In the first place, if D fulfills (6), then D must admit a unique leaf, denoted by L(D). Indeed, if there were two leaves l 1 and l 2 , we would have H(l 1 ) = H(l 2 ) = 0 but also C(l 1 ) = C(l 2 ) = ∅, which would violate (6). Let r 1 , . . . , r k be the vertices in D that have no parent. We define D 1 , . . . , D k as the subDAG rooted respectively in r 1 , . . . , r k . Then, we define T i = R −1 (D i ) and F = {T 1 , . . . , T k }. The T i 's are well defined as all vertices in D (consequently in D i ) have a different multiset of children, and therefore compress distinct subtrees -i.e. F fulfills (2), therefore F ∈ F. Moreover, D = R(F). 2 3 2 v ψ(v) 0 1 2 3 4 5 C ψ (v) 0 00 000 1 211 Red arrows indicate the roots of the trees of the forest that is compressed by D. In the sequel of the article, we shall only consider FDAGs. Consequently, from Proposition 2.4, they admit a unique topological ordering ψ satisfying both constraints (4) and (5), called canonical ordering. Thus, for any FDAG D, the associated canonical ordering ψ will be implicitly defined. The vertices will be numbered accordingly to their ordering, i.e D = (v 0 , . . . , v n ) with ψ(v i ) = i. Finally, as a consequence of constraints (4) and (5), note that D can be partitioned in subsets of vertices with same height, each of them containing only consecutive numbered vertices. Figure 3 provides an example of a FDAG and its canonical ordering. Expansion rules Reverse search techniques implies finding reduction rules, and then inverse them. Equally, we will define instead three expansion rules, of which inverse will be reduction rules. An expansion rule takes a FDAG and create a new DAG, that is "expanded" in the sense of having either more vertices or more arcs. Our rules are analysed at the end of the subsection, where notably we prove in Proposition 2.10 that expansion rules preserve the canonicalness. Moreover, we show in Proposition 2.11 that they are "bijective": any FDAG is in the image of a unique FDAG through the expansion rules. We begin with a preliminary definition. Definition 2.5. Let D be a FDAG, with D = (v 0 , . . . , v n ). We define the two following alphabets A = = {ψ(v) : v ∈ D, H(v) = H(v n )} = {p + 1, . . . , n}, A < = {ψ(v) : v ∈ D, H(v) < H(v n )} = {0, . . . , p}, where p ∈ [[0, n − 1]] and ψ(·) is the canonical ordering of D. In other words, A = contains the indices of all vertices that have the same height as the vertex with the highest index according to ψ, and A < the indices of all vertices that have an inferior height. The FDAG presented in Figure 3 will serve as a guideline example all along this subsection. Here, we have A = = {4, 5} and A < = {0, 1, 2, 3}. The three expansion rules are now introduced. Let D = (v 0 , . . . , v n ). Each of these rules is associated with an explicit symbol, which may be used, when necessary, to designate the rule afterward. It is worth noting that all of these rules will operate according to the vertex of highest index, v n . Branching rule This rule adds an arc between v n and a vertex below. The end vertex of the new arc is chosen such that C ψ (v n ) remains a decreasing word. In Figure 4, is applied on our guideline example. Definition 2.6. Let C ψ (v n ) = a 0 · · · a m . Choose a m+1 ∈ A < such that a m ≥ lex. a m+1 and add an arc between ψ −1 (a m+1 ) and v n . 2 3 3 v . . . ψ(v) . . . 5 C ψ (v) . . . 2111 (a) 2 3 2 v . . . ψ(v) . . . 5 C ψ (v) . . . 2110 (b) Elongation rule This rule adds a new vertex v n+1 such that H(v n+1 ) = H(v n )+1. Consequently, the alphabets change and become A = = {n + 1} and A < = {0, . . . , n}. Note that after using this rule, it is not possible to ever add a new vertex at height H(v n ). See Figure 5 for an illustration of this rule on the guideline example. Definition 2.7. Add new vertex v n+1 such that C ψ (v n+1 ) = a 0 ∈ A = . Widening rule This rule adds a new vertex v n+1 at height H(v n ). The vertex is added with children that respects the canonicalness of the DAG, that is, such that C ψ (v n+1 ) > lex. C ψ (v n ) -as in condition (5). In other terms, denoting Λ < the language of decreasing words on alphabet A < , and with w = C ψ (v n ), C ψ (v n+1 ) must be chosen in Λ w < -see Definition 2.2. However, this set is infinite, so we restrict C ψ (v n+1 ) to be chosen among the minimal words of Λ w < . It follows from the definition of suffix-cut operator SC(·) that, by inverting the said operator, the other words in Λ w < can be obtained by performing repeated operations. Finally, this new vertex is added to A = . Add new vertex v n+1 such that 2 3 2 v . . . ψ(v) . . . 5 6 C ψ (v) . . . 211 4 (a) 2 3 2 v . . . ψ(v) . . . 5 6 C ψ (v) . . . 211 5 (b)C ψ (v n+1 ) ∈ w ∈ Λ w < : w is a minimal word of Λ w < with w = C ψ (v n ). From Proposition 2.3 we now that such minimal words exist. We prove in the upcoming lemma that, as claimed, H(v n+1 ) = H(v n ). Lemma 2.9. Any element of Λ w < defines a new vertex v n+1 such that H(v n+1 ) = H(v n ). Proof. From the definition of H(·) -(1), it suffices to prove that v n+1 admits at least one child at height h = H(v n ) − 1. Let us denote b 0 and a 0 the first letter of, respectively, C ψ (v n+1 ) and C ψ (v n ). Denoting v = ψ −1 (b 0 ) and u = ψ −1 (a 0 ), we already know that H(u) = h -as ψ respects (5) and C ψ (v n ) is a decreasing word. Therefore, as by construction C ψ (v n+1 ) > lex. C ψ (v n ), either (i) b 0 = a 0 and therefore v = u, either (ii) b 0 > lex. a 0 . In the latter, as ψ respects (4) and (5), H(v) ≥ H(u) = h. But, as b 0 ∈ A < , H(v) < H(v n ) = h + 1. In both cases, H(v) = h. Figure 6 illustrates the use of the widening rule on the followed example. It should be noted that the possible outcomes of are obtained by using Algorithm 2, applied to w = C ψ (v n ) and pwith A < = {0, . . . , p}. 2 3 2 v . . . ψ(v) . . . 5 6 C ψ (v) . . . 211 3 (a) 2 3 2 2 v . . . ψ(v) . . . 5 6 C ψ (v) . . . 211 22 (b) 2 3 2 3 v . . . ψ(v) . . . 5 6 C ψ (v) . . . 211 2111 (c) 2 3 2 2 v . . . ψ(v) . . . 5 6 C ψ (v) . . . 211 2110 (d) Analysis of the rules Since our goal is to enumerate FDAGs, it is required that the expansion rules indeed construct FDAGs. This is achieved by virtue of the following proposition. Let a be the letter added to w = C ψ (v n ). As wa > lex. w > lex. C ψ (v n−1 ), the ordering is unchanged. (5) is also still met. The new vertex v n+1 is such that H(v n+1 ) > H(v n ), so condition (4) is still met. The new vertex v n+1 is chosen so that H(v n+1 ) = H(v n ) and C ψ (v n+1 ) > lex. C ψ (v n ), so condition Therefore, any DAG obtained from D is still a FDAG. Secondly, since our goal is to provide the FDAGs space with an enumeration tree, which will be explored via the expansion rules, it is important that these expansion rules are "bijective" in the following sense: for any FDAG D, there exists a unique FDAG D such that D is obtained from D via one of the three rules , or . Such D can be constructed via Algorithm 3 as shown in upcoming Proposition 2.11. Conditional expressions applied to D are used to determine which modification should be applied to construct D . The gray symbol (in the algorithm) next to these modifications indicates which expansion rule allows to retrieve D from D . Proposition 2.11. Algorithm 3 applied to any FDAG constructs the unique antecedent of this FDAG. Proof. Let D = (v 0 , . . . , v n ) be a FDAG. Let w = C ψ (v n ) and w = C ψ (v n−1 ). Two cases can occur: (i) either v n is the only vertex at height H(v n ), (ii) or it is not. Algorithm 3: ANTECEDENT Input: D = (v 0 , . . . , v n ); w = C ψ (v n ); w = C ψ (v n−1 ) 1 if v (i) It is clear in this case that D can not be obtained from any FDAG via the rule -otherwise v n would not be alone at its height. Concerning and , let us look at the number of children of v n . (a) If v n admits only one child, it must come from an step, since would imply that #w ≥ 2. Therefore, in this case, D can be retrieved among the outcomes of rule applied to D = (v 0 , . . . , v n−1 ). (b) Otherwise, when #w > 1, D can not come from an step, and must therefore come from . Denoting v n the vertex with list of children SC(w) -see (3), D is one of the outcomes of D = (v 0 , . . . , v n−1 , v n ) via . (ii) following the same logic as (i), D can not be obtained via . We discrimine between rules and be comparing w and w . If w is a minimal word of Λ w < , then D can not be obtained from -this would break the canonical order. Therefore, in this case, D is an outcome of rule applied to D = (v 0 , . . . , v n−1 ). Otherwise, if w is not a minimal word, then it can not be obtained from , and must come from a step, applied to D = (v 0 , . . . , v n−1 , v n ) where C ψ (v n ) = SC(w). Whatever the case among those evoked, they correspond exactly to the conditional expressions of the Algorithm 3, which therefore constructs the correct antecedent of D, which is unique by virtue of the previous discussion. Enumeration tree In this subsection, we construct the enumeration tree of FDAGs derived from the expansion rules of Subsection 2.3. As aimed, their inverse is indeed a reduction rule. (ii) The sequence of general term f k (D) is made of discrete objects whose size is strictly decreasing, therefore the sequence is finite and reaches D 0 . The associated expansion rule is exactly, in light of Proposition 2.11, the union of the three expansion rules , and . Since D 0 , the DAG with one vertex and no arcs, is a FDAG, by virtue of what precedes and with Algorithm 1 -here with g(·) = , we just defined an enumeration tree covering the whole set of FDAGs, whose root is D 0 . A fraction of this enumeration tree is shown in Figure 7, illustrating the path from the root D 0 to the FDAG of Figure 3. Unexplored branches are ignored, but are still indicated by their respective root. Growth of the tree In this section, we analyse the enumeration tree defined in Section 2. In Subsection 3.1, we exhibit a bijection -Theorem 3.1 -between FDAGs and a class of combinatorial objects from the literature, allowing us to obtain an asymptotic expansion of the growth of the tree. In Subsection 3.2, we show that any FDAG has a linear number of children in that tree in Theorem 3.3, and that the time complexity to construct those children is quadratic -see Proposition 3.5. Finally, Theorem 3.6 states that our algorithm runs with polynomial delay [15]. Asymptotic growth In this subsection, we show that FDAGs are in bijection with a set of particular matrices, whose combinatorial properties are known and give us access to an asymptotic expansion of the enumeration tree growth. Let us denote E k the set of all FDAGs that are accessible from D 0 in exactly k steps in the enumeration tree -with E 0 = {D 0 }; then Table 2 depicts the values of #E k for the first nine values of k 3 . Actually, the terms of Table 2 coincide with the first terms of OEIS sequence A158691 4 , which counts the number of row-Fishburn matrices, that are upper-triangular matrices with at least one nonzero entry in each row. The size of such a matrix is equal to the sum of its entries. Proof. The proof lies in Appendix A. This connection is to our advantage since Fishburn matrices (in general) are combinatorial objects widely explored in the literature as they are in bijection with many others -see [13, Section 2] for a general overview. Notably, the asymptotic expansion of the number of row-Fishburn matrices has been conjectured first by Jelínek [14] and then proved by Bringmann et al. [7]. Branching factor Given the overall structure of FDAGs, it is no surprise that the enumeration tree grows extremely fast. However, despite this combinatorial explosion, we show in this subsection that the branching factor, i.e., the outdegree of the nodes in the enumeration tree, is controlled. Actually, we prove that any FDAG has a linear number of successors 5 in the enumeration tree. Proof. Let D = (v 0 , . . . , v n ) be a FDAG. We denote C ψ (v n ) = a 0 · · · a m . Depending on the rule chosen: a m+1 belongs to A < = {0, . . . , p}, so the maximum number of successors is at most p + 1, and at least 1, depending on the condition a m ≥ lex. a m+1 . The child of the new vertex is taken from A = = {p + 1, . . . , n} so the number of successors is exactly n − p. Following Proposition 2.3, the number of successors is exactly #A < = p + 1. Combining everything, the number of successors is at least n + 2 and at most n + p + 2 ≤ 2n + 1 (as p ≤ n − 1, with equality for FDAGs obtained just after using rule). In the previous proof, we have shown that the number of successors of a FDAG with n vertices is between n + 1 and 2n − 1. Figure 8 illustrates that these boundaries are tight, on 1 000 randomly generated FDAGs. A random FDAG is constructed as follows. Definition 3.4 (Random FDAG). Let k ≥ 0. Starting from D 0 -the root, construct iteratively D i as a successor of D i−1 in the enumeration tree, picked uniformly at random. We stop after k steps, and keep D k . In Figure 8, we have generated 10 random FDAGs for each k ∈ {1, . . . , 100}. It is indeed a suitable property that any FDAG admits a linear number of successors; but it would be of little use if the time required to compute those successors is too important. We demonstrate in the following proposition that temporal complexity is manageable. There are two possible strategies: (i) one can keep the enumeration tree in memory, and store on each node only the increment allowing to construct a FDAG from its predecessor; or (ii) one can explicitly build the successors by copying the starting FDAG, so that the tree can be forgotten. Depending on whether one wants to build the tree itself or only the FDAGs that compose it, one will choose either strategy. Proof. Let D = (v 0 , . . . , v n ) be a FDAG with n + 1 vertices, with C ψ (v n ) = a 0 · · · a m , A < = {0, . . . , p} and A = = {p + 1, . . . , n}. Although the alphabets A < and A = can be retrieved in linear time, it is more efficient to maintain the pair (n, p) during enumeration; how to update these indices has already been presented in Subsection 2.3, when introducing each expansion rule. The explicit construction of the successors in case (ii) requires to copy the vertices of D and their children, leading to a complexity in the order of n i=0 (1 + deg(v i )), which can be roughly bounded by (n + 1)(deg(D) + 1). Depending on the expansion rule, the complexity for computing the new vertex or new arc varies: The last letter of C ψ (v n ) determines the number of successors -but it is no more than p + 1. In case (i), although we could just store the information of the new letter, it is better to copy C ψ (v n ) and add the new letter and store the result. Indeed, this allows to always have the knowledge of C ψ (v n ) in the enumeration tree. The complexity for case (i) is therefore bounded by deg(v n )(p + 1); whereas it is p + 1 in case (ii) since C ψ (v n ) is already copied. Each successor is obtained by picking one element of A = = {p + 1, . . . , n}. The complexity is exactly (up to a constant) n − p in both cases. The successors are obtained by Algorithm 2, involving copying subwords of C ψ (v n ) -the overall complexity is bounded by (p + 1) deg(v n ). The overall complexity is therefore of the order of 2(p + 1) deg(v n ) + n − p in case (i) and of (n+1)(deg(D)+1) [(p + 1)(deg(v n ) + 1) + n − p] in case (ii). Using rough bounds, with deg(v n ) ≤ deg(D) and p ≤ n, we end up with the stated complexity. Whereas Figure 8 shows the number of successors of 1,000 random FDAGs, we measured the time needed to compute explicitly -i.e., implying copy, which is case (ii) in the previous Proposition 3.5 -these successors. The results are depicted in Figure 9, where we plotted (in blue) the total time t D for computing all successors of a given FDAG D, and (in red) what we call amortized time, i.e. t D /(#D deg(D)) 2 . As expected from Proposition 3.5, one can observe an asymptotic quadratic behaviour for the total time (in blue); concerning amortized time (in red), despite some variability, the upper bound seems to be constant. Polynomial delay Let E ≤K+1 = k≤K+1 E k be the set of all FDAGs reachable in at most K + 1 steps from the root of the FDAG enumeration tree. In this subsection, we show that the time complexity for enumerating E ≤K+1 can be expressed as a function of the cardinality of E ≤K and has polynomial delay. Proof. We adopt the configuration where we keep the enumeration tree in memory and where each node contains the incremental information to construct a FDAG from its predecessor. We first observe the following: as D 0 , the root, has one vertex and no arcs, and since the rules of expansion can only add one vertex and/or increase the degree of the last vertex by one, for any D ∈ E k , it follows naturally that #D ≤ k + 1 and deg(D) ≤ k. It implies that the time complexity for generating the successors of a FDAG in E k is O(k 2 ), according to case (i) of Proposition 3.5. Thus, enumerating all the elements of E k+1 requires a time complexity of O(k 2 #E k ). It follows that we have a complexity of O k≤K k 2 #E k for enumerating E ≤K+1 . Since k ≤ K and k≤K #E K = #E ≤K , we end up with the announced complexity. As such, our algorithm has a polynomial delay, which is desirable for this kind of enumeration [15]. Variations on the enumeration tree In this section, two variants of the enumeration tree presented in Section 2 are introduced. Subsection 4.1 proposes a way to enumerate forests in their classical sense, i.e., where redundancies within the forest are accepted, by adding an extra step following the previous enumeration. Finally, options to constrain the enumeration tree -on maximum number of vertices, height or outdegreeand making it finite are proposed in Subsection 4.2. Extension to forests with repetitions The enumeration tree constructed in Section 2 only allows to enumerate, in their compressed form, irredundant forests, where no tree can be a subtree of (or equal to) another. In this subsection, we propose a method to enumerate forests in the usual sense, without this non-redundancy restriction. Let F be a forest in the classical sense, i.e., where some trees may be identical to or subtrees of other trees. If we compute D = R(F), by definition, all these redundancies will be lost: the trees which are subtrees of another will be compressed with these subtrees in the obtained DAG, and those which are identical will be compressed in the same source of the DAG. This is why we specified in Subsection 1.2 that DAG compression is lossless if and only if the forest is irredundant. We can preserve the information lost by the compression if we keep, in addition, a presence vector. Let us rewrite D = (v 0 , . . . , v n ) according to the canonical order. Each tree T ∈ F is associated with an index i ∈ {0, . . . , n} such that T = R −1 (D[v i ]). The presence vector π F : {0, . . . , n} → N is constructed such that π F (i) counts how many times the tree R −1 (D[v i ]) appears in F. Thus, the couple (D, π f ) completely characterizes the forest F. To enumerate all (redundant) forests, it is therefore sufficient to enumerate both all FDAGs (corresponding to irredundant forests) and the presence vectors that may be associated with them. Let D be an FDAG constructed in the FDAG enumeration tree. We define π D as the presence vector associated to the (irredundant) forest R −1 (D). This vector can be computed in a linear traversal of D, where the sources of D are assigned a value of 1 and the other vertices are assigned a value of 0. Adding redundancies in a forest means incrementing the presence vector, each +1 resulting in a new tree, whether it is equal to an existing tree or a subtree of it. Our strategy is to enumerate, from π D , all presence vectors corresponding to forests whose DAG reduction would be exactly D. To do so, we use a reverse search structure, with the following expansion rule (E). Let j be the index of the last increment, initialized to j = 0. Definition 4.1 (E). Choose any index j ≥ j. Increase π(j ) by one and set j ← j . This rule allows to get any presence vector from π D in a unique way, i.e., each index must be increased to its desired final value before moving to the next index. This defines an enumeration tree of presence vectors. If we implement this tree in such a way that each node contains only the new index j , we obtain an algorithm that enumerates each presence vector from its parent in constant time and space. The growth of this (infinite) new enumeration tree is given by the following proposition. Proof. We first notice that the expected number is exactly the same as the number of presence vectors constructible in at most k steps from π D . We then notice that if the current node (in the presence vector enumeration tree) has index j, then it has n + 1 − j successors by the expansion rule (E) of Definition 4.1. For instance, since the starting index is 0, for k = 1, we obtain the indices 0, . . . , n in one copy each. We denote by n p (j) the number of times the index j appears in the nodes obtained in exactly p steps from the origin. Thus, n 1 (j) = 1 by the above. Each index j ≤ j existing at step p − 1 will induce a successor with index j at step p, so that n p (j) = j j =0 n p−1 (j ). We establish by induction on p that n p (j) = k−1+j j , using the so called hockey-stick identity m r=0 n+r r = n+1+m m [16]. Since the number of presence vectors that can be constructed in at most k ≥ 1 steps from π D is given by k p=1 n j=0 n p (j), we obtain the expected result after applying twice the hockey-stick identity. We can merge the enumeration tree of repetitions with the enumeration tree of FDAGs, to form a single enumeration tree, which enumerates forests in the classical sense (and in compressed form), as follows: the nodes of the enumeration tree carry a couple (FDAG, presence vector), and the available expansion rules are , , and (E). However, successors created with the last rule produce branches where it becomes the only rule available. In other words, once one chooses repetition, one can not modify any longer the topology of the FDAG -this is to ensure that each forest can only be enumerated in a unique way. Constraining the enumeration In [20], the authors propose an algorithm to enumerate all trees with at most n vertices. They simply check whether the current tree has n vertices or not, and as their expansion rule adds one vertex at a time, they decide to cut a branch in the enumeration tree once they have reached n vertices. Similarly, adding a vertex to a tree can only increase its height or outdegree, so we can proceed in the same way to enumerate all trees with maximal height H and maximal outdegree d. This property also holds with the approach presented in Section 2: following one of the three expansion rules, we can only increase the height, outdegree or number of vertices of the FDAG. So, it makes sense to define similar constraints on the enumeration. However, for this constrained enumeration to generate a finite number of FDAGs, constraints must be chosen wisely, as shown in the following proposition. (i) maximum number of vertices n and maximum outdegree d, (ii) maximum height H and maximum outdegree d. Proof. As allows to add arcs indefinitely without changing the numbers of vertices, constraining on the maximum outdegree is mandatory in both cases. As the two others rules add vertices, constraining by the number of vertices leads to a finite enumeration tree -(i) is proved. To conclude, we only need to prove that can not be repeated an infinite number of times, i.e. there is only a finite number of new vertices that can be added at a given height, up to the maximum outdegree. This is achieved by virtue of the upcoming lemma. Let H > 2 and d ≥ 1. Let D be the FDAG constructed so that for each 0 ≤ h ≤ H, D has the maximum possible number n h of vertices of height h and with maximum outdegree d. Initial values are n 0 = 1 and n 1 = d. Lemma 4.4. ∀2 ≤ h ≤ H, n h = d k=1 k + n h−1 − 1 k d − k + n 0 + · · · + n h−2 d − k . Let h ≥ 2 be fixed. To lighten the notation, let n = n h−1 and m = n 0 + · · · + n h−2 . Let v be a vertex to be added at height h. For any vertex v i at height h − 1, let x i be the multiplicity of v i in C(v) - 0 if v i / ∈ C(v). Similarly, for any vertex v h with H(v j ) ≤ h − 2, y j is the multiplicity of v j in C(v) - possibly 0. By definition of H(·) -see (1), at least one x i is non-zero. Therefore, there exist k ∈ [[1, d]] such that: x 1 + · · · + x n = k y 1 + · · · + y m ≤ d − k Remark 4.5. In the constrained enumeration proposed in [20], all the trees with n vertices are the leaves of the enumeration tree. To get all trees with n + 1 vertices, it suffices to add to the enumeration all children of these leaves, i.e. trees obtained by adding a single vertex to them. This property -moving from one parameter value to the next by enumerating just one step further -does not hold anymore as soon as our set of constraints involve the maximum outdegree d, both for trees and FDAGs. For instance, from a FDAG of height H, one can obtain FDAG of height H + 1 by using once and repeating up to d − 1 times. In this section we define forests of subtrees, which will be our substructures. Compressed as FDAGs, these objects will be called subFDAGs. We then address the problem of enumerating all subFDAGs appearing in an FDAG D -similar as the one of enumerating all subtrees of a tree. Forests of subtrees can be directly constructed from FDAGs, as shown by the upcoming proposition. Let D be a FDAG, and V be a subset of vertices of D. Forests of subtrees Proposition 5.2. If ∀v ∈ V, C(v) ⊆ V, then V defines a FDAG ∆, such that R −1 (∆) is a forest of subtrees of R −1 (D). Proof. We recall from Subsection 1. . Therefore we have proved that ∀t ∈ f, ∃T ∈ F, t ∈ S(T ). We say that the FDAG ∆ is a subFDAG 6 of D. Figure 10 provides an example of such a construction. One can spot that t 1 ∈ S(T 1 ), t 2 ∈ S(T 2 ) so f is a forest of subtrees of F, and ∆ a subFDAG of D. Enumeration of subFDAGs We now solve the following enumeration problem: given a forest F, find all forests of subtrees of F. Equally, given a FDAG D, find all subFDAGs of D. To address this, we make extensive use of the reverse search technique, adapting the one presented in Section 2. Since a subFDAG is also a FDAG, it admits successors in the enumeration tree defined in Section 2. We are interested in those of these successors who are also subFDAGs (if any). In fact, since a subFDAG can be defined from a set of vertices, all one has to do is determine which new vertex can be chosen to expand an existing subFDAG -corresponding to a or step.The covering of all added new arcs is implicit in this construction and corresponds to some steps of . Let ∆ be a subFDAG of D and v its last inserted vertex -it is also the vertex with the largest ordering number in ∆. We denote by S(∆) the set of all vertices v ∈ D that can be added to ∆ to expand it to a new subFDAG. Let us call S(∆) the set of candidate vertices of ∆. More precisely: Lemma 5.3. S(∆) is the set of vertices v ∈ D that satisfies both: (i) C(v ) ⊆ ∆ (ii) ψ(v ) > ψ(v) where ψ(·) is the canonical ordering of D. Proof. (i) This condition is necessary so that ∆ = ∆ ∪ {v } fulfill the requirements for Proposition 5.2. (ii) This condition is necessary so that ∆ remains a FDAG. As ψ(v ) > ψ(v), either H(v ) = H(v) + 1 -then it is a step -or H(v ) = H(v) and C ψ (v ) > lex. C ψ (v) -for a step. Algorithm 4: HEIRS Input: D, ∆, S(∆) 1 Set L to the empty list 2 for s ∈ S(∆) do 3 Let S be a copy of S(∆) 4 S ← S \ {v ∈ S : C ψ (v ) ≤ lex. C ψ (s)} 5 S ← S ∪ v ∈ D : s ∈ C(v ) ⊆ ∆∪{s} 6 Add ∆ ∪ {s}, S to L 7 return L When S(∆) is not empty, picking s ∈ S(∆) ensure that ∆ = ∆ ∪ {s} is a subFDAG of D. With respect to the enumeration tree of Section 2, ∆ is an ancestor of ∆ -but not necessarily its parent, since the steps of are implicit. ∆ is called an heir of ∆. We can in turn calculate S(∆ ), by updating S(∆): (i) remove from S(∆) all vertices v such that ψ(s) > ψ(v ); (ii) in D, look only after the vertices v such that s ∈ C(v ) ⊆ ∆ ∪ {s} and add them to S(∆ ). If ∆ is an heir of ∆, then by removing the last inserted vertex of ∆ , one can retrieve ∆. This define a reduction rule f, and therefore an enumeration tree. Algorithm 4 is meant to construct the set f −1 (∆). Applying Algorithm 1 together with it, and starting from ∆ = L(D) 7 -in this case, S(∆) is the set of parents of L(D) of height 1 -permits to enumerate all subFDAGs of D. Figure 11 provide an example by enumerating all subFDAGs of the FDAG of Figure 3. Frequent subFDAG mining problem Using the reverse search formalism defined in Subsection 1.1, the frequent pattern mining problem can be formulated as follows: from a dataset X = {s 1 , . . . , s n } with s i ∈ S, and a fixed threshold σ, find 7 where L(D) designates the leaf of D, i.e. the only vertex without children. all elements s ∈ S that satisfy freq(s, X) ≥ σ, where freq(·) is a function that counts the frequency of appearance of s in the dataset X. This problem can be solved using Algorithm 1 with the function g(s, X, σ) = (freq(s, X) ≥ σ), which is trivially anti-monotone. We emphasize here that each possible definition of "s appears in X" leads to a different data mining problem. The choice of this definition is therefore of prime importance. In particular, this choice should induce a way of calculating freq(s, X) that reflects the specificity of the reduction rule f, so that {s ∈ f −1 (s 0 )\|g(s) = } can be constructed directly, instead of first generating f −1 (s 0 ) and then filtering according to the value of g. Indeed, if g is too restrictive, and f −1 (s 0 ) too large, one would have to enumerate objects that are not relevant to the enumeration problem, which is not desirable. In this article, the problem we consider is the following: given a set of trees X = {T 1 , . . . , T n }, account for forests of subtrees that appear simultaneously in different T i 's. In other words, if we denote F i the set of all forests of subtrees appearing in the forest formed by {T i }, we are interested in the study of ∩ i∈Iσ F i where I σ ⊆ [ [1, n]], such that #I σ ≥ σ · n. A first, naive strategy would be to first build the F i 's, e.g. by using Algorithm 4 on R(T i ), and then construct ∩ i∈Iσ F i for all possible choices of I σ . Obviously, this approach has its weaknesses: (i) many subFDAGs will be enumerated for nothing or in several copies, and (ii) it does not take into account that X is itself a forest. Our aim is to propose a variant of Algorithm 4 that, applied to R(X), would enumerate only subFDAGs appearing in the R(T i )'s with a large enough frequency. Given a forest F = {T 1 , . . . , T n } and its DAG compression D = R(F), we have to retrieve, for each vertex in D, their origin in the dataset, that is, from which tree they come from. This issue has already been addressed in a previous article [5,Section 3.3], and has led to the concept of origin. For any vertex v ∈ D, the origin of v is defined as o(v) = i ∈ [[1, n]] : R −1 (D[v]) ∈ S(T i ) ,v ∈ D(v), i ∈ o(v ) -as R −1 (D[v ]) ∈ S(R −1 (D[v])). Let ∆ be a subFDAG of D. For ∆ to compress a forest of subtrees of a tree T i , it is necessary that i ∈ o(v) for all v ∈ ∆. Therefore, the set of trees for which ∆ compress a forest of subtrees -the origin of ∆, denoted by o(∆) -is equal to Let S be a copy of S(∆) o(∆) = v∈∆ o(v). If ∆ = ∆ ∪ {s}5 S ← S \ {v ∈ S : C ψ (v ) ≤ lex. C ψ (s)} 6 S ← S ∪ v ∈ D : s ∈ C(v ) ⊆ D 0 ∪ {s} 7 Add ∆ ∪ {s}, S , o(∆) ∩ o(s) to L 8 return L We stated earlier that we wanted to avoid generating unnecessary or multiple copies of subFDAGs, which is achieved with Algorithm 5. We now empirically study what we have gained from this, by comparing the use of Algorithm 5 on D = R(F), with the use of Algorithm 4 on each R(T i ). As in Subsection 3.2, we generated 1 000 random FDAGs with parameter σ = 0 when using Algorithm 5. The results are provided in Figure 12. Despite a rather marked variability, there is a general trend of decreasing as the number of vertices increases. We obtain fairly low quotients, around 20%, quite quickly. Given the combinatorial explosion of the objects to be enumerated, such an advantage is of the greatest interest. Implementation The treex library for Python [3] is designed to manipulate rooted trees, with a lot of diversity (ordered or not, labeled or not). It offers options for random generation, visualization, edit operations, conversions to other formats, and various algorithms. The enumeration of Section 2 and the algorithms of Section 5 and 6 have been implemented as a module of treex so that the interested reader can manipulate the concepts discussed in this paper in a ready-to-use manner. Installing instructions and the documentation of treex can be found from [3]. FDAG ↔ Reduced adjacency matrix Let D = (v 0 , . . . , v n ) be a FDAG constructed in k steps from D 0 in the enumeration tree defined in Subsection 2.5. The adjacency matrix of D is defined as A = (A i,j ) i,j∈[[n,0]] 2 where, if m is the multiplicity of v j in C(v i ), then A i,j = m -possibly 0 if v j / ∈ C(v i ) . By construction of D, as v n is the last inserted vertex, it has no parents, so A n,· is a column of zeros; and as v 0 is a leaf, it has no children, so A ·,0 is a row of zeros. We define the reduced adjacency matrix M as the matrix A deprived of this column and this row. Therefore, M = (A i,j ) i∈[[n,1]],j∈[[n−1,0]] . As a vertex can not be a parent to any vertex introduced after it, we have A i,j = 0 for all i ≤ j -so that M is an upper-triangular matrix. In addition, as all vertices except v 0 have at least one child, there is at least one non-zero entry in each row of M. Therefore, M is a row-Fishburn matrix. However, we have no guarantee that this matrix verifies size(M) = k. Reduced adjacency matrix → Incremental adjacency matrix Let D = (v 0 , . . . , v n ) be a FDAG, and M its reduced adjacency matrix. Let M i be the row of M corresponding to C ψ (v i ). The incremental adjacency matrixM is defined as: M 1 = M 1 M i+1 = M i+1 M i where the operation is defined as follow: given two rows a 0 · · · a n and b 0 · · · b n , then denoting j = min{i : a i = b i }, and c = a j − b j , a 0 · · · a j−1 a j a j+1 · · · a n b 0 · · · b j−1 b j b j+1 · · · b n = 0 · · · 0 c a j+1 · · · a n . We claim that this new matrixM is a row-Fishburn matrix of size k, if D ∈ E k . Actually, since M was already a row-Fishburn matrix, we just have to check that the size is correct. Let us consider v i and v i+1 . The vertex v i+1 has been constructed from v i by using either or , and potentially several after that -let us say p ≥ 0 times. Therefore, if the claim is correct, the sum overM i+1 should be exactly p + 1. Consider the operation by which v i+1 was added in the first place: C ψ (v i+1 ) is reduced to a single element a, such that a > lex. C ψ (v i ). Therefore, the index j of the first non-zero coefficient of M i+1 is ahead of the one of M i so that the coefficient c of is equal to the j-th coefficient of M i+1 minus zero. Since the rule adds children to respect decreasing words, the p extra coefficients are added to the right of the j-th coefficient (including it) and therefore they are kept unchanged in the operation. Eventually, the sum over M i+1 is p + 1 and so is the sum overM i+1 . C ψ (v i+1 ) is built from C ψ (v i ) with Algorithm 2, and therefore they (i) share a common prefix, possibly empty and (ii) then differ by a single letter. The index of that letter in M i+1 corresponds to the index j defined in . Therefore, the coefficient c is -before any -equal to one. The argument of letters being added to the right of j still hold and therefore the sum overM i+1 is also p + 1. To conclude the proof, we have to exhibit the inverse function of the mapping we just defined. This will prove that this mapping is indeed a bijection, and then the theorem holds. Incremental adjacency matrix → Reduced adjacency matrix Let M andM be constructed as before. FromM, we can define a matrix M as: M 1 =M 1 M i+1 = M i ⊕M i+1 where the ⊕ operation is defined as follow: given two words a 0 · · · a n and b 0 · · · b n , then denoting j = min{i : b i = 0}, and c = a j + b j , a 0 · · · a j−1 a j a j+1 · · · a n ⊕ b 0 · · · b j−1 b j b j+1 · · · b n = a 0 · · · a j−1 c b j+1 · · · b n . By construction, ⊕ is the inverse operation of , so that we have the following lemma: Lemma A.1. The following properties hold: • M i ⊕ (M i+1 M i ) = M i+1 • (M i ⊕M i+1 ) M i =M i+1 Thefore, M = M . The FDAG of Figure 3 is reproduced below to illustrates the stages of the proof. This FDAG is constructed in 7 steps, that are (in this order): , , , , , and . The matrices A, M andM are given in Figure 13. One can see thatM is of size 7, as expected. Dots represent zeros corresponding to A i,j elements with i ≤ j. Remark A.2. It should be noted that (general) Fishburn matrices, with at least one non-zero entry on each row and column, are in bijection with FDAGs compressing forests made of a unique tree. Indeed, via the bijection above, as such FDAG have a unique root, it must be the last inserted vertex, and therefore, each column admits at least one non-zero entry (otherwise it would be another root). Remark A.3. It is possible to enumerate row-Fishburn matrices by using the previous bijection and the FDAGs enumeration tree together. Nevertheless, things are a little simpler in this case and the equivalent of the operations and can be merged, giving two rules for matrix expansion: (R1) Increase one coefficient to the (inclusive) left of the rightmost nonzero coefficient of the top row by 1. Figure 2 : 2The DAG on the left admits five topological orderings, which are shown in the table. by decreasing order w.r.t. the lexicographical order. In other words, C ψ (v) is a decreasing word -see Definition 2.1 -on the alphabet A = [[0, #D − 1]]. Figure 3 : 3A FDAG D (left) and its canonical ordering ψ (right). Vertices that are at the same height are enclosed in the table between the dashed lines. Figure 4 : 4Branching rule applied to the FDAG of Figure 3. As C ψ (v 5 ) = 211, the only letters a we can pick from A < = {0, 1, 2, 3}, satisfying a ≤ lex. 1, are 0 and 1. The only two possibles outcomes of are the words (a) 2111 and (b) 2110. Figure 5 : 5Elongation rule applied to the FDAG of Figure 3. As A = = {4, 5}, there are only two choices leading to (a) C ψ (v 6 ) = 4 and (b) C ψ (v 6 ) = 5. The alphabets become A < = {0, . . . , 5} and A = = {6}. Definition 2.8. Figure 6 : 6We apply to the FDAG of Figure 3. Here, A < = {0, 1, 2, 3} and w = 211. As seen in Subsection 2.1, the minimal words of Λ w < are 3, 22, 2111 and 2110. Therefore, there are 4 ways to add a new vertex v 6 via the widening rule, that are such that (a) C ψ (v 6 ) = 3, (b) C ψ (v 6 ) = 22, (c) C ψ (v 6 ) = 2111 or (d) C ψ (v 6 ) = 2110. Finally, we update A = to be equal to {4, 5, 6}. Proposition 2. 10 . 10The expansion rules preserve the canonicalness property.Proof. Let D = (v 0 , . . . , v n ) be a FDAG. The proposition follows naturally from the definitions: n is the only vertex of height H(v n ) ·) is the suffix-cut operator defined in(3). Theorem 2. 12 . 12Algorithm 3 is a reduction rule, as defined in Subsection 1.1. Proof. Let us denote f(D) the output of Algorithm 3 applied to a FDAG D. We need to prove that: (i) f(D) is a subgraph of D and (ii) for any D = D 0 , there exists an integer k such that f k (D) = D 0 , where D 0 is the FDAG with one vertex and no arcs. (i) Since Algorithm 3 deletes either one vertex and its leaving arcs, or just one arc, f(D) is indeed a subgraph of D. Figure 7 : 7The path (in bold) in the FDAGs enumeration tree leading to the FDAG ofFigure 3. The unexplored branches are only displayed by their root, which are shown partially transparent. The order of insertion of the vertices of each FDAG is always the same, and follows the color code (in the order of insertion):, , , , , and . With respect to the canonical ordering, they are numbered 0 to 6 in the same order. Theorem 3. 1 . 1There exists a bijection Φ between the set of FDAGs and the set of row-Fishburn matrices, such that if D is a FDAG and M = Φ(D), then D ∈ E k ⇐⇒ size(M) = k. Proposition 3. 2 ( 2Jelínek, Bringmann et al.). As k → ∞, e π 2 /24 = 1.29706861206 . . . . Theorem 3. 3 . 3Any FDAG D has Θ(#D) successors in the FDAG enumeration tree. Figure 8 : 8Numbers of successors of 1,000 random FDAGs in the enumeration tree, according to their number of vertices. Orange lines have equations y = n + 1 and y = 2n − 1. Proposition 3. 5 . 5Computing the successors of any FDAG D has complexity:(i) O(#D deg(D)) if the construction is incremental from D;(ii) O (#D deg(D)) 2 if the construction involves copying D. Figure 9 : 9The computations have been made on a Mac-Book Pro (2014) with an Intel Core i7 2.8 GHz processor and 16 GB of RAM. Total computation time (in blue) and amortized time (in red) for the explicit construction of the successors of the 1,000 random FDAGs ofFigure 8, according to their number of vertices. Theorem 3. 6 . 6Enumerating E ≤K+1 has time complexity O(K 2 #E ≤K ). Proposition 4. 2 . 2The number of redundant forests that can be constructed in at most k ≥ 1 steps from R −1 (D) -following expansion rule (E) -is given by n+1+k k − 1. Indeed, the number of trees satisfying those constraints is finite [4, Appendix D.2]. Proposition 4. 3 . 3The enumeration tree of FDAGs is finite if at least one of those set of constraints is chosen: By virtue of the stars and bars theorem, for a fixed k, there are k+n−y j . Summing upon all values for k proves the claim. Similarly to forest being tuple of trees, forests of subtrees are tuple of subtrees, satisfying (2). Formally: Definition 5.1. Let F and f be two forests. f is a forest of subtrees of F if and only if ∀t ∈ f, ∃T ∈ F, t ∈ S(T ). 2 that the notation D[v] stands for the subDAG of D rooted in v composed of v and all its descendants D(v). The notation R −1 (D[v]) stands for the tree compressed by D[v]. The demonstration is in two steps. (i) Remove from D the vertices that does not belong to V; as there are no arcs that leave V by hypothesis, end up with a FDAG. Let us call ∆ this FDAG. (ii) Let ρ be a root of ∆. By construction, ρ is also a vertex in D. Among all roots of D, there exists a root r such that ρ ∈ D(r). Therefore, D[ρ] is a subDAG of D[r], and then t = R −1 (D[ρ]) is a subtree of T = R −1 (D[r]) -with T ∈ F = R −1 (D). As ∆[ρ] and D[ρ] are isomorphic, t ∈ f = R −1 (∆) Figure 10 : 10Construction of a forest of subtrees from FDAG. (a) A FDAG D. The set V is circled in red. (b) The FDAG ∆ (c) The forest f compressed by ∆. (d) The forest F compressed by D. Figure 11 : 11Enumeration tree of the subFDAGs of the FDAG ofFigure 3, using both Algorithm 1 and Algorithm 4. The indices of the vertices correspond to the canonical ordering defined inFigure 3. In each vertex, the upper part corresponds to the current subFDAG ∆ whereas the lower part stands for the set S(∆). Numbers in red indicate what changes for an heir compared to its parent. is an heir of ∆ -as defined earlier, then o(∆ ) = o(∆) ∩ o(s). Algorithm 4 can therefore be refined so that ∆ should be ignored if o(∆ ) = ∅ -as ∆ does not anymore compresses any forest of subtrees actually present in the trees of F. So far we neglected the threshold σ. We only want to keep subFDAGs that appear in at least σ% of the data. If # o(∆)/#F < σ, then the successors of ∆ are not investigated. Indeed, as o(·) is a decreasing function, successors of ∆ can not exceed the threshold again.We can finally introduce Algorithm 5 that solves the frequent subFDAG mining problem for trees. With the notations of Subsection 1.1, this algorithm builds the set {∆ ∈ f −1 (∆)\|freq(∆ , F) ≥ σ}, with freq(∆ , F) = # o(∆ )/#F. The set is also built directly, without any posterior filtering, which is suitable as discussed at the beginning of the present subsection.Algorithm 5: FREQUENTHEIRS Input: D = R(F), ∆, S(∆), o(∆) , σ 1 Set L to the empty list 2 for s ∈ S(∆) do 3 if o(∆) ∩ o(s) = ∅ and #(o(∆) ∩ o(s)) ≥ σ · #F then 4 D k , 10 repetitions for each k ∈ {1, . . . , 100}, creating D k as in Definition 3.4. We assume D k = R(f) where f = {R −1 (D k [r]) : r source of D k }. For each D k , we have computed the quotient Q(D k ) = number of subFDAGs of D k enumerated via Algorithm 5 r source of D k number of subFDAGs of D k [r] enumerated via Algorithm 4 Figure 12 : 12Quotient Q(D) according to the number of vertices of D. Here 1 000 random FDAGs are displayed. Figure 13 : 13The FDAG of Figure 3 reproduced (top left), its adjacency matrix A (top right), its reduced adjacency matrix M (bottom left) and its incremental adjacency matrixM (bottom right). ( R2 ) R2Increase the dimension of the matrix by 1 (to the left and top), all new coefficients set to zero. Set one coefficient of the top row to 1. B Index of frequent notations Trees & DAGs v designates indifferently a vertex of a tree T or a DAG D. C(v) children of v: all vertices connected to an arc leaving v deg(v) outdegree of v: number of children of v D(v) descendants of v: children of v, their children, and so on H(v) height of v: length of the longest path from v to a leaf #T, #D number of vertices T [v], D[v] subtree/subDAG rooted in v and composed of v and D(v) L(T ), L(D) leaves: vertices without any children deg(T ), deg(D) outdegree: maximum outdegree among all vertices S(T ) the set of all distinct subtrees of T T the set of all treesF the set of all forests, i.e. sets of trees such that no tree is a subtree of another DAG reduction D designates a FDAG, F a forest.R(F) DAG reduction of the forest F R −1 (D) the forest F compressed by D, so that R(F) = D R −1 (D[v]) the tree T compressed by D[v], so that R({T }) = D[v]Canonical FDAGs Let D be a fixed FDAG, v any vertex of D and v n the vertex with highest index in the canonical ordering.ψ(·) canonical topological ordering of D C ψ (v) C(v) sorted by decreasing order on the indices defined by ψ(·) A = the indices of all vertices of D with same height as v n A < the indices of all vertices of D with strictly inferior height as v n Λ < the set of all decreasing words on A < , i.e. where each letter is greater than or equal to those who follow, w.r.t. the lexicographical order Λ w < the set of all decreasing words bounded by w, i.e. all words in Λ < that are greater than or equal to w, w.r.t. the lexicographical order SC(w) suffix-cut operator: the word w deprived of its last letter Enumeration tree D 0 the FDAG with one vertex and no arcs E k the set of FDAGs that are accessible in exactly k steps from D 0 in the FDAGs enumeration tree π D presence vector: π D (i) counts how many times the tree R −1 (D[v i ]) appears in the forest R −1 (D), for any FDAG D = (v 0 , . . . , v n ) and i ∈ {0, . . . , n}. Forests of subtrees Let ∆ be a subFDAG of D, and v any vertex. Let F = R −1 (D). S(∆) candidate vertices of ∆: the set of all vertices v of D so that ∆ ∪ {v } is still a subFDAG of D o(v) origin of v: the set of indices i so that R −1 (D[v]) is a subtree of T i ∈ F. o(∆) origin of ∆: the set of indices i so that R −1 (∆) is a forest of subtrees of T i ∈ F. Definition 1.2. A set {T 1 , . . . , T n } of trees is an irredundant forest if and only if v) the set of children of v. H(v) and deg(v) are inherited as well. Similarly to trees, we denote by D[v] the subDAG rooted in v composed of v and all its descendants D(v). Note that since D[v]has a unique root v, it compresses a forest made of a single tree. For the simplicity of notation, we use R −1(D[v]) to designate the tree compressed by D[v] -instead of the singleton. Table 2 : 2Number of FDAGs accessible from D 0 in k steps in the enumeration tree. where R −1(D[v]) designates the tree compressed by the subDAG D[v] rooted in v. In other words, o(v) represents the set of trees for which R −1(D[v]) is a subtree. We state in[5, Proposition 3.4] that origins can be iteratively computed in one exploration of D. The proof lies in the property that if i ∈ o(v), then for all A rooted tree is a connected graph without cycles such that there exists a special vertex called the root that has no parent, and the other vertices have exactly one parent. Trees are called ordered or unordered whether the order among siblings is important or not. See Subsection 1.2. The notation # is used in this paper to denote both (i) the cardinality #S of any set S, and (ii) the number of vertices #G of any graph G. These numbers were obtained numerically (cf. "Implementation" at the end of the article).4 OEIS Foundation Inc. (2021), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A158691. 5 "successor" in the sense of "children in the enumeration tree". We make the distinction to avoid confusion with the children denoted by C(·). Enumeration of forests of subtreesOnce the reverse search scheme has been set up to enumerate a certain type of structure, it is natural to move to a finer scale by using the same scheme to enumerate substructures. However, the notion of "substructure" is not obvious to derive from the main structure, as several choices are possiblee.g. for trees one can think of subtrees[24,5], subset trees[8], etc. From a practical point of view, the enumeration of substructures permits to solve the frequent pattern mining problem -which will be tackled in Section 6. Not to be confused with subDAG, introduced in Subsection 1.2. A subDAG admits a single root and therefore compresses a single tree, whereas a subFDAG admits several roots and compresses a forest. AcknowledgmentsThis work has been supported by the European Union's H2020 project ROMI. 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[ "Observational constraints on f (T ) theory", "Observational constraints on f (T ) theory" ]	[ "Puxun Wu \nCenter for Nonlinear Science\nDepartment of Physics\nNingbo University\n315211NingboZhejiangChina\n\nDepartment of Physics\nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education\nHunan Normal University\n410081ChangshaHunanChina\n", "Hongwei Yu \nCenter for Nonlinear Science\nDepartment of Physics\nNingbo University\n315211NingboZhejiangChina\n\nDepartment of Physics\nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education\nHunan Normal University\n410081ChangshaHunanChina\n" ]	[ "Center for Nonlinear Science\nDepartment of Physics\nNingbo University\n315211NingboZhejiangChina", "Department of Physics\nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education\nHunan Normal University\n410081ChangshaHunanChina", "Center for Nonlinear Science\nDepartment of Physics\nNingbo University\n315211NingboZhejiangChina", "Department of Physics\nKey Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education\nHunan Normal University\n410081ChangshaHunanChina" ]	[]	The f (T ) theory, which is an extension of teleparallel, or torsion scalar T , gravity, is recently proposed to explain the present cosmic accelerating expansion with no need of dark energy.In this Letter, we first perform the statefinder analysis and Om(z) diagnostic to two concrete f (T ) models, i.e., f (T ) = α(−T ) n and f (T ) = −αT (1 − e pT 0 /T ), and find that a crossing of phantom divide line is impossible for both models. This is contrary to an existing result where a crossing is claimed for the second model. We, then, study the constraints on them from the latest Union 2 Type Ia Supernova (Sne Ia) set, the baryonic acoustic oscillation (BAO), and the cosmic microwave background (CMB) radiation. Our results show that at the 95% confidence level Ω m0 = 0.272 +0.036 −0.032 , n = 0.04 +0.22 −0.33 for Model 1 and Ω m0 = 0.272 +0.036 −0.034 , p = −0.02 +0.31 −0.20 for Model 2. A comparison of these two models with the ΛCDM by the χ 2 M in /dof (dof: degree of freedom) criterion indicates that ΛCDM is still favored by observations. We also study the evolution of the equation of state for the effective dark energy in the theory and find that Sne Ia favors a phantom-like dark energy, while Sne Ia + BAO + CMB prefers a quintessence-like one.	10.1016/j.physletb.2010.08.073	[ "https://arxiv.org/pdf/1006.0674v5.pdf" ]	118,505,411	1006.0674	9d87780f77b88e2eddf2549e482a159923cbcda5	Observational constraints on f (T ) theory 10 Oct 2010 Puxun Wu Center for Nonlinear Science Department of Physics Ningbo University 315211NingboZhejiangChina Department of Physics Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education Hunan Normal University 410081ChangshaHunanChina Hongwei Yu Center for Nonlinear Science Department of Physics Ningbo University 315211NingboZhejiangChina Department of Physics Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education Hunan Normal University 410081ChangshaHunanChina Observational constraints on f (T ) theory 10 Oct 2010arXiv:1006.0674v5 [gr-qc]PACS numbers: 04 The f (T ) theory, which is an extension of teleparallel, or torsion scalar T , gravity, is recently proposed to explain the present cosmic accelerating expansion with no need of dark energy.In this Letter, we first perform the statefinder analysis and Om(z) diagnostic to two concrete f (T ) models, i.e., f (T ) = α(−T ) n and f (T ) = −αT (1 − e pT 0 /T ), and find that a crossing of phantom divide line is impossible for both models. This is contrary to an existing result where a crossing is claimed for the second model. We, then, study the constraints on them from the latest Union 2 Type Ia Supernova (Sne Ia) set, the baryonic acoustic oscillation (BAO), and the cosmic microwave background (CMB) radiation. Our results show that at the 95% confidence level Ω m0 = 0.272 +0.036 −0.032 , n = 0.04 +0.22 −0.33 for Model 1 and Ω m0 = 0.272 +0.036 −0.034 , p = −0.02 +0.31 −0.20 for Model 2. A comparison of these two models with the ΛCDM by the χ 2 M in /dof (dof: degree of freedom) criterion indicates that ΛCDM is still favored by observations. We also study the evolution of the equation of state for the effective dark energy in the theory and find that Sne Ia favors a phantom-like dark energy, while Sne Ia + BAO + CMB prefers a quintessence-like one. I. INTRODUCTION Various cosmological observations, including the Type Ia Supernova [1], the cosmic microwave background radiation [2] and the large scale structure [3,4], et al., have revealed that the universe is undergoing an accelerating expansion and it entered this accelerating phase only in the near past. This unexpected observed phenomenon poses one of the most puzzling problems in cosmology today. Usually, it is assumed that there exists, in our universe, an exotic energy component with negative pressure, named dark energy, which dominates the universe and drives it to an accelerating expansion at recent times. Many candidates of dark energy have been proposed, such as the cosmological constant, quintessence, phantom, quintom as well as the (generalized) Chaplygin gas, and so on. However, alternatively, one can take this observed accelerating expansion as a signal of the breakdown of our understanding to the laws of gravitation and, thus, a modification of the gravity theory is needed. One of the most popular modified gravity models is obtained by generalizing the spacetime curvature scalar R in the Einstein-Hilbert action in general relativity to a general function of R. The theory so obtained is called as the f (R) theory (see [5] for recent review). Recently, a new modified gravity by extending the teleparallel gravity [6] is proposed to account for the present accelerating expansion [7][8][9][10]. Differing from general relativity using the Levi-Civita connection, in teleparallel gravity, the Weitzenböck connection is used. As a result, the spacetime has only torsion and thus is curvature-free. Similar to general relativity where the action is a curvature scalar, the action of teleparallel gravity is a torsion scalar T . In analogy to the f (R) theory, Bengochea and Ferraro suggested, in Ref. [7], a new model, named f (T ) theory, by generalizing the action of teleparallel gravity, and found that it can explain the observed acceleration of the universe. Let us also note here that models based on modified teleparallel gravity may also provide an alternative to inflation [11,12]. Another advantage the generalized f (T ) torsion theory has is that its field equations are second order as opposed to the fourth order equations of f(R) theory. More recently, Linder proposed two new f (T ) models to explain the present cosmic accelerating expansion [8] and found that the f (T ) theory can unify a number of interesting extensions of gravity beyond general relativity. In this Letter, we plan to first perform a statefinder analysis and an Om diagnostic to these models and then discuss the constraints on them from the latest observational data, including the Type Ia Supernovae released by the Supernova Cosmology Project Collaboration, the baryonic acoustic oscillation from the spectroscopic Sloan Digital Sky Survey, and the cosmic microwave background radiation from Wilkinson Microwave Anisotropy Probe seven year observation. We find that for both models the crossing of the −1 line is impossible. This is consistent with what obtained in Ref. [10], but in conflict with the result obtained in Ref. [8] where a crossing is found for the exponential model. II. THE f (T ) THEORY In this section, following Refs. [7,8], we briefly review the f (T ) theory. We start with teleparallel gravity where the action is the torsion scalar T defined as T ≡ S µν σ T σ µν ,(1) where T σ µν is the torsion tensor T σ µν ≡ e σ A (∂ µ e A ν − ∂ ν e A µ ) ,(2) and S µν σ ≡ 1 2 (K µν σ + δ µ σ T αν α − δ ν σ T αµ α ) .(3) Here e A µ is the orthonormal tetrad component, where A is an index running over 0, 1, 2, 3 for the tangent space of the manifold, while µ, also running over 0, 1, 2, 3, is the coordinate index on the manifold. The spacetime metric is related to e A µ through g µν = η AB e A µ e B ν ,(4) and K µν σ is the contorsion tensor given by K µν σ = − 1 2 (T µν σ − T νµ σ − T µν σ )(5) By assuming a flat homogeneous and isotropic Friedmann-Robertson-Walker universe which is described by the metric ds 2 = dt 2 − a 2 (t)δ ij dx i dx j ,(6) where a is the scale factor, one has, from Eq. (1), T = −6H 2 ,(7) with H =ȧa −1 being the Hubble parameter. In order to explain the late time cosmic accelerating expansion without the need of dark energy, Linder, following Ref. [7], generalized the Lagrangian density in teleparallel gravity by promoting T to be T + f (T ). The modified Friedmann equation then becomes H 2 = 8πG 3 ρ − f 6 − 2H 2 f T ,(8)(H 2 ) ′ = 16πGP + 6H 2 + f + 12H 2 f T 24H 2 f T T − 2 − 2f T ,(9) where a prime denotes a derivative with respect to ln a, ρ is energy density and P is the pressure. Here we assume that the energy component in the universe is only matter with radiation neglected, thus P = 0. From Eqs. (8,9), we can define an effective dark energy, whose energy density and the equation of state can be expressed, respectively, as ρ ef f = 1 16πG (−f + 2T f T )(10)w ef f = − f /T − f T + 2T f T T (1 + f T + 2T f T T )(f /T − 2f T ) .(11) Some models are proposed in Refs. [7,8] to explain the present cosmic accelerating expansion, which satisfy the usual condition f /T → 0 at the high redshift in order to be consistent with the primordial nucleosynthesis and cosmic microwave background constraints. Here we consider two models proposed by Linder [8]: • Model 1 f (T ) = α(−T ) n .(12) Here α and n are two model parameters. Using the modified Friedmann equation, one can obtain α = (6H 2 0 ) 1−n 1 − Ω m0 2n − 1 ,(13) where Ω m0 = 8πGρ(0) 3H 2 0 is the dimensionless matter density today. Substituting above expression into the modified Friedmann equation and defining E 2 = H 2 /H 2 0 , one has E 2 (z) = Ω m0 (1 + z) 3 + (1 − Ω m0 )E 2n .(14) Let us note that this model has the same background evolution equation as some phenomenological models [13,14] and it reduces to the ΛCDM model when n = 0, and to the DGP model [15] when n = 1/2. When n = 1, the Friedmann equation (Eq. (8)) can be rewritten as H 2 = 8πG 3(1−α) ρ, which is the same as that of a standard cold dark matter (SCDM) model if we rescale the Newton's constant as G → G/(1 − α). Therefore, in order to obtain an accelerating expansion, it is required that n < 1. • Model 2 f (T ) = −αT (1 − e pT 0 /T ) ,(15) which is similar to a f (R) model where an exponential dependence on the curvature scalar is proposed [16,17]. Using the modified Friedmann equation again, we have α = 1 − Ω m0 1 − (1 − 2p)e p ,(16) and E 2 (z) = Ω m0 (1 + z) 3 + (1 − Ω m0 ) E 2 − E 2 e p/E 2 + 2pe p/E 2 1 − (1 − 2p)e p .(17) It is easy to see that p = 0 corresponds to the case of ΛCDM. III. STATEFINDER ANALYSIS AND Om DIAGNOSTIC In order to discriminate different dark energy models from each other, Sanhi et al. proposed a geometrical diagnostic method by adding higher derivatives of the scale factor [18]. In this method, two parameters (r, s), named statefinder parameters, are used, which are defined, respectively, as r ≡ ... a aH 3 ,(18)s ≡ r − 1 3(q − 1/2) ,(19) where q ≡ − 1 H 2ä a is the decelerating parameter. Apparently, ΛCDM model corresponds to a point (1, 0) in (r, s) phase space. The statefinder diagnostic can discriminate different models. For example, it can distinguish quintom from other dark energy models [19]. The Om(z) is a new diagnostic of dark energy proposed by Sahni et al. [20]. It is defined as Om(z) ≡ E 2 (z) − 1 (1 + z) 3 − 1 .(20) Apparently, this diagnostic only depends on the first derivative of the luminosity D L (z) (see Eq. (A3)). Thus, its advantage, as opposed to the equation of state of dark energy, is that it is less sensitive to the observational errors and the present matter energy density Ω m0 . One can use this diagnostic to discriminate different dark energy models by examining the slope of Om(z) even if the value of Ω m0 is not exactly known, since the positive, null, or negative slopes correspond to w < −1, w = −1 or w > −1, respectively. Here, we perform the statefinder and Om diagnostics to two f (T ) models, i.e., Model 1 and Model 2 given in the previous section. In Figs. (1) and (2), we show the diagnostic As demonstrated in Ref. [22], for a simple power law evolution of the scale factor a(t) ≃ t 2/3γ , one has r = (1 − 3γ)(1 − 3γ/2) and s = γ. Thus, a phantom-like dark energy corresponds to s < 0, a quintessence-like dark energy to s > 0, and an evolution from phantom to quintessence or inverse is given by a crossing of the point (1, 0) in (r, s) phase plane. A crossing of phantom divide line is also represented by a crossing of the red solid line (ΛCDM) in middle panels ((r, q) plane) of Figs. (1, 2). Therefore, we find, from the left and middle panels of Figs. (1, 2), that n > 0 (Model 1) or p < 0 (Model 2) f (T ) corresponds to a quintessence-like dark energy model, while n < 0 (Model 1) or p > 0 (Model 2) corresponds to a phantom-like one. A crossing of the phantom divide line is impossible for Model 1 and Model 2. These results are also confirmed by the Om(z) analysis given in the right panels. In order to further confirm our results, we redo our analysis with other values of Ω m0 , such as Ω m0 = 0.2 or 0.5, and find that the result remains unchanged. Thus, we conclude that the phantom divide line is not crossed for both models. This is in conflict with what given in Ref. [8] where a crossing of the phantom line is found for Model 2. For the Sne Ia data, we use the Union 2 compilation released by the Supernova Cosmology Project collaboration recently [23]. Calculating the χ 2 Sne , we find that, for Model 1, the best fit values occur at Ω m0 = 0.302, n = −0.18 with χ 2 M in = 543.953, whereas, for Model 2, Ω m0 = 0.279, p = 0.08 with χ 2 M in = 543.369. Then, we consider the constraints from the BAO data. The parameter A given by the BAO peak in the distribution of SDSS luminous red galaxies [4] Fig. (3). Furthermore, the CMB data is added in our analysis. The CMB shift parameter R [26,27] is used. The constraints from Sne Ia + BAO + CMB are given by χ 2 all = χ 2 Sne +χ 2 BAO +χ 2 CM B . Fig. (4) shows the results. We find that, at the 95% confidence level, Ω m0 = 0.272 +0.036 With the observational data considered above, we also discuss the constraints on the ΛCDM and the results are Ω m0 = 0.277 +0.040 −0.038 with χ 2 M in = 543.400 (Sne Ia + BAO) and Ω m0 = 0.276 +0.032 −0.036 with χ 2 M in = 543.745 (Sne Ia + BAO + CMB) at the 95% confidence level. A summary of constraint results on Model 1, Model 2 and ΛCDM is given in Table (1). From Figs. (3, 4) and Table (1), one can see that the ΛCDM (corresponding to n = 0 for Model 1 and p = 0 for Model 2) is consistence with the observations at the 68% confidence level, while the DGP model (corresponds to n = 1/2 for Model 1) is ruled out at the 95% confidence level. Meanwhile, using the χ 2 M in /dof (dof: degree of freedom) criterion, we find that the ΛCDM is favored by observations. In addition, we study the evolution of the equation of state for the effective dark V. CONCLUSION Recently, the f (T ) gravity theory is proposed to explain the present cosmic accelerating expansion without the need of dark energy. In this Letter, we discuss firstly the statefinder geometrical analysis and Om(z) diagnostic to the f (T ) gravity. Two concrete f (T ) models proposed by Linder [8] are studied. From the Om(z) diagnostic and the phase space analysis of the statefinder parameters (r, s) and pair (s, p), we find that, for both Model 1 and Model 2, a crossing of the phantom divide line is impossible, which conflicts with the result obtained in Ref. [8] where a crossing is found for Model 2. We We also find that the ΛCDM (corresponds to n = 0 for Model 1 and p = 0 for Model 2) is consistence with observations at 1 − σ confidence level and it is favored by observation through the χ 2 M in /dof (dof: degree of freedom) criterion. However, the DGP model, which corresponds to n = 1/2 for Model 1, is ruled out by observations at the 95% confidence level. Finally, we study the evolution of the equation of state for the effective dark energy in the f (T ) theory. Our results show that Sne Ia favors a phantom-like dark energy, while Sne Ia + BAO + CMB prefers a quintessence-like one. The analysis of the current paper also indicates that the f (T ) theory can give the same background evolution as other models such as ΛCDM, although they have completely different theoretical basis. Thus, it remains interesting to study other aspects of f (T ) theory, such as the matter density growth, which may help us distinguish it from other gravity theories. 2Bµ 0 + C with A = 1/σ 2 u,i , B = [µ obs (z i ) − 5 log 10 D L ]/σ 2 u,i and C = [µ obs (z i ) − 5 log 10 D L ] 2 /σ 2 u,i , one can find thatχ 2 Sne has a minimum value at µ 0 = B/A, which is given by χ 2 Sne = C − B 2 A .(A4) Thus, we can minimize χ 2 Sne instead ofχ 2 Sne to obtain constraints from Sne Ia. Baryon Acoustic Oscillation For BAO data, the parameter A given by the BAO peak in the distribution of SDSS luminous red galaxies [4] is used. The results can be obtained by calculating: χ 2 BAO = [A − A obs ] 2 σ 2 A (A5) where A obs = 0.469(n s /0.98) −0.35 ± 0.017 with the scalar spectral index n s = 0.963 from the WMAP 7-year data [25] and the theoretical value A is defined as A ≡ Ω 1/2 m0 E(z b ) −1/3 1 z b z b 0 dz ′ E(z ′ ) 2/3(A6) with z b = 0.35. Cosmic Microwave Background Since the CMB shift parameter R [26,27] contains the main information of the observations of the CMB, it is used in our analysis. The WMAP7 data gives the observed value of R to be R obs = 1.725 ± 0.018 [25]. The corresponding theoretical value is defined as R ≡ Ω 1/2 m0 z CM B 0 dz ′ E(z ′ ) ,(A7) where z CM B = 1091.3. Therefore, the constraints on model parameters can be obtained by fitting the observed value with the corresponding theoretical one of parameter R through the following expression χ 2 CM B = [R − R obs ] 2 σ 2 R .(A8) FIG. 1 : 1The evolutionary curves of statefinder pair (r, s) (left), pair (r, q) (middle) and Om(z) (right) for Model 1 with Ω m0 = 0.278. FIG. 2 : 2The evolutionary curves of statefinder pair (r, s) (left), pair (r, q) (middle) and Om(z) (right) for Model 2 with Ω m0 = 0.278. results with Ω m0 = 0.278, which is the best fit value obtained from Sne Ia and BAO with a model independent method[21] and is also consistent with the result in the next section of the present Letter. The left panels show the evolutionary curves of statefinder pair (r, s), the middle panels are the evolutionary curves of pair (r, q), and the right panels are the Om(z) diagnostic. Although, both Model 1 and Model 2 evolve from the SCDM to a de Sitter (dS) phase as one can see from the middle panels of these figures, the effective dark energy for Model 2 with p = 0 is similar to a cosmological constant both in the high redshift regimes and in the future, while for Model 1 with n = 0 this similarity occurs only in the future. on model parameters of Model 1 and Model 2 will be discussed, respectively, in this section. Three different kinds of observational data, i.e., the Type Ia supernovae (Sne Ia), the baryonic acoustic oscillation (BAO) from the spectroscopic Sloan Digital Sky Survey (SDSS) and the cosmic microwave background (CMB) radiation from Wilkinson Microwave Anisotropy Probe (WMAP), will be used in order to break the degeneracy between the model parameters. The fitting methods are summarized in the Appendix. − 0 . 0032 , n = 0.04 +0.22 −0.33 with χ 2 M in = 543.168 for Model 1 and Ω m0 = 0.272 +0.036 −0.034 , p = −0.02 +0.31 −0.20 with χ 2 M in = 543.631 for Model 2. FIG. 3 :FIG. 4 : 34The constraints on Model 1 (left) and Model 2 (right) from Sne Ia + BAO. The red and blue+red regions correspond to 1 − σ and 2 − σ confidence regions, respectively. The constraints on Model 1 (left) and Model 2 (right) from Sne Ia + BAO + CMB.The red and blue+red regions correspond to 1 − σ and 2 − σ confidence regions, respectively. FIG. 5 : 5energy. The results are shown in Fig. (5). The dashed, dotdashed and solid lines show the evolutionary curves with the model parameters at the best fit values from Sne Ia, Sne Ia+BAO, and Sne Ia+BAO+CMB, respectively. Apparently, Sne Ia favors a phantom-like dark energy, while Sne Ia + BAO + CMB favor a quintessence-like one. The evolutionary curves of the equation of state for the effective dark energy from Model1 (left) and Model2 (right). The model parameters are set at the best fit values. The dashed, dotdashed and solid lines correspond to the constraints from Sne Ia, Sne Ia + BAO, and Sne Ia + BAO + CMB, respectively. then consider the constraints on Model 1 and Model 2 from the latest Union 2 Type Ia Supernova set released by the Supernova Cosmology Project collaboration, the baryonic acoustic oscillation observation from the spectroscopic Sloan Digital Sky Survey Data Release galaxy sample, and the cosmic microwave background radiation observation from the seven-year Wilkinson Microwave Anisotropy Probe result. We find that at the 95% confidence level, for Model 1, Ω m0 = 0.272 +0.036 −0.032 , n = 0.04 +0.22 −0.33 with χ 2 M in = 543.168 and for Model 2, Ω m0 = 0.272 +0.036 −0.034 , p = −0.02 +0.31 −0.20 with χ 2 M in = 543.631. is used. The constraints from Sne Ia+BAO are given by minimizing χ 2Sne +χ 2 BAO . The results are Ω m0 = 0.279 +0.050 −0.047 , n = −0.01 +0.31 −0.54 (at the 95% confidence level) with χ 2 M in = 542.978 for Model 1 and Ω m0 = 0.278 +0.050 −0.045 , p = 0.02 +0.48 −0.24 (at the 95% confidence level) with χ 2 M in = 543.383 for Model 2. The contour diagrams are shown in TABLE I : ISummary of the constraint on model parameters and χ 2 M in /dof . In the table S+B+C represents Sne Ia + BAO + CMB.Model 1 Model 2 ΛCDM Ω m0 n χ 2 M in /dof Ω m0 p χ 2 M in /dof Ω m0 χ 2 M in /dof Sne+BAO 0.279 +0.050 −0.047 −0.01 +0.31 −0.54 0.974 0.278 +0.050 −0.045 0.02 +0.48 −0.24 0.975 0.277 +0.040 −0.038 0.973 S+B+C 0.272 +0.036 −0.032 0.04 +0.22 −0.33 0.975 0.272 +0.036 −0.034 −0.02 +0.31 −0.20 0.976 0.276 +0.032 −0.036 0.974 Appendix A: Data and Fitting MethodType Ia SupernovaeRecently, the Supernova Cosmology Project collaboration[23]released the Union2 compilation, which consists of 557 Sne Ia data points and is the largest published and spectroscopically confirmed Sna Ia sample today. We use it to constrain the theoretical models in this paper. 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[ "Photophoretic transport of hot minerals in the solar nebula", "Photophoretic transport of hot minerals in the solar nebula" ]	[ "A Moudens \nInstitut de Physique de Rennes\nUMR 6251\nCNRS\nUniversité de Rennes 1\nCampus de Beaulieu35042RennesFrance\n\nInstitut UTINAM\nUMR 6213\nCNRS\nObservatoire de Besançon\nBP 161525010Besançon CedexFrance\n", "O Mousis \nInstitut UTINAM\nUMR 6213\nCNRS\nObservatoire de Besançon\nBP 161525010Besançon CedexFrance\n", "J.-M Petit \nInstitut UTINAM\nUMR 6213\nCNRS\nObservatoire de Besançon\nBP 161525010Besançon CedexFrance\n", "G Wurm \nFaculty of Physics\nUniversity of Duisburg-Essen\nLotharstr. 147048DuisburgGermany\n", "D Cordier \nInstitut de Physique de Rennes\nUMR 6251\nCNRS\nUniversité de Rennes 1\nCampus de Beaulieu35042RennesFrance\n\nEcole Nationale Supérieure de Chimie de Rennes\nUMR 6226, Avenue du Général Leclerc\nCNRS\n50837, 35708Rennes Cedex 7CSFrance\n", "S Charnoz \nEquipe AIM\nCentre de l'Orme Les Merisiers\nUniversité Paris Diderot/CEA/CNRS\nCEA/SAp\n91191Gif-Sur-Yvette CedexFrance\n" ]	[ "Institut de Physique de Rennes\nUMR 6251\nCNRS\nUniversité de Rennes 1\nCampus de Beaulieu35042RennesFrance", "Institut UTINAM\nUMR 6213\nCNRS\nObservatoire de Besançon\nBP 161525010Besançon CedexFrance", "Institut UTINAM\nUMR 6213\nCNRS\nObservatoire de Besançon\nBP 161525010Besançon CedexFrance", "Institut UTINAM\nUMR 6213\nCNRS\nObservatoire de Besançon\nBP 161525010Besançon CedexFrance", "Faculty of Physics\nUniversity of Duisburg-Essen\nLotharstr. 147048DuisburgGermany", "Institut de Physique de Rennes\nUMR 6251\nCNRS\nUniversité de Rennes 1\nCampus de Beaulieu35042RennesFrance", "Ecole Nationale Supérieure de Chimie de Rennes\nUMR 6226, Avenue du Général Leclerc\nCNRS\n50837, 35708Rennes Cedex 7CSFrance", "Equipe AIM\nCentre de l'Orme Les Merisiers\nUniversité Paris Diderot/CEA/CNRS\nCEA/SAp\n91191Gif-Sur-Yvette CedexFrance" ]	[]	Context. Hot temperature minerals have been detected in a large number of comets and were also identified in the samples of Comet Wild 2 that were returned by the Stardust mission. Meanwhile, observations of the distribution of hot minerals in young stellar systems suggest that these materials were produced in the inner part of the primordial nebula and have been transported outward in the formation zone of comets. Aims. We investigate the possibility that photophoresis provides a viable mechanism to transport high-temperature materials from the inner solar system to the regions in which the comets were forming. Methods. We use a grid of time-dependent disk models of the solar nebula to quantify the distance range at which hot minerals can be transported from the inner part of the disk toward its outer regions as a function of their size (10 −5 to 10 −1 m) and density (500 and 1000 kg m −3 ). These models will also yield information on the disk properties (radius of the inner gap, initial mass, and lifetime of the disk). The particles considered here are in the form of aggregates that presumably were assembled from hot mineral individual grains ranging down to submicron sizes and formed by condensation within the hottest portion of the solar nebula. Our particle-transport model includes the photophoresis, radiation pressure, and gas drag. Results. Depending on the postulated disk parameters and the density of particles, 10 −2 to 10 −1 m aggregates can reach heliocentric distances up to ∼35 AU in the primordial nebula over very short timescales (no more than a few hundred thousand years). 10 −3 m particles follow the same trajectory as the larger ones but their maximum migration distance does not exceed ∼26 AU and is reached at later epochs in the disks. On the other hand, 10 −5 to 10 −4 m aggregates are continuously pushed outward during the evolution of the solar nebula. Depending on the adopted disk parameters, these particles can reach the outer edge of the nebula well before its dissipation.Conclusions. Our simulations suggest that irrespective of the employed solar nebula model, photophoresis is a mechanism that can explain the presence of hot temperature minerals in the formation region of comets. Comets probably had the time to trap the dust transported from the inner solar system either in their interior during accretion or in the form of shells surrounding their surface if they ended their growth before the particles reached their formation location.	10.1051/0004-6361/201116476	[ "https://arxiv.org/pdf/1105.6259v1.pdf" ]	56,564,338	1105.6259	cccda0a3cab83bfa88436d939faeec58c1e225ff	Photophoretic transport of hot minerals in the solar nebula January 26, 2013 A Moudens Institut de Physique de Rennes UMR 6251 CNRS Université de Rennes 1 Campus de Beaulieu35042RennesFrance Institut UTINAM UMR 6213 CNRS Observatoire de Besançon BP 161525010Besançon CedexFrance O Mousis Institut UTINAM UMR 6213 CNRS Observatoire de Besançon BP 161525010Besançon CedexFrance J.-M Petit Institut UTINAM UMR 6213 CNRS Observatoire de Besançon BP 161525010Besançon CedexFrance G Wurm Faculty of Physics University of Duisburg-Essen Lotharstr. 147048DuisburgGermany D Cordier Institut de Physique de Rennes UMR 6251 CNRS Université de Rennes 1 Campus de Beaulieu35042RennesFrance Ecole Nationale Supérieure de Chimie de Rennes UMR 6226, Avenue du Général Leclerc CNRS 50837, 35708Rennes Cedex 7CSFrance S Charnoz Equipe AIM Centre de l'Orme Les Merisiers Université Paris Diderot/CEA/CNRS CEA/SAp 91191Gif-Sur-Yvette CedexFrance Photophoretic transport of hot minerals in the solar nebula January 26, 2013Received ??; accepted ??Astronomy & Astrophysics manuscript no. phot˙editor c ESO 2013planetary systems -protoplanetary disks -comets: general -comets: individual (81P/Wild 2) -Kuiper Belt: general - Oort Cloud Context. Hot temperature minerals have been detected in a large number of comets and were also identified in the samples of Comet Wild 2 that were returned by the Stardust mission. Meanwhile, observations of the distribution of hot minerals in young stellar systems suggest that these materials were produced in the inner part of the primordial nebula and have been transported outward in the formation zone of comets. Aims. We investigate the possibility that photophoresis provides a viable mechanism to transport high-temperature materials from the inner solar system to the regions in which the comets were forming. Methods. We use a grid of time-dependent disk models of the solar nebula to quantify the distance range at which hot minerals can be transported from the inner part of the disk toward its outer regions as a function of their size (10 −5 to 10 −1 m) and density (500 and 1000 kg m −3 ). These models will also yield information on the disk properties (radius of the inner gap, initial mass, and lifetime of the disk). The particles considered here are in the form of aggregates that presumably were assembled from hot mineral individual grains ranging down to submicron sizes and formed by condensation within the hottest portion of the solar nebula. Our particle-transport model includes the photophoresis, radiation pressure, and gas drag. Results. Depending on the postulated disk parameters and the density of particles, 10 −2 to 10 −1 m aggregates can reach heliocentric distances up to ∼35 AU in the primordial nebula over very short timescales (no more than a few hundred thousand years). 10 −3 m particles follow the same trajectory as the larger ones but their maximum migration distance does not exceed ∼26 AU and is reached at later epochs in the disks. On the other hand, 10 −5 to 10 −4 m aggregates are continuously pushed outward during the evolution of the solar nebula. Depending on the adopted disk parameters, these particles can reach the outer edge of the nebula well before its dissipation.Conclusions. Our simulations suggest that irrespective of the employed solar nebula model, photophoresis is a mechanism that can explain the presence of hot temperature minerals in the formation region of comets. Comets probably had the time to trap the dust transported from the inner solar system either in their interior during accretion or in the form of shells surrounding their surface if they ended their growth before the particles reached their formation location. Introduction Hot-temperature minerals have been detected in a large number of comets (Campins & Ryan 1989;Crovisier et al. 2000;Sitko et al. 2004;Wooden et al. 2000Wooden et al. , 2004Wooden et al. , 2010 and were also identified in the samples of Comet 81P/Wild 2 that were returned by the Stardust mission (Brownlee et al. 2006). These minerals include crystalline silicates that presumably condensed in the 1200-1400 K temperature range in the solar nebula (Hanner 1999) and calcium, aluminum-rich inclusions (CAIs), which are the record of a very hot (1400-1500 K) stage of nebular evolution because they are composed of the first minerals to condense from a gas of solar composition (Grossman 1972;Jones et al. 2000). On the other hand, observations of young stellar systems show that the abundance of crystalline silicates is much higher in the inner disk than in the outer disk, but that even the outer disks show more crystalline silicates than the interstellar medium (Tielens et al. 2005). These observations then suggest that crystalline silicates, and probably also CAIs, were produced in the inner part of the primordial nebula and have been transported outward in the formation zone of comets. A number of mechanisms has been invoked to account for the origin of these high-temperature minerals in comets. It has been proposed that shock waves in the outer solar nebula could anneal the amorphous silicates to crystallinity in situ prior to their incorporation in comets (Harker & Desch 2002). However, the isotopic composition, minor element composition, and even the range of Fe/Si ratios measured in the dust that was returned by the Stardust spacecraft from Comet 81P/Wild 2 appear to be inconsistent with an origin by annealing of interstellar silicates in the primordial nebula (Brownlee et al. 2006). An alternative possiblity is the radial mixing induced by turbulence which is responsible for the angular momentum transport within the primitive nebula (Shakura & Sunyaev 1973). This turbulence favors the rapid diffusion of the different gaseous compounds and gas-coupled solids throughout the nebula. One-dimensional (vertically averaged) diffusive transport of particles in the disk (Bockelée-Morvan et al. 2002) or through its surrounding layers (Ciesla 2007(Ciesla , 2009) has therefore been proposed to account for the presence of hot temperature minerals in the formation zone of comets. It is uncertain however whether turbulent transport suffices to explain the observations, or whether alternative physical processes are also needed. On the other hand, Hughes & Armitage (2010) recently studied the outward transport of particles in the nebula via a combination of advection (inward drift of particles though interaction with gas) and turbulent diffusion in an evolving disk. These authors found that the advection of solids within the gas flow significantly reduces the outward transport efficiency for larger particles (typically a few millimeters), thereby limiting the extent of mixing uniformity that is achievable within the disk via turbulent diffusion. An alternative transport mechanism to turbulent diffusion whose effects have been investigated in the last years in the solar nebula is photophoresis (Krauss & Wurm 2005;Wurm & Krauss 2006;Krauss et al. 2007; Mousis et al. 2007;Wurm et al. 2010). This effect is based on a radiation-induced temperature gradient on the surface of a particle and the consequential nonuniform interaction with surrounding gas. When the existence of an inner gap is postulated in the disk, this latter becomes optically thin enough for particles to see the proto-Sun, but still has a reasonable gas content, which enables the photophoretic force to push dust grains outward ). This process provides a mechanism to transport high-temperature material from the inner solar system to the regions in which the comets were forming. Eventually, the dust driven outward in this manner will reach a region where the gas pressure and irradiation are so low that the combined outward forces of radiation pressure and photophoresis can only balance the inward drift of particles. In this work, we use a grid of time dependent models of the solar nebula to quantify the distance range at which particles (i.e hot minerals) can be transported from the inner part of the disk toward its outer regions as a function of their size and density as well as of the disk properties (radius of the inner gap, initial mass, and lifetime of the disk). The grid of models used here allows us to consider the full range of thermodynamic conditions that might have taken place during the solar nebula's evolution. The particles considered in our model are in the form of hot mineral aggregates with sizes ranging between 10 −5 and 10 −1 m. The trajectories of particles with lower sizes are generally influenced by radiation pressure while those of particles with larger sizes begin to be mostly affected by gas drag. The aggregates are presumed to have been assembled from hot mineral individual grains ranging down to submicron sizes. We consider that these hot minerals have formed by condensation within the hottest portion of the solar nebula, well inside 1 AU (Chick & Cassen 1997). We also show that the determination of the dust size distribution within rings observed in young circumstellar disks and their position relative to the parent star is likely to bring some constraint on the lifetime and eventually the initial mass of the disk from which they originate. Section 2 is devoted to the description of our modeling approach, detailing the particle transport and solar nebula models employed in this work. In Section 3 we detail the disk and particle parameters employed in our different models. In Section 4 we present and analyze the trajectories of particles determined in the frame of these models. In Section 5 we show that calculations of particle trajectories induced by photophoresis can be used as a tool to determine some physical parameters of circumstellar disks. Section 6 is devoted to the discussion of the assumptions of our model. Model The photophoretic force Any particle embedded in gas and heterogeneously heated by light feels a photophoretic force, which usually pushes it away from the light source (Krauss & Wurm 2005;Wurm & Krauss 2006). The force is strongly pressure-dependent and can be stronger than radiation pressure and the gravity of the Sun by orders of magnitude in the solar nebula. This mechanism induces the migration of particles ranging from micron to centimeter sized in the solar nebula under the combined action of photophoresis, radiation pressure, and gas drag, provided that the disk is sufficiently transparent . Following the approach developed by Krauss et al. (2007), we assume here that the disk's gas flow conditions are described by the Knudsen number, Kn, which is defined as Kn = l/a, where l is the mean free path of the gas molecules and a is the radius of the particle. If the mean free path of the gas molecules is large compared to the considered particle sizes, i.e., for Kn > 1, then the gas flow is in the free molecular flow regime. In the contrary case, the gas flow is in the continuum regime. In these conditions, the photophoretic force F ph on a spherical particle, valid for both flow regimes, can be expressed as follows (Beresnev et al. 1993): F ph = π 3 a 2 I J 1 πm g 2kT 1/2 α E ψ 1 α E + 15ΛKn(1 − α E )/4 + α E Λψ 2 ,(1) where I is the light flux (power incident per area), m g is the average mass of the gas molecules (3.89 × 10 −27 kg), T is the gas temperature, and k the Boltzmann constant. J 1 is the asymmetry factor that contains the relevant information on the distribution of heat sources over the particle's surface upon irradiation. In the following calculations, we assume J 1 = 0.5, which corresponds to the case where the incident light is completely absorbed on the illuminated side of the particle. The energy accommodation coefficient α E is the fraction of incident gas molecules that accommodate to the local temperature on the particle surface and, thus, contribute to the photophoretic effect. Here, we assume complete accommodation, i.e, α E = 1. The thermal relaxation properties of the particle are summarized in the heat exchange parameter Λ = λ e f f /λ g , where λ g is the thermal conductivity of the gas and λ e f f the effective thermal conductivity of the particle. For λ g , we adopt values for molecular hydrogen for temperatures above 150K (as tabulated by Incropera & DeWitt 2002). On the other hand, because helium has a higher thermal conductivity than hydrogen for lower temperatures, we assume that this species determines the thermal conductivity of the gas and use values taken from the compilation of Bich et al. (1990). The expression of Λ includes the conduction of heat through the particle and the thermal emission from the particle's surface, according to λ e f f = λ p + 4 σT 3 a.(2) where λ p is the thermal conductivity of the particle supposed here to be 10 −3 W m −1 K −1 ), its emissivity assumed to be 1, and σ the Stefan-Boltzmann constant. On the other hand, the functions ψ 1 and ψ 2 in Eq. (1) depend only on Kn in the form ψ 1 =Kn As noted by Krauss et al. (2007), an additional photophoretic force arises if the accommodation coefficients vary over the surface of the dust grain (Cheremisin et al. 2005), but we restrict the treatment to the "classical" photophoretic force as given in Eq. (1). In the present work, we assume that Eq. (1) is valid for all parts of the solar nebula and all particles. Balistic transport In a protoplanetary disk where the gas pressure (in the midplane) decreases with distance from the star, the gas is supported by a pressure gradient and rotates slower than the Keplerian velocity (Weidenschilling 1977). Solid particles are only stable on a Keplerian orbit. Therefore, interaction with the gas leads to an inward drift of solids toward the star. For particles that couple to the gas flow on timescales short compared to an orbital period, the problem reduces to a one-dimensional (radial) calculation. The inward drift is then induced by the fraction of gravity (residual gravity), which is not balanced by the circular motion with the sub-Keplerian gas velocity. The force, F res , acting on a particle of mass m p due to residual gravity is given as F res = m p ρ g d p dr ,(4) where ρ g is the density and p the pressure of the gas. In addition, radiation pressure has also to be considered for at least micron-sized particles (Krauss & Wurm 2005). The radiation pressure force can then be expressed as follow: F rad = πa 2 I c light ,(5) where c light is the speed of light. The sum of the outward forces (Eq. (1) and Eq. (5)) and the inward force (Eq. (4)) gives the drift force F dri f t . We treat the problem as being purely radial here because we are mostly interested in the small particles. These small particles couple to the gas on timescales much shorter than the orbital timescale, which justifies the radial treatment as outlined in Wurm & Krauss (2006). The radial drift velocity with respect to the nebula is then estimated to be v dr = F ph + F rad + F res m p τ, where τ is the gas grain coupling time and m p the mass of the considered particle. As larger dust aggregates drift outward, they pass from a region where the continuum flow regime is valid to a region where the free molecular flow regime applies. Hence, as with the photophoretic force, we have to consider an equation describing the gas grain friction time in both regimes. It is given by τ = m p 6πηa C c ,(7) where η is the dynamic viscosity of the gas. This assumes Stokes friction, which is justified because the Reynolds numbers for the drift of particles smaller than 10 cm are well below 1. The Cunningham correction factor, C c , accounts for the transition between the different flow regimes (Cunningham 1910) and is given as (Hutchins 1995) C c = 1 + Kn 1.231 + 0.47e −1.178/Kn .(8) To close the set of equations, we need to determine the dynamic viscosity η and the mean free path l. In the framework of the classical kinetic theory for dilute gases (see e.g. Reif 1972), these quantities are given by η = 1 3 nm g 8kT m g π l (9) and l = 1 √ 2nσ (10) where n is the molecule number density in the gas, and σ the collisional cross section of the gas molecules. The latter is very difficult to obtain. It is easier to find the value of the dynamic viscosity at a given temperature for H 2 and then use the functional form of η in Eq. 9 to determine its value at any temperature. We use η 0 = 9.0 × 10 −6 Pa s at T 0 = 300 K (Lide 2007). Finally, inverting Eq. 9, one obtains the mean free path. We note in passing that Beresnev et al. (1993) used a normalising factor of 1/2 in Eq. 9 instead of the 1/3 that applies for three-dimensional gases, and this may slightly modify the numerical constants in Eq. 1. However, the change is likely smaller than the uncertainties because of all the approximations made to solve the conservation equations in their model. Our description of the radial transport of particles in the disk includes their drag back toward the central star by the infalling nebula flow that moves at the velocity of v ac . In our disk model, the accretion speed ranges from a few tens of cm/s in the inner part to below one cm/s at larger distance in the early stages and substantially decreases later on. In the simplified solar nebula model presented in Section 2.3, the accretion velocity is esti- mated to be v ac = r 2t vis ,(11) where r is the distance from the Sun, t vis = 1 3α r 2 H 2 1 Ω is the typical local viscous time, H is the local height of the nebula, Ω is the local Keplerian frequency given by Ω 2 = GM /r 3 and α is the viscosity parameter of the disk described in Section 2.3. Finally, the position of particles is integrated from the inner edge of the disk at time t = 0 to a position r(t disk ) at the age of the disk t disk via r(t disk ) = t disk 0 (v dr (r(t), t) − v ac (r(t), t)) dt.(12) The protoplanetary disk The structure and evolution of the protoplanetary disk is modeled as a non irradiated, 1+1D turbulent disk, following the method originally presented in Papaloizou & Terquem (1999) and also developed by Alibert et al. (2005). The diffusion equation (see Lynden-Bell & Pringle 1974;Papaloizou & Lin 1995) describing the evolution of the gas surface density Σ is consequently solved as a function of time t and distance r to the star: dΣ dt = 3 r ∂ ∂r r 1/2 ∂ ∂r νΣr 1/2 +Σ w (r),(13) where Σ is the surface density of mass in the gas phase in the nebula and ν the mean (vertically averaged) turbulent viscosity. Compared to the original equation, the photo-evaporation term,Σ w (r), was added and is taken to be the same as in Veras & Armitage (2004). The mean turbulent viscosity is determined from the calculation of the vertical structure of the nebula: for each radius, r, the vertical structure is calculated by solving the equation for hydrostatic equilibrium together with the energy equation and the diffusion equation for the radiative flux (see Papaloizou & Terquem 1999). The local turbulent viscosity (as opposed to that averaged in the vertical direction) is computed using the standard Shakura & Sunyaev (1973) formalism: ν = αC 2 s /Ω, where α is a free parameter and C s the local speed of sound determined by the equation of state. Using this procedure, we derived the midplane pressure and temperature as well as the mean turbulent viscosity as a function of r and Σ. These laws are finally used to solve the diffusion equation (Eq. (13)) and to calculate the pressure-and temperature-dependant forces on dust grains. Figure 1 represents the temperature, pressure and surface density profiles in the midplane of the disk characterized by a mass of 0.03 M and a lifetime of 6 Myr (see Sec. 3 for fore details) at different epochs of its evolution. Following the approach of Mousis et al. (2007), we consider that the disk is not optically thin and that Rayleigh scattering from molecular hydrogen is the dominant dimming effect in the nebula (Mayer & Duschl 2005) for temperatures below 1500 K and at wavelengths shorter than a few µm. This condition is fulfilled only after 10 5 yr and beyond 0.5 AU in all the solar nebula models used in our calculations. For H 2 , i.e, the dominant molecule, the Rayleigh scattering cross section is σ(λ) = 8.49 × 10 −45 /λ 4 (cm 2 ) (Vardya 1962). Assuming the illuminating light follows a black body spectrum, the Planck mean cross section as a function of the black body temperature T B is found to be σ(T B ) = 1.54 × 10 −42 T 4 B (cm 2 ) (Dalgarno & Williams 1962). Note that, in our case, T B is not the temperature of the nebula, but rather the effective temperature of the illuminating source, the Sun. With a disk's mean molar mass of 2.34 g/mol, the mass absorption coefficient is found to be σ m (T B ) = 3.96 × 10 −19 T 4 B (cm 2 /g). The effective temperature and the luminosity of the early Sun were taken from the ZAMS (Zero Age Main Sequence) model computed by Pietrinferni et al. (2004), which is available in the BaSTI database (http://albione.oa-teramo.inaf.it). We chose the parameters relevant for the Sun, i.e, a solar mixture of heavy elements, no overshooting, a metallicity Z= 0.0198, and a helium content Y= 0.273 (Z and Y together in the mass fraction). In this model, the surface temperature of the early Sun is 5652 K and its initial luminosity is 2.716 × 10 26 W. We derived σ m (5770) = 4.0 × 10 −4 (cm 2 /g) from the adopted effective temperature of the early Sun. Light becomes extinguished close to the star as a result of the high gas density, while the outer regions play only a minor role in the extinction. Choice of parameters We constructed a grid of nine disk models encompassing the range of thermodynamic conditions that might have taken place during the solar nebula's evolution. The three initial disk masses were fixed to 0.01, 0.03 and 0.1 M respectively, with 0.01 M corresponding to the minimum mass solar nebula (hereafter MMSN) defined by Hayashi (1981). The initial mass of each disk is integrated between 0.25 and 50 AU and the initial gas surface density is given by a power law Σ ∝ r −3/2 , with an initial value taken to be Σ(5.2AU) = 100, 300, and 1000 g cm −2 at 5.2 AU for disk masses of 0.01, 0.03 and 0.1 M , respectively. Here, the lifetime of the disk is governed both by viscosity and photoevaporation by the Sun or nearby stars. On the other hand, the viscosity parameter rules the accretion velocity of the disk (Eq. 12) but this latter is found to be low compared to the velocities due to photophoresis and gas drag for particles larger than 10 −4 m (see Fig. 2 for an example of particle velocities due to photophoresis, radiation pressure, residual gravity and accretion flow along their trajectories in the nebula). Here the viscosity parameter is fixed to 7 × 10 −3 , i.e, a value adopted in works aiming at synthesizing different populations of planets around other stars (Mordasini et al. 2009a(Mordasini et al. , 2009b and the photoevaporation rate is varied to obtain the appropriate disk lifetimes (1, 3, and 6 Myr for each selected mass). In each case, the lifetime corresponds to the time taken for the mass of the disk (integrated until 50 AU) to decrease to 1% of its initial value. Mousis et al. (2007) have calculated the optical depth of the disk at 30 AU as a function of time. They found that even at late epochs, only ∼0.1% of the Sun's radiation is available in this region. As a result, these authors found that the high extinction induced by H 2 Rayleigh scattering limits the outward transport of particles only to very short heliocentric distances (typically a few AU) when they are released from the innermost regions. On the other hand, particle transport can be enhanced at larger heliocentric distances when a gap is formed in the inner disk. In particular, there is a growing body of observational evidence for the existence of disks whose inner few AU are cleared or are strongly depleted of gas (D'Alessio et al. 2005;Sicilia-Aguilar et al. 2006;Espaillat et al. 2008;Pontoppidan et al. 2008;Thalmann et al. 2010). For this reason, and similar to Mousis et al. (2007), we assume here the presence of 1 and 2 AU inner gaps within the nebula during the course of its viscous evolution. Gaps are prescribed in a way independent of the structure of the disk models used in this work and their sizes remain constant with time. As shown in Sect. 4, such an inner hole is large enough to leave a reasonable fraction of the incoming light to let photophoresis work even in the outer solar system. Particles considered in our simulations have sizes ranging between 10 −5 and 10 −1 m and are assumed to be spherical and composed of olivine, with a variable porosity. Density of aggregates is varied between 500 and 1000 kg m −3 . The first value corresponds to the random deposition of irregular olivine particles with density of 3300 kg m −3 , with a 15% filling factor (Blum & Schräpler 2004). The second value corresponds to the average density measured in cometary interplanetary dust particles (Joswiak et al. 2007). We do not consider particles with sizes lower than 10 −5 m because their path in the nebula is essentially controlled by radiation pressure. Moreover, for objects larger than about 1 m, the radial treatment we apply does no longer hold because the gas grain friction times become comparable to the orbital period. Outward transport of hot temperature aggregates All our calculations are based on the assumption that the disk opacity is essentially caused by Rayleigh scattering and not to dust, implying that the dust size distribution in the nebula is dominated by large particles instead of small particles. In the contrary case, smallest aggregates (here 10 −5 m) would create a prominent opacity in the disk, implying that larger aggregates could only migrate outward in the wake of the small ones. Figures 3-6 represent the trajectories of 10 −5 to 10 −1 m aggregates in the solar nebula that were computed using the defined particle densities and a set of six disk models that are expected to encompass the range of plausible thermodynamic conditions within the solar nebula (disk masses of 1 MMSN, 3 MMSN, 10 MMSN with lifetimes of 1 or 6 Myr). At the beginning of each Myr. The density of particles is 500 kg m −3 and the radius of the inner gap is 2 AU. Position of larger particles essentially corresponds to the balance between photophoresis and residual gravity velocities. When the disk opacity is prominent, i.e., at early epochs, the position of smaller particles is mainly driven by the balance between photophoresis and accretion flow velocities. At later epochs, the position of these particles becomes ruled by the balance between all velocities. computation, the particles start their migration within the disk from the outer edge of the inner gap. Figure 3 shows that 10 −2 -10 −1 m particles with densities of 500 kg m −3 that migrate within a disk with a 1 AU inner gap can reach heliocentric distances ranging between ∼24 and 28.1 AU, depending on the choice of the initial mass and lifetime of the nebula. Each of these positions corresponds to an equilibrium reached at the position where the outward drift of aggregates just balances the accretion flow and in no more than a few hundred thousand years. With time, the location of these particles slightly rebounds toward the Sun until the dissipation of the disk. The figure also shows that 10 −3 m particles follow the same trajectory as the larger ones but their equilibrium position is reached at lower heliocentric distance (∼20-22.8 AU) and at later epochs in disks owning similar input parameters. Interestingly enough, the position of smaller aggregates (10 −5 -10 −4 m) continuously progresses outward during the evolution of the disks. 10 −5 m particles can even be pushed beyond the outer edge (∼50 AU) of the nebula if one selects a low-mass disk (1 MMSN) with a long lifetime (6 Myr). This is because of the strong decrease of the gas density and opacity in this model that enables the radiation pressure to push the particles at higher heliocentric distance. Figure 4 represents the trajectories of the same particles as in Fig. 3, but for disk models with inner gaps fixed to 2 AU. Because the Rayleigh scattering through H 2 is strongly diminished here, all particles reach higher heliocentric distances than in the cases considered in Fig. 3, but for similar migration timescales. Thus, 10 −2 -10 −1 m particles reach heliocentric distances as high as ∼27-35.1 AU, depending on the adopted parameters of the disk. In similar conditions, 10 −3 m particles are also able to reach the ∼23.9-26.3 AU distance range within the nebula. Moreover, 10 −5 and even 10 −4 m particles reach the edge of the nebula for low mass (1 MMSN) and long lifetime (6 Myr) disk. Figures 5 and 6 show the trajectories of 10 −5 to 10 −1 m aggregates with densities of 1000 kg m −3 within disk models with inner gaps of 1 and 2 AU, respectively. Migration timescales remain similar to the previous cases: larger particles migrate very rapidly toward a maximum heliocentric distance while smaller ones continuously drift outward during the evolution of the disk. Because (i) the inward drift linearly depends on the mass of the aggregate (see Eq. 4) and (ii) the radial drift velocity v dr is inversely proportional to this quantity (see Eq. 6), all particles here migrate at lower heliocentric distances than in cases of disks based on similar parameters. Indeed, 10 −2 -10 −1 m particles do not exceed ∼20.2-23.6 AU (∼22.7-29.7 AU) in the case of disks with 1 AU (2 AU) inner gaps. The maximum migration distance reached by intermediary size particles (10 −3 m) becomes ∼20 AU (∼23.3 AU) in the case of disks with 1 AU (2 AU) inner gaps. In every case, the maximum migration distance of 10 −4 m particles is several AU smaller than those of same size particles with densities of 500 kg m −3 . Now only 10 −5 m particles reach the outer edge of the nebula for low mass (1 MMSN) and long lifetime (6 Myr) disk, irrespective of the gap size. Probing the dissipation of circumstellar disks Particle transport through the combination of photophoresis and radiation pressure has been invoked to explain the presence of ring-shaped dust distributions in young circumstellar disks such as the one around HR 4796A (Krauss & Wurm 2005). Here we show that in some cases, the determination of the dust size distribution within rings and their position relative to the parent star is likely to bring some constraints on the lifetime and eventually on the initial mass of the circumplanetary disk from which they originate. Indeed, Figure 7 represents the settling distances reached by particles of different sizes and with densities of 500 kg m −3 at the end of the solar nebula evolution. The figure shows, for example, that ring-like structures essentially composed of 10 −5 m particles and located in the ∼1.8-12 AU (3-21 AU) distance range from the star could have formed in disks with 1 AU (2 AU) inner gap, which have short or intermediary lifetimes (here 1-3 Myr), irrespective of the initial disk's mass. In addition, same size particles located at long distance to the star, i.e., ∼50 AU (upper limit owing to the truncation of our model) or farther, could have formed in disks with long lifetimes (6 Myr) and low initial masses (1 MMSN), irrespective of the size of the inner gap. To a lesser extent, one can also identify in Fig. 7 a relationship between the position (in the 15-25 AU range) of ring-like structures dominated by the settling of 10 −3 to 10 −1 m particles and the disk's lifetime and inner gap size. Discussion Disk's structure and evolution One could argue that the existence of an inner gap at early epochs within the nebula remains questionable. Indeed, gaps are often found in disks (i.e., transition disks) that are millions of years old. In this context, a significant offset might exist between the times of the different models used in this work and the chronology of the solar system formation and evolution that is testified by meteorite measurements or by the age of disks as inferred from luminosity studies of protostars. For these reasons, the less massive disk models used in this work that are associated to inner gaps correspond to cases that are the most consistent with the structure of transition disks. Moreover, our calculations are based on the assumption that the nebula is essentially devoid of dust, i.e., that the dust opacity is negligible. Indeed, if we assume that the smallest aggregates (10 −5 m) have created a prominent opacity in the nebula, larger aggregates could only follow the small ones and reach the formation zone of comets toward the end of the disk evolution. Despite these caveats, the use of a set of disk models covering the whole range of plausible thermodynamic conditions that took place in the primordial nebula allows us to show that hot-temperature minerals can drift up to heliocentric distances reaching ∼34 AU for the largest particles and 50 AU or beyond for the smallest ones provided that i) the existence of an inner gap is postulated within the nebula and ii) the opacity of the smallest dust particles remains negligible inside the photophoretic transport front. These simulations suggest that, irrespective of the employed solar nebula model, photophoresis is a mechanism that can explain the presence of hot-temperature minerals at early epochs of the disk's evolution in the formation region of comets (from 10 to 30 AU according to the different scenarios -see, e.g., Horner et al. 2007 for a review). Because comets have presumably accreted within a few hundred thousand years (Weidenschilling 1997), i.e., a timescale shorter than the one probably required to form an inner gap in the disk and to allow the photophoretic transport of particles formed close to the Sun, they probably had the time to essentially trap the dust transported from the inner solar system in the form of shell surrounding their surface if their accretion ended before particles reached their formation location. Because photophoresis works heterogeneously, depending on the individual properties of a dust aggregate (composition, size, thermal and optical properties), one would expect the bulk of comets to be laden with particles of size and/or composition that would vary as a function of their accretion distance in the solar nebula. It is important to note that our calculations were made with the typical values of 0.5 and 1 for the asymmetry factor J 1 and the emissivity . Assuming lower but still reasonable values for these parameters would not alter our conclusions. For example, if one assumes J 1 = 0.4 and = 0.8 in the disk model characterized by a mass of 3 MMSN and a lifetime of 6 Myr, 10 −1 -10 −2 m particles still migrate up to ∼ 25.4-25.9 AU (29.5-30 AU) and 10 −5 -10 −4 m particles up to ∼ 11-23.8 AU (18.5-33.3 AU) in the nebula owning a 1 AU (2 AU) inner gap. Role of turbulence The influence of turbulence on the particle motion has also to be considered in comparison to photophoresis. Two different cases have to be discussed. In the first case, the disk has an inner clear region and an optically thick outer region where opacity is provided by dust grains. In principle photophoresis is capable of moving the edge between optically thin and thick parts outward, thus clearing the disk inside-out from solids (see Krauss et al. 2007 for details). In this scenario, turbulent inward diffusion might counteract the outward motion of the edge. However, Krauss et al. (2007) discuss that turbulence will not prevent the outward motion of the edge. The edge will finally reach a lower heliocentric distance than in a similar situation where turbulence is negligible. The position of the edge will depend on the effective inward transport of particles by the turbulence. The second case would be more like the situation discussed in this paper. In a disk where solid particles are treated as test particles and opacity is essentially generated by the gas (Rayleigh scattering) and not by dust extinction, turbulence is not an issue. It only broadens any ring-like particle concentration because photophoresis is a directed force while turbulence is diffusive and acts statistically in both directions. Influence of particle rotation Particle rotation is an interesting topic for photophoretic forces. Photophoresis depends on the fact that a temperature gradient is established across the particle. Usually, an illuminated particle is warmer on the bright side than on the dark side. This temperature gradient always needs a certain time to adjust to changes in the illumination. Therefore, if the particle rotates, the temperature gradient might be different from the simple case where the particle is considered at rest. Rotations might be divided into two classes. The first class corresponds to rotations around the direction of the incident radiation. This rotation does not change the front and back side of the particle. The temperature gradient in a reference frame fixed to the particle always stays the same. While the main component of photophoresis will still point away from the light source, the subtleties of particle morphology and composition will induce sideward components of photophoresis as well. If a particle only rotates around the direction of light, the sideward motion will oscillate but the component along the direction of light will remain constant. The second class corresponds to rotations around axes perpendicular to the direction of light. These rotations change front and back of the particle. For slow rotations, the temperature gradient can follow the rotation. For somewhat faster spin the gradient lags behind, meaning that the cold side will trail into the warm side and vice versa. In total this results in a certain temperature gradient directed in a direction perpendicular to the direction of light. On a given orbit, these directed forces along the orbit can accelerate or slow down the particle and make it drif inward or outward. This is a photophoretic analog to the Yarkovsky effect, which works by radiation pressure, but the photophoretic effect can be several orders of magnitudes larger. This rotation would change the calculations given here one way or the other. Wurm et al. (2010) present first experimental evidence showing this effect in microgravity experiments for rotating particles. However, this second class of rotations can only occur temporarily and is not of general importance. To see this, we remind the reader that the idea of photophoresis is founded on the fact that particles in protoplanetary disks are embedded in a gaseous environment. This point might explicitly be noted in view of particle rotation. This simple fact is important because it implies a fundamental difference to particles in the current solar system where particles move in vacuum. While a particle in the solar system retains any arbitrary rotation state for a long time, random rotation of small particles in protoplanetary disks is rapidly damped away. For example, if a particle rotates through collisions with other particles, this rotation is gone on the order of one gas-grain friction time, which for bodies smaller than a meter is much shorter than the orbital timescale. Typical values for dust particles in the inner disk would be on the order of seconds, depending on the model and specific location in detail. Therefore, to retain a rotation, a constant torque around a given axis has to be applied to the particle. There are only two effects that can lead to torques: 1. Particle motion with respect to the gas, i.e., radial drift. In analogy to particles in Earth's atmosphere (like snowflakes), particle rotation will be around the drift axis. For the small particles considered here, this relative motion can be re-garded as purely radial and transversal motion is not important (Weidenschilling 1977). A windmill might be an appropriate visualization for this. 2. Radiation-induced torques. In principle this might be regarded in analogy to gas drag (assuming photons instead of gas molecules) and any rotation is around the direction of illumination. In both cases, a torque around another axis might initially be present but this will align the particles and only the systematic torques around the drift direction or radiation direction pertain. In both cases a potential rotation axis is oriented toward the star. This will always adjust during the orbit. This rotation does not change the front and back side and does not influence radial photophoresis. Details with quantitative estimates can be found in Krauss et al. (2007). Prospects A more realistic description of the disk's structure requires one to account for irradiation by the Sun at the disk atmosphere's surface (D'Alessio et al. 1998;Hueso & Guillot 2005;Garaud & Lin 2007;Cabral et al. 2010). Comparisons between irradiated and non irradiated models show that, for similar disk parameters, the midplane temperature in the outer part of the disk becomes substantially higher (up to a few dozen of K) in the first series of cases (Garaud & Lin 2007;Cabral et al. 2010). In order to estimate the influence of irradiation on photophoretic transport, we need to know not only the temperature profile, but also the pressure and volume density in the midplane of the disk. As a first attempt, we increased the temperature by 30 K or up to 100 K for any initial temperature lower than 100 K in our nominal disk model, to mimic the difference in temperature caused by irradiation. The trajectories of larger particles (10 −3 -10 −1 m) remain almost similar to those plotted in Figs. 3-6, except for the maximum distances that are 2 to 5 AU closer to the Sun. The trajectories of small particles (10 −5 m) are more affected by the temperature difference because their maximum migration distance is about half that shown in Figs. 3-6. Modifying the midplane density to keep the product T × ρ constant roughly doubles the effect. However, both the photophoretic force and gas drag depend on the gas pressure and density and a fully consistent irradiated disk model will be needed to investigate the real influence of irradiation on the trajectories of transported particles. An interesting evolution of this work would also be to consider explicitly the motion of dust particles in both radial and vertical directions. Indeed, the disk's upper layers are more transparent than those close to the midplane (because of dust sedimentation), and are therefore a perfect place for photophoresis to be effective in the earliest phase of the disk's evolution and prior to the "transition disk" phase. At this epoch turbulence driven by active magnetorotational instability (MRI) may even be a useful ingredient because it would help a fraction of particles to be maintained above the photosphere and be transported outward thanks to both photophoeresis and radiation pressure. Indeed, computations of turbulence in MRI disks (Turner et al. 2010) have shown that turbulence is increasingly effective with scale-height, and consequently, may help to maintain particles high above. This may be a potentially interesting mechanism that would help photoporesis to be effective even in the youngest ages of the disk. CNES. G.W. acknowledges support by the DFG (SPP 1385). We acknowledge an anonymous Referee whose useful comments allowed us to strenghten our manuscript. Fig. 2 . 2Velocities of particles due to photophoresis (a), radiation pressure (b), residual gravity (c), and accretion flow (d) represented as a function of time and size in the midplane of the disk characterized by a mass of 0.03 M and a lifetime of 6 Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 . 3456Position of particles of size 10 −5 to 10 −1 m, as a function of time for disks with masses of 1, 3, or 10 MMSN and lifetimes of 1 or 6 Myr. The density of particles is 500 kg m −3 and the radius of the inner gap is 1 AU. Same as inFig. 3, but for an inner gap radius of 2 AU. Position of particles of size 10 −5 to 10 −1 m, as a function of time for disks with masses of 1, 3, or 10 MMSN and lifetimes of 1 or 6 Myr. The density of particles is 1000 kg m −3 and the radius of the inner gap is 1 AU. Same as inFig. 5but for an inner gap radius of 2 AU.Moudens et al.: Photophoretic transport of hot minerals in the solar nebula 9 Fig. 1 . 1Temperature, pressure and surface density profiles in the midplane of the disk characterized by a mass of 0.03 M and a lifetime of 6 Myr. From top to bottom in each panel, times are 1 Fig. 7 . 7Heliocentric distance reached by aggregates at the end of the solar nebula evolution as a function of the disk parameters and of the particle sizes. Density of all particles is 500 kg m −3 and the inner gap of the disk is 1 AU (top) and 2 AU (bottom). Acknowledgements. We thank Y. 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[ "Prepared for submission to JHEP D 0 →pe + Form Factors in LCSR", "Prepared for submission to JHEP D 0 →pe + Form Factors in LCSR" ]	[ "Anshika Bansal [email protected] \nPhysical Research Laboratory\n380009AhmedabadIndia\n\nIndian Institute of Technology\n382424GandhinagarIndia\n" ]	[ "Physical Research Laboratory\n380009AhmedabadIndia", "Indian Institute of Technology\n382424GandhinagarIndia" ]	[]	Baryon number is conserved in the Standard Model (SM) of particle physics. Baryon Number Violation (BNV) is one of the criteria to explain the matter anti-matter asymmetry of the universe. Various beyond the SM (BSM) scenarios motivate BNV. In that view, it becomes important to look for various BNV decays. In this work, we consider the BNV decay of heavy charmed meson to anti-proton and positron. This decay proceeds via the dim-6 SM effective field theory (SMEFT) operator and involves 12 independent form factors. We give estimates for these form factors in the framework of Light Cone Sum Rules (LCSR). As the interpolation current for the proton state is not unique, we have considered different forms of proton interpolation current and discuss the effect of the choice of the interpolation current.	null	[ "https://arxiv.org/pdf/2205.13564v1.pdf" ]	249,151,799	2205.13564	6d05f466a241ea5ca2856bfdd007d785305c94a4	Prepared for submission to JHEP D 0 →pe + Form Factors in LCSR 26 May 2022 Anshika Bansal [email protected] Physical Research Laboratory 380009AhmedabadIndia Indian Institute of Technology 382424GandhinagarIndia Prepared for submission to JHEP D 0 →pe + Form Factors in LCSR 26 May 20221 Corresponding author. Baryon number is conserved in the Standard Model (SM) of particle physics. Baryon Number Violation (BNV) is one of the criteria to explain the matter anti-matter asymmetry of the universe. Various beyond the SM (BSM) scenarios motivate BNV. In that view, it becomes important to look for various BNV decays. In this work, we consider the BNV decay of heavy charmed meson to anti-proton and positron. This decay proceeds via the dim-6 SM effective field theory (SMEFT) operator and involves 12 independent form factors. We give estimates for these form factors in the framework of Light Cone Sum Rules (LCSR). As the interpolation current for the proton state is not unique, we have considered different forms of proton interpolation current and discuss the effect of the choice of the interpolation current. Introduction According to Sakharov's conditions [1], baryon number violation (BNV) is an important criterion to explain the matter-anti-matter asymmetry of the universe. But the baryon number turns out to be an accidental symmetry in the Standard Model (SM) of particle physics. This leads us to look for the physics beyond the SM (BSM). In order to do that, one can look for the BNV processes at the experimental facilities as they will be a clear signature of new physics. Till date, we have not observed any BNV process, but the experiments put very stringent bounds on the decay widths of several BNV processes like proton decay, decays of heavy mesons to baryons, etc [2]. Theoretically, BNV processes are well motivated in various BSM theories like Grand Unification Theories (GUTs), super Symmetry (SUSY), models of baryogenesis, etc (see for eg. [3][4][5][6][7][8], and reference therein). In these scenarios, the BNV processes are possible via the exchange of heavy gauge vector or scalar bosons like leptoquarks. In the framework of SM effective field theory (SMEFT), these processes can be studies using BNV higher dimension operators which can be obtained by integrating out these heavy particles. Out of all the BNV processes, proton decay has got the most attention so far, theoretically (see for eg. [9][10][11] and references therein) as well as experimentally (see for eg. [12,13] and references therein) . With the advances in the experimental facilities, it becomes important that one pays proper theoretical attention to -1 -other processes as well. Very recently, BESIII collaboration updated the upper limit on the branching fraction of D 0 →pe + to be < 1.2 × 10 −6 [14]. To the best of our knowledge, there are no theoretical estimates for this decay mode so far. In order to get the theoretical estimates, one needs an input for the form factors (FFs) involved in this process. In the present work, we attempt to get an estimate for these FFs using the method of Light Cone Sum Rules (LCSR). The rest of the paper is organised as follows; In Section-2, we describe the general parameterisation of the decay amplitude for the process in terms of the FFs. In Section-3, we derive these FFs in the framework of LCSR and provide the numerical analysis in Section-4. Finally, we conclude our findings and discuss the results in Section-5. This article includes 4 Appendices. In Appendix-A, we provide the useful identities and integrals involved. In Appendix-B, we provide the analytic expressions of the correlation functions involved for a general interpolation current for the proton state. Finally, we collect the numerical values of all the parameters involved in the calculations in Appendix-C. Amplitude parametrisation As discussed above, BNV processes can be calculated in the BSM scenarios using the higher dimensional effective operators. In the SMEFT, there are 4 types of dimension-6 operators which can lead to BNV processes. These operators respect the SM gauge group but violate the baryon number which is an accidental symmetry of the SM. These BNV operators can be written as [15][16][17] O duq ijkl = abc αβ d a i Cu b j q αc k C β l , O qque ijkl = abc αβ q αa i Cq βb j (u c k Ce l ) O qqq ijkl = abc αβ γδ q αa i Cq betab j q γc k C δ l , O duue ijkl = abc d a i Cu b j (u c k Ce l ) (2.1) Here, {i, j, k, l} represent the flavour indices, {a, b, c} are the color indices, {u, d} represent the right-handed up and down quarks and {q, } represent the left-hand doublets of quarks and leptons. The Einstein's convention of summation over repeated indices is adopted. Using these operators and the generalised Fierz transformations [18], the baryon number violating lagrangian which will contribute to D 0 →pe + can be written as, L (6) / B = Γ,Γ c A ΓΓ O A ΓΓ = ΓΓ c A ΓΓ ijk d T i CP Γ Γ A u j e T CP Γ Γ A c k (2.2) Here, C = iγ 2 γ 0 is the charge conjugation matrix, superscript T represents the transpose, P Γ and P Γ are the chirality projection operators with {Γ, Γ } ∈ {L, R} and Γ A ∈ {1, γ µ , σ µν } with A ∈ {S, V, T }. c A ΓΓ are the Wilson coefficients 1 . The transition amplitude for D 0 →pe + is defined as the matrix element of this Lagrangian between the initial and the final states as A(D 0 (p D ) →p(p p )e + (p e )) = Γ,Γ c A ΓΓ e + (p e )p(p p ) O A ΓΓ D 0 (p D ) (2.3) 1 The Lagrangian is assumed to be extracted in terms of the physical fields at the charm scale and thus, c A ΓΓ 's also include all the flavour and usual RG running effects. -2 - The leptonic and the hadronic parts can be factorised here such that, A(D 0 (p D ) →p(p p )e + (p e )) = Γ,Γ c A ΓΓ v c e H A ΓΓ v p (p p ) (2.4) where,v c e is the spinor corresponding to the positron and H A ΓΓ v p (p p ) is the hadronic object of interest and is given by H A ΓΓ v p (p p ) = p(p p ) ijk d T i CΓ A P Γ u j Γ A P Γ c k D 0 (p D ) (2.5) This hadronic object can be generally parameterized as, H A ΓΓ v p (p p ) = P Γ F A,0 ΓΓ (p 2 e ) + / vF A,1 ΓΓ (p 2 e ) v p (p p ) (2.6) Here, F A,n ΓΓ (p 2 e ) are the form factors with A ∈ {S, V, T } and n ∈ {0, 1}. As D-meson comprises of a heavy quark, we will treat it in the framework of heavy quark effective theory (HQET) [19,20]. v is the velocity of the D-meson such that p µ D = m D v µ with v 2 = 1. The parity conservation in QCD leads to the following relations between these FFs, F A,n LL = F A,n RR F A,n LR = F A,n RL (2.7) Hence, there are effectively 12 independent FFs. We will compute these FFs in the framework of the light cone sum rules (LCSR). Form Factors in LCSR LCSR is a technique to compute the hadronic objects of interest using the analytic properties of the correlation function involved in the process. Any LCSR computation involves four basic tools: dispersion relation, light cone operator product expansion, quark hadron duality and Borel transformation. We will not discuss them in detail here and suggest the interested readers to refer to [21][22][23][24]. To obtain the correlation function involved here, we first need to opt for an interpolation current for the anti-proton state. The interpolation current for the proton is not unique [25,26]. We choose a general form for the interpolation current as χ(x) = χ 1 (x) + tχ 2 (x) (3.1) where, χ 1 (x) and χ 2 (x) are defined as, χ 1 (x) = lmn u T l (x)Cγ 5 d m (x) u n (x), χ 2 (x) = lmn u T l (x)Cd m (x) γ 5 u n (x) (3.2) where, {l, m, n} are the color indices. The interpolation current is defined such that p(p p )\|χ(0)\|0 = m p λ p v p (p p ) where, λ p is a measure of the strength with which this current couples with the proton/antiproton state. -3 - We interpolate the proton state using this current in Eq.(2.5) and get the relevant correlation function which read as Π A ΓΓ = i d 4 x e ipe.x 0 T {χ(0)Q A ΓΓ (x)} D 0 (v) (3.3) where, T denotes the time ordering, Q A ΓΓ (x) = ijk d T i CP Γ Γ A u j P Γ Γ A c k andχ(0) = χ † γ 0 . According to the matching condition of the LCSR, Π A,had ΓΓ (p p , p e ) = Π A,QCD ΓΓ (p p , p e ) (3.4) where, Π A,had ΓΓ (p p , p e ) and Π A,QCD ΓΓ (p p , p e ) are the hadronic and quantum chromodynamics (QCD) parameterizations of the correlation function, respectively. Here, the hadronic parameterization can be obtained by inserting the complete set of intermediate states. Once we separate the pole contribution coming from the proton state, we get, Π A,had ΓΓ = −m p λ pvp (p p ) [H ΓΓ ] v p (p p ) + . . . = iP Γ Π A,S ΓΓ (p 2 p , p 2 e ) + Π A,V ΓΓ (p 2 p , p 2 e )/ v + Π A,P ΓΓ (p 2 p , p 2 e ) / p p m p + Π A,V P ΓΓ (p 2 p , p 2 e ) / v / p p m p (3.5) The contributions coming from the continuum and the heavier states are represented here by ellipses. Π A,r ΓΓ are the scalar function of p 2 p and P 2 e = −p 2 e with r = {S, V, P, V P }. Using the dispersion relation, these functions can be written in terms of the spectral densities which can inturn be related to the imaginary part of the correlation function (for details look at [11]). Separating the ground state pole contribution coming from the proton state and the continuum and heavier states contributions, these spectral densities can be written as ρ A,r,had ΓΓ (s, P 2 e ) = λ p m 2 p δ(s − m 2 p )F A,r ΓΓ (s, P 2 e ) + 1 π Im Π A,r,had ΓΓ (s, P 2 e ) (3.6) Here, F A,r ΓΓ (s, P 2 e ) are related to F A,n ΓΓ (s, P 2 e ) for the on-shell proton, i.e. s = m 2 p , as − F A,S ΓΓ = F A,P ΓΓ = F A,0 ΓΓ − F A,V ΓΓ = F A,V P ΓΓ = F A,1 ΓΓ (3.7) The final dispersion relation for the correlation function in Eq.(3.5) reads as Π A,r,had ΓΓ (p 2 p , P 2 e ) = λ p m 2 p F A,r ΓΓ m 2 p − p 2 p + ∞ s h 0 ds 1 π Im Π A,r,had ΓΓ (s, P 2 e ) s − p 2 p (3.8) where, s h 0 is the continuum threshold. In order to get the QCD parameterization of the correlation function, we first need the time -4 - ordered product of Q A ΓΓ (x) andχ(0) which is given by T χ(0)Q A ΓΓ (x) = − lmn ijk P Γ Γ A c i (x) ū l (0)γ 5S d mj (x)P Γ Γ A S u nk (x) +ū l (0)Tr γ 5S d mj (x)P Γ Γ A S u nk (x) +t ū l (0)S d mj (x)P Γ Γ A S u nk (x)γ 5 +ū l (0)γ 5 Tr S d mj (x)P Γ Γ A S u nk (x) (3.9) Here,Γ = CΓC −1 and S q ij (x) is the quark propagator at the light like separations and is given by S q ij (x) = i/ x 2π 2 x 4 − qq 12 δ ij + . . . (3.10) considering quark to be mass-less. qq represents the quark condensates and the ellipses represents the higher terms involving one or more gluons and are not considered in the present analysis. Secondly, we need the matrix element of the quark bilinear which can be defined in terms of the distribution amplitudes of D-meson as [27] 0 \|ū α (0)[x, 0]c β (x)\| D 0 (v) = −if D m D 4 ∞ 0 dw e iwv.x (1 + / v) φ D + (w) − φ D + (w) − φ D − (w) 2v.x / x γ 5 βα (3.11) Here, f D is the decay constant of D-meson, φ D + (w) and φ D − (w) are the light cone distribution amplitudes (LCDAs) of D-meson. We use the exponential mode parameterization for the LCDAs of the D-meson [28] which reads as φ + D (w) = 1 w 2 0 e −w/w 0 , φ − D (w) = 1 w 0 e −w/w 0 (3.12) where, w 0 is a model input parameter. Using the above definitions and the integrals collected in Appendix-A, the correlation function in QCD reads as, Π A,QCD ΓΓ = iP Γ Π A,S,QCD ΓΓ (p 2 p , p 2 e ) + Π A,V ΓΓ (p 2 p , p 2 e )/ v + Π A,P ΓΓ (p 2 p , p 2 e ) / p p m p + Π A,V P ΓΓ (p 2 p , p 2 e ) / v / p p m p (3.13) Here, Π A,S,QCD ΓΓ (p 2 p , p 2 e ) with r = {S, V, P, V P } are the analytic functions of p 2 p and p 2 e . The expressions for these functions are rather lengthy and hence we collect then in Appendix-B for both ΓΓ = LL and ΓΓ = LR cases. Now to derive the sum rules, we make use of the quark hadron duality to write the contribution of continuum and heavier states in the spectral density in terms of QCD calculated spectral functions as, ∞ s h 0 ds 1 π Im Π A,r,had ΓΓ (s, P 2 e ) s − p 2 p ≈ ∞ s 0 ds 1 π Im Π A,r,QCD ΓΓ (s, P 2 e ) s − p 2 p (3.14) -5 - where, s 0 is the continuum threshold and is not necessarily equals to s h 0 . It is a free parameter in the sum rule calculation. Finally, to get the final sum rule, we put all the things together and perform the Borel transformation to suppress the effect of continuum and heavier states. The statement of final sum rule reads as, F ΓΓ (P 2 e , s 0 , M 2 ) = − e m 2 p M 2 m 2 p λ p s 0 0 dse −s M 2 1 π Im Π A,r,QCD ΓΓ (s, P 2 e ) (3.15) where, M is the Borel mass and is another free parameter in the sum rule calculations. We will discuss how to choose the values of M and s 0 in the next section. Results The BNV process D 0 →pe + involves 12 independent form factors (FFs). We have studied these FFs as a function of P 2 e = −p 2 e and M 2 in the framework of LCSR using the distribution amplitudes of D-meson. As already discussed, the interpolation current for the proton state is not unique. To see the dependence of the FFs on the choice of interpolation current, we perform our numerical analysis for two different currents, χ IO (see Eq.(A.4) and Fig.(1 and 2)) and χ LA (see Eq.(A.5) and Fig.(3 and 4)) and provide the analytical form for the general current in Appendix-B. χ IO and χ LA are the general choices of the proton interpolation current in LCSR and lattice QCD calculations, respectively. In the sum rule, we have two independent parameters: the continuum threshold, s 0 and the Borel mass, M . The values of these are parameters are to be to be chosen such that the sum rule is saturated with the ground state contribution and the contribution coming from the continuum and the higher resonances should be well suppressed such that they do not contribute more than 30% to the result. In order to do that, we choose s 0 = (1.44 GeV) 2 , the Roper resonance. This is the next resonance state after proton with the quantum numbers of the proton state. We have also shown the effect of varying s 0 on different FFs by taking three different values of s 0 (See Fig.(1, 2, 3 and 4)). The Borel mass, M should be chosen such that the form factors are stable with the variation in M for a certain range of M called the 'Borel window'. We have found the FFs to be almost stable for M 2 > 2 GeV 2 . We have shown the stability of the FFs in the Borel window for, M 2 = (2 − 5) GeV 2 (See Fig.(1, 2, 3 and 4)). Each form factor can be calculated from two F A,r ΓΓ as given in Eq.(3.7). It has been found that some of these F A,r ΓΓ get contribution only from the condensate. Due to the presence of only condensate contributions in some F A,r ΓΓ , we found the difference in the extraction of the FFs using different combinations of F A,r ΓΓ . We tabulate these form factors in Table-1 and Table-2 for χ IO and χ LA , respectively at P 2 e = 0.5 GeV 2 . In Figs.(1,2,3 and 4), we have labelled these different combinations with (C) and (NC+C) for having only the condensate contribution and having condensate as well as non-condensate contributions, respectively. The two FFs, F T,0 LR and F T,1 LR are found to be explicitly zero. We have also calculated the errors in the FFs associated with the uncertainties in the values of the parameters used for the numerical analysis. We have found that the errors are very large (even upto 200%) in some cases as can be seen from Table-1 and Table-2. In Fig.(5) and Fig.(6), we show some representative graphs showing the variation of error in F A,r ΓΓ with P 2 e for the proton interpolations currents, χ IO and χ LA , respectively. The errors are found to be dominated by the uncertainty in w 0 which is a model input parameter in the LCDAs of D-meson. F V,0 LL F V,S LL 0.360 ± 0.467 F V,0 LR F V,S LR 0.277 ± 0.227 F V,P LL 0.099 ± 0.036 F V,P LR 0.043 ± 0.077 F V,1 LL F V,V LL 0.271 ± 0.097 F V,1 LR F V,V LR −0.251 ± 0.341 F V,V P LL −0.067 ± 0.154 F V,V P LR 0.078 ± 0.067 F T,0 LL F T,S LL 1.114 ± 0.812 F T,0 LR F T,S LR 0 F T,P LL 0.550 ± 0.256 F T,P LR 0 F T,1 LL F T,V LL 1.378 ± 0.629 F T,1 LR F T,V LR 0 F T,V P LL 0.156 ± 0.133 F T,V P LR 0 Conclusions and Discussion The form factors involved in the BNV process D 0 →pe + are calculated in the framework of light cone sum rules using the distribution amplitudes of D-meson. We have found that this process involves 12 independent FFs, each of which can be extracted from two F A,r ΓΓ . The extraction from the two does not match completely. There might be two reasons for that. Firstly, some of F A,r ΓΓ consists of only condensate contribution while others have both condensate and non-condensate contributions. The case where both the combinations have condensate as well as non-condensate contributions, like for F S,0 LL and F S,1 LL for χ LA case (see Fig.(3)), the extractions from the two combinations are close to each other. Secondly, as the LCSR predictions are more trustworthy at large P 2 e , the extractions from different combinations might be different at low P 2 e . At large P 2 e , they seem to be approaching each other. The similar analysis using different methods like lattice QCD will shed some light on the validity of different extractions. The higher order effects are also required to be included to get better clarity in future works. Also, the errors are found to be very large which are mainly dominated by the uncertainty in the model input parameter, w 0 in the LCDAs of D-meson. Better understanding of these LCDAs are required to required to get better understanding of these FFs. In this work we have taken a first step in computing FFs for baryon number violating decays of the heavy charm meson: D 0 →pe + . Other modes like D 0 →Λe + can be studied straightforwardly using the same method. As experimental searches improve, it is required to have first estimates of these non-perturbative inputs. Even though FFs extracted from two combinations do not match in some cases, they are numerically within the error bars of each other and thus provide a reasonable estimate. These can be used in a specific model framework where c A ΓΓ 's are known functions of heavy particle masses and couplings to obtain the bounds on the parameters of the theory. Acknowledgments I thank Namit Mahajan for fruitful discussions. A Useful definitions and integrals A.1 Proton interpolation current As discussed in Section-3, the interpolation current, χ(x) for the proton state is not unique. The general form for χ(x) is adopted in Eq.(3.1) as χ(x) = χ 1 (x) + tχ 2 (x) (A.1) with χ 1 (x) and χ 2 (x) defined in Eq.(3.2) as χ 1 (x) = lmn u T l (x)Cγ 5 d m (x) u n (x), χ 2 (x) = lmn u T l (x)Cd m (x) γ 5 u n (x) (A. 2) In general, the interpolation current used in LCSR calculation is given as χ IO (x) = 2 (χ 2 (x) − χ 1 (x)) = lmn u T l (x)Cγ µ u m (x) γ 5 γ µ d n (x) (A.3) such that, p(p p )\|χ IO \|0 = m p λ p1 v p (p p ). This current is popularly known as the Ioffe current. This current can be obtained by taking t = −1 and multiplying χ(x) by -2 i.e. χ IO (x) = −2χ(x) with t = −1. (A.4) However, the usual form of interpolation current used for lattice QCD computations, χ LA (x) is given by χ(x) with t = 0 i.e. χ LA (x) = χ(x) with t = 0 (A.5) such that p(p p )\|χ LA \|0 = m p λ p2 v p (p p ). A.2 Useful Integrals In this section we collect all the useful integrals required for the derivation of the sum rule for the FFs involved or simply the calculation of the correlation function in QCD. The general formula for the integrals in D-dimension which usually appear in the sum rule calculations is [22] d D xe ip.x 1 (x 2 ) n = (−i) (−1) n 2 (D−2n) π D/2 −p 2 n−D/2 Γ (D/2 − n) Γ(n) (A.6) for n ≥ 1 ,p 2 < 0. The desired integrals for the present case can be obtained by differentiating it with respect to the four momentum p α . d 4 x e ip.x x α x 6 = −π 2 4 p α ln(−p 2 ), d 4 x e ip.x 1 x 6 = −iπ 2 8 p 2 ln(−p 2 ) d 4 x e ip.x x α x 4 = 2π 2 p α p 2 , d 4 x e ip.x 1 x 2 = −4iπ 2 p 2 d 4 xe ipx x α x β x 8 = −iπ 2 48 p 2 g αβ + 2p α p β ln(−p 2 ) (A.7) -9 - The divergent terms, proportional to p 2 are omitted here as they vanishes upon Borel transformation. B Correlation Functions In this section we collect all the correlation functions calculated in QCD for different combinations of Γ, Γ and A with P 2 = (p e + wv) 2 = ((w + m D )v − p p ) 2 = w(w + m D ) − ws m D − w + m D m D P 2 e (B.1) where, s = p 2 p and P 2 e = −p 2 e . Also, as v 2 = 1 (v.P ) = −(v.p p − (w + m D )) = 2w + m D 2 − s + P 2 e 2m D (B.2) B.1 Case-1: P Γ = P Γ = P L • For Γ A = 1 Π S,S LL = f D m D 8 ∞ 0 dw (t − 1) 4π 2 (w + m D )Φ D ± (w) + P 2 φ D + (w) ln(−P 2 ) + qq (t − 1) 3 (w + m D )φ D + (w) P 2 + Φ D ± (w) P 2 (B.3) Π S,V LL = f D m D 8 ∞ 0 dw 3(t + 1) 8π 2 (w + m D )Φ D ± (w) + P 2 φ D + (w) ln(−P 2 ) − qq (t − 1) 3 (w + m D )φ D + (w) P 2 + Φ D ± (w) P 2 (B.4) Π S,P LL = −m p f D m D 8 ∞ 0 dw 3(t + 1) 8π 2 Φ D ± (w)ln(−P 2 ) − qq (t − 1) 3 φ D + (w) P 2 (B.5) Π S,V P LL = −m p f D m D 8 ∞ 0 dw (t − 1) 4π 2 Φ D ± (w)ln(−P 2 ) + qq (t − 1) 3 φ D + (w) P 2 (B.6) • For Γ A = γ µ Π V,S LL = f D m D 4 ∞ 0 dw (t − 1) 8π 2 (w + m D )Φ D ± (w) + P 2 φ D + (w) ln(−P 2 ) + qq 3 (t − 1)Φ D ± (w) P 2 + (3 + t)(w + m D )φ D + (w) P 2 − 4(v.P )φ D + (w) P 2 (B.7) Π V,V LL = − f D m D qq 12 ∞ 0 dw (3 + t)Φ D ± (w) P 2 − (w + m D ) P 2 2φ D − (w) − (t + 1)φ D + (w) (B.8) Π V,P LL = −m p f D m D qq 12 ∞ 0 dw 1 P 2 2φ D − (w) − (t + 1)φ D + (w) (B.9) Π V,V P LL = −m p f D m D 4 ∞ 0 dw (t − 1) 8π 2 Φ D ± (w)ln(−P 2 ) + qq (t + 3) 3 φ D + (w) P 2 (B.10) • For Γ A = σ µν Π T,S LL = f D m D qq (t − 1) 6 ∞ 0 dw 1 P 2 4φ D + (w)(v.P − (w + m D )) + 3Φ D ± (w) (B.11) Π T,V LL = f D m D 2 ∞ 0 dw (t + 1) 4π 2 φ D + (w) 6 4(w + m D )(v.P ) − P 2 + 3 2 Φ D ± (w)(w + m D ) ln(−P 2 ) − qq (t − 1) 3 (w + m D )(φ D + (w) + 2φ D − (w)) P 2 − Φ D ± (w) P 2 (B.12) Π T,P LL = −m p f D m D 2 ∞ 0 dw (t + 1) 4π 2 3 2 Φ D ± (w) + 2(v.P ) 3 φ D + (w) ln(−P 2 ) − qq (t − 1) 3 (φ D + (w) + 2φ D − (w) P 2 (B.13) Π T,V P LL = m p f D m D qq (t − 1) 6 ∞ 0 dw φ D + (w) P 2 (B.14) B.2 Case-2: P Γ = P L and P Γ = P R • For Γ A = 1 Π S,S LR = f D m D 8 ∞ 0 dw 3(t + 1) 8π 2 (w + m D )Φ D ± (w) + P 2 φ D + (w) ln(−P 2 ) − qq (t − 1) 3 (w + m D )φ D + (w) P 2 + Φ D ± (w) P 2 (B.15) Π S,V LR = f D m D 8 ∞ 0 dw (t − 1) 4π 2 (w + m D )Φ D ± (w) + P 2 φ D + (w) ln(−P 2 ) + qq (t − 1) 3 (w + m D )φ D + (w) P 2 + Φ D ± (w) P 2 (B.16) Π S,P LR = −m p f D m D 8 ∞ 0 dw (t − 1) 4π 2 Φ D ± (w)ln(−P 2 ) + qq (t − 1) 3 φ D + (w) P 2 (B.17) Π S,V P LR = −m p f D m D 8 ∞ 0 dw 3(t + 1) 8π 2 Φ D ± (w)ln(−P 2 ) − qq (t − 1) 3 φ D + (w) P 2 (B.18) -11 - • For Γ A = γ µ Π V,S LR = f D m D qq 12 ∞ 0 dw φ D + (w) P 2 {(w + m D ) − (t + 3)(v.P )} − (t + 2)Φ D ± (w) P 2 (B.19) Π V,V LR = − f D m D 4 ∞ 0 dw (t − 1) 4π 2 (w + m D )Φ D ± (w) + φ D + (w) 6 P 2 + 2(v.P )(w + m D ) ln(−P 2 ) − qq 3 (w + m D ) P 2 φ D − (w)(t + 3) + φ D + (w)(t + 1) − Φ D ± (w) P 2 (B.20) Π V,P LR = m p f D m D 4 ∞ 0 dw (t − 1) 4π 2 φ D + (w)(v.P ) 3 + Φ D ± (w) ln(−P 2 ) − qq 3P 2 φ D − (w)(t + 3) + φ D + (w)(t + 1) (B.21) Π V,V P LR = −m p f D m D qq 6 ∞ 0 dw φ D + (w) P 2 (B.22) • For Γ A = σ µν : All the correlation functions are zero. Extracting the imaginary of part of these correlation functions is rather easy as, Im 1 P 2 − m 2 = −πδ(P 2 − m 2 ) ln(−x) = ln\|x\| − iπθ(x) (B.23) where, m is some mass scale and is zero our case. Hence, to get the imaginary part, just replace ln(−P 2 ) by −πθ(P 2 ) and 1 P 2 by −πδ(P 2 ) in the correlation functions. C Numerical Values of parameters used In this appendix, we collect all the numerical values of the parameters used during numerical analysis 2 . S.No. Parameter Value Used Reference 1. Proton mass (m p ) 0.938 GeV [2] 2. Quark condensate ( qq ) −((256 ± 2)MeV) 3 [10] 3. D-meson decay constant, f D (0.212 ± 0.001) GeV 2 [2] 4. D-meson mass, m D 1.864 GeV [2] 5. λ p1 (−0.027 ± 0.009) GeV 3 [29] 6. λ p2 (−0.013 ± 0.004) GeV 3 [26] 7. w 0 (0.45 ± 0.3) GeV [30] 2 The decay constant for D 0 meson is not known. We have used the decay constant for D + meson here. -12 - 1 : 1Tabulation of all the 12 independent FFs at P 2 e = 0.5 GeV 2 for s 0 = (1.44 GeV) 2 and M = 2 GeV calculated using the proton interpolation current χ IO . The errors are associated with the errors in the parameters used for the numerical analysis. Table 2 : 2Tabulation of all the 12 independent FFs at P 2 e = 0.5 GeV 2 for s 0 = (1.44 GeV) 2 and M = 2 GeV calculated using the proton interpolation current χ LA . The errors are associated with the errors in the parameters used for the numerical analysis. FFFFFFFFFFFFFigure 1 :FFFFFFFFigure 2 :FFFFFFFFFFFFigure 3 :FFFFFFFFigure 4 :Figure 5 :Figure 6 : 123456S,P LL C -F S,S LL NC + C S,P LL C -F S,S LL NC + C S,VP LL NC + C -F S,V LL C S,VP LL NC + C -F S,V LL C V ,P LL C -F V ,S LL NC + CC V ,P LL C -F V ,S LL NC + C V ,VP LL NC + C -F V ,V LL C V ,VP LL NC + C -F V ,V LL C T,P LL NC + C -F T,S LL C T,P LL NC + C -F T,S LL C T,VP LL C -F T,V LL NC + C T,VP LL C -F T,V LL NC + C The FFs, F A,n LL with A ∈ {S, V, T } and n ∈ {0, 1} are extracted from different combinations of F A,r LL with r ∈ {S, V, P, V P } using the proton interpolation current χ IO . Left panel: We plot F A,n LL vs P 2 e for s 0 = (1.4 GeV) 2 (dashed), s 0 = (1.44 GeV) 2 (solid) and s 0 = (1.5 GeV) 2 (dotted) with M = 2GeV . Right panel: We plot F A,n LL vs M 2 for P 2 e = 0.1 GeV 2 (dashed), P 2 e = 0.5 GeV 2 (solid) and P 2 e = 1 GeV 2 (dotted) with s 0 = (1.44GeV) 2 . -15 -F S,P LR NC + C -F S,S LR C S,P LR NC + C -F S,S LR C S,VP LR C -F S,V LR NC + C S,VP LR C -F S,V LR NC + C V ,P LR NC + C -F V ,S LR C V ,P LR NC + C -F V ,S LR C V ,VP LR C -F V ,V LR NC + C V ,VP LR C -F V ,V LR NC + C Same as Fig.(1) for F A,n LR extracted from F A,r LR . -16 -F S,P LL NC + C -F S,S LL NC + C S,P LL NC + C -F S,S LL NC + C S,VP LL NC + C -F S,V LL NC + C S,VP LL NC + C -F S,V LL NC + C V ,P LL C -F V ,S LL NC + C V ,P LL C -F V ,S LL NC + C V ,VP LV C -F V ,V LL NC + C V ,VP LL C -F V ,V LL NC + C T,P LL NC + C -F T,S LL C T,P LL NC + C -F T,S LL C T,VP LL C -F T,V LL NC + C T,VP LL C -F T,V LL NC + C Same as Fig.(1) but for interpolation current, χ LA . -17 -F S,P LR NC + C -F S,S LR NC + C S,P LR NC + C -F S,S LR NC + C S,VP LR NC + C -F S,V LR NC + C S,VP LR NC + C -F S,V LR NC + C V ,P LR NC + C -F V ,S LR C V ,P LR NC + C -F V ,S LR C V ,VP LR C -F V ,V LR NC + C V ,VP LR C -F V ,V LR NC + C Same as Fig.(3) for F A,n LR extracted from F The representative graphs showing the variation of errors with P 2 e for some F A,r ΓΓ functions calculated using the proton interpolation current, χ IO . 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[]	[ "Daniel Le And ", "Bao V Le Hung " ]	[]	[]	We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mézard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of Qp with small regular Hodge-Tate weights.	null	[ "https://arxiv.org/pdf/2203.02045v1.pdf" ]	247,244,689	2203.02045	a728dbdfdd0b0796d2f1b71af3e279d093b792fb	3 Mar 2022 Daniel Le And Bao V Le Hung 3 Mar 2022SERRE WEIGHTS, GALOIS DEFORMATION RINGS, AND LOCAL MODELS We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mézard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of Qp with small regular Hodge-Tate weights. Introduction The study of congruences between automorphic forms has a long and rich tradition. A paradigm shift occurred when Deligne's construction of Galois representations attached to classical holomorphic Hecke eigenforms opened the door to the study of congruences of automorphic forms through congruences of Galois representations. In fact, conjectures of Fontaine-Mazur-Langlands and Serre suggest that these are really two sides of the same coin. Let p be a prime. Recall that Serre's conjecture asserts that every continuous, odd, and irreducible Galois representation ρ : G Q → GL 2 (F p ) of the absolute Galois group of Q is modular, i.e. arises from the reduction of a Galois representation attached to a classical modular form. It furthermore asserts a refinement which specifies the minimal weight and level at which one may find such a modular form in terms of local properties of ρ. Turning the perspective around, if one begins with a modular ρ, then the refinement predicts congruences between modular forms of many different levels and weights. While the level part generalizes quite naturally, the weight part is subtler, because it turns out to be inextricably linked to integral p-adic Hodge theory. The goal of this survey is to describe the circle of ideas surrounding recent developments on the weight part of Serre's conjecture. 1.1. Overview. In §2, we articulate the main questions of interest in the context of cohomological automorphic forms, especially motivating the representation theoretic perspective on congruences. In §3 and 4, we briefly discuss background on modular representation theory and Galois representations. This will be necessary to state the higher dimensional generalizations of the weight part of Serre's conjecture. In §5, we state generalizations of Serre's conjecture due to Ash, Gee, Herzig, and Savitt among many others which provide conjectural answers to these questions in many cases. In §5.3, we narrow our focus to two cases, definite unitary groups in joint work with B. Levin and S. Morra and GL n over CM fields, where we established some conjectures of [GHS18] when ρ is tame and sufficiently generic at p. In the proofs, the Kisin-Taylor-Wiles method plays a vital role in reducing a global problem to a local one. As the internal details of the Kisin-Taylor-Wiles method are orthogonal to our goals, we have chosen to axiomatize its essential output and explain how it is used. Finally, §6 summarizes the key results on local models for local deformation rings which are the essential ingredients to prove the results on the weight part of Serre's conjecture. 1.2. Acknowledgments. This survey is based on two talks given by the authors at the International Colloquium on Arithmetic Geometry at TIFR in January 2020. We thank the organizers for the invitation to this wonderful mathematical and cultural event. D.L. was supported by the National Science Foundation under agreement No. DMS-1703182. B.LH. acknowledges support from the National Science Foundation under grant Nos. DMS-1128155, DMS-1802037 and the Alfred P. Sloan Foundation. Cohomology of arithmetic manifolds Let G be a connected reductive group over Q. Let A • ∞ be the connected component of the group of R-points of a maximal Q-split torus in the center of G, and let K • ∞ be a maximal connected compact subgroup of G(R). For a compact open subgroup K f ⊂ G(A), consider the adelic double quotient Y (K f ) def = G(Q)\G(A)/A • ∞ K • ∞ K f . In various places, we require technical properties of K f that can always be attained by passing to a finite index subgroup. Furthermore, these required properties are preserved under passing to a finite index subgroup (away from a finite set of places). We assume throughout that K f is sufficiently small (cf. [LLHLM20a, §9.1]). In particular, K f is neat so that Y (K f ) is naturally a real manifold. Moreover, Y (K f ) can be rewritten as a finite disjoint union of quotients of the symmetric space G(R)/A • ∞ K • ∞ by subgroups of G(Q) which are discrete and of finite covolume in G(R). Finally, Y (K f ) is homotopy equivalent to its Borel-Serre compactification (cf. [BS73]), a compact real manifold with corners and, in particular, a finite CW complex. 2.1. Rational cohomology. Let V be an algebraic representation of G over Q. Then let V Q be the Q-local system G(Q) (G(A)/A • ∞ K • ∞ K f ) × V (Q) , where G(Q) acts diagonally. Then the (finite-dimensional) sheaf cohomology groups H * (Y (K f ), V Q ) have a convolution action by the double coset (Hecke) algebra Q[K f \G(A f )/K f ]. (We will consider sequences of cohomology groups as objects in appropriate bounded derived categories. In most instances, little is lost if H * (Y (K f ), V Q ) is replaced by ⊕ i∈Z H i (Y (K f ), V Q ).) The significance of these cohomology groups stems from the fact that the Hecke module H * (Y (K f ), V Q ) ⊗ Q C can be computed by automorphic forms by work of Matsushima and Franke [Mat67,Fra98]. This is essentially Hodge theory for the locally symmetric manifolds Y (K f ). Suppose that A • ∞ acts on V (C) through a character. If we let A(K f ) denote the space of automorphic forms on Y (K f ), then (2.1) H * (Y (K f ), V Q ) ⊗ Q C ∼ = H * (g, p, (A(K f ) ⊗ C V (C)) A • ∞ ), where g is the Lie algebra of the group of real points of the intersection of all kernels of rational characters of G. Let V C be the local system (G(Q)\G(A)/K f ) × V (C) A • ∞ K • ∞ , where the right action of A • ∞ K • ∞ is diagonal with the inverse of the left action on V (C). Then the bijection G(A) × V (C) → G(A) × V (C) (2.2) (g, v) → (g, g −1 ∞ v) (2.3) induces an isomorphism V Q ⊗ Q C ∼ → V C . This is the first step in establishing (2.1). We will assume that we can write K f as K Σ K Σ where Σ is a finite set of finite places and K Σ = ℓ / ∈Σ K ℓ where for all ℓ / ∈ Σ, K ℓ is a hyperspecial subgroup of G(Q ℓ ) (in particular, we assume that G is unramified at places ℓ / ∈ Σ). Then T Σ Q def = Q[K Σ \G(A Σ )/K Σ ] is commutative (see §4.2). Since H * (Y (K f ), V Q ) are finite dimensional Q-vector spaces, the eigenvalues of the T Σ Q -action on the part of A(K f ) which contributes to H * (Y (K f ), V C ) (the cohomological automorphic forms) are algebraic numbers. 2.2. Classical modular forms. If G = GL 2 , then Y (K f ) is a modular curve and has the additional structure of a variety defined over Q. Any irreducible algebraic representation of GL 2 is of the form V (a, b) def = Sym a−b (Q 2 ) ⊗ Q det b . Let T ℓ ∈ Q[GL 2 (Z ℓ )\GL 2 (Q ℓ )/GL 2 (Z ℓ )] be the double coset operator GL 2 (Z ℓ ) ℓ 0 0 1 GL 2 (Z ℓ ). A well-known incarnation of (2.1) is that there is a normalized Hecke eigenform f (z) = ∞ n=0 a n q n , with q = e 2πiz (i.e. a 1 = 1) of weight k ≥ 2 and level K f (or K f ∩ SL 2 (Z)) if and only if there is a T Σ Q -eigenvector in H 1 (Y (K f ), V(b + k − 2, b)) such that T ℓ acts by ℓ b a ℓ for all ℓ / ∈ Σ. It is well-known that the space of modular forms has a basis with integral q-expansions whose Z-span is Hecke stable. In particular, (a ℓ ) ℓ are not just algebraic numbers, but are in fact algebraic integers. This gives one way to make the notion of congruences between eigenforms precise: one asks for a congruence between the (integral) Fourier coefficients. It turns out that there are a lot of congruences between q-expansions of integral Hecke eigenforms. A basic example comes from the Eisenstein series G k (z) def = − B k 2k + n≥1 σ k−1 (n)q n , k ≥ 4 even, where B k is the k-th Bernoulli coefficient. Fixing a (rational) prime p, the mod p q-series G k (mod p) depends only on k (mod p − 1). Another well-known example is the congruence (2.4) ∆(z) def = q ∞ m=1 (1 − q n ) 24 ≡ q ∞ m=1 (1 − q n ) 2 (1 − q 11n ) 2 (mod 11) between the unique normalized cuspforms of level Γ(1) and weight 12 and of level Γ 0 (11) and weight 2, respectively. The above notion of congruences between eigenforms is essentially equivalent to congruences between the system of Hecke eigenvalues on rational cohomology, and thus can also be detected by contemplating the action of (suitably integral) Hecke operators on cohomology with integral coefficients. It turns out that this shift of perspective from q-expansion to integral cohomology (initiated by Ash-Stevens) will give a systematic mechanism to explain congruences between automorphic forms via representation theory. 2.3. Integral structure. Fix a prime p and suppose that K f factors as the product K p f K p . We fix an algebraic closure Q p of Q p and let E be a subfield of Q p of finite degree over Q p . By replacing E if necessary, we will assume that E is sufficiently large. Let O be the ring of integers of E with uniformizer ̟ and F be the residue field. We define V E to be the nonarchimedean analogue (G(Q)\G(A)/A • ∞ K • ∞ K p f ) × V (E) K p of V C in §2.1, where K p acts diagonally (using the natural right action on G(Q)\G(A)/A • ∞ K • ∞ K p f and the inverse of the natural left action on V (E)). Then as before, the map G(A) × V (E) → G(A) × V (E) (2.5) (g, v) → (g, g −1 p v) (2.6) induces an isomorphism V Q ⊗ Q E ∼ → V E . As K p is a compact group, there exists a K p -stable O-lattice W in V (E). If we let (2.7) W def = (G(Q)\G(A)/A • ∞ K • ∞ K p f × W )/K p , then the map (2.8) H * (Y (K f ), W) → H * (Y (K f ), V E ) ∼ = H * (Y (K f ), V Q ) ⊗ Q E gives a natural integral structure on H * (Y (K f ), V Q ) ⊗ Q E.H * (Y (K f ), W)[ 1 p ] survives in the localization H * (Y (K f ), W) m [ 1 p ] if and only if its (automatically integral) system of Hecke eigenvalues lifts the mod p system given by m. However, for a fixed m, there may be various local systems W for which H * (Y (K f ), W) m is nonzero. Indeed, we saw in §2.2 that if m G k corresponds to (the system of Hecke eigenvalues of) the Eisenstein series G k (mod p) for 4 ≤ k ≤ p + 1 and W(a, b) corresponds to the lattice W (a, b) def = Sym a−b Z 2 p ⊗ det b for a ≥ b, then H * (Y (GL 2 ( Z)), W(k ′ − 2, 0)) m G k is nonzero for all k ′ ≡ k (mod p − 1). (Since GL 2 ( Z) is not neat, these cohomology groups should be interpreted as the cohomology groups of an orbifold.) Furthermore, if m ∆ corresponds to the Ramanujan Delta function mod 11, then both H * (Y (K f ), Z p ) m ∆ and H * (Y (GL 2 ( Z)), W(10, 0)) m ∆ are nonzero where K f corresponds to the congruence subgroup Γ 0 (11). The upshot of our discussion above is that congruences between eigenforms can be thought as the non-vanishing of localized cohomology for many different coefficient sheaves. Thus a complete classification of such congruences is equivalent to the following question: Question 2.4.1. Given a mod p Hecke eigensystem m, for which O-local systems W on Y (K f ) is H * (Y (K f ), W) m nonzero? Serre studied this question extensively in the case of GL 2 [Ser87]. This perspective of cohomology actually gives a natural explanation for the congruences for GL 2 in §2.2. We explain how it naturally leads to considerations in modular representation theory. From the short exact sequence 0 → W ·p → W → W ⊗ Zp F p → 0, we see that H * (Y (K f ), W) m is nonzero if and only if H * (Y (K f ), W ⊗ Zp F p ) m is. While W (a, b) ⊗ Zp Q p is an irreducible GL 2 (Z p )-module, W (a, b) ⊗ Zp F p is irreducible if and only if a−b ≤ p−1, in which case, we let F (a, b) def = W (a, b)⊗ Zp F p . All (absolutely) irreducible GL 2 (Z p )- modules over F p arise in this way, and F (a, b) ∼ = F (c, d) if and only if a − c = b − d ∈ (p − 1)Z. Let F(a, b) be the corresponding local system. Let us first revisit the congruences between Eisenstein series. For any a ′ > b, the submodule of W (a ′ , b) ⊗ Zp F p generated by a (nonzero) highest weight vector is isomorphic to F (a, b) where a is the unique integer such that 0 < a − b < p and a ≡ a ′ (mod p − 1). This gives a map H * (Y (GL 2 ( Z)), W(k − 2, 0) ⊗ Zp F p ) m G k → H * (Y (GL 2 ( Z)), W(k ′ − 2, 0) ⊗ Zp F p ) m G k where 2 < k < p + 2 and k ′ > 2 with k ′ ≡ k (mod p − 1). It can be shown (for example by applying Hida's ordinary projector) that these maps are injective in each degree. This illustrates how modular representation theory can be used to produce infinite families of congruences between Hecke eigensystems. The congruence (2.4) between cuspforms is simpler. Recall that here p = 11 and that G = GL 2 . Then the fact that m ∆ is non-Eisenstein implies that for all local systems W, H * (Y (K f ), W) m ∆ is zero unless * = 1. First, Shapiro's lemma now implies that H * (Y (K f ), W) m ∆ ∼ = H * (Y (GL 2 ( Z)), W ′ ) m ∆ , where W ′ corresponds to the principal series representation Ind GL 2 (Fp) B(Fp) W . Second, the functor W → H 1 (Y (GL 2 ( Z)), W) m ∆ is an exact functor from the category of finite Z 11 [[GL 2 (Z 11 )] ]-modules to the category of finite Z 11 -modules. Now Ind GL 2 (F 11 ) B(F 11 ) 1 is naturally identified with the space of F 11 -valued functions on P 1 (F 11 ) and decomposes as Sym 10 F 2 11 ⊕ 1. Then the injection H * (Y (GL 2 ( Z)), F(10, 0)) m ∆ ֒→ H * (Y (K f ), F p ) m ∆ provides the desired congruence. This example illustrates the important phenomenon of how the weight and level of modular forms can interact mod p. With these representation theoretic arguments in mind, Ash, Stevens, and others have suggested that one should narrow the focus of Question 2.4.1 to when K p is a maximal compact open subgroup and W is an irreducible F-local system. Question 2.4.2 (The weight part of Serre's conjecture). Suppose that K p is a maximal compact open subgroup. Given a mod p Hecke eigensystem m, for which irreducible F-local systems W on Y (K f ) is H * (Y (K f ), W) m nonzero? While this is a substantial reduction since there are only finitely many such F-local systems up to isomorphism, little is expected to be lost as we now explain. The following proposition is immediate. Proposition 2.4.3. If H * (Y (K f ), W) m is nonzero, then H * (Y (K f ), F) m is nonzero for some ir- reducible subquotient F of W ⊗ O F. The converse to Proposition 2.4.3 holds in non-Eisenstein cases for G a Weil restriction of GL n if expected vanishing conjectures hold (see §4.6). These vanishing conjectures generalize the vanishing outside of degree 1 for G = GL 2 . Question 2.4.2 turns out to be quite subtle. Nonisomorphic irreducible F-local systems may contain the same Hecke eigensystems, i.e. not all congruences arise from modular representation theory. For example, with p = 23, both H * (Y (GL 2 ( Z)), F(10, 0)) m ∆ and H * (Y (GL 2 ( Z)), F(21, 11)) m ∆ are nonzero. If we write ∆(z) = ∞ n=1 τ (n)q n , then T ℓ acts on H * (Y (GL 2 ( Z)), F(21, 11)) m ∆ by ℓ 11 τ (ℓ) for all primes ℓ = 23 by (2.1) (see §2.2). This implies that (2.9) τ (n) ≡ n 23 τ (n) (mod 23) for all n coprime to 23. In other words, 23 \| τ (n) if n 23 = −1. 3. An interlude on representation theory ]]-module factors through the finite quotient K p /K p (1), which can often be arranged to be a finite group of Lie type. In this section, we discuss the (modular) representation theory of these groups. Let G be a connected reductive group over F p which splits over F. (We will eventually take G to be the mod p reduction of an integral model of G.) An isomorphism class of a simple F[G(F p )]module is known as a Serre weight for G(F p ). Our goal now is to describe the (finite) set of Serre weights for G(F p ). Let B be an F p -rational Borel subgroup in G with Levi subgroup T . We denote by W the Weyl group N (T )/T , which has a Bruhat partial order with a unique longest element w 0 . We write X(T ) for the character group of T , which has an action of W and an induced action from the relative Frobenius F acting on T . This group has a subset For any character λ ∈ X(T ), we can consider the algebraic induction W (λ) def = Ind G B w 0 λ (also known as the dual Weyl module), which is nonzero if and only if λ is dominant with respect to B. We let L(λ) denote the socle of W (λ), which is the simple submodule generated by a nonzero highest weight vector. Then we have the following result about Serre weights for G(F p ). Theorem 3.1.1. The map X 1 (T ) (F − 1)X 0 (T ) → {Serre weights for G(F p )} (3.1) λ → L(λ)(F)\| G(Fp) (3.2) is a bijection. We denote L(λ)(F)\| G(Fp) by F (λ). (To avoid conflicts with the F -action on X(T ), we will write this action without parentheses.) 3.2. Deligne-Lusztig representations. While Question 2.4.2 only involves F-local systems, we will see that it is inextricably linked to O-torsion free local systems. It is then natural to ask for a classification of irreducible G(F p )-representations in characteristic 0. We now recall such a classification, provided by work of Deligne and Lusztig [DL76]. For an element w ∈ W , there exists g w ∈ G(F p ) such that g −1 w F (g w ) ∈ N (T )(F p ) represents w. Then we let T w be the F -stable torus g w T g −1 w . Let W denote the extended affine Weyl group which is the semidirect product X(T ) ⋊ W . For an element w = (µ, w) ∈ W , we define θ w : T w (F p ) → E × to be the restriction of the character T w (F p ) → Q × p g → [µ](g −1 w gg w ) to T w (F p ) (here, [µ] denotes the Teichmüller lift of µ). To a character θ w of a maximal rational torus T w (F p ) of G(F p ), Deligne and Lusztig associate a virtual (Deligne-Lusztig) representation over E which they denote ǫ G ǫ Tw R θ w Tw . We will instead denote this virtual representation by R( w) and say that w is a presentation for R( w). The map w → R( w) is not injective-two elements map to the same virtual representation if and only if they lie in the same orbit of the action of W on itself given by (ν, s) · (µ, w) = (sµ + F ν − swF (s) −1 (ν), swF (s) −1 ). The simplest case of the above construction occurs when w is the identity. Then T w = T , θ w is a character of T (F p ) and by inflation a character of B(F p ), and R( w) is the principal series representation Ind G(Fp) B(Fp) θ w . Nonuniqueness of presentations can be seen from the existence of intertwiners between principal series representations. The group W acts on X(T ) in the usual way-W acts on X(T ) by group automorphisms and X(T ) acts on itself by translation. Let m be a nonnegative integer and let 0 ∈ X(T ) denote the trivial character. We say that w ∈ W is (lowest alcove) m-generic if w(0), α ∨ > m for all simple roots α and w(0), α ∨ < p − m for all roots α ∨ . We say that a Deligne-Lusztig representation R is m-generic if R = R( w) for some m-generic w. An m-generic w or R exists only if mh < p, where h denotes the Coxeter number of G. [DL76, Proposition 10.10] implies that if R is 0-generic, then R is in fact a genuine representation. Let R denote the semisimplification of the reduction of any G(F p )-stable O-lattice in a genuine G(F p )-representation R over E (R does not depend on the choice of lattice). In relation to Question 2.4.2, it is important to have an understanding of R for Deligne-Lusztig representations R. This is provided by Jantzen's formula for the reductions of Deligne-Lusztig representations in terms of virtual linear combinations of dual Weyl modules [Jan81]. If R is sufficiently generic, then the Jordan-Hölder factors of R admit the following description in terms of alcove geometry, which is in a sense independent of p. For convenience, we assume that G admits a twisting element η ∈ X(T ), defined up to X 0 (T ), which by definition has the property that η, α ∨ = 1 for all simple roots α. The existence of an η can always be arranged by passing to a central extension of G by G m (see [BG14, Proposition 5.3.1(a)]). We write · for the p-dot action so that (ν, w) · λ = w(λ + η) − η + pν. See [LLHLM20a, §2] for any unexplained notation below. F (π −1 ( w 1 ) · ( w w −1 2 (0) − η)) with w 1 ∈ W restricted and dominant, w 2 ∈ W dominant, and w 1 ↑ (η, w 0 ) w 2 . Remark 3.2.2. Of course, the description in Proposition 3.2.1 does not depend on the choice of twisting element η and could in fact be rephrased without any reference to η. Relations to Galois representations In order to address Question 2.4.2 (and to explain the congruence (2.9)), we introduce some conjectures and results concerning the relationship between cohomological automorphic forms and Galois representations. We follow the approach in [Gro99], which seems to be more standard when G is a general linear group or a unitary group. For a more canonical approach to conjectures concerning Galois representations attached to cohomological automorphic forms, see [BG14]. 4.1. Twisting element. For a field F , let G F denote the absolute Galois group Gal(F sep /F ) where we fix some separable closure F sep . Fix a maximal torus T and Borel subgroup B in G /Q . We assume now that G has a twisting element η which is by definition an element of X(T ) G Q such that η, α ∨ = 1 for all simple roots α. If G is GL n , we can take η to be (n − 1, n − 2, . . . , 1, 0). As before, a twisting element always exists if we replace G by a central extension of G by G m . The effect of this on the constructions and questions in §2 is minimal. A twisting element is only unique up to X 0 (T ) G Q . Satake parameters. Fix a prime ℓ and suppose that there is a reductive model G over Z ℓ for G such that G(Z ℓ ) = K ℓ . Let T ⊂ B ⊂ GS : Z[1/ℓ][K ℓ \G(Q ℓ )/K ℓ ] ∼ → Z[1/ℓ][T (Z ℓ )\T (Q ℓ )/T (Z ℓ )] Ws normalized using the choice of twisting element η as in [Gro98, Proposition 3.6], where W s is the Weyl group of the maximal split torus in T . If T is split, then T (Z ℓ )\T (Q ℓ )/T (Z ℓ ) ∼ = Y (T ) (here Y (T ) denotes the cocharacter group of T ) and E-valued characters of Z[1/ℓ][K ℓ \G(Q ℓ )/K ℓ ] ∼ = Z[1/ℓ][Y (T )] W ∼ = O( T / /W ) are in bijection with semisimple conjugacy classes in the dual group G(E) for any coefficient field E of characteristic not equal to ℓ. In general, E-valued characters χ of Z[1/ℓ][K ℓ \G(Q ℓ )/K ℓ ] are in bijection with semisimple conjugacy classes C χ of L G(E), where L G denotes the Langlands dual of G (see [Gro99, §16]). 4.3. Conjectures on Galois representations associated to cohomological automorphic forms. We fix a prime p and a sufficiently large subfield E ⊂ Q p of finite degree over Q p . Conjecture 4.3.1. Let V (λ) E denote the irreducible representation of G /E of highest weight λ. Suppose that p ⊂ T Σ Q ⊗ Q E is a maximal ideal such that the p-torsion H * (Y (K f ), V(λ) E )[p] is nonzero. Then there exists a continuous homomorphism ρ : G Q → L G(E) such that (1) the composition of ρ with the projection L G(E) → G Q is the identity on G Q ; (2) for ℓ / ∈ Σ and ℓ = p, ρ is unramified at ℓ and ρ(Frob ℓ ) is in C χ where χ is the character There is an analogous conjecture with torsion coefficients. As before, let F denote the residue field of E. Z[1/ℓ][K ℓ \G(Q ℓ )/K ℓ ] ⊂ T Σ Q ⊗ Q E → (T Σ Q ⊗ Q E)/p ∼ = E andConjecture 4.3.2. If W is a finite F[[K p ]]-module, let W denote the F-local system on Y (K f ). Suppose that m ⊂ T Σ O is a maximal ideal such that H * (Y (K f ), W) m is nonzero. Then there exists a continuous homomorphism ρ : G Q → L G(F) such that (1) the composition of ρ with the projection L G(F) → G Q is the identity on G Q ; (2) for ℓ / ∈ Σ and ℓ = p, ρ is unramified at ℓ and ρ(Frob ℓ ) is in C χ where χ is the character (1) One also expects that ρ satisfies a compatibility with the conjectural local Langlands correspondence at places in Σ. Z[1/ℓ][K ℓ \G(Q ℓ )/K ℓ ] → T Σ Q ⊗ Q F → T Σ Q ⊗ Q F/m ∼ = F and (2) Comparing the two conjectures, observe that there is no property at p for ρ. Such a property would be closely related to Questions 2.4.1 and 2.4.2. (3) It is clear that ρ and ρ determine p and m, respectively. On the other hand, the properties of ρ described in Conjectures 4.3.1 and 4.3.2 do not characterize ρ in general. When G is a Weil restriction of GL n , the first two properties characterize the isomorphism class of the semisimplification of ρ by the Chebotarev density theorem and the Brauer-Nesbitt theorem. But even for tori, these properties do not characterize ρ (see [ (2) the Weil restriction of GL n over a CM field [Sch15]. In these cases, the attached Galois representations are determined up to semisimplification by Remark 4.3.3(3). We say that ρ (or m) is non-Eisenstein if ρ does not factor through a proper parabolic subgroup after any finite extension of F. 4.4. Modular Serre weights. Fix p and E as before, and let F be the residue field of E. Suppose from now on that G /Qp has an integral model G /Zp . Having classified irreducible representations of G(Z p ) and introduced Galois representations, we now revisit Question 2.4.2 through that lens. We let K p be G(Z p ). An irreducible G(Z p )-representation over F factors through the reduction map G(Z p ) ։ G(F p ) (whose kernel is pro-p), i.e. is the inflation of a Serre weight for G(F p ). For a Serre weight σ, let F σ denote the corresponding F-local system on Y (K f ). Fix an F-valued Hecke eigensystem m. To understand Question 2.4.2 is to understand the following (finite) set. Definition 4.4.1. Let W (m) be the set of isomorphism classes of Serre weights σ for G(F p ) for which H * (Y (K f ), F σ ) m is nonzero. If σ ∈ W (m), then we say that σ is a modular Serre weight (for m). If there is a Galois representation ρ : G Q → L G(F) satisfying the properties in Conjecture 4.3.2 for m, then we also write W (ρ) for W (m) and say that σ is a modular Serre weight for ρ when σ ∈ W (ρ). 4.5. The case of GL 2 . Fix p and E as before. The first result on Conjecture 4.3.1 for nonabelian G was work of Deligne [Del71] in the case of G = GL 2 (building on work of Eichler and Shimura). In this case there is no torsion in cohomology, and so If H 1 (Y (K f ), F(a, b)) m is nonzero, then so is H 1 (Y (K f ), W(a, b)) m . A necessary condition for F (a, b) to be in W (ρ) is that ρ is the reduction of a representation ρ : G Q → GL 2 (E) which is unramified outside of Σ and p and is crystalline at p of Hodge-Tate weights a + 1 and b. (Since ρ is irreducible, a G Q -invariant O-lattice in ρ is unique up to scaling.) In particular, the restriction ρ\| G Qp is the reduction of (a lattice in) a crystalline representation ρ p : G Qp → GL 2 (E) of Hodge-Tate weights a + 1 and b. The following result, known as the weight part of Serre's conjecture, is a local-global principle (i.e. W (ρ) only depends on ρ\| G Qp ) that asserts that the necessary condition is in fact sufficient. Remark 4.5.2. (1) The above formulation is slightly different, albeit equivalent, from Serre's original formulation in [Ser87]. In loc.cit., the recipe for W (ρ) is completely explicit in terms of the "inertial weights" of ρ\| G Qp when ρ\| G Qp is semisimple, with additional modfications in terms of ramification properties of an extension class in general. In particular, in the semisimple case, there is a simple combinatorial formula for a and b solely in terms of the inertial weights. Early generalizations of Serre's conjecture beyond GL 2/Q , e.g. [ADP02, Conjecture 3.1], involved similar formulas (see [GHS18,§7] for more recent formulas for a larger list of Serre weights). On the other hand, while [BDJ10] also contains formulasà la Serre, it emphasizes the above "crystalline lifts" perspective. (2) Theorem 4.5.1 was generalized to (the Weil restriction of) the unit groups in quaternion algebras over totally real fields split at no more than one archimedean place and definite unitary groups over a totally real field when p > 2 (and under mild additional hypotheses) in a series of works by Gee, Newton, Kisin, Liu, and Savitt [New14, GK14,GLS14]. Some of these build on earlier work of Gee that introduced the Taylor-Wiles method to produce proofs rather different from the original proofs of Theorem 4.5.1. We now revisit the mod 23 congruence for the Ramanujan Delta function. Let ρ : G Q → GL 2 (F 23 ) be the associated Galois representation. In this case, ρ\| I Q 23 ∼ = ω 11 ⊕1, where ω denotes the reduction of the 23-adic cyclotomic character χ and I Q 23 ⊂ G Q 23 denotes the inertial subgroup. Then ρ\| G Q 23 is the reduction of both χ 11 ⊕ 1 and χ 11 ⊕ χ 22 (up to unramified twists). Theorem 4.5.1 implies that {F (10, 0), F (21, 11)} ⊂ W (ρ). In fact, this is an equality. The behavior of the Ramanujan Delta function modulo 23 illustrates a rare phenomenon. For example, if we instead take p = 19 and ρ : G Q → GL 2 (F 19 ) the corresponding representation, then ρ\| I Q 19 is a nontrivial extension of 1 by ω 11 . Moreover, ρ\| G Q 19 is the reduction of a crystalline representation which after restriction to inertia is a nontrivial extension of 1 by χ 11 . However, any nontrivial extension of χ 18 by any unramified twist of χ 11 is not crystalline. In fact, we have {F (10, 0)} = W (ρ) in this case. One expects more generally that W (ρ) is larger when ρ\| G Qp is semisimple. Indeed, if ρ\| G Qp is the reduction of an O-lattice in r : G Qp → GL 2 (E), then there is another O-lattice (possibly after enlarging E) whose reduction is the semisimplification of ρ\| G Qp . One approach to Theorem 4.5.1 is as follows. If ρ\| G Qp admits a local crystalline lift of Hodge-Tate weights a + 1 and b, show that ρ admits a global lift ρ which is crystalline (at p) of Hodge-Tate weights a + 1 and b. Then show that ρ comes from a modular form as predicted by the Fontaine-Mazur conjecture [FM95]. These steps can be executed using a combination of tools in the Taylor-Wiles method independently discovered by Khare-Wintenberger and Gee [KW09,Gee11]. 4.6. Vanishing conjectures for cohomology. In the previous section, we were in the advantageous situation where H * (Y (K f ), W) m vanished outside of degree one for all O-local systems W when m is non-Eisenstein. While this is not true in general, this property does nevertherless admit a conjectural generalization. Let d Y be the dimension of Y (K f ). Define the integers ℓ 0 def = rk G(R) − rk A • ∞ K • ∞ and q 0 = 1 2 (d Y − ℓ 0 ). (The group G admits discrete series if and only of ℓ 0 = 0.) Conjecture 4.6.1 ([CG18]). Suppose that m is non-Eisenstein. If H i (Y (K f ), W) m = 0, then i ∈ [q 0 , q 0 + ℓ 0 ]. [ GN] shows that if Conjecture 4.6.1 holds, then so does the converse to Proposition 2.4.3 when G is a Weil restriction of GL n , so that, as in our discussion above, it suffices to analyze Question 2.4.1 for irreducible mod p local systems (note that this is far from obvious in general, as it is a priori possible for the cohomology complex of irreducible local system to cancel each other out when they spread to several cohomological degrees). More seriously, Conjecture 4.6.1 plays a prominent role in the Taylor-Wiles patching method, which is the main tool to attack Question 2.4.1 in general, cf. Remark 5.1.2 below. Unfortunately, there are few cases where Conjecture 4.6.1 is known. One trivial case is that of groups which are anisotropic modulo their center, e.g. definite unitary groups, when Y (K f ) is a finite set of points. [CS17] have shown that for certain unitary groups (ℓ 0 = 0) Conjecture 4.6.1 holds under some additional hypotheses. The case of the Weil restriction of GL n (for n > 1) over a number field F (even CM fields) is open beyond the case n = 2 and F either totally real or imaginary quadratic. 5. Conjectures and results on the weight part of Serre's conjecture 5.1. Taylor-Wiles patching. Suppose for the moment that we are in a context where ℓ 0 = 0 and Conjectures 4.3.1 and 4.6.1 hold (e.g., definite unitary groups). We fix p and E as before. We assume that a reductive integral model G /Zp of G /Qp exists and continue to let K p = G(Z p ). If F (λ) ∈ W (ρ), then H * (Y (K f ), W(λ)) m is nonzero (where W (λ) is an O-lattice in the irreducible algebraic representation V (λ)). In particular, ρ (attached to m) is the reduction of a crystalline representation of Hodge-Tate cocharacter λ + η. In light of Theorem 4.5.1, it is tempting to guess that the converse holds. However, counterexamples to this have been found for definite unitary groups in three variables [ [H * (Y (K f ), F(λ)) m ] = W c W [H * (Y (K f ), W) m ]. Since the F-vector spaces on the right hand side lift, their dimensions can in principle be computed in characteristic zero. However, they are of a global nature, and thus difficult to access. In contrast, we still expect that W (ρ) depends only on ρ\| G Qp . The Taylor-Wiles method "patches" together cohomology functors (or rather, the total cohomology complex computing) H * (Y (K f ), F − ) m (for varying K f ) to obtain a functor M ∞ (−) that plausibly depends (roughly speaking) only on ρ\| G Qp . Moreover, a control theorem guarantees that M ∞ (σ) is nonzero if and only if H * (Y (K f ), F σ ) m is. For a Galois representation r : G Qp → L G(F), let R r denote the (framed) deformation ring parametrizing lifts r : G Qp → L G(R) of r for complete local Noetherian O-algebras R with residue field F. Building on work of Kisin [Kis08] in the case when G /Qp is a Weil restriction of GL n , Balaji [Bal12] in particular defined a family of (reduced) semistable deformation rings R λ+η,τ r whose Q ppoints correspond to potentially semistable Galois representations of Hodge-Tate cocharacter λ + η and Galois type τ . In certain contexts where (enough of) an inertial local Langlands correspondence is known, one can define a finite dimensional locally algebraic E[K p ]-module σ(λ, τ ) def = V (λ)⊗ E σ(τ ). For example, when τ is tame, σ(τ ) can be taken to be a certain combinatorially defined Deligne-Lusztig representation. For a ring A, let A-mod fg denote the full subcategory of A-mod of finitely generated A-modules. (1) It may be impossible to arrange for the stringent rank conditions in Axiom 5.1.1, but the ranks can still be controlled and many arguments below can be successfully modified. (2) If Conjecture 4.6.1 holds, then it is often possible to use the Taylor-Wiles method to construct a functor M ∞ (−) satisfying the first three items of Axiom 5.1.1 with r def = ρ\| G Qp [CEG + 16, GN], at least after adding formal variables to R λ+η,τ r which we ignore. In general, the total (localized) cohomology complex may have non-vanishing cohomology groups in several degrees. The role of Conjecture 4.6.1 is to guarantee that after Taylor-Wiles patching, these complexes becomes concentrated into a single cohomological degree, i.e. they turn into usual modules. This concentration effect relies on the "numerical coincidence" that powers the Taylor-Wiles method. (3) It seems to be difficult to guarantee the last item (for all choices of (λ, τ )) except when G = GL 2 [Kis09]. Indeed, it is essentially equivalent to a modularity lifting result with very general p-adic Hodge theoretic hypotheses. However, there are some instances of specific (λ, τ ) where the final item follows from the third: when R λ+η,τ r is zero and when R λ+η,τ r is a domain and M ∞ (σ(λ, τ ) • ) is nonzero for some O-lattice σ(λ, τ ) • ⊂ σ(λ, τ ). Admitting Axiom 5.1.1 for the moment, there is the following strategy for determining W (ρ). Let d = dim E G /E + dim E G /E /B /E where B /E ⊂ G /E is a(5.2) Z(M ∞ (σ)) = (λ,τ ) c λ,τ Z(R λ+η,τ r ⊗ O F), where Z(−) corresponds to taking the d-dimensional support cycle of the R r -module (note that all modules involved have support of dimension ≤ d). Since the right hand side of (5.2) only depends on r def = ρ\| G Qp , so does the left hand side. In particular, this implies W (ρ), being the set of σ with Z(M ∞ (σ)) = 0 by the Cohen-Macaulay property, depends only on r as expected. Another feature of the situation is that there are many possible choices of expressions (5.1), even if we restrict to the case when c λ,τ = 0 for all λ = 0. Since the left hand side of (5.2) involves only σ, we get the surprising conclusion that the right hand side must be independent of the choice of expressions (5.1). In other words, there are many non-trivial relations between cycles of special fibers of potentially crystalline deformation rings. We summarize the above arguments as the following conjecture. Conjecture 5.1.3. (1) [BM02,EG14] The left hand side of (5.2) is independent of the presentation in (5.1). (2) [GHS18] For any presentation as in (5.1), σ ∈ W (ρ) if and only if the right hand side of (5.2) is nonzero. Remark 5.1.4. Conjecture 5.1.3(1) is purely local in the sense that it only involves G /Qp and r while (2) is global since it involves ρ. However, both follow from Axiom 5.1.1. 5.2. Herzig's recipe. The complexity of Conjecture 5.1.3 suggests that Question 2.4.2 does not admit a simple answer. Indeed, the case of GL 2 is perhaps misleading because of the simplicity of the geometry and representation theory involved. However, the class of semisimple representations G Q → L G(F) admits an essentially combinatorial classification, and so one could ask for a combinatorial description of W (ρ) when ρ\| G Qp is semisimple which generalizes Serre's recipe (see Remark 4.5.2). Herzig's recipe gives such a (conjectural) description. We assume in this section that G /Qp is unramified, with reductive integral model G /Zp . We let K p be G(Z p ). As before, we fix a twisting element η of G. A regular Serre weight is a Serre weight F (λ) with 0 ≤ λ, α ∨ < p − 1 for all simple roots α. We define an involution R on the set of regular Serre weights F (λ) by the formula R(F (λ)) = F ((−w 0 (η), w 0 ) · λ). If r : G Qp → L G(F) is semisimple, then the restriction r\| I Qp to the inertial subgroup factors through a torus. Its Teichmüller lift [r\| I Qp ] : G Qp → L G(E) is then a tame inertial type. Recall that to a tame inertial type τ (defined over E), one can attach through a tame inertial local Langlands a Deligne-Lusztig representation σ(τ ) of G(F p ) (defined over E). Then K p acts on σ(τ ) by inflation, and we let σ(τ ) denote the semisimplification of the reduction of any K p -stable O-lattice in σ(τ ). We have the following conjecture. Suppose now that G /E is a product of GL n over a finite set J . For w ∈ W , w(0) is a tuple of elements in Z n indexed by J . We write w(0) j ∈ Z n for the element corresponding to j ∈ J . We say that a semisimple r : G Qp → L G(F) is sufficiently generic at p if σ([r\| I Qp ]) = R( w) where 0 ≤ w(0), α ∨ ≤ p for all simple roots α and p ∤ P ( w(0) j ) for an implicit polynomial P ∈ Z[X 1 , . . . , X n ] which depends only on n (and not on p or j). If r is not semisimple, we say that it is sufficiently generic if its semisimplification is. In a precise sense, most r's are sufficiently generic as p → ∞. (1) [LLHLM20a, Corollary 8.5.2] Suppose that G /Qp is an unramified Weil restriction of GL n . Then Conjecture 5.1.3(1), restricted to presentations coming from Deligne-Lusztig representations, holds for sufficiently generic r. (2) [LLHLM20a, Theorem 9.1.6] Suppose that G is the Weil restriction of a definite unitary (over a nontrivial totally real extension of Q) and that ρ\| G Qp is sufficiently generic and semisimple. Then under mild Taylor-Wiles hypotheses, Conjectures 5.1.3(2) (for the restricted presentations in the previous item) and 5.2.1 hold. (3) [LLH] Suppose that G is the Weil restriction of GL n over a CM field and that ρ\| G Qp is sufficiently generic and semisimple. Suppose moreover that A critical tool to prove Theorem 5.3.1 is the following. 1(1)-(3), with G a Weil restriction of a definite unitary group, such that M ∞ (σ(τ ) • ) is nonzero if R η,τ r is nonzero for a sufficiently generic tame inertial type τ . H * (Y (K f ), W) m ⊗ O E is Remark 5.3.3. (1) Theorem 5.3.2(2) combines the modularity of obvious weights [LLHL19] and the coherence conjecture for local models of Shimura varieties [Zhu14]. (2) In fact, Theorem 5.3.2 also holds for any λ + η with λ dominant, though the implicit polynomial defining genericity depends on λ. See Theorem 6.1.3. As alluded to in Remark 5.1.2(3), Theorem 5.3.2 implies the existence of a functor M ∞ satisfying Axiom 5.1.1(1)-(4) restricted to cases when λ = 0 and τ is a generic tame inertial type. The argument from §5.1 shows that Theorem 5.3.2 implies Theorem 5.3.1(1) for sufficiently generic semisimple r. The nonsemisimple case follows from a simple argument using the global geometry of the Emerton-Gee stack relying on the fact that there is a semisimple r on every component (see Remark 6.2.3(1)). Moreover, Theorem 5.3.1(2) follows from Axiom 5.1.1(2). Remark 5.3.4. The final part of Theorem 5.3.1 is a bit more subtle. When ℓ 0 = 0, one expects H * (Y (K f ), W) m to have little torsion. In contrast when ℓ 0 > 0, one expects cohomology to be dominated by torsion and characteristic 0 classes to be rare. This means that the lifting hypothesis, i.e. that H * (Y (K f ), W) m ⊗ O E is nonzero in Theorem 5.3.1(3), is quite restrictive. This condition could be removed if one knew Conjecture 4.6.1 (see Remark 5.1.2). In lieu of this, the lifting hypothesis can be used to make an argument with Euler characteristics of the functor M ∞ , whose image is a priori an object in D b (R ρ\| G Qp -mod fg ), adopting Taylor's Ihara avoidance trick to this setting [ACC + 18]. Local models for potentially crystalline deformation rings The heart of the proof of Theorem 5.3.1 reduces, via the Taylor-Wiles method, to understanding the support of the patched modules M ∞ (σ) in Axiom 5.1.1, and ultimately to geometric properties of the potentially crystalline deformation rings R λ,τ r . We achieve this by introducing and analyzing certain (finite type) group-theoretic moduli spaces which algebraize these deformation rings. Fix a prime p. In this section, we will restrict to the case G /Qp = Res K/Qp GL n for an unramified extension K = Q p f of Q p . In particular, we have a reductive integral model G /Zp = Res O K /Zp GL n of G. Recall from §2.3 that E is a sufficiently large finite extension of Q p with ring of integers O, uniformizer ̟, and residue field F. 6.1. Potentially crystalline deformation rings. Let r : G K → GL n (F) be a mod p local Galois representation. Recall that we have R r the (framed) deformation ring that classifies lifts of r, and the Q p -points of R r correspond to p-adic Galois representations of G K (lifting r). Given a Hodge-Tate cocharacter λ and inertial type τ , one has the potentially crystalline deformation ring R λ,τ r which is characterized as the unique reduced p-flat quotient of R r whose Q p -points correspond to lifts r : G K → GL n (Q p ) which are potentially crystalline of type (λ, τ ) (i.e. the Hodge-Tate weights of r are given by λ and WD(r) induces the inertial type τ ). In the setting of GL n , these rings were first constructed by Kisin, who also established their basic properties [Kis08]. Theorem 6.1.1 (Kisin). ( 1) R λ,τ r [ 1 p ] is regular. (2) dim O R λ,τ r = dim E G /E + dim E G /E /P λ/E , where P λ is the parabolic subgroup corresponding to λ. In particular dim O R λ,τ is a constant d as λ varies over regular dominant cocharacters. When λ is regular dominant, the rings R λ,τ r play a pivotal role in the Taylor-Wiles method: they act on patched spaces of automorphic forms M ∞ (σ(λ − η, τ )), which govern questions about modularity and congruences (cf. Axiom 5. 1.1(3)). Even better, they are maximal Cohen-Macaulay modules, and hence must be supported on a union of irreducible components. For global applications, it is essential to understand global properties of R λ,τ r such as irreducibility. This turns out to be a notoriously difficult problem. There are roughly two reasons for this. • Outside some special cases, R λ,τ r , being characterized by its Q p -points, has no known moduli interpretation. This is related to the fact that integral p-adic Hodge theory is much less well understood than rational p-adic Hodge theory. • The internal structure of R λ,τ r is intrinsically complicated in general. Thus, one can not expect to have completely explicit descriptions for all λ and τ . The second point is best illustrated by the Breuil-Mézard conjecture, which quantifies the complexity of the special fibers of R λ+η,τ r as λ and τ vary in terms of the mod p representation theory of GL n (O K ) (we shift from λ to λ + η for the rest of this subsection to be consistent with §5.1). We let Z(R λ+η,τ r /̟) denote the d-dimensional cycle of R λ+η,τ r /̟, which counts the irreducible components with appropriate multiplicities. For a GL n (O K )-representation V over E, recall that V denotes the GL n (O K )-representation over F which is the semisimplification of the reduction modulo ̟ of any GL n (O K )-stable O-lattice in V . The following is a reformulation of Conjecture 5.1.3(1). Conjecture 6.1.2 (Breuil-Mézard, Emerton-Gee). There exist d-dimensional cycles Z σ (r) in Spec R r /̟ for each irreducible GL n (O K )-representation σ over F (i.e. a Serre weight for G(F p )) such that for all τ and λ, Z(R λ+η,τ r /̟) = σ m λ,τ (σ)Z σ (r), where m λ,τ (σ) denotes the multiplicity of σ in σ(λ, τ ). In other words, the special fibers R λ+η,τ r /̟ are built out of a finite list of basic cycles Z σ (r), with multiplicities governed by the purely representation theoretic quantities m λ,τ (σ). Conjecture 6.1.2 is known when n = 2 and λ = 0 by work of Gee and Kisin [GK14]. When τ is a generic tame type, m 0,τ (σ) = 1 for 2 f Serre weights σ and is zero otherwise. In general, the quantities m λ,τ are very complicated: if λ = 0 and τ is tame, m λ,τ (σ) computes the multiplicities of a mod p Deligne-Lusztig representation, which for generic τ is given by periodic Kazhdan-Lusztig polynomials. In particular, as the rank of G grows, the special fibers R λ+η,τ r /̟ tend to be highly non-reduced. As explained in §5.1, to prove Theorem 5.3.1, one needs to establish Axiom 5.1.1, particularly the main bottleneck (4). We do this by proving Theorem 5.3.2. Theorem 6.1.3. [LLHLM20a, Theorem 7.3.2(2)] Assume that r is semisimple and τ is a tame inertial type which is sufficiently generic relative to λ (in the sense of §5.3). Then R λ+η,τ r is a domain (or zero). Remark 6.1.4. (1) Explicit computations that suggest that Theorem 6.1.3 is false without the tameness assumption on r when n > 2 unless n = 3 and λ = 0. (2) If R λ+η,τ r = 0, then sufficient genericity of τ implies that of r and vice versa (generally with different choices of implicit polynomials). Because of this, the conclusion of Theorem 6.1.3 also holds if we let r be tame and sufficiently generic but impose no genericity hypothesis on τ . 6.2. The Emerton-Gee stack. In [EGa], Emerton-Gee constructed the moduli stack X n over Spf O of rank nétale (ϕ, Γ)-modules. By its construction, X n interpolates framed deformation rings in the sense that the set X n (F p ) is in bijection with the set of continuous representations r : G K → GL n (F p ), and framed deformation rings R r are versal rings (in the sense of [EGb, Definition 2.2.9]) for X n . Furthermore, for a Hodge-Tate cocharacter λ and an inertial type τ , they construct a p-flat p-adic formal algebraic closed substack X λ,τ which is characterized by the property that its points over any finite flat O-algebra correspond to potentially crystalline representations r of type (λ, τ ). Thus X λ,τ interpolates the potentially crystalline deformation rings R λ,τ r as r varies. The basic properties of these stacks are as follows: Theorem 6.2.1 (Emerton-Gee). (1) [EGa,Corollary 5.5.18] X n is a Noetherian formal algebraic stack. (2) [EGa, Theorem 4.8.12] X λ,τ is a p-flat p-adic formal algebraic stack of dimension dim G /E /P λ/E . (3) [EGa, Theorem 6.5.1] The irreducible components of the underlying reduced stack X n,red are in bijection with the Serre weights of G(F p ). We let C σ be the irreducible component labelled by σ. Let Z λ+η,τ denote the top dimensional cycle of X λ+η,τ /̟, which has dimension independent of λ since λ + η is regular dominant. One has the following interpolation of the Breuil-Mézard conjecture over X n : Conjecture 6.2.2 (Conjecture 8.2.2 [EGa]). For each Serre weight σ, there exists an effective top-dimensional cycle Z σ on X n,red such that for all λ and inertial types τ , we have Z λ+η,τ = σ m λ,τ (σ)Z σ . Remark 6.2.3. (1) Conjecture 6.2.2 recovers Conjecture 6.1.2 by taking versal rings at r. Conversely, knowledge of Conjecture 6.1.2 at sufficiently many r would imply Conjecture 6.2.2. This gives a mechanism to deduce Conjecture 6.1.2 for more general r from a few "basic r". This allows us to reduce Theorem 5.3.1(1) to the case of semisimple r. (2) In [LLHLM20a, §8], using Taylor-Wiles patching, we constructed cycles Z σ for sufficiently generic σ, which satisfies a (finite) subset of the equations postulated in Conjecture 6.2.2. As the conjectural cycles in Conjecture 6.2.2 is expected to be compatible with Taylor-Wiles patching, the cycles constructed in loc. cit. should be the "correct" ones. (3) (cf. [LLHLM20a, Remark 1.4.11]) One expects Z σ to contain the irreducible component C σ with multiplicity one. That is, one should have a decomposition Z σ = σ ′ b σ ′ ,σ C σ ′ with b σ ′ ,σ ≥ 0 and b σ,σ = 1. This is indeed true in the cases studied in [LLHLM20a,§8] and [GK14]. For example, in the setting of [GK14], the cycles Z σ = C σ , unless σ is a twist of the Steinberg weight (in particular such σ would be non-generic), in which case Z σ is C σ + C σ ′ for a suitable σ ′ (cf. [EGa,Theorem 8.6.2]). For n > 3, it is quite difficult to compute b σ ′ ,σ , and one does not expect Z σ = C σ in general, even for generic σ. This is analogous to the situation of the locally analytic Breuil-Mézard conjecture studied in [BHS19]. The quotient L + G\LG is represented by an ind-proper O-ind-scheme Gr G . This is a mixed characteristic version of the degeneration of affine Grassmannians introduced by Gaitsgory: indeed its generic fiber Gr G,E is isomorphic to an affine Grassmannian, while the special fiber Gr G,F is isomorphic to the affine flag variety Fl. The affine Grassmannian has the affine Schubert stratification Gr G, E = λ L + G E \L + G E (v + p) λ L + G E ,+ G E \L + G E (v + p) λ L + G E in Gr G , cf. [PZ13]. Let a ∈ O n . We now consider the condition (⋆) v dA dv A −1 + ADiag(a)A −1 ∈ 1 v + p Lie L + G for A ∈ LG(R) . This is an approximation to the monodromy condition coming from p-adic Hodge theory. This condition clearly descends to a closed condition on Gr G . + G E \L + G E (v + p) λ L + G E . Note that right multiplication by the constant diagonal torus T ∨ preserves (⋆). (Here, T ∨ is a maximal torus in GL n which is the dual group G ∨ of the group G = GL n which appeared in §3.1.) Thus, M (λ, ∇ a ) inherits a T ∨ -action compatible with the T ∨ -action on M (≤λ). By contemplating the interaction of condition (⋆) with the affine Schubert stratification, one observes: • [LLHLM20a, Proposition 4.1.1] M (λ, ∇ a ) /E is isomorphic to P λ \GL n , hence is smooth and irreducible. • [LLHLM20a, Theorem 4.2.4] Provided a mod p sufficiently regular, the locus cut out by (⋆) in each open Schubert cell I\I wI ⊂ Fl is an affine space, with dimension combinatorially determined by w. Thus M (λ, ∇ a ) is a degeneration of a partial flag variety, and one has control over its reduced special fiber. Example 6.3.2. (1) For example, when n = 2 and λ = (1, 0), condition (⋆) is empty, and M (λ, ∇ a ) = M (≤λ) is a degeneration of P 1 into a union of two P 1 crossing transversely at a point. More generally, one has M (λ, ∇ a ) = M (≤λ) if and only if λ is minuscule. (2) Suppose n = 3 and λ = (2, 1, 0), and a mod p is sufficiently regular. Then dim M (≤λ) = 4, whereas dim M (λ, ∇ a ) = dim B\GL 3 = 3. The special fiber M (λ, ∇ a ) F is reduced and has 9 irreducible components, six of which are isomorphic to the flag variety B\GL 3 , while the remaining three are more complicated rational smooth varieties. Already in this case, the behavior of the intersections among the irreducible components is somewhat elaborate, cf. [LHLM22]. Remark 6.3.3. Around each point z ∈ Fl, one can write down an explicit open neighborhood U ( z) of Gr G using the theory of the "big cell". This allows us to in principle give explicit coordinate charts for M (λ, ∇ a ): the coordinate charts parametrize matrices A with polynomial entries whose degrees are bounded in terms of z, and one then imposes elementary divisor conditions dictated by λ together with the explicit equation (⋆) and takes the p-saturation of the result. It is the p-saturation operation that makes this description rather difficult to work with. In order to establish the connection between the above models to Galois deformation theory, we have to understand the behavior of M (λ, ∇ a ) under completion. The essential difficulty is that an irreducible variety may break up into formal branches in some complicated way after completions: its singularities may not be unibranch. Unfortunately, M (λ, ∇ a ) fails to be unibranch in general, and in such situations it is difficult to control the subset of the formal branches that are related to Galois deformation theory. Fortunately, it turns out there is supply of special points where this difficulty does not manifest: Theorem 6.3.4. ([LLHLM18, Theorem 3.7.1]) There exists a nonzero polynomial P ∈ Z[X 1 , . . . , X n ] such that if P (a) = 0 mod p, then for any T -fixed point x ∈ M (λ, ∇ a )(F p ), the completed local ring O ∧ M (λ,∇a),x is a domain (i.e., M (λ, ∇ a ) is unibranch at its T -fixed points). This key result, whose proof we now sketch, underlies everything else. One first observes that the theorem holds (under a mild assumption on the characteristic) for the equal characteristic analogues of M (λ, ∇ a ) where E is replaced by F((t)). In this function field setting, there is an additional symmetry: there is an extra G m -action given by "loop rotation" which scales t. This implies that the T -fixed points look like cone points, i.e. the fixed point of an attracting torus action, and one observes that cone points are unibranch. We then deduce the mixed characteristic case by a spreading out argument. The essential point here is that unibranch can be phrased in terms of connectedness of fibers of the normalization map, and normalization is preserved by generic base change. This explains the occurence of the universal polynomial P : its vanishing locus is the obstruction to certain properties being preserved under base change. 6.4. Local models and Emerton-Gee stacks. Recall that we fixed a finite unramified extension K/Q p . Let k be the residue field of K. Let J be the set of embeddings Hom Qp (K, Q p ) which we identify with Hom Qp (K, E) = Hom(k, F) using the inclusion E ⊂ Q p . To any tame inertial type τ for I K , one can associate a collection a τ = (a τ,j ) j∈J , where a τ,j ∈ O n records the inertial weights of τ . In the "lowest alcove" principal series case, a τ is defined so that τ is the direct sum n i=1 j∈J j • ω a (i) τ,j (2) When λ = η, one has λ ′ ≤λ λ ′ reg. dom. X λ ′ ,τ = X η,τ . Since potentially crystalline deformation rings of type (η, τ ) are versal rings to X η,τ , we see that they appear (up to smooth modifications) as the completion of local rings of M (η, ∇ aτ ) at closed points. In particular, we deduce the irreducibility of the potentially crystalline deformation rings of Theorem 6.1.3 from the unibranch property of the local models (at the appropriate points). This completes the proof of Theorem 5.3.1(1) and the first half of Theorem 5.3.1(2) (see Remark 6.2.3(1)). (3) Combining the theorem with Remark 6.3.3, we get an algorithm to write down explicit presentations of (unions of) R λ,τ r (for regular λ). We now give a slightly simplified outline of the proof of Theorem 6.4.1. The construction of the stacks X λ,τ comes in two steps: • Using integral p-adic Hodge theory, one can attach Breuil-Kisin modules to lattices in (potentially) crystalline representations for G K . Thus the first step is to construct a moduli stack of Breuil-Kisin modules Y ≤λ,τ with tame descent data of type (λ, τ ). • As not all Breuil-Kisin modules come from lattices in (potentially) crystalline representations, one needs cut down Y λ,τ by appropriate conditions to get X λ,τ . Accordingly, the proof is divided into two steps: • In the first step, we show that Y ≤λ,τ is locally modelled by the Pappas-Zhu model M (≤λ). This is not surprising, as Breuil-Kisin modules are a projective O K [[u]]-modules with certain semi-linear structures, and thus are closely related to points of Gr G . Using the open cover Gr G = z U ( z) (cf. Remark 6.3.3), we get an analogue of the local model diagram (6.1) for Y ≤λ,τ and induced open affine covers on every object in sight. • After the first step, we get two closed substacks of Y ≤λ,τ ( z): the substack X ≤λ,τ ( z) and the substack X ≤λ,τ,⋆ ( z) induced by the p-adic completion of λ ′ ≤λ M (λ ′ , ∇ aτ ) along the local model diagram for Y ≤λ,τ . They are genuinely different substacks, because condition (⋆) is only the "first order term" of the condition cutting out X ≤λ,τ inside Y ≤λ,τ . However, the two substacks are p-adically close, and using the smoothness of the generic fiber of M (λ, ∇ a ), one can produce a non-canonical embedding X ≤λ,τ ( z) ֒→ X ≤λ,τ,⋆ ( z). Since both stacks turn out to have the same dimension, the maximal dimensional part X ≤λ,τ reg ( z) of X ≤λ,τ ( z) embeds into the maximal dimension part of X ≤λ,τ,⋆ ( z). Now, using the results of [LLHL19] (which ultimately uses Taylor-Wiles patching, and hence automorphic forms), one obtains a lower bound on the number of irreducible components (of the spectrum of the structure sheaf) of the former, while Theorem 6.3.4 gives the same upper bound for the number of irreducible components (of the spectrum of the structure sheaf) of the latter. Thus the two maximal dimension parts are (non-canonically) isomorphic to each other, which concludes the proof. As the above outline suggests, the arrows in the local model diagram are produced by Hensel-type lifting arguments, and thus are highly non-canonical. However, modulo p this issue disappears, and the local model diagrams on the open cover produced by Theorem 6.4.1 glue together. Consequently, the analysis of irreducible components of the special fibers of local models implies the following. Theorem 6.4.3. For τ sufficiently generic (with respect to λ): (1) X λ+η,τ red = ∪ σ C σ , where the union runs over all Serre weights σ ∈ JH(σ(λ, τ )). (2) There is a natural bijection between the irreducible components of M (λ + η, ∇ aτ ) and the Jordan-Hölder factors of σ(λ, τ ). (3) For each σ ∈ JH(σ(λ, τ )), we have a mod p local model diagram: (6.2) C σ & & ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ C σ M (λ + η, ∇ aτ ) σ where M (λ + η, ∇ aτ ) σ is the irreducible component of M (λ + η, ∇ aτ ) labelled by σ and both arrows are torsors for the torus (T ∨ ) J (with respect to different (T ∨ ) J -actions). Remark 6.4.4. (1) The proof of Theorem 6.4.3 does not go through Theorem 6.4.1. Because of that it holds under much milder genericity conditions compared to our other theorems: if σ(τ ) = R, we only require that R is m-generic for m sufficiently large depending on λ (larger than both 2 λ, α ∨ + 2 and 4n + λ, α for all roots α). (2) Over F, equation (⋆) becomes the equation cutting out a deformed affine Springer fiber in Fl, cf. [FZ10]. Thus the irreducible components M (λ + η, ∇ aτ ) σ are irreducible components of a (product of) deformed affine Springer fiber(s). (It is immediate from the aforementioned [LLHLM20a, Theorem 4.2.4] that these irreducible components are rational varieties.) Theorem 6.4.3 then allows us to get a handle on the irreducible component C σ of the reduced Emerton-Gee stack for generic σ. In particular, one can get a description of the semisimple points on C σ , and this is the critical ingredient for the verification of Herzig's recipe (Theorems 5.3.1(2) and 5.3.1(3)). (3) The irreducible components M (λ + η, ∇ aτ ) σ are fairly easy to implement on computer algebra systems such as Macaulay2. For any given n, this allows us to probe the structure of C σ in a purely algorithmic manner. Example 6.4.5. (1) (Fontaine-Laffaille components) For σ in the lowest alcove, the corresponding irreducible component of the deformed affine Springer fiber is isomorphic to a product of flag varieties (B\GL n ) J . We deduce from this that C σ = [(N \GL n ) J /T J ] where N is the subgroup of unipotent upper triangular matrices, and T J acts via shifted conjugation: (t j ) · (N g j ) = (N t j g j t −1 j•ϕ ), where •ϕ denotes pre-composition with Frobenius on K. In particular, for n = 2, all components C σ for generic σ are of this form. (2) When n = 3, there are two types of irreducible components at each factor j ∈ J : the flag variety B\GL 3 or a more complicated rational and smooth variety (this can be extracted, for example, from the description of minimal primes in [LHLM22, Table 3]). In particular, there are 2 f types of C σ (for generic σ) which correspond to the possible p-alcoves containing the highest weight of σ. (3) For n = 4, there are generic σ for which C σ is singular (e.g. for σ with highest weight in the highest p-restricted alcove). Thus the smoothness of C σ appears to be a low rank coincidence. = {λ ∈ X(T ) : 0 ≤ λ, α ∨ ≤ p − 1, for all simple α} which plays an important role in the modular representation theory of G. We let X 0 (T ) def = {λ ∈ X(T ) : λ, α ∨ = 0 for all roots α}. ] Let h be the Coxeter number of G. If w ∈ W is 2h-generic, then the Jordan-Hölder factors of R( w) are precisely the Serre weights of the form Frob ℓ is an arithmetic Frobenius element at ℓ; (3) ρ\| G Qp is de Rham with Hodge-Tate cocharacter λ+η (see [BG14, §2.4]; in our normalization the cyclotomic character corresponds to the cocharacter id Gm ) and is moreover crystalline if p / ∈ Σ; and (4) ρ is odd in the sense of [Gro99, Conjecture 17.2(a)]. Frob ℓ is an arithmetic Frobenius element at ℓ; and (3) ρ is odd in the sense of [Gro99, Conjecture 17.2(a)]. constructed ρ satisfying all the properties of Conjecture 4.3.1 except the third, which follows from subsequent work of Faltings on p-adic comparison theorems. (We set η = (1, 0) here.) Let K p = GL 2 (Z p ). Fix a mod p Hecke eigensystem m. Assume that m is non-Eisenstein, i.e. that the attached Galois representation ρ : G Q → GL 2 (F) is absolutely irreducible. (The data of m is equivalent to that of the isomorphism class of ρ by Remark 4.3.3(3).) Since cohomology groups outside degree 1 do not admit non-Eisenstein mod p Hecke eigensystems, the functor W → H 1 (Y (K f ), W) m is exact, as explained in §2.4. In particular, Question 2.4.1 reduces to Question 2.4.2, namely an investigation of W (ρ). Theorem 4.5.1 ([Gro99,Edi92,CV92]). Suppose that m is non-Eisenstein. Then F (a, b) ∈ W (ρ) if and only if ρ\| G Qp is the reduction of a crystalline representation of Hodge-Tate weights a + 1 and b. There is an exact functor M ∞ (−) : O[[K p ]]-mod fg → R ρ\| G Qp -mod fg with the following properties.(1) For a Serre weight σ, M ∞ (σ) is a maximal Cohen-Macaulay module on R λ+η,τ r ⊗ O F for some λ and τ .(2) M ∞ (σ) is nonzero if and only if H * (Y (K f ), F σ ) m is nonzero. (3) If σ(λ, τ ) • is an O-lattice in σ(λ, τ ), then M ∞ (σ(λ, τ ) • )is a maximal Cohen-MacaulayR λ+η,τ r -module of generic rank at most 1. (4) If σ(λ, τ ) • is an O-lattice in σ(λ, τ ), then M ∞ (σ(λ, τ ) • )[1/p] is a generically free R λ+η, Conjecture 5.2.1 ([Her09,GHS18]). The subset of regular Serre weights in W (ρ) is R(JH(σ([ρ\| I Qp ]))).Conjectures 5.1.3 and 5.2.1 are of a rather different nature. For one thing, Conjecture 5.2.1 only applies to ρ which are semisimple locally at p, which is when one expects W (ρ) is largest. However, Conjecture 5.2.1 is rather more explicit than Conjecture 5.1.3 when combined with Proposition 3.2.1.5.3.Results on the weight part of Serre's conjecture. Conjecture 5.2.1 and a weakened version of Conjecture 5.1.3 is known when G is GL 2 , the Weil restriction of the unit group in a quaternion algebra which is indefinite at no more than one archimedean place, or the Weil restriction of a definite unitary group in two variables under mild hypotheses (see Remark 4.5.2). Similar results[LLHLM18,LLHLM20b,LHLM22] are known for the Weil restrictions of definite unitary groups in three variables that are unramified at p under an additional genericity hypothesis. Theorem 5 .3. 2 . 52Suppose that r : G Qp → L G(F) is semisimple.(1) [LLHLM20a, Theorem 7.3.2(2)] If τ is a sufficiently generic tame inertial type, then R η,τ r is an integral domain (if it is nonzero).(2) [LLHLM20a, Proposition 6.2.7] There exists a functor M ∞ (up to adding formal variables) satisfying Axiom 5.1. 6. 3 . 3Local models and their geometric properties. Let LG be the loop group, which is the ind-group scheme given by LG(R) = GL n (R((v+p))) for any O-algebra R. Consider the positive loop group scheme L + G over O sending an O-algebra R to the subgroup of GL n (R[[v + p]]) consisting of matrices that are upper triangular mod v. Note that when p is invertible in R, L + G(R) = GL n (R[[v + p]]) is the positive loop group for GL n , whereas when p = 0 in R, L + G(R) = I(R), the standard Iwahori group scheme. where λ runs over dominant coweights of GL n . Similarly, the affine flag variety Fl = w I\I wI, where w runs over the extended affine Weyl group W . For dominant λ, the Pappas-Zhu local model M (≤λ) is the Zariski closure of L Definition 6 .3. 1 . 61The local model M (λ, ∇ a ) is the Zariski closure in M (≤λ) of the locus cut out by (⋆) in L In fact, the definition (2.7) makes sense for any O[[K p ]]-module and defines a functor from O[[K p ]]-modules to local systems on Y (K f ). We caution that (2.8) may not be injective in any given degree. Indeed, H * (Y (K f ), W) may contain torsion, and in fact this torsion is expected to be abundant and to play an important role in connecting cohomological automorphic forms and Galois representations. 2.4. Congruences between Hecke eigensystems. Let T Σ O be the Hecke algebra O[K Σ \G(A Σ )/K Σ ], which acts naturally on H * (Y (K f ), W). As H * (Y (K f ), W) is a finite O-module, there are only finitely many maximal ideals m ⊂ T Σ O for which the localization H * (Y (K f ), W) m is nonzero. These localized modules record congruences between systems of Hecke eigenvalues: a Hecke eigenclass be a maximal torus and Borel subgroup, respectively. We have the Satake isomorphism (see e.g. [Car79, §4.2]) The existence of torsion cohomology classes means that Conjecture 4.3.1 does not immediately imply Conjecture 4.3.2. A version of Conjecture 4.3.2 for all torsion coefficients implies Conjecture 4.3.1 (except for the third property) by taking a limit. There are two cases, relevant to what follows, when both conjectures are known: (1) the Weil restriction of a definite unitary group relative to a CM extension of a totally real field not equal to Q [Kot92, HT01, Lab99, Shi11, CH13], andBG14, Remark 3.2.4]). (4) LLHLM18, Proposition 7.18]. The reason is that in contrast to the case of GL 2 , W (λ) may include many Jordan-Hölder factors other than F (λ). In fact, F (λ) as a O[[K p ]]module often does not lift to an O-torsion-free module, which makes it more difficult to use the (expected) p-adic Hodge theoretic properties of Galois representations attached to automorphic forms. However, F (λ) does lift virtually. Suppose that [F (λ)] = W c W [W ] in the Grothendieck group of F[[K p ]]-modules, where each W in the sum lifts to characteristic 0 (for example, one can take W running over the reductions of various O[G(F p )]-modules using [Ser77, Theorem 33]). Then exactness gives us Borel subgroup. By a result of Kisin (see Theorem 6.1.1), d is the dimension of R λ+η,τ r over O for any λ and τ . For each Serre weightσ, we write a presentation (5.1) [σ] = (λ,τ ) c λ,τ [σ(λ, τ )] in the Grothendieck group of F[[K p ]]-modules. Then Axiom 5.1.1 guarantees that Kwhere ω K : I K → k × is the reduction of the Lubin-Tate character I K → O × K . Set λ = (λ j ) j∈J ∈ (Z n ) J , a Hodge-Tate cocharacter. 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[ "Spectroscopic and DFT studies of graphene intercalation systems on metals", "Spectroscopic and DFT studies of graphene intercalation systems on metals" ]	[ "Yu S Dedkov [email protected] \nSPECS Surface Nano Analysis GmbH\nVoltastraße 513355BerlinGermany\n\nFachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany\n", "E N Voloshina \nInstitut für Chemie\nHumboldt-Universität zu Berlin\n10099BerlinGermany\n" ]	[ "SPECS Surface Nano Analysis GmbH\nVoltastraße 513355BerlinGermany", "Fachbereich Physik\nUniversität Konstanz\n78457KonstanzGermany", "Institut für Chemie\nHumboldt-Universität zu Berlin\n10099BerlinGermany" ]	[]	Intercalation of different species under graphene on metals is an effective way to tailor electronic properties of these systems. Here we present the successful intercalation of metallic (Cu) and gaseous (oxygen) specimens underneath graphene on Ir(111)and Ru(0001), respectively, that allows to change the charge state of graphene as well as to modify drastically its electronic structure in the vicinity of the Fermi level. We employ ARPES and STS spectroscopic methods in combination with state-of-the-art DFT calculations in order to illustrate how the energy dispersion of graphenederived states can be studied in the macro-and nm-scale experiments.	10.1016/j.elspec.2016.11.012	[ "https://arxiv.org/pdf/1611.07198v1.pdf" ]	99,730,373	1611.07198	6bebaca2beac136dd533209b337b1b186a712137	Spectroscopic and DFT studies of graphene intercalation systems on metals 22 Nov 2016 Yu S Dedkov [email protected] SPECS Surface Nano Analysis GmbH Voltastraße 513355BerlinGermany Fachbereich Physik Universität Konstanz 78457KonstanzGermany E N Voloshina Institut für Chemie Humboldt-Universität zu Berlin 10099BerlinGermany Spectroscopic and DFT studies of graphene intercalation systems on metals 22 Nov 2016(Dated: October 6, 2018)* Corresponding author. 1 Intercalation of different species under graphene on metals is an effective way to tailor electronic properties of these systems. Here we present the successful intercalation of metallic (Cu) and gaseous (oxygen) specimens underneath graphene on Ir(111)and Ru(0001), respectively, that allows to change the charge state of graphene as well as to modify drastically its electronic structure in the vicinity of the Fermi level. We employ ARPES and STS spectroscopic methods in combination with state-of-the-art DFT calculations in order to illustrate how the energy dispersion of graphenederived states can be studied in the macro-and nm-scale experiments. I. INTRODUCTION Intercalation compounds on the basis of different layered host materials attract a lot of attention in the last decades. The most common examples, which appeared naturally during studies of the transport properties of graphite, are the graphite intercalation compounds (GICs) [1], which can be formed in different ways via incorporation of atoms of metals or nonmetals or big molecules between single graphite layers. Further studies lead to the discovery of intercalation compounds on the basis of layers of CoO 2 (Li x CoO 2 , Na x CoO 2 , etc.) [2], transition metal dichalcogenides (Cu x TiSe 2 , Li x MoS 2 , etc.) [3] and many others. All these studies demonstrate that chemical and physical properties of the intercalant hosts might be modified via controllable tailoring their electronic band structure, sometimes leading to the observation of new phenomena, which were previously not observed for the parent material, like superconductivity in alkali-metal GICs [4] or Cu x TiSe 2 [5]. In case of intercalation compounds on the basis of graphite or transition metal dichalcogenides, inserting intercalant layers leads to the decoupling of 2D layers of the host material from each other allowing to perform studies of the electronic structure of the single layers, which is modified by the presence of intercalant. It is interesting that despite the 3D crystallographic structure of these materials and change of the periodicity in the system upon intercalation, the valence band structure of the resulting materials has a symmetry of the host material [6][7][8]. This effect was nicely demonstrated in the angle-resolved photoelectron spectroscopy (ARPES) experiments on La-GIC compound [6,8], where band structures of a 2D host graphene layers and the inserted La layer, which have different symmetries, give the photoemission signals in the Brillouine zone of the respective symmetry. It was shown that if the symmetry of the sub-systems is different and the interaction between atoms in the sub-unit is stronger than between units of the crystal, then the folded bands are not observed in the photoemission experiment. Similar consideration is also valid for the local spectroscopic experiments, such as scanning tunneling spectroscopy (STS) or transport measurements, where effect of additional periodicity can change the transport coefficients dramatically. Graphene (gr), as a pure 2D material, is a subject of the intensive studies after its extraordinary transport properties, caused by the linear dispersion of the electronic states in the vicinity of the Fermi level (E F ), were discovered [9,10]. Besides the perspectives for graphene to be used in different technological applications, e. g. in gas sensors [11], in touch screens [12], or as a protective anticorrosion layer [13][14][15][16], it is a perfect material to study different physical phenomena in 2D materials that in many cases leads to the observation of the new fascinating effects. For example, if exfoliated graphene flake is placed on the h-BN substrate, then the small missalignment of two sublattices leads to the formation of the moiré lattices in this system that influences the electronic spectrum of graphene and the so-called Hofstadter butterfly spectrum of electrons, which movement is influenced by simultaneously acting periodic potential and the external magnetic field [17,18]. Further example is the graphene/metal system [19][20][21][22], where graphene is usually synthesised via chemical vapour deposition (CVD) technique. Here the interaction between graphene and the metallic substrate can be modified by placing different species (atoms of metals or non-metals, molecules, like CO, H 2 O, or C 60 ) underneath graphene [23][24][25][26][27][28][29][30][31]. The main goal of such studies is to decouple graphene from the metallic substrate, that in most cases changes the doping level of graphene (and even its sign) and to restore the original linear dispersion of the graphene-derived states in the vicinity of the Fermi level. Here ARPES and STS techniques allow to perform electronic structure studies on the macroor/and nm-scale giving information about doping level, gap opening, band dispersion and renormalization, and also providing new physics, which shed light on the new phenomena in graphene [22,[32][33][34][35][36]. In the present work we demonstrate the application of space-integrated (ARPES) and local (STS) spectroscopic methods in combination with density-functional theory (DFT) calculations for investigation of the electronic band structure of the graphene-based intercalation systems on metals: gr/Cu/Ir(111) and gr/O/Ru(0001). Both systems show the electronic properties (doping level and band dispersions) which are different from those for the parent systems, demonstrating the effective ways to tailor physical and chemical properties of the graphene/metal interface. This work is an extended review of the previously published results [30,36]. II. EXPERIMENTAL DETAILS Graphene in both experiments was prepared on hot metallic substrates (preliminary cleaned by cycles of Ar + -sputtering/annealing) via decomposition of C 2 H 4 at a partial pres-sure of p = 1 × 10 −7 mbar. Intercalation of Cu in gr/Ir(111) was performed via annealing of the Cu/gr/Ir(111) system with a nominal thickness of Cu-layer slightly more than 1 ML. The process of intercalation was monitored in the live-XPS (x-ray photoelectron spectrosocpy) experiments. Oxygen intercalation in gr/Ru(0001) was performed with a stainless steel pipe, which end was placed in the close vicinity of the sample surface, at the relatively high gas pressure (p O 2 = 1.5×10 −4 mbar) and sample temperature of 150 • C. Homogeneity and cleanness of the systems before and after intercalation were controlled in the respective STM, XPS, and ARPES experiments. The more detailed description of the sample preparation procedures can be found elsewhere [30,36]. All prepared samples were characterized at room temperature by means of STM/AFM performed with SPM Aarhus 150 equipped with KolibriSensor TM . In these measurements the sharp W-tip was used, which was cleaned in situ via Ar + -sputtering. In the presented STM images the tunnelling bias voltage, U T , is applied to the sample. Low-temperature STM measurements were performed in the SPECS JT-STM at the sample and tip temperature of 1 K. dI/dV spectroscopy and mappings were performed at low temperatures using the lock-in-technique with a modulation voltage of U mod = 10 mV and a modulation frequency The DFT calculations were carried out with the VASP program [37] using the projector augmented wave (PAW) method [38], a plane wave basis set, and the generalised gradient approximation as parameterised by Perdew et al. [39]. The plane wave kinetic energy cutoff was set to 400 eV. The long-range van der Waals interactions were accounted for by means of the DFT-D2 approach [40]. The supercells used to model the graphene-substrate interfaces are constructed from a slab of 5 metal layers with a graphene layer adsorbed from one side and a vacuum region of approximately 20Å. In the case of graphene/Ir(111) and graphene/Cu/Ir(111) this supercell has a (9×9) lateral periodicity with respect to metal and a (10×10) lateral periodicity with respect to graphene. In the case of graphene/O/Ru(0001) this supercell has a (12 × 12) lateral periodicity with respect to metal and a (13 × 13) lateral periodicity with respect to graphene. To avoid interactions between periodic images of the slab, a dipole correction is applied. During the structure relaxation, the positions of the carbon atoms as well as those of the top two layers of metal atoms (as well as oxygen atoms in the case of graphene/O/Ru(0001)) are allowed to relax. In the total energy calculations and during the structural relaxations the k-meshes for sampling the supercell Brillouin zone are chosen to be as dense as 6 × 6 and 3 × 3, respectively, and centred at the Γ-point. The band structures calculated for the studied systems were unfolded to the graphene (1 × 1) primitive unit cell according to the procedure described in Refs. [41,42] using the BandUP code. III. RESULTS AND DISCUSSIONS A. Systems preparation and characterization. between graphene π and Ir 5d z 2 states [43,44]. For graphene on Ru(0001) the direct imaging contrast is prevailed in STM images for low bias voltages due to the domination of the topographic contribution in the STM imaging [30,[45][46][47]. Intercalation of Cu and oxygen in gr/Ir(111) and gr/Ru(0001) (Figure 1, bottom row), respectively, conserves the periodicity of the moiré lattices (pseudomorphic growth of Cu at the gr/Ir interface [36] and (2 × 1)-O structure at the gr/Ru interface [30]), but leads to the dramatic changes in the STM imaging contrast. After intercalation the imaging contrast is direct for gr/Cu/Ir(111) and the STM contrast for gr/O/Ru(0001) becomes extremely flat. Such changes in the imaging contrast for both systems are the reflection of the respective modifications in the electronic structure of graphene on the corresponding support and they are the subjects of the further ARPES and STS studies of the respective systems. In order to control the formation of the gr/Cu/Ir(111) system, which STM images might be, at some imaging conditions, similar to those of gr/Ir(111), the process of Cu intercalation under graphene on Ir(111) was monitored in the live-XPS measurements, where C 1s and Ir 4f core levels were acquired as functions of time with sample temperature ramped from room temperature to 850 K. Fig. 2 shows the high-resolved C 1s and Ir 4f XPS spectra taken before and after intercalation (bottom and top rows) as well as the respective photoemission intensity maps as a function of binding energy and sample temperature (middle row). From these intensity maps one can see that until ≈ 820 K there are only gradual changes in the spectra: small shift of the C 1s line to the higher binding energy and the weak decrease of intensity of the interface component of the Ir 4f line (low binding energy shoulder). At T ≈ 830 K drastic changes are observed: (i) C 1s line shifts stepwise to the higher binding energy by ≈ 0.55 eV and (ii) the peak intensity of the "interface" components for the Ir 4f lines is reduced by ≈ 30% (at the same time the peak intensity of the "bulk" components is reduced by ≈ 18%). Both changes as well as the absence of any C 1s signal at the binding energies corresponding to gr/Ir(111) indicates the complete intercalation of the Cu layer and formation of the gr/Cu/Ir(111) intercalation system. Before and after Cu intercalation the C 1s XPS line can be fitted by one (284.18 eV) and two (284.69 eV and 285.01 eV) components, respectively, reflecting stronger interaction between graphene and Cu/Ir (111) with smaller gr/metal distance compared to gr/Ir(111), and that two distinct areas in the STM images are observed. For the Ir 4f spectra the energy splitting between "bulk" and "interface" components is reduced from 537 meV to 463 meV for systems before and after Cu intercalation, respectively. Intensity of the "interface" part is strongly reduced after Cu intercalation in gr/Ir(111). B. ARPES and DFT of gr/Ir(111) and gr/Cu/Ir(111) Figure 3 compiles the results on the electronic structure studies of graphene on Ir(111) and Cu/Ir(111). ARPES intensity map for gr/Ir(111) shows a clear dispersion of the graphenederived π band along the k direction perpendicular to Γ−K in the Brillouine zone [ Fig. 3(a)]. According to the recent theoretical works [36,43,48], the interaction between graphene and Ir (111) (111) covered by graphene [52,53]. Intercalation of Cu in gr/Ir(111) leads to the significant changes in the observed ARPES picture [ Fig. 3(b)]. For the formed intercalation system, the energy shift of the Dirac point is observed, resulting in the n-doping of graphene, E D − E F = −0.69 ± 0.01 eV. The most important observation is the opening of the energy gap of (0.36 ± 0.01) eV for the graphene π states directly at the Dirac point. Similar behavior was also observed for the other intercalation systems with Cu, Au, and Ag used as intercalants [54][55][56]. It is interesting to note that a substantial hybridization between graphene π state and Cu 3d states is observed in the binding energy range of ≈ 2−4 eV. This effect is reflected in the opening of the energy gaps at the energy and wave-vector values, where π states are intersected by Cu 3d bands. As will be shown later this effect of hybridization and space-overlap of the graphene-derived π states with the Cu 3d states of different symmetry has an important implication on the spectral function of graphene π states at the Dirac point. [36,59] shows that, in case of graphene on the close-packed surface of a d metal, p z orbitals of two carbon atoms in the graphene unit cell overlap with d orbitals of the underlying metal atom, which have different symmetry. In general case [59] we can say that the following hybrid states are formed: p z (C top ) + d z 2 and p z (C f cc ) + d xz,yz , where C top and C f cc are carbon atoms in the unit cell occupying top and f cc high-symmetry adsorption positions above the close-packed surface. In case of the free-standing graphene, where carbon atoms are identical, the electronic states of two carbon sublattices are degenerate at the Dirac point that leads to the zero density of states at this point. If we consider graphene on d-metal, then symmetry between two carbon sublattices is broken and electronic states in the vicinity of the Dirac point have different symmetry as discussed above. This effect leads to the lifting of the degeneracy and opening of the energy gap directly at the Dirac point, which width is determined by the hybridization strength between graphene π and metal d states [36,59]. C. STS and DFT of gr/O/Ru(0001) The crystallographic structure of gr/O/Ru(0001) was studied in details by means of STM and non-contact atomic force microscopy (AFM) in Ref. [30] and it was shown that graphene in this system is extremely flat with the corrugation of the moiré structure of only 0.1Å. The electronic structure of this nearly free-standing graphene was studied by means of STS on the local scale. The results of these experiments are compiled in Fig. 5. The STM image of the gr/O/Ru(0001) system [ Fig. 5(a)] shows drastic changes in the morphology of the system compared to the strongly corrugated parent gr/Ru(0001) [30,60,61] [ Fig. 1]. Intercalation of oxygen in gr/Ru(0001) leads to the significant reduction of the corrugation of the graphene moiré lattice from 1.27Å for the parent system [47] to 0.12Å for gr/O/Ru(0001) [30]. This effect together with the previously observed by ARPES strong p-doping of graphene in the later system [62] indicates the decoupling of graphene from the substrate and restoring of the nearly free-standing character of the electronic states of graphene. In order to obtain information about electronic structure of graphene in the intercalation system we performed local scale STS experiments on gr/O/Ru(0001). In the first approach we collected a series of dI/dV maps at different bias voltages (U T ). One of such map acquired at U T = +50 mV is shown in Fig. 5 thus, forming the moiré lattice. Such structure was found to be adequate for the description of this system, giving the accurate description of the crystallographic and electronic structure of gr/Ru(0001) [30,47]. In our analysis we use the structure where after intercalation of O 2 in gr/Ru(0001), the oxygen atoms form the p(2 × 1) structure with 0.5 ML coverage with respect to Ru(0001) at the interface between graphene and Ru. It is supported by the experimental observations as well as by the calculated doping level [30,62,66,67]. Although the complete band structure has a "spaghetti-like" view due to the folding of the bands, the main graphene-derived bands, π and σ, can be easily recognised. One can clearly see that after intercalation of oxygen in gr/Ru(0001), graphene becomes nearly freestanding and its electronic states are completely decoupled from the Ru substrate (compared to gr/Ru(0001) [30,62,68] The FFT maps obtained in the experiment shown in Fig. 5(c-f) can be compared with the respective theoretical results shown in Fig. 6(d-g) and good agreement between both sets of data is found. IV. CONCLUSIONS In the present manuscript we demonstrate the successful application of the space-resolved (local) and space-integrated spectroscopic methods for the investigation of the electronic structure of the graphene intercalation systems formed on metals. All experimental data were reproduced in the DFT calculations allowing to understand the mechanisms leading to the formation of the spectrum of the charge carriers in graphene (doping level, gap openings, etc.). Two cases, of metal (Cu) and non-metal (oxygen) intercalation, were considered. In Symmetry analysis of the respective states around E D allows to draw the mechanism, which is responsible for the modifications of the band dispersion around the Dirac point. After intercalation of oxygen in gr/Ru(0001), a strong p-doping of graphene was found, which was confirmed in our DFT calculations. Band dispersion in the vicinity of E F of the graphene π states for the obtained intercalation system was extracted in the analysis of the FFT-STS maps and the main results were reproduced in the framework of the JDOS approach for the strongly p-doped nearly free-standing graphene. f mod = 684.7 Hz. ARPES experiments were performed in the UHV station equipped with SPECS PHOI-BOS 150/2D-CCD analyzer and Ar/He UV-light source. The sample was placed on a 5-axis motorized manipulator, allowing for its precise alignment in the k space. The sample was azimuthally pre-aligned in such a way that the polar scans were performed along the Γ − K direction of the graphene-derived BZ with the photoemission intensity on the channelplate images acquired along the direction perpendicular to Γ − K. The final 3D data sets of the photoemission intensity as a function of kinetic energy and two emission angles were transformed in the respective data sets for the reciprocal space, I(E B , k x , k y ) for the careful analysis, where E B is the binding energy of electrons and k x,y are two orthogonal components of the wave vector in BZ. Part of ARPES and XPS experiments on the intercalation process studies was performed at BESSY II (HZB Berlin). Figure 1 1summarizes the results of the samples preparation and characterization. Original, hexagonally packed clean surfaces of 4d or 5d metals, hcp Ru(0001) or fcc Ir(111) (top row), are used for the preparation of the high-quality graphene layers, which in these cases form the so-called moiré structures with several nm periodicities (middle row). Here several high-symmetry adsorption positions for carbon atoms on the close-packed surfaces can be identified: ATOP, FCC, HCP (they are marked by the respective capital letters in Fig. 1, middle row). They are determined with respect to the corresponding adsorption places of the metal surface, which are surrounded by the carbon ring of a graphene layer. For low bias voltages, graphene on Ir(111) is imaged in the inverted contrast, when the topographically highest ATOP place of a graphene layer is imaged as a dark spot in the STM image as opposite to the HCP and FCC positions. Such an effect was explained by the formation of the corresponding interface states between graphene and Ir(111) as a result of an overlap is mainly governed by the van der Waals forces and the hybridization between graphene-derived and Ir valence band states is relatively small. This leads to the observation of the linear dispersion of the graphene π states around E F , with the position of the Dirac point at approximately 100 ± 10 meV above E F (graphene is p-doped). The influence of the Ir(111) substrate is manifested via observation of the so-called replica bands in the ARPES intensity map and opening of the mini-gaps at E − E F ≈ −0.76 eV and E − E F ≈ −2.1 eV according to the avoid-crossing mechanism [44, 49, 50]. These effects are due to the additional moiré lattice periodicity (of ≈ 25Å) observed for this system. The relatively weak interaction between graphene and Ir(111) does not destroy the original Rashba-split surface state observed on Ir(111) around the Γ point [Fig. 3(c); here data were acquired with non-monochromatized Ar I line (Ar Iα/Ar Iβ, hν = 11.83 eV/11.62 eV), leading to two sets of parabolas separated by energy of 0.21 eV]. Adsorption of graphene on Ir(111) leads only to the upward energy shift of the surface state from E −E F = −0.34 eV at Γ for Ir(111) to E − E F = −0.21 eV at Γ for gr/Ir(111), that can be assigned to the stronger localization of the surface state wave function upon graphene adsorption. This behavior is similar to the one observed for the same system[51] as well as for Au(111) and Ag Figure 4 4shows the calculated band structure of a graphene layer on Ir(111) and Cu/Ir(111) for the corresponding supercell and then unfolded to the primitive (1 × 1) unit cell of graphene (band structures are presented along the respective high-symmetry directions of the graphene Brillouine zone shown as an inset in panel (a)). In both panels the dispersion of the graphene-derived π states in the vicinity of the Fermi level can be easily distinguished. For gr/Ir(111) [Fig. 4(a)] a clear p-doping of a graphene layer is detected with the calculated position of the Dirac point of E D − E F = 0.11 eV. Closer analysis of the region around the Dirac point shows the existence of the energy gap of 0.22 eV in the energy dispersion of the π band. Such gap with the width of ≈ 100 meV was previously observed in the ARPES experiments on the electron-doping of gr/Ir(111) via K-adsorption [57]. The appearance of this gap might be explained by the effect of hybridization between graphene states of the p z symmetry (π band) and the Ir d z 2 surface state localized around the K-point, which was recently detected in the experiment [58]. In case of the gr/Cu/Ir(111) intercalation system, a graphene layer has an n-doping with a position of the Dirac point at E D − E F = −0.49 eV. At the same time the energy gap of 0.11 eV is opened directly at the Dirac point. Taking into account the size of the studied system, the agreement between experimental and theoretical values of doping and the width of the band gap is rather good. There is also a series of the energy gaps in the energy range E − E F = −1.5... − 5 eV, which appear at energies and k-vector values where graphene π band is crossed by the Cu 3d bands leading to the formation of the hybrid states. The deep analysis performed in Refs. (b). Different mechanisms for the scattering of the electron waves in graphene lead to the formation of the interference picture, which can be identified in such dI/dV map together with the signals originating from the atomic graphene and moiré lattices. Fast Fourier Transformation (FFT) analysis of such dI/dV maps [Fig. 5(c-f)] allows to extract the energy dispersion of the carriers involved in the scattering processes. In the FFT image several spots can be identified. First set, marked by the white rectangle in Fig. 5(f), is due to the reciprocal lattice of graphene (here the main central spot is surrounded by six spots originating from the moiré lattice of the system). The second set of spots in the FFT image, marked by the white circle, is placed at the positions of the ( √ 3 × √ 3)R30 • structure of the reciprocal lattice, i. e. they are centred around the Kpoints of the graphene Brillouine zone. This ring-shape structure is formed by the intervalley scattering of electrons between two equivalent points in BZ, K and K . The radius of this structure 2k can be used for the calculation of the wave vector of the scattering electron wave at the energy E = eU T [30, 52, 53, 63-65]. Such analysis performed for the series of dI/dV images collected at different U T allows to extract the energy dispersion of the carriers in graphene around the K-point of BZ. The results are shown in Fig. 5(g) by solid circles. Linear fit of these data leads to E D = (610 ± 20) meV and v F = (1.06 ± 0.04) · 10 6 m/s. The extracted value of E D was compared with the one obtained from the single dI/dV curves collected along the line in the STM image [inset of Fig. 5(g)]. The intensity dip around 550 meV [Fig. 5(h,i)] was assigned to the position of E D for graphene on O/Ru(0001) and this value is in rather good agreement with the one obtained in the previous analysis. Experimental STM/STS results for gr/O/Ru(0001) were analysed in the framework of DFT. The parent gr/Ru(0001) system is modelled in the slab geometry, where a graphene layer with a (13×13) periodicity is placed on a Ru(0001) surface with a (12×12) periodicity, Figure 6 ( 6a) shows the calculated band structure of the graphene/O/Ru(0001) system in the supercell geometry described earlier and unfolded for the (1 × 1) unit cell of graphene. . Analysis of the electronic structure in the vicinity of the K-point of graphene in this system shows that the π band has a linear dispersion with a position of the Dirac point E D − E F = +536 meV and v F = 1.01 · 10 6 m/s, indicating the strong p-doping of graphene. These values are in rather good agreement with experimental data presented above.In order to demonstrate the effects observed in the STM/STS experiments we use the position of the Dirac point of E D − E F = +600 meV, which is very close to the one obtained in the experiment (the difference of 10 meV does not dramatically alter the obtained results).The band structure of graphene in the vicinity of the Dirac point in the tight-binding (TB) approach can be expressed asE ± = ±t 3 + f (k), where f (k) = 2 cos( √ 3k y a) + 4 cos( √ 3/2k y a) cos(3/2k x a) [10] (t ≈ 2.8 eV is the nearest-neighbour hopping energy; a = 1.42Å is the distance between carbon atoms; k x,y are components of the wave vector). FFT maps presented in Fig. 5(c-f) were obtained at bias voltages ±50 meV and ±150 meV, corresponding to E − E F = −450, −550, −650, −750 meV for free-standing graphene. The respective constant energy cuts (CECs) for the p-doped graphene around the K-point are shown as an inset of Fig. 6(a) with the corresponding U T values marked in the figure. From these data sets the clear trigonal warping of the π bands with increasing of E is visible that can be compared with experimental data. Calculated CECs at different energies with respect to E D (and respectively at different U T ) can be used for modelling of the FFT scattering maps in STS experiments. As was shown in Refs. [63, 69], the power spectrum in the FFT map can be build on the basis of a joint-density-of-states (JDOS) approach, where one simply counts the number of pairs of ( − → k , − → k ), giving a scattering vector − → q with the same length and direction. For simplicity one can consider such calculations as a self-correlation procedure for the particular CEC image taken at the certain energy. Fig. 6 (b) and (c) show the calculated CEC and FFT-STS maps obtained in the JDOS approach for the free-standing graphene corresponding to the energy of E − E F = −750 meV (corresponds to the value of the bias voltage of U T = −150 meV in the STS experiments on gr/O/Ru(0001)), where ring-like structure around the K-points is assigned to the intervalley scattering of the electron waves between equivalent K and K . both cases the reversing of the doping level of graphene compared to the parent systems was observed. In the ARPES studies of gr/Ir(111) and gr/Cu/Ir(111) we followed the changes in the electronic structure and found n-doped graphene in the later case, with the position of the Dirac point of E D − E F = −0.69 eV and the band gap of 0.36 eV directly at E D . FIG. 2 : 2XPS spectra of gr/Ir(111) (bottom) and gr/Cu/Ir(111) (top): C 1s (left column) and Ir 4f (right column). The respective fitting components are shown by the shaded areas. Middle row shows the evolution of the C 1s and Ir 4f intensity as a function of the annealing temperature resulting in formation of gr/Cu/Ir(111) at 840 K. FIG. 4: Unfolded to the primitive (1 × 1) graphene unit cell the band structures of a graphene layer on (a) Ir(111) and (b) Cu/Ir(111).FIG. 6: (a) Band structure of gr/O/Ru(0001) along the Γ − M − K − Γ direction of BZ. Inset shows TB-calculated CECs around the K-point corresponding to the experimental values of U T (marked for every curve) and the Dirac point of E D − E F = +600 meV. (b,c) CEC and the calculated FFT-STS map for energy E−E F = −750 meV for free-standing graphene, corresponding to U T = −150 meV used in the experiment. (d-g) Series of the calculated FFT-STS structures around the K-point corresponding to the experimental values of U T marked in every image. Scale bar is 1 nm −1 . ) AcknowledgementsThe authors thank H. Vita Intercalation compounds of graphite. M Dresselhaus, G Dresselhaus, Adv. Phys. 51M. 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FIG. 1: Preparation steps (from top to bottom) of intercalation systems on the basis of graphene/metal. bottom row: STM images of gr/Cu/Ir(111) and gr/O/Ru(0001) intercalation systemsFIG. 1: Preparation steps (from top to bottom) of intercalation systems on the basis of graphene/metal. Top row: atomically resolved STM images of Ir(111) and Ru(0001); middle row: STM images of gr/Ir(111) and gr/Ru(0001) moiré structures; bottom row: STM images of gr/Cu/Ir(111) and gr/O/Ru(0001) intercalation systems. Data were obtained with photon energy of hν = 40.8 eV. (c) ARPES intensity map [I(E B , k y )] around the Γ-point and the respective constant energy cuts [I(k x , k y )] showing interface state in gr/Ir(111). Data were acquired with non-monochromatized Ar I line (Ar Iα/Ar Iβ, hν = 11.83 eV/11.62 eV). (d) APRES intensity map along Γ − K for gr/Cu/Ir(111) obtained with photon energy hν = 65 eV. ARPES intensity maps. around the K-point and the respective constant energy cuts [I(k x , k y )] for (a) gr/Ir(111) and (b) gr/Cu/Ir(111). Inset shows the sketch of the hexagonal Brillouine zone with marked k x , k y directionsFIG. 3: ARPES intensity maps [I(E B , k y )] around the K-point and the respective constant energy cuts [I(k x , k y )] for (a) gr/Ir(111) and (b) gr/Cu/Ir(111). Data were obtained with photon energy of hν = 40.8 eV. (c) ARPES intensity map [I(E B , k y )] around the Γ-point and the respective constant energy cuts [I(k x , k y )] showing interface state in gr/Ir(111). Data were acquired with non-monochromatized Ar I line (Ar Iα/Ar Iβ, hν = 11.83 eV/11.62 eV). (d) APRES intensity map along Γ − K for gr/Cu/Ir(111) obtained with photon energy hν = 65 eV. Inset shows the sketch of the hexagonal Brillouine zone with marked k x , k y directions. Combined STM/STS measurements on gr/O/Ru(0001): (a) z(x, y) and (b) dI/dV. x, yFIG. 5: Combined STM/STS measurements on gr/O/Ru(0001): (a) z(x, y) and (b) dI/dV (x, y). U mod = 10 mV. (c-f) Series of the FFT images obtained after transformation of the respective dI/dV maps acquired at U T marked in every image. White circles mark the intervalley scattering features used in the analysis. (g) Extracted from the FFT maps energy dispersion of the carriers in graphene on. 20 × 20 nm 2 , U T = +50 mV, I T = 200 pA, f mod = 684.7 Hz. Scanning parameters: 20 × 20 nm 2 , U T = +50 mV, I T = 200 pA, f mod = 684.7 Hz, U mod = 10 mV. (c-f) Series of the FFT images obtained after transformation of the respective dI/dV maps acquired at U T marked in every image. White circles mark the intervalley scattering features used in the analysis. (g) Extracted from the FFT maps energy dispersion of the carriers in graphene on solid circles -extracted points, solid line -the respective linear fit with E D and v F marked in the figure. (h) Single dI/dV spectra acquired along the line marked in the STM image shown as an inset in panel (g). (i) Extracted from dI/dV spectra in. O/Ru, h) the position of E DO/Ru(0001): solid circles -extracted points, solid line -the respective linear fit with E D and v F marked in the figure. (h) Single dI/dV spectra acquired along the line marked in the STM image shown as an inset in panel (g). (i) Extracted from dI/dV spectra in (h) the position of E D .	[]
[ "Controlled photon transfer between two individual nanoemitters via shared high-Q modes of a microsphere resonator", "Controlled photon transfer between two individual nanoemitters via shared high-Q modes of a microsphere resonator" ]	[ "S Götzinger ", "L De ", "S Menezes ", "A Mazzei ", "O Benson ", "S Kühn ", "V Sandoghdar ", "\nNano-Optics\nLaboratory of Physical Chemistry\nHumboldt University\nHausvogteiplatz 5-7D-10117BerlinGermany\n", "\nETH Zurich\nCH-8093ZurichSwitzerland\n" ]	[ "Nano-Optics\nLaboratory of Physical Chemistry\nHumboldt University\nHausvogteiplatz 5-7D-10117BerlinGermany", "ETH Zurich\nCH-8093ZurichSwitzerland" ]	[]	We realize controlled cavity-mediated photon transfer between two single nanoparticles over a distance of several tens of micrometers. First, we show how a single nanoscopic emitter attached to a near-field probe can be coupled to high-Q whispering-gallery modes of a silica microsphere at will. Then we demonstrate transfer of energy between this and a second nanoparticle deposited on the sphere surface. We estimate the photon transfer efficiency to be about six orders of magnitude higher than that via free space propagation at comparable separations.Typeset by REVT E X 1If two dipolar emitters are separated by a distance r much less than the transition wavelength λ, they can undergo strong coherent dipole-dipole coupling, leading to sub-and superradiance 1,2 . If their transitions are broadened, dipole-dipole coupling becomes incoherent as in the case of Fluorescence Resonant Energy Transfer (FRET), where the energy from a "donor" is transferred to an "acceptor", provided there is sufficient overlap between the former's emission spectrum and the latter's absorption line. The efficiency of FRET 3 is proportional to (1 + (r/r 0 ) 6 ) −1 and falls to 50% already at r = r 0 ∼ 10 nm. For r > λ, optical communication between the two emitters takes place via propagating photons, while the coupling drops as 1/r 2 . At a distance of 50 µm, the efficiency of one emitter absorbing	10.1021/nl060306p	[ "https://export.arxiv.org/pdf/quant-ph/0604081v1.pdf" ]	20,932,948	quant-ph/0604081	347607a9d823cdc219f4dd3ba720828defc47f9e	Controlled photon transfer between two individual nanoemitters via shared high-Q modes of a microsphere resonator 11 Apr 2006 S Götzinger L De S Menezes A Mazzei O Benson S Kühn V Sandoghdar Nano-Optics Laboratory of Physical Chemistry Humboldt University Hausvogteiplatz 5-7D-10117BerlinGermany ETH Zurich CH-8093ZurichSwitzerland Controlled photon transfer between two individual nanoemitters via shared high-Q modes of a microsphere resonator 11 Apr 2006arXiv:quant-ph/0604081v1 Typeset by REVT E X 1 We realize controlled cavity-mediated photon transfer between two single nanoparticles over a distance of several tens of micrometers. First, we show how a single nanoscopic emitter attached to a near-field probe can be coupled to high-Q whispering-gallery modes of a silica microsphere at will. Then we demonstrate transfer of energy between this and a second nanoparticle deposited on the sphere surface. We estimate the photon transfer efficiency to be about six orders of magnitude higher than that via free space propagation at comparable separations.Typeset by REVT E X 1If two dipolar emitters are separated by a distance r much less than the transition wavelength λ, they can undergo strong coherent dipole-dipole coupling, leading to sub-and superradiance 1,2 . If their transitions are broadened, dipole-dipole coupling becomes incoherent as in the case of Fluorescence Resonant Energy Transfer (FRET), where the energy from a "donor" is transferred to an "acceptor", provided there is sufficient overlap between the former's emission spectrum and the latter's absorption line. The efficiency of FRET 3 is proportional to (1 + (r/r 0 ) 6 ) −1 and falls to 50% already at r = r 0 ∼ 10 nm. For r > λ, optical communication between the two emitters takes place via propagating photons, while the coupling drops as 1/r 2 . At a distance of 50 µm, the efficiency of one emitter absorbing a photon radiated by the other is merely 3 × 10 −13 , considering a typical room temperature absorption cross section 4 of σ A ≈ 10 −16 cm 2 . In order to enhance the coupling between two emitters, one could funnel the energy from one to the other by using optical elements such as lenses and waveguides. However, this process remains limited because each photon flies by the atom only once. Thus, it is interesting to exploit resonant structures to provide longer effective interaction times. In addition, resonators can influence radiative processes by modifying the density of states 5,6 . Furthermore, if the cavity is made very small, the field per photon becomes large, resulting in a much stronger coupling between the emitter and the photon field. These effects depend on the three-dimensional locations of the donor and acceptor molecules in a decisive manner. Enhancement of the energy transfer rate in microcavities has been previously demonstrated for systems containing ensembles of molecules distributed over a volume (area) much larger than λ 3 (λ 2 ) in microdroplets 7,8 or polymer microcavities 9 . However, the ideal case where two single emitters couple via photon transfer through a single mode of a high finesse microcavity remains a great experimental challenge. In this Letter we report on a major step toward this goal. We present experimental results on the controlled optical coupling and photon transfer between two individual subwavelength emitters at large distances via high-Q modes of a microresonator. Silica microspheres melted at the end of a fiber support very high-Q Mie modes known as whispering-gallery modes (WGMs) 6 . WGMs are characterized by the radial number n and angular numbers l and m which determine their resonance frequencies and spatial intensity distributions 6 . Each mode can have polarizations TE or TM corresponding to radial magnetic and electric fields, respectively. For a sphere of radius R and refractive index N, an increment in l shifts the spectrum by one free spectral range F SR = c/(2πRN). The modes with n = 1 and l = \|m\| are called the "fundamental" modes and yield the most confined WGM with the largest electric field at the sphere surface. The quantities n and l − \|m\| + 1 give the number of intensity maxima along the sphere radius and perpendicular to the equator, respectively 10 . We first discuss the realization of an on-command coupling between a dye-doped nanoparticle and the WGMs of a microsphere. As depicted in Fig. 1a, our strategy has been to attach the nanoparticle to the end of a fiber tip (see inset) and use a home-built Scanning Near-field Optical Microscope (SNOM) stage to manipulate the emitter in the vicinity of the microsphere surface. The recipe for the production of such probes is discussed by Kühn et al. 11 while the alignment and characterization of the microspheres and their WGMs are described in Refs 12,13 . During all that follows, the quality factor Q of the sphere could be measured by direct spectroscopy using a narrow-band diode laser at λ = 670 nm. The fluorescent nanoparticle at the tip was excited through the fiber, and a prism was used to extract part of the particle emission that coupled to the WGMs 14 . Alternatively, a microscope objective (NA=0.75) allowed us to collect both the free space components of the nanoparticle fluorescence and the scattering from the sphere 14 . (Red Fluorescent, Molecular Probes, Inc.) of 500 nm in diameter was placed within a few nanometers of the microsphere. The spectrum shows a FSR of 0.85 nm expected for a sphere used in this measurement of 2R ≈ 96 µm . By using polarized detection, we identified the two dominant peaks in each FSR as TM and TE modes (see Fig. 1b). Since the small numerical aperture of our detection path via the prism was optimized for coupling to low n modes 15 , we attribute these resonances to n = 1 and the weaker ones to higher n modes. Now, we examine the position dependence of the bead's coupling to the WGMs. To do this, we tuned the emitter's angular coordinate θ to the sphere equator 13 and then varied its radial separation to the sphere surface using the SNOM distance stabilization and scanning machinery (see insets in Figs. 2a and b). The fluorescence emitted into the WGMs was collected through the prism coupler and detected by a photomultiplier tube. Figure 2a shows a characteristic decrease expected for the evanescent coupling between the bead and the WGMs of the microsphere. Next, we fixed the particle-sphere separation to 5 − 10 nm and scanned the bead about the equator along θ. The symbols in Fig. 2b show that the fluorescence signal detected through the prism drops quickly within 5 • or equivalently 4 µm about the equator. We note that at room temperature the broad spectrum of a molecule couples to many modes of different m. However, because WGMs with higher l − \|m\| values have larger mode volumes and therefore lower electric fields at the equator, their coupling Next, we discuss photon transfer between a donor and an acceptor nanoparticle via the WGMs. In this experiment a sphere of 2R = 35 µm was coated with a solution of acceptor beads (Crimson, Molecular Probes, Inc.), 200 nm in diameter. After coating there were a total of less than 10 particles on the surface of the microsphere and the under-coupled Q of the fundamental mode was measured to be 3 × 10 7 . Then we retracted the prism to avoid losses due to output coupling and imaged the location of the acceptor beads on the microsphere using a CCD camera 14 . We located a single nanoparticle close to the sphere equator and centered it in the confocal detection path of the spectrometer (see Fig. 3a). A single donor bead (Red Fluorescent, Molecular Probes, Inc.) of 500 nm in diameter was attached to a tip and was excited through the fiber with a laser power of ≈ 20 µW . The black and green curves in Fig. 3b show reference fluorescence spectra of the donor and acceptor beads, respectively, recorded on a cover glass. Finally, we approached the donor to the sphere and recorded the spectrum of the single acceptor bead through the microscope objective. The red curve in Fig. 3b plots the spectrum obtained from the location of the acceptor. The fast spectral modulations provide a direct evidence of coupling to high-Q WGMs 6,16,17,18 . Comparison of this spectrum with the black and green spectra reveals the coexistence of contributions from the donor and the acceptor fluorescence. We remark that although our confocal detection efficiently discriminates against light emitted at the donor location, it is possible for this emission to couple to the WGMs and get scattered into our collection path by the acceptor bead. To take this into account, we subtracted the donor fluorescence spectrum from the recorded (red) spectrum after normalizing their short wavelength parts. Furthermore, to rule out the possibility of direct excitation of the acceptor by the laser light, we retracted the donor from the sphere, photobleached it with an intense illumination of the excitation light and approached it again to the sphere. The signal at the acceptor position was then collected under exactly the same conditions and is shown in Fig. 3b. This contribution is clearly negligible compared to the total emission (red curve), verifying that the acceptor fluorescence has been almost entirely pumped by the donor emission. After subtracting this small contribution, we arrive at the brown curve in Fig. 3c which coincides very well with the fluorescence spectrum of the acceptor (also shown in 3c for convenience). We note in passing that we have also checked that bleaching the acceptor bead would result in the disappearance of the longer wavelength part of the red spectrum in Fig. 3b. These measurements show, to our knowledge, the first experimental realization of photon exchange between two well-defined nanoemitters via shared high-Q modes of a microresonator. Below, we discuss the underlying physical phenomena. Let us define the transfer efficiency η i as the probability β i of the donor emitting a photon into the i th WGM and subsequently for this photon to get absorbed by the acceptor. Then the efficiency η i for a photon that is emitted by the donor to be absorbed by the acceptor can be written as η i = β i σ A,abs σ A,abs + σ D,sca + σ D,abs + σ i,Q ,(1) where the quotient stands for the probability of a cavity photon being absorbed by the acceptor before getting lost in other channels. Note that because the emission and absorption between the two quantities can be notable, but it has been shown that it remains well within a factor of 3 even for strongly scattering silver particles of diameter 200 nm at plasmon resonance 20 . Thus, in our case it is appropriate to use the conventional values of the cross sections for obtaining an order of magnitude estimate. The absorption cross section of the acceptor particle can be taken as σ A,abs ≈ 10 −11 cm 2 , assuming σ abs ≈ 10 −16 cm 2 per molecule 4 and 10 5 molecules per particle. Since due to the Stokes shift of molecular fluorescence the donor does not absorb its own emission very efficiently, we can neglect σ D,abs . In addition, we obtain σ Q = 2 × 10 −12 cm 2 for a fundamental mode based on Q = 3 × 10 7 . Since in our experiment the measured Q remained unchanged as the tip approached the microsphere, we conclude that σ D,sca was negligible compared to σ Q 21 . We find, therefore, that for a fundamental mode, the quotient in Eq. (1) is about 10 −4 considering a single molecule acceptor. In the weak coupling regime, the spontaneous emission rate Γ can be written as Γ = respectively. The red curve shows the recorded spectrum when a donor is brought close to the sphere's surface. The blue curve shows the same measurement after bleaching of the donor. c) The brown curve plots the net emission spectrum of the acceptor as a result of photon transfer. The green curve shows the normalized spectrum of a free acceptor from (b) for comparison. d) Calculated dependence of β 0 on emitter's separation from the microsphere. are proportional to the projection of each mode's field intensity \|E i \| 2 at the sphere surface onto the dipolar axis. In what follows we calculate β i for the fundamental mode and use its scaling with \|E i \| 2 to evaluate the contribution of other WGMs to the energy transfer process. Since the seminal work of Purcell, it is known that spontaneous emission of a narrow-band dipole is enhanced when it is coupled to a resonator mode 5 . This enhancement is reduced if the linewidth of the dipole is broadened to Γ b , greater than the cavity linewidth Γ cav , as is the case for Γ cav = 6 × 10 −5 nm and Γ b ≈ 20 nm in our experiment. The ratio β, of the emission into the cavity mode to the total emitted power is thus, given by β ≈ β 0 (Γ cav /Γ b ) 22 whereby β 0 represents the fraction for a narrow-band emitter. Note that since β 0 ∝ Q, in this case β becomes independent of Q. To calculate β 0 for a fundamental mode, we calculated the power radiated into this mode 23 by approximating the emission of a randomly oriented dipole by a spherical wave and calculating its overlap with the mode 24 . The finite Q of the sphere was accounted for by using a complex N 25 . Figure 3d shows the result as a function of the distance between the emitter and the sphere's surface. We find that β 0 = 0.5 at a distance of 50 nm from the sphere surface, leading to β i = 1.5 × 10 −6 . For other WGMs, higher n and l −\|m\| result in an increase of the mode volume and lower conclude that η is about 10 × 40 × 20 × 2 ≈ 2 × 10 4 times larger than that of a single fundamental mode. Putting all the above-mentioned information together, we find that η ≈ (1.5 × 10 −6 ) × 10 −4 × 2 × 10 4 ≈ 10 −6 for a single molecule acceptor which is more than six orders of magnitude larger than the free-space rate for absorbing a photon emitted at a distance of about 35 micrometers. We have demonstrated that the application of scanning probe techniques allows one to achieve an on-command and efficient evanescent coupling between a nanoscopic emitter and a WGM resonator. In the current system we have coupled the broad-band emission of dye molecules to a large number of resonator modes in a dissipative energy transfer process. Under these conditions, the role of the high cavity Q in the enhancement of spontaneous emission has been negligible. Instead, the importance of the high Q has been to circulate each photon a large number of times, increasing its chance of interaction with the acceptor. However, at low temperatures emitter transitions as narrow as a few tens of MHz can be achieved, enhancing the photon transfer efficiency between single emitters by an additional five orders of magnitude. Furthermore, one could get around the inhomogeneous distribu- Figure 1b FIG. 1 1b1shows part of the fluorescence spectrum recorded via the prism when : a) The schematics of the experimental setup. The inset shows an SEM image of a single 500 nm bead attached to a glass tip. b) Spectrum recorded via the prism when the bead was close to the sphere's surface. Colored triangles indicate the theoretical positions of the resonances assuming a sphere of diameter 96 µm and an index of refraction of N=1.45724. The fundamental modes are marked in black (TE) and blue (TM). Modes with n = 2 are marked in red (TE) and green (TM). The first label denotes n and the second corresponds to the l-number. FIG. 2 : 2Total fluorescence intensity detected through the prism coupler as a function of the spherebead distance (a) and the particle's lateral position (b). The thick red line in b) is a fit using the first ten WGMs. The weighted intensity distribution of only the first five WGMs is plotted for clarity. efficiencies to both the bead fluorescence and to the prism is reduced. The red curve inFig. 2bdisplays a fit to the data, accounting for the contributions of the first ten WGMs of different m. The profiles of the first five are plotted under the experimental data where the blue curve represents the fundamental mode, and the profile heights reflect the respective weighting factors in the fit procedure. The data inFig. 2demonstrate the local and controlled coupling of a single nanoemitter to the WGMs of a microsphere. processes are independent here (in contrast to ordinary FRET), a simple multiplication of probabilities is appropriate19 . The parameter σ A,abs denotes the absorption cross section of the acceptor whereas cross sections σ D,sca and σ D,abs quantify losses out of the mode due to scattering and absorption of a photon by the donor. Finally, σ i,Q is a cross section signifying all losses associated with the measured Q of the microsphere, including those caused by scattering from the acceptor. A small mode volume is therefore, important for the enhanced emission rate of photons from the donor into the sphere and enters the quantity β i . Furthermore, the high Q and hence small σ i,Q result in the enhanced absorption probability of photons by the acceptor. Both effects are absent if light is transferred merely by a waveguide or optical fiber.The cross sections used in Eq. (1) are, strictly speaking, defined for evanescent illumination and different from those commonly quoted for plane wave excitation. The deviation 2π h 2 FIG. 3 2π23\| e \|E.D\| g \| 2 ρ(ω) where E is the fluctuating vacuum field at the location of the emitter, D is the dipole operator associated with the optical transition at hand, and ρ is the density of photon states. Hence, the strength of emission into different WGMs and consequently β i , : a) Scheme of the cavity-mediated photon transfer measurement. b) The black and the green curves display the reference fluorescence spectra of the naked donor and the acceptor beads, tion of resonance frequencies by local application of electric fields 2 and use our experimental arrangement to achieve controlled coherent coupling of individual quantum emitters mediated by a high-Q microcavity 27 . Indeed, cryogenic efforts have already demonstrated the coupling of single molecules to WGMs 28 and their manipulation at the end of a tip 29 . \|E i \| 2 values. By computing the mode functions of the various WGMs, we have determined \|E i \| 2 on the sphere surface normalized to its value for the fundamental mode. Furthermore, by calculating the diffraction limited Q for various n 26 , we have determined the dependence of the quotient in Eq. (1) on this parameter. Combining these results we find that for each l the contribution to η = η i of modes with n > 10 drops by an order of magnitude. We also find that the first 40 modes with different l − \|m\| values account for half the contribution of all modes. Considering that the fluorescence spectrum of a bead spans about 20 FSRs (F SR = 2.3 nm for 2R = 35 µm) and taking into account both TE and TM modes, we We acknowledge funding from the DFG (SP113) and the Swiss National Foundation. We acknowledge funding from the DFG (SP113) and the Swiss National Foundation. 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[ "Mechanisms for tuning clustering and degree-correlations in directed networks", "Mechanisms for tuning clustering and degree-correlations in directed networks" ]	[ "G Kashyap \nIndian Institute of Science Education and Research\n411008PuneIndia\n", "G Ambika \nIndian Institute of Science Education and Research\n411008PuneIndia\n" ]	[ "Indian Institute of Science Education and Research\n411008PuneIndia", "Indian Institute of Science Education and Research\n411008PuneIndia" ]	[]	With complex networks emerging as an effective tool to tackle multidisciplinary problems, models of network generation have gained an importance of their own. These models allow us to extensively analyze the data obtained from real-world networks, study their relevance and corroborate theoretical results. In this work, we introduce methods, based on degree preserving rewiring, that can be used to tune the clustering and degree-correlations in directed networks with random and scale-free topologies. They provide null-models to investigate the role of the mentioned properties along with their strengths and limitations. We find that in the case of clustering, structural relationships, that are independent of topology and rewiring schemes are revealed, while in the case of degree-correlations, the network topology is found to play an important role in the working of the mechanisms. We also study the effects of link-density on the efficiency of these rewiring mechanisms and find that in the case of clustering, the topology of the network plays an important role in determining how link-density affects the rewiring process, while in the case of degree-correlations, the link-density and topology, play no role for sufficiently large number of rewiring steps. Besides the intended purpose of tuning network properties, the proposed mechanisms can also be used as a tool to reveal structural relationships and topological constraints.	10.1093/comnet/cnx057	[ "https://export.arxiv.org/pdf/1705.01689v1.pdf" ]	21,126,658	1705.01689	cef76e5925c3ca3893c6ee7f4e42716d0d355e37	Mechanisms for tuning clustering and degree-correlations in directed networks G Kashyap Indian Institute of Science Education and Research 411008PuneIndia G Ambika Indian Institute of Science Education and Research 411008PuneIndia Mechanisms for tuning clustering and degree-correlations in directed networks (Dated: February 27, 2022) With complex networks emerging as an effective tool to tackle multidisciplinary problems, models of network generation have gained an importance of their own. These models allow us to extensively analyze the data obtained from real-world networks, study their relevance and corroborate theoretical results. In this work, we introduce methods, based on degree preserving rewiring, that can be used to tune the clustering and degree-correlations in directed networks with random and scale-free topologies. They provide null-models to investigate the role of the mentioned properties along with their strengths and limitations. We find that in the case of clustering, structural relationships, that are independent of topology and rewiring schemes are revealed, while in the case of degree-correlations, the network topology is found to play an important role in the working of the mechanisms. We also study the effects of link-density on the efficiency of these rewiring mechanisms and find that in the case of clustering, the topology of the network plays an important role in determining how link-density affects the rewiring process, while in the case of degree-correlations, the link-density and topology, play no role for sufficiently large number of rewiring steps. Besides the intended purpose of tuning network properties, the proposed mechanisms can also be used as a tool to reveal structural relationships and topological constraints. I. INTRODUCTION Large-scale global connectivity has become an indispensable part of our lives and as a consequence, complex networks have gained an enormous importance in multiple fields of research and applications [1][2][3][4]. With rapid advances in technology, abundant data about these systems has been made available and this is redefining how we perceive and understand complex networks. Analyses of these datasets have revealed a multitude of properties associated with the respective networks. Though some of these properties could be causal or consequential, some structural and functional and others independent or interrelated, it is clear that our understanding of complex networks depends crucially on our understanding of their associated properties, origins and relationships. Although, in a lot of cases, we lack proper understanding of the origins of these properties, we are nevertheless interested in studying how they affect each other and the overall structure and functioning of the network. To this end, models of network generation, both growth mechanisms [5][6][7][8][9][10][11][12] and static mechanisms [13][14][15][16], have become an integral aspect of the study of complex networks. In this regard, considerable work has been done on generating scale-free (SF) networks, using both growth and static methods. Further properties, like degreecorrelations and clustering have also been incorporated into static methods [17][18][19][20][21][22] and growth models [23][24][25][26][27]. Despite the availability of extensive literature, we find that the bulk of it is limited mostly to undirected networks. In spite of their abundant occurrence and rich structural variety, directed networks have been given significantly less attention [28][29][30][31][32] or have been approximated to the undirected case. * e-mail: [email protected] In this work, we focus our attention on directed networks, with 2 important properties: 2-node degreecorrelations and clustering. We use existing models to generate directed SF and random networks. We then propose Degree Preserving Rewiring (DPR) mechanisms to introduce and tune the properties of interest in these networks. Methods based on DPR allow us to isolate the role of the degree distribution and related intrinsic structural properties and focus on the aspects that are unique to the network under consideration [32][33] [34]. We compare and contrast the performance of these mechanisms on the 2 types of networks and investigate their effects and side-effects. We also study the effect of link density on the performance of these techniques. II. DPR METHODS FOR TUNING OF CLUSTERING In networks, the tendency for nodes to organize into well-knit neighborhoods or form small cliques, is referred to as clustering, and is quantified by the Mean Clustering Coefficient (MCC). These substructures, occur spontaneously in most networks and play an important role in the spreading of epidemics in large communities and also effect network flow in local neighborhoods. The MCC of a network is defined as the average of Local Clustering Coefficients (LCCs) of all nodes in the network. The LCC of a node is the ratio of number of pairs of connected neighbors of the node to the number of pairs of neighbors of the node. In other words, it measures the extent to which the neighbors of a node form closed triplets with the node of interest. In directed networks, as a consequence of the directionality of edges, 4 different types of simple closed triplets are possible [28]. The 4 types of triangles, namely, Cycles, Middleman-triangles (Mids), In-triangles (Intri) and C out = 1 N N i=1 (AAA T ) ii d out i (d out i − 1) (1d) To improve the amount of clustering in the network, we employ mechanisms that identify a suitable chain of connected nodes, that can then be reconfigured to give a closed triplet of the desired form, while preserving the degrees of the nodes involved. While the mechanisms are distinct for different types of triplets, they only differ very little from each other based on the triplet of interest. To improve the amount of clustering w.r.t cycles, we use the following procedure: Step-1: Calculate the initial value of C cyc using (1a). Select a node i, whose neighborhood can be suitably modified. While there is no fixed manner in which i can be chosen, the only condition that needs to be satisfied is that i must have at least 1 incoming link and 1 outgoing link. In other words, the in and out degrees of i must be greater than or equal to 1. It is also worth mentioning that i can be chosen either uniformly randomly or in a weighted manner, where the weights could be LCC of i, in-degree of i or out-degree of i. Although a weighted selection of i would seem to be more effective, we find that, for a given number of rewiring steps, the networks undergo approximately the same amount of change in clustering. Therefore, for the purpose of this work, i is chosen uniformly randomly. Step-2: From the list of incoming neighbors of i, we randomly select a node j, and from the list of outgoing neighbors of i, we select k. Step-3: From the list of incoming neighbors of j, we randomly select a node m, and from among the outgoing neighbors of k, we select n. If it is not possible to make either of the selections, we go back to step-1. Step-4: If nodes m, n and i are distinct, and edges (m,n) and (k,j) do not exist, then edges (m,j) and (k,n) are rewired to (k,j) and (m,n). In this process, a cyclic triplet (j,i,k) is created. Step-5: Calculate C cyc of the network. If there is an overall increase from the initial value, then retain the change and go back to step-1. If there is a decrease in clustering, then reject the rewiring and go back to step-1. The above process is iterated for a predetermined number of steps or until a predetermined value of clustering coefficient is reached. To improve the amount of clustering w.r.t mids, we use the following procedure: Step-1: Calculate the initial value of C mid from (1b). Randomly select a node i, with both in and out-degrees greater than or equal to 1. Step-2: Select node j from among the incoming neighbors of i and node k from among the outgoing neighbors of i. Step-3: Further, select node m from among the outgoing neighbors of j and node n from the incoming neighbors of k. If any one of them is not possible, then go back to step-1. Step-4: If m, n and i are distinct and edges (j,k) and (n,m) do not exist, then edges (j,m) and (n,k) are rewired to (j,k) and (n,m), forming a middleman triangle (j,i,k). Step-5: Calculate C mid of the network. If there is an overall decrease in clustering, then reject the rewiring and go back to step-1, and in case of an overall increase, retain the rewiring and go to step-1. The above process is iterated for a predetermined number of steps or until a predetermined value of clustering is reached. To improve the amount of clustering w.r.t intri, we use the following procedure: Step-1: Calculate the initial value of C int using (1c). Randomly select a node i, with in-degree greater than 1. Step-2: Select nodes j and k from among the incoming neighbors of i. Step-3: Further, select node m from the neighbors (in and out) of j and node n from the neighbors (in and out) of k. If any one of them is not possible, then go back to step-1. Step-4: If m, n and i are distinct and edges (j,k) and (n,m) do not exist and (j,m) and (n,k) exist, then edges (j,m) and (n,k) are rewired to (j,k) and (n,m), to form an in-triangle (j,i,k). Else, if nodes m, n and i are distinct, and edges (m,n) and (k,j) do not exist and (m,j) and (k,n) exist, then edges (m,j) and (k,n) are rewired to (k,j) and (m,n), once again forming an in-triangle (j,i,k). Step-5: Calculate C int of the network. If there is an overall decrease in clustering, then reject the rewiring and got back to step-1, and in case of an overall increase, retain the rewiring and go to step-1. The above process is iterated for a predetermined number of steps or until a predetermined value of clustering is reached. To improve the amount of clustering w.r.t outri, we use the following procedure: Step-1: Calculate the initial value of C out using (1d). Randomly select a node i, with out-degree greater than 1. Step-2: Select nodes j and k from among the outgoing neighbors of i. Step-3: Further, select node m from the neighbors (in and out) of j and node n from the neighbors (in and out) of k. If any one of them is not possible, then go back to step-1. Step-4: If m, n and i are distinct and edges (j,k) and (n,m) don't exist and (j,m) and (n,k) exist, then edges (j,m) and (n,k) are rewired to (j, k) and (n, m), to form an out-triangle (j,i,k). Else, if nodes m, n and i are distinct, and edges (m, n) and (k, j) do not exist and (m,j) and (k,n) exist, then edges (m,j) and (k,n) are rewired to (k,j) and (m,n), once again forming an out-triangle (j,i,k). Step-5: Calculate C out of the network. If there is an overall decrease in clustering, then reject the rewiring and go back to step-1, and in case of an overall increase, retain the rewiring and go to step-1. The above process is iterated for a predetermined number of steps or until a predetermined value of clustering is reached. III. RESULTS FOR TUNING OF CLUSTERING We test the performance of the proposed mechanisms, presented in the previous section, on a directed SF network and a directed random network and study the effectiveness of these mechanisms in generating the properties of interest. We use the directed configuration model [16] to generate a random network, while the SF network is generated using the model given by Bollobas et al. in [9]. The working of the mechanisms are first studied for a given parameter value of the networks and then a further investigation is conducted for different parameter values of the respective networks. In the SF network model, the link-density(k) is parametrized by β. We see from [9] that in the N→ ∞ limit,k = β/(1-β). This allows us to compare performances of the rewiring mechanisms in SF networks and ER networks of approximately the same link-density. To study the generation of any type of triangles, we generate 2 ensembles, each consisting of 100 networks. The first ensemble contains directed ER (random) networks of size N = 10 3 and link-density equal to 5 and the second ensemble contains SF networks of size N = 10 3 and β = 0.8. The DPR method for the clustering of interest is applied iteratively to these ensembles of net-works and the average results are shown in fig.6. From fig.6, we observe that for a given N and approximately the same average degree, the mechanisms affect a substantially greater change in the ER network compared to the SF network, for the same number of rewiring steps. This is particularly pronounced in the case of C out , where DPR introduces twice the amount of change in ER networks than in the SF networks. The MCCs show a rapid initial increase following which they remain more or less constant or show a very slow increase. Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Cycles Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Ú Cycles Ê Mids Ï Intri ‡ Outri Cycles Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ 0 10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 Mids Rewirings Hx10 4 L ÚÚ Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ 0 10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 Mids Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Intri Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Intri Rewirings We also notice how certain structural side-effects of the DPR mechanisms manifest themselves, independent of the network topology. For example, when a network is rewired to increase C out , there is noticeable increase in C int and C mid . This can be explained by analyzing fig.5. When the node i is selected and the links between its first and second neighbors are rewired, as a consequence of the rewiring, 2 complementary triangles, w.r.t its first neighbors, are also created. We also notice that there is almost equal increase in both C int and C mid . A similar effect is observed in the case of C mid , where C int and C out show an increase. In the case of C int , there is an increase in both C out and C mid , but unlike in the former 2 cases, the amount of change in MCC w.r.t the complementary triangles is not equal. There is greater increase in C mid than C out . Also, this is observed only in the case of SF networks. However, none of this is observed in the case of C cyc as the complementary triangles formed are also cycles. From fig.7, we see that in SF networks, different MCCs show the same qualitative behavior but different quantitative behaviors. They get rewired to different values for the same number of rewiring steps. This requires further investigation to ascertain if it is a structural limitation. C out reaches a value of 0.6 for β = 0.8 while C int reaches only as high as 0.3 and C mid and C cyc go higher upto 0.4. The behavior of all 4 MCCs is consistent with increasing values MCCs for gradually increasing values of β. In the ER network, the final values of MCCs are not as smoothly varying as in the SF network. C cyc and C mid have values clustered close together fork = 2, 3, 4, 5 while the values for C int and C out are more smoothly distributed. Overall, on comparing the results in both the network types, we find that all the DPR mechanisms for clustering are affected to some extent by the network topology. Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ 0 10 20 30 40 50 0.2 0.4 0.6 0.8 1.0 Outri Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ IV. DPR METHODS FOR TUNING OF CORRELATIONS Degree-correlations measure the tendency of nodes to connect with other nodes with similar or dissimilar degrees. While it is not completely understood whether these correlations are the cause or consequence of other properties/processes, the importance of their effect on network structure and dynamics cannot be ignored. While correlations can be studied w.r.t any enu-merative property (color, race etc.) or scalar property (age, income etc.) of the nodes, correlations between node-degrees gain further importance because of the interplay between the two structural properties involved. In directed networks, since each node has an in-degree and an out-degree, 4 types of 2-node degree-correlations can be defined, namely, In-In, In-Out, Out-In and Out-Out degree correlations. Further in this paper, we will refer to a particular type of correlation as p-q, where p,q ∈ {In,Out} and p and q are associated with the source and target nodes respectively. The traditional metric used to quantify degree correlations is the Pearson correlation coefficient, r p q , as introduced in [17]. But, as was shown in [35][36] [37], r p q scales with the network size N and therefore, in the case of distributions with heavy tails, it converges to a non-negative number as N→ ∞. As a result, it does not lend itself well to capture the dissortativity in large SF networks. For this reason, in this work, we turn to the Spearmans rank correlation(2), to quantify these dependencies. ρ p q = 12 e R p e R q e − 3M (M + 1) 2 M 3 − M(2) where R p e and R q e are the ranks of source and target nodes associated with edge e, based on their p and qdegrees, respectively, and ρ p q is referred to as the Spearman's Rho [37]. Both ρ p q and r p q are bound in the range [-1,1], with the values being positive when nodes of similar degrees are connected by an edge and negative when nodes of dissimilar degrees are connected. The value becomes 0 when there is no net bias. In order to design a mechanism to tune degree correlations, we start with the rewiring rule from [33] and adapt it to the case of directed edges, where the identities of the source and target nodes need to be additionally preserved. We modify the procedure in [33] in the following manner. Given p,q ∈ {in, out}, we randomly choose 2 links, (a,b) and (c,d). From among the 2 source nodes, a and c, we select the node with higher p-degree and from the target nodes, b and d, we select the node with higher q-degree. If they aren't identical or already connected by a link, then the existing links are deleted and new links are placed between the 2 selected nodes and the 2 remaining nodes. This is done to prevent the appearance of multi-edges and self-edges during the course of rewiring. Repeated iteration of the rewiring step ( fig.8 (top)) generates a network that is assortative in p-q type of correlations. To generate a network with dissortative p-q correlations, 2 links (a,b) and (c,d) are randomly chosen. From among the source nodes, the node with higher p-degree is selected and from the target nodes, the node with smaller q-degree is selected. Existing links are deleted and new links are placed between the 2 selected nodes and the 2 remaining nodes, while simultaneously ensuring that no multi-edges or self-loops are created. This rewiring a b c d Max (! " # , ! $ # ) Min (! " # , ! $ # ) Max (! % & , ! ' & ) Min (! % & , ! ' & ) a b c d Max (! " # , ! $ # ) Min (! " # , ! $ # ) Max (! % & , ! ' & ) Min (! % & , ! ' & ) FIG. 8. Schematic representation of DPR mechanisms for assortative rewiring (top) and dissortative rewiring (bottom). step (Fig.8 (bottom)) is iterated over to incrementally increase the dissortativity in the network. V. RESULTS FOR TUNING OF CORRELATIONS To study the generation of degree-correlations, we generate 2 ensembles, each containing 100 networks of size N = 10 4 . The SF networks are generated with β = 0.8 and the ER networks withk = 5. The relevant assortative and dissortative rewiring mechanisms are iterated over on these ensembles and the results, that are ensemble averages, are shown in fig.9. In ER networks, all variants of the mechanisms work perfectly and the networks show only the relevant pq correlations (p,q ∈ {in,out}) that correspond to the respective rewiring mechanism and nothing else ( fig.9 (right)). However, the SF networks present a more interesting scenario ( fig.9 (left)). Although the p-q correlation of interest is introduced to the largest extent, other variants also arise with considerable magnitude. Since the same mechanisms do not result in a similar behavior in ER networks, it is evident that this behavior is not the result of rewiring itself but some other topological property. On examining the 2 types of networks for topological differences besides the distribution of degrees, we find them to be identical in all aspects except the case of 1node correlation. We find that, in ER networks, there is no correlation between the in and out degrees of a given node, while in the case of SF networks, there is a very strong correlation. This can be traced back as an artifact of the construction process in [9], where nodes appearing early in the growth process, have higher in and out degrees. To confirm 1-node correlations as the cause for the observed behavior, we introduce 1-node in-out correlations in ER networks in a systematic manner. Fig.10 shows the change in behavior of 2-node Out-Out correlation when the 1-node correlation is gradually increased. In-In In-In In-In In-In In-Out In-Out In-Out In-Out Out-In Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Ú In-In Ê In-Out Ï Out-In ‡ Out-OutÚ Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Out-In Rewirings Out-In Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Out-In Rewirings Out-Out Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Out-Out Rewirings Out-Out Also, for the same number of rewiring steps, the absolute value of ρ p q is smaller in the case of assortative rewiring and higher in the dissortative case. Another important feature is that, in the case of in-out and out-in correlations, the results for assortative and dissortative rewiring show qualitatively similar behavior with the 4 types of correlations showing 3 distinct values. This symmetric behavior is not seen in the case of in-in and out-out correlations, where the correlations taking the intermediate values further split up, so that the 4 variants take 4 distinct values. Further, this splitting only occurs in the case of assortative rewiring and not in the dissortative case, leading to further questions about 2-node and 1-node structural relationships. Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Ú In-In Ê In-Out Ï Out-In ‡ Out-OutHaL Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡HbL Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ HbL Rewirings Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡HcL Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ HcL Rewirings In ER networks, assortative and dissortative rewirings are studied fork = 2, 3, 4, 5. The behavior is identical in both cases, with higher link-densities showing slower rate of change ( fig.11 (right)). We also see that for sufficiently large number of rewiring steps, the absolute values of ρ p q , corresponding to allk, converge to a large value close to 1. The results are similar in the case of dissortative rewiring in SF networks, for corresponding values of β ( fig.11 (left)). In assortative rewiring, however, values of ρ p q do not quite converge for increasing values of β. The network reaches lower values of ρ p q for higher values of β, for the same number of rewiring steps. In-In Hx10 4 L Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡HdL Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ÏÚ Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ú b = 2ê3 Ê b = 3ê4 Ï b = 4ê5 VI. CONCLUSIONS To summarize, we have presented DPR mechanisms to tune the amount of degree-correlations and clustering in directed networks with random and SF topologies. These mechanisms allow us to explore the relevant properties, independent of the topology, in our attempt to understand their role in the structural organization and functioning of the networks. They provide alternate ways to introduce and tune properties, especially when growth mechanisms fail, due to our lack of knowledge about the processes or mechanisms leading to these properties. Having mechanisms that can artificially tune the amount of clustering makes it easier to study the role of clustering in information dissemination in computer or social networks and rumor-spreading in online or offline communities. It also helps in designing and testing efficient mitigation strategies to contain epidemics in contact networks. In this regard, the ability to tune 2-node degree-correlations also plays an important role. Tuning correlations also helps us to study their effect on the robustness of networks under attack and during failure. We find that, for both correlations and clustering, the density of links in the network does not affect any qualitative change in the working of the mechanisms. The general observation is that higher link-densities slow down the effects of rewiring. We also conclude that the topology itself affects the qualitative behavior of the mechanisms. There exist structural side-effects that are inherent in the definitions of the properties, as a result of which some properties cannot be manipulated in isolation. Basically, the results emphasize on the structural relationships present in directed networks, particularly in SF networks. Finally, although we set out to find mechanisms to tune clustering and degree-correlations in directed networks, we find that the same mechanisms double up as tools to explore further structural relationships in the networks. FIG. 2 . 2Schematic representation of the DPR mechanism for tuning clustering w.r.t cycles. FIG. 3 . 3Schematic representation of the DPR mechanism for tuning clustering w.r.t mids. FIG. 4 . 4Schematic representation of DPR mechanisms for tuning clustering w.r.t intri. FIG. 5 . 5Schematic representation of DPR mechanisms for tuning clustering w.r.t outri. FIG. 6 . 6(Color Online) MCC values for the different types of clustering generated in ER networks (right) and SF networks (left) as a function of rewiring steps. The ER networks are generated with N = 10 3 andk =k in =k out = 3 and the SF networks are generated with N = 10 3 and β = 0.8. Online) Different MCC values generated in ER networks (right) and SF networks (left) for multiple parameter values. ER networks are studied fork = 2, 3, 4, 5 and SF networks for β = 0.5, 0.66, 0.75, 0.8. The values shown are obtained after 10 6 rewiring steps. Rewirings Hx10 4 L FIG. 9. (Color Online) Results of DPR mechanisms, associated with the 4 different types of degree-correlations, in ER networks(right) and SF networks(left). 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[ "Spin squeezing in nonlinear spin coherent states", "Spin squeezing in nonlinear spin coherent states" ]	[ "Xiaoguang Wang \nInstitute of Physics and Astronomy\nAarhus University\nDK-8000Denmark\n" ]	[ "Institute of Physics and Astronomy\nAarhus University\nDK-8000Denmark" ]	[]	We introduce the nonlinear spin coherent state via its ladder operator formalism and propose a type of nonlinear spin coherent state by the nonlinear time evolution of spin coherent states. By a new version of spectroscopic squeezing criteria we study the spin squeezing in both the spin coherent state and nonlinear spin coherent state. The results show that the spin coherent state is not squeezed in the x, y, and z directions, and the nonlinear spin coherent state may be squeezed in the x and y directions.	10.1088/1464-4266/3/3/304	[ "https://export.arxiv.org/pdf/quant-ph/0103061v1.pdf" ]	21,243,036	quant-ph/0103061	6b030c15cf999d94765449d9e9648546a0729606	Spin squeezing in nonlinear spin coherent states arXiv:quant-ph/0103061v1 12 Mar 2001 Xiaoguang Wang Institute of Physics and Astronomy Aarhus University DK-8000Denmark Spin squeezing in nonlinear spin coherent states arXiv:quant-ph/0103061v1 12 Mar 2001(March 31, 2022)Spin squeezing, nonlinear spin coherent states PACS numbers: 4250Dv We introduce the nonlinear spin coherent state via its ladder operator formalism and propose a type of nonlinear spin coherent state by the nonlinear time evolution of spin coherent states. By a new version of spectroscopic squeezing criteria we study the spin squeezing in both the spin coherent state and nonlinear spin coherent state. The results show that the spin coherent state is not squeezed in the x, y, and z directions, and the nonlinear spin coherent state may be squeezed in the x and y directions. I. INTRODUCTION In nonlinear systems such as optical Kerr medium [1], squeezed states of the radiation field [2] have been extensively studied. The spin squeezed states are also studied in nonlinear systems [3] and proved to be useful to enhance spectroscopic resolution [4], e.g., in atomic clocks. The generation of the spin squeezed states has been studied by many ways, such as by interaction of atoms with squeezed light [4][5][6][7][8], quantum nondemolition measurement of atomic spin states [9], and atomic collisional interactions [10]. The spin squeezed states can be characterized in many different ways [3,4,11]. In our study we employ the criteria of spin squeezing recently proposed by Sørensen et al. [12] as a new version of spectroscopic squeezing [4,13]. The squeezing parameter is defined as [12] ξ 2 n1 = 2j(∆J n1 ) 2 J n2 2 + J n3 2 ,(1) where J n = n · J, n i (i = 1, 2, 3) are orthogonal unit vectors and J is the spin-j angular momentum operator. The states with ξ 2 n < 1 are spin squeezed in the direction n. We will show a interesting feature that the squeezing parameters ξ 2 x = ξ 2 y = ξ 2 z = 1 for spin coherent states(SCS) [14], which indicates that the SCS is not squeezed in the x, y and z directions. In this paper we introduce the nonlinear SCS and consider the spin squeezing in it. Sec. II gives the definition of the nonlinear SCS via its ladder operator formalism and propose a type of nonlinear SCS by the time evolution of the SCS under nonlinear Hamiltonian. In Sec. III we first prove that the SCS exhibits no squeezing in the x, y, and z directions and then study the spin squeezing in the nonlinear SCS. A conclusion is given in Sec. IV. II. NONLINEAR SCS We work in a (2j + 1)-dimensional angular momentum Hilbert space {\|j, m ; m = −j, ..., +j}. It is convenient to define a number operator N = J z + j, and the 'number states' \|n which satisfy \|n ≡ \|j, −j + n , N \|n = n\|n . ( The SCS is given by [14], \|η = (1 + \|η\| 2 ) −j 2j n=0 2j n 1/2 η n \|n ,(3) where η is complex. It is easy to check that the SCS satisfies the following equation J − \|η = η(2j − N )\|η .(4) where the operators J ± = J x ± iJ y . This is a ladder operator formalism of the SCS. By recalling the definition of the bosonic nonlinear coherent state [15] and su(1,1) nonlinear coherent states [16], it is natural to define the nonlinear SCS as f (N )J − \|η nl = η(2j − N )\|η nl ,(5) where f (N ) is a nonlinear function of the number operator N . Eq.(5) is of the ladder operator form. Now we propose a type of nonlinear SCS. The SCS under the evolution of the nonlinear Hamiltonian F (N ) is directly given by \|η, t = e −itF (N ) \|η = (1 + \|η\| 2 ) −j 2j n=0 2j n 1/2 η n e −itF (n) \|n .(6) From Eqs. (4) and (6) we find the state \|η, t satisfies e it[F (N +1)−F (N )] J − \|η, t = η(2j − N )\|η, t .(7) According to the definition of the nonlinear SCS, the above state is a nonlinear SCS with the nonlinear func- tion e it[F (N +1)−F (N )] . For F (N ) = N 2 − N , Eq.(7) reduces to e i2N t J − \|η, t = η(2j − N )\|η, t .(8) Further let t = π/2, the above equation reduces to ΠJ − \|η, π/2 = η(2j − N )\|η, π/2 ,(9) where Π = (−1) N is the parity operator. Next we study the spin squeezing of the SCS \|η and the nonlinear SCS \|η, t . III. SPIN SQUEEZING We first give a proof that there is no spin squeezing in the SCS along the x, y, and z directions, and then study the spin squeezing in the nonlinear SCS. A. The SCS In order to calculate the squeezing parameter (1) we need to know the expectation values N k and J k − (k = 1, 2). It is convenient to calculate N k by the generation function method. The generation function of the SCS is given by G(λ) = η\|λ N \|η = (1 + λ\|η\| 2 ) 2j (1 + \|η\| 2 ) 2j ,(10) from which the factorial moments follow F (k) = d k G(λ) d k λ \| λ=1 = \|η\| 2k (2j)! (1 + \|η\| 2 ) k (2j − k)! .(11) The factorial moments immediately give the expectation values of the operators N and N 2 and the variance of N N = F (1) = 2j\|η\| 2 1 + \|η\| 2 ,(12)N 2 = F (2) + F (1) = 2j\|η\| 2 + 4j 2 \|η\| 4 (1 + \|η\| 2 ) 2(13)(∆N ) 2 = 2j\|η\| 2 (1 + \|η\| 2 ) 2 .(14) From Eq.(3) the expectation value J k − are obtained as J k − = η k (2j)! (1 + \|η\| 2 ) k (2j − k)!(15) Now we calculate the squeezing parameter ξ 2 z , which is rewritten as ξ 2 z = 2j(∆N ) 2 \| J − \| 2 .(16) Then substituting Eqs. (14) and (15) into Eq. (16), we immediately obtain ξ 2 z = 1. To calculate ξ 2 x and ξ 2 y we need the identities J 2 x = 1 4 [2j(2N + 1) − 2N 2 + J 2 + + J 2 − ],(17)J 2 y = 1 4 [2j(2N + 1) − 2N 2 − J 2 + − J 2 − ],(18) which gives the expectation values of J 2 x and J 2 y in terms of N , N 2 and J 2 ± . From Eq. (15) and the relation J z = N − j, we get J x = j(η + η * ) 1 + \|η\| 2 , J y = j(η * − η) i(1 + \|η\| 2 ) , J z = j(\|η\| 2 − 1) 1 + \|η\| 2(19) From Eqs. (12)(13), (15), and (17)(18)(19), the variances of J x and J y are expressed as ∆J x 2 = j(1 + \|η\| 4 − η 2 − η * 2 ) 2(1 + \|η\| 2 ) 2 = J y 2 + J z 2 2j ,(20)∆J y 2 = j(1 + \|η\| 4 + η 2 + η * 2 ) 2(1 + \|η\| 2 ) 2 = J z 2 + J x 2 2j .(21) The above two equations directly lead to ξ 2 x = ξ 2 y = 1. Thus we have shown that the squeezing parameters ξ 2 x = ξ 2 y = ξ 2 z = 1 for the SCS. That is to say, the SCS exhibits no squeezing in the x, y and z direction, irrespective of the complex η. We expect that the spin squeezing exists in the nonlinear SCS. B. The nonlinear SCS We examine the spin squeezing in the nonlinear SCS \|η, t . The expectation values N , N 2 and the variance (∆N ) 2 are time independent and given by Eqs. (12), (13), and (14), respectively. From Eq. (6), we obtain the expectation value of J k − on the state \|η, t as J k − = η k (1 + \|η\| 2 ) −2j (2j)! (2j − k)! × 2j−k n=0 2j − k n \|η\| 2n e it[F (n)−F (n+k)] .(22) Of course Eq.(22) reduces to Eq.(15) at t = 0. By substituting Eqs. (14) and (22) into (16), the squeezing parameter ξ 2 z is given by ξ 2 z = 1 1 (1+\|η\| 2 ) 2j−1 2j−1 n=0 Using this inequality in Eq.(23), we find that ξ 2 z ≥ 1. (25) So a general conclusion is made that no squeezing occurs in the z direction for arbitrary nonlinear function F (N ). However spin squeezing may exist in the x or y directions. Next we make numerical calculations to show the spin squeezing. 1 gives the squeezing parameters ξ 2 α (α = x, y) as a function of time t for different nonlinear Hamiltonians F (N ) = N k (k = 2, 3, 4). For small time t we observe that the state is squeezed in the x direction other than the y direction. As k increases, the frequency of occurrence of spin squeezing increases. In most of the time we also see that the spin squeezing alternatively appears in the x and y directions, i.e., when the state is squeezed in the x(y) direction, it is not squeezed in the y(x) direction. The state can show no spin squeezing in both the x and y directions in some small ranges of t, but it can not show spin squeezing at the same time in the two directions. x and ξ 2 y , respectively. We plot ξ 2 α for η = 0.1, ξ 2 α + 1 for η = 0.2, and ξ 2 α + 2.5 for η = 0.3.The parameter j = 5. It is interesting to consider the squeezing in the nonlinear Hamiltonian H = sin(aN ) which can be realized in physical systems [17]. The numerical results are shown in Fig.2. For η = 0.1 we observe that the spin squeezing in the x and y directions appears alternatively in the beginning of the time evolution. For small time t, the state is squeezed in the y direction other than x direction in contrast to Fig.1. We also observe that the time range of spin squeezing decreases as the parameter η increases, i.e., the squeezing does not occur in most of the time as η is large. IV. CONCLUSIONS In conclusion we have given the definition and proposed an example of the nonlinear SCS. We have studied the spin squeezing in both the SCS and the nonlinear SCS. The main results are as follows: 1. The squeezing parameters ξ 2 x = ξ 2 y = ξ 2 z = 1 for the SCS. That is to say, the SCS is not squeezed in the x, y and z directions, irrespective of the parameter η. 2. The nonlinear SCS shows no spin squeezing in the z direction for arbitrary nonlinear Hamiltonian F (N ). 3. The nonlinear SCS may be squeezed in the x and y directions. In most of the time the squeezing appears alternatively in the x and y directions as time goes on. Also we observe that the state can not be squeezed at the same time in the two directions. The spin squeezing originates from the nonlinearity of the nonlinear SCS. Then we expect that the spin squeezing exists in other nonlinear SCS with difference nonlinear functions. For M + 1 complex quantities c i (i = 0...M ), there is an inequality \|c 0 + c 1 + ... + c M \| ≤ \|c 0 \| + \|c 1 \| + ... + \|c M \| parameters ξ 2 α (α = x, y) as a function of time t. We plot ξ 2 α for the nonlinear Hamtonian F (N ) = N 2 , ξ 3 α + 0.6 for F (N ) = N 3 , and ξ 2 α + 1.2 for F (N ) = N 4 . Solid line and dashed line correspond to ξ 2x and ξ 2 y , respectively. The parameters η = 0.1 and j = 5. Fig. Fig.1 gives the squeezing parameters ξ 2 α (α = x, y) as a function of time t for different nonlinear Hamiltonians F (N ) = N k (k = 2, 3, 4). For small time t we observe that the state is squeezed in the x direction other than the y direction. As k increases, the frequency of occurrence of spin squeezing increases. In most of the time we also see that the spin squeezing alternatively appears in the x and y directions, i.e., when the state is squeezed in the x(y) direction, it is not squeezed in the y(x) direction. The state can show no spin squeezing in both the x and y directions in some small ranges of t, but it can not show spin squeezing at the same time in the two directions. FIG. 2 . 2Squeezing parameters ξ 2 α (α = x, y) as a function of time t for the nonlinear Hamtonian F (N ) = sin(2N ). Solid line and dashed line correspond to ξ 2 ACKNOWLEDGMENTSThe author thanks for many helpful discussions with Klaus Mølmer, Anders Sørensen, and Bin Shu. 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[ "Rapid Spectral Variability during the 2003 Outburst of V4641 Sgr (= SAX", "Rapid Spectral Variability during the 2003 Outburst of V4641 Sgr (= SAX" ]	[ "Dipankar Maitra \nYale University\n06511New HavenCTUSA\n", "Charles Bailyn \nYale University\n06511New HavenCTUSA\n" ]	[ "Yale University\n06511New HavenCTUSA", "Yale University\n06511New HavenCTUSA" ]	[]	The black hole candidate V4641 Sgr (= SAX J1819.3-2525) went through a brief outburst during 2003 Aug 01 to Aug 08. During the outburst, activity was noted in optical, radio as well as X-rays. Here we present results of Spectral and Temporal analysis of a pointed Rossi X-ray Timing Explorer (RXTE) observation of the source during the outburst. During this pointing we observed flaring activities with associated X-ray luminosity variations over factors of 5 or more in timescales of few tens of seconds. The observed flares during are intrinsically different in their spectral and temporal properties. We see evidence of variable column density during one of the flares. The spectral and temporal analyses of the data suggest occasional outflow/mass ejection.	null	[ "https://arxiv.org/pdf/astro-ph/0510277v1.pdf" ]	119,035,455	astro-ph/0510277	913069556fb1a6006bd16438549d42bdd0b4bf55	Rapid Spectral Variability during the 2003 Outburst of V4641 Sgr (= SAX arXiv:astro-ph/0510277v1 10 Oct 2005 Dipankar Maitra Yale University 06511New HavenCTUSA Charles Bailyn Yale University 06511New HavenCTUSA Rapid Spectral Variability during the 2003 Outburst of V4641 Sgr (= SAX arXiv:astro-ph/0510277v1 10 Oct 2005accretionaccretion disks -starsblack holes -X-raysbinaries - individual (V4641 Sgr) PACS: 0130Cc9585Nv9780Jp9760Lf9862Mw9710Gz0470-s The black hole candidate V4641 Sgr (= SAX J1819.3-2525) went through a brief outburst during 2003 Aug 01 to Aug 08. During the outburst, activity was noted in optical, radio as well as X-rays. Here we present results of Spectral and Temporal analysis of a pointed Rossi X-ray Timing Explorer (RXTE) observation of the source during the outburst. During this pointing we observed flaring activities with associated X-ray luminosity variations over factors of 5 or more in timescales of few tens of seconds. The observed flares during are intrinsically different in their spectral and temporal properties. We see evidence of variable column density during one of the flares. The spectral and temporal analyses of the data suggest occasional outflow/mass ejection. The compact binary system V4641 Sgr (= SAX J1819.3-2525) has been known to exhibit short, irregular outbursts, with highly variable lightcurve profile. The rapid time variability [1,2] in the lightcurve as well as observation of moving radio structure [3] during an outburst in 1999 led to its classification as a microblazar [4]. During early Aug 2003, signs of enhanced activity were reported by the VS-NET 2 group and shortly thereafter by the Small and Moderate Aperture Research Telescope System consortium telescopes at Cerro Tololo Inter-American Observatory (CTIO) in Chile [5]. In Fig. 1 we show the V band lightcurve of Email address: [email protected] (Dipankar Maitra). 1 This work was supported by the National Science Foundation grant AST 0407063. 2 http://ooruri.kusastro.kyoto-u.ac.jp/mailman/listinfo/vsnet-alert V4641 Sgr during this outburst, as observed by the SMARTS 1.3m telescope. We used the CTIO observations to trigger the series of target-of-opportunity (ToO) X-ray observations. During four pointed RXTE observations we saw activity from the source. The observed X-ray spectrum during all these 4 pointings were hard in nature, showing large contribution of hard X-rays. Strong Fe Kα fluorescent emisssion line near 6.5 keV was detected with large equivalent widths in the range of 700 -1000eV. In this article we concentrate, in particular, on second dwell where we observed two flaring activities with flare timescales of tens of seconds. In panel (a), (b), (c) and (d) of Fig. 2 we show the evolution of the source lightcurve, hardness ratio, color-magnitude diagram and color-color plane respectively during dwell (2). The distinctively opposite characters of the two flares are evident. While the first flare at ∼ 250 seconds is essentially hard, the second flare during ∼ 450 second is predominantly soft. Since the Iron line dominates the low energy spectrum, estimating the continuum becomes difficult. We modelled the X-ray spectrum in the 10-25 keV region using the pexrav model [6] which calculates an exponentially cut off power law spectrum reflected from neutral material. The spectrum during the first 14-94 seconds is well fit by this model and a warm absorber with its column density (n H ) set to the standard value of 2.3 × 10 21 atoms/cm −2 for this source [7]. The fit model, when extended to softer energies, fails to account for the Fe Kα line near 6.5 keV, but matches the continuum at the softest bands as shown in the left panel of Fig. 3. However, during the hard flare, the fits to high energy regimes of the spectrum (> 10 keV) largely overestimates the counts in softer energies as shown in the right panel of Fig. 3. In fact, we were unable to find reasonable fits with any physically motivated models with n H fixed to its standard value. Allowing n H results in much better fits. In Fig. 4 we show the variation of the fit parameter n H during the hard flare. Conclusions • V4641 Sgr went through a short lived, hard flaring outburst between 2003 Aug 01 through Aug 08, during which X-ray fluctuations by a factor of 10 on timescales of tens of seconds were seen. • Compton reflection features like strong Fe Kα line near 6.5 keV and curvature in spectrum at high energies were present in all observations. • Spectral nature of some flares separated in time only by few minutes were intrinsically different. • The thick warm envelope/outflow scenario [1,8] can explain the variation of column density during the second pointing although other scenarios like variation of partial covering fractions cannot be ruled out. Fig. 1 . 1Optical V band lightcurve showing the 2003 outburst of V4641 Sgr. Four vertical lines on the top labelled 1-4 are the times when significant X-ray activity was observed by RXTE. The horizontal dotted line represents the mean quiescent brightness of 13.7 magnitude. The data were taken by the CTIO 1.3m telescope operated by the SMARTS Consortium. Fig. 2 . 2RXTE/PCA observations of V4641 starting MJD 52857.37 are shown. The data during the hard flare are shown by , those during soft flare are shown by • and the rest of the dwell are shown by + symbols. Panel (a): The 3-20 keV lightcurve showing the rapid variability of this source that sets it apart from typical X-ray binaries. Panel (b): Variation of hardness-ratio with time. Hard band is 10.3-20.4 keV, medium band is 5.3-10.3 keV, soft band is 2.0-5.3 keV. Panel (c): Color-Luminosity plot showing the different regions occupied by the soft and hard flares. Panel (d): Color-color plot, also shows the while the soft flare occupies the most soft spectral state (lower-left corner), the hard flare essentially occupies the extreme hard state in the upper-right corner of the color-color diagram. Fig. 3 . 3Variation of n H during the hard flare in dwell (2). Panel (a): The energy spectrum during 14-94 seconds. Panel (b): The spectrum during 222-254 seconds when the source was going through a hard flare. The solid histogram in both panels is a fit to the 10.0-25.0 keV spectrum with n H = 2.3 × 10 21 atoms/cm 2 . Note that the extrapolated fit matches the continuum at lowest energies for panel (a) whereas it grossly overestimates the counts during the flare as shown in panel (b). Fig. 4 . 4Variation of the fit parameter n H during the hard flare. The dashed horizantal line represents the standard adopted value of n H for this source. . M Revnivtsev, M Gilfanov, E Churazov, R Sunyaev, A&A. 3911013Revnivtsev, M., Gilfanov, M., Churazov, E., & Sunyaev, R. 2002, A&A, 391, 1013 . R Wijnands, M Van Der Klis, ApJ. 52893Wijnands, R. & van der Klis, M. 2000, ApJ, 528, L93 . R M Hjellming, ApJ. 544977Hjellming, R. M., et al. 2000, ApJ, 544, 977 . I F Mirabel, L F Rodriguez, Nature. 37146Mirabel, I. F., & Rodriguez, L. F. 1994, Nature, 371, 46 M Buxton, D Maitra, C Bailyn, L Jeanty, D Gonzalez, The Astronomer's Telegram. 1701Buxton, M., Maitra, D., Bailyn, C., Jeanty, L., & Gonzalez, D. 2003, The Astronomer's Telegram, 170, 1 . P Magdziarz, A A Zdziarski, MNRAS. 273837Magdziarz, P., & Zdziarski, A. A. 1995, MNRAS, 273, 837 . J M Dickey, F J Lockman, ARA&A. 28215Dickey, J. M., & Lockman, F. J. 1990, ARA&A, 28, 215 . L Titarchuk, C Shrader, ApJ. 623362Titarchuk, L., & Shrader, C. 2005, ApJ, 623, 362	[]
[ "MOMENT-SEQUENCE TRANSFORMS", "MOMENT-SEQUENCE TRANSFORMS" ]	[ "Alexander Belton \nTo Gadadhar Misra\n\n", "Dominique Guillot \nTo Gadadhar Misra\n\n", "Apoorva Khare \nTo Gadadhar Misra\n\n", "Mihai Putinar \nTo Gadadhar Misra\n\n" ]	[ "To Gadadhar Misra\n", "To Gadadhar Misra\n", "To Gadadhar Misra\n", "To Gadadhar Misra\n" ]	[]	We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.Date: 8th September 2021. 2010 Mathematics Subject Classification. 15B48 (primary); 30E05, 44A60, 26C05 (secondary).	10.4171/jems/1145	[ "https://arxiv.org/pdf/1610.05740v7.pdf" ]	7,117,542	1610.05740	2b670cd12fdab805f0331a9d5d6635ea5a02d146	MOMENT-SEQUENCE TRANSFORMS 9 Sep 2021 Alexander Belton To Gadadhar Misra Dominique Guillot To Gadadhar Misra Apoorva Khare To Gadadhar Misra Mihai Putinar To Gadadhar Misra MOMENT-SEQUENCE TRANSFORMS 9 Sep 2021 We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.Date: 8th September 2021. 2010 Mathematics Subject Classification. 15B48 (primary); 30E05, 44A60, 26C05 (secondary). Introduction The ubiquitous encoding of functions or measures into discrete entities, such as sampling data, Fourier coefficients, Taylor coefficients, moments, and Schur parameters, leads naturally to operating directly on the latter 'spectra' rather than the original. The present article focuses on operations which leave invariant power moments of positive multivariable measures. To put our essay in historical perspective, we recall a few similar and inspiring instances. The characterization of positivity preserving analytic operations on the spectrum of a self-adjoint matrix is due to Loewner in his groundbreaking article [33]. Motivated by the then-novel theory of the Gelfand transform and the Wiener-Levy theorem, in the 1950s Helson, Kahane, Katznelson, and Rudin identified all real functions which preserve Fourier transforms of integrable functions or measures on abelian groups [23,28,38]. Roughly speaking, these Fourier-transform preservers have to be analytic, or even absolutely monotonic. The absolute-monotonicity conclusion was not new, and resonated with earlier work of Bochner [9] and Schoenberg [42] on positive definite functions on homogeneous spaces. Later on, this line of thought was continued by Horn in his doctoral dissertation [25]. These works all address the question of characterizing real functions F which have the property that the matrix (F (a ij )) is positive semidefinite whenever (a ij ) is, possibly with some structure imposed on these matrices. Schoenberg's and Horn's theorems deal with all matrices, infinite and finite, respectively, while Rudin et al. deal with Toeplitz-type matrices via results of Herglotz and Carathéodory. In this article, we focus on functions which preserve moment sequences of positive measures on Euclidean space, or, equivalently, in the one-variable case, functions which leave invariant positive semidefinite Hankel kernels. As we show, these moment preservers are quite rigid, with analyticity and absolute monotonicity again being present in a variety of combinations, especially when dealing with multivariable moments. We state in detail in Section 2 our results for one-variable functions and domains and for moment sequences of measures on them, but first we present in Section 1.1 tabulated lists of our results in one and several variables. The first significant contribution below is the relaxation to a minimal set of conditions, which are very accessible numerically, that characterize the positive definite Hankel kernel transformers in one variable. Specifically, Schoenberg proved that a continuous map F : (−1, 1) → R preserves positive semidefiniteness when applied to matrices of all dimensions, if and only if F is analytic and has positive Taylor coefficients [42]. Later on, Rudin was able to remove the continuity assumption [38]. In our first major result, we prove that a map F : (−1, 1) → R preserves positive semidefiniteness of all matrices if and only if it preserves this on Hankel matrices. Even more surprisingly, a refined analysis reveals that preserving positivity on Hankel matrices of rank at most 3 already implies the same conclusion. Our result can equivalently be stated in terms of preservers of moment sequences of positive measures. Thus we also characterize such preservers under various constraints on the support of the measures. Furthermore, we examine the analogous problem in higher dimensions. In this situation, extra work is required to compensate for the failure of Hamburger's theorem in higher-dimensional Euclidean spaces. Our techniques extend naturally to totally non-negative matrices, in parallel to their natural connection to the Stieltjes moment problem. We prove that the entrywise transformations which preserve total non-negativity for all rectangular matrices, or all symmetric matrices, are either constant or linear. Furthermore, we show that the entrywise preservers of totally non-negative Hankel matrices must be absolutely monotonic on the positive semi-axis. The class of totally non-negative matrices was isolated by M. Krein almost a century ago; he and his collaborators proved its significance for the study of oscillatory properties of small harmonic vibrations in linear elastic media [18,19]. Meanwhile this chapter of matrix analysis has reached maturity and it continues to be explored and enriched on intrinsic, purely algebraic grounds [13,14]. We conclude by classifying transformers of tuples of moment sequences, from which a new concept emerges, that of a piecewise absolutely monotonic function of several variables. In particular, our results extend original theorems by Schoenberg and Rudin. For more on the wider framework within which this article sits, we refer the reader to the survey [4]. Besides the classical works cited above delineating this area of research, we rely in the sequel on Bernstein's theory of absolutely monotone functions [7,54], a related pioneering article by Lorch and Newman [32] and Carlson's interpolation theorem for entire functions [11]. The study of positive definite functionals defined on * -semigroups, with or without unit, led Stochel to a series of groundbreaking discoveries, complementing the celebrated Naimark and Sz. Nagy dilation theorems and, in particular, putting multivariate moment problems in a wider, more flexible framework [47,48,49]. A byproduct of his studies is a classification of positive definite functionals on the multiplicative semigroup (−1, 1) [48], culminating with a similar conclusion to our main one-dimensional result: these positive functionals are absolutely monotonic on (0, 1) with possibly discontinuous derivatives, of any order, at the origin. As a final remark, we note that entrywise transforms of moment sequences were previously studied in a particular setting motivated by infinite divisibility in probability theory [26,50]. The study of entrywise operations which leave invariant the cone of all positive matrices has also recently received renewed attention in the statistics literature, in connection to the analysis of big data. In that setting, functions are applied entrywise to correlation matrices to improve properties such as their conditioning, or to induce a Markov random-field structure. The interested reader is referred to [3,21,22] and the references therein for more details. A companion to the present article [5] was recently completed, which extends the work here with definitive classifications of preservers of totally positive and totally non-negative kernels, and together with kernels having additional structure, such as those of Hankel [53] or Toeplitz [43] type, or generating series, such as Pólya frequency functions and sequences. 1.1. Summary of main results. Tables 1.1 and 1.2 below summarize the results proved in this article. The notation used below is explained in the main body of the article; see also the List of Symbols following this subsection. In the one-variable setting, we have identified the positivity preservers acting on (i) all matrices, and (ii) all Hankel matrices, in the course of classifying such functions acting on (iii) moment sequences, i.e., all Hankel matrices arising from moment sequences of measures supported on [−1, 1]. Characterizations for all three classes of matrices are obtained with the additional constraint that the entries of the matrices lie in (0, ρ), (−ρ, ρ), and [0, ρ), where ρ ∈ (0, ∞]. We then extend each of the results in Table 1.1 to apply to functions acting on tuples of positive matrices or moment sequences: see Table 1.2. [0, ρ) Proposition 9.8 Proposition 9.8 Theorem 9.5 (−ρ, ρ) Theorem 9.11 Theorem 9.11 Theorem 9.11 (see [16] for ρ = ∞) In the one-variable setting, we do more than is recorded in Table 1.1, since our results cover various classes of totally non-negative matrices (Section 5), as well as the closedinterval settings of [0, ρ] and [−ρ, ρ] for ρ < ∞ (Section 8). The multivariable case may contain products of open and closed intervals, but it would be rather cumbersome, and somewhat artificial, to consider them all. We do not pursue this direction in the present work. In all of the above contexts, with the exception of functions on [0, ρ) m (i.e., the (2, 3) entry in both tables), the characterizations are uniform: all such positivity preservers are necessarily analytic on the domain and absolutely monotonic on the closed positive orthant. The converse result holds trivially by the Schur product theorem. The one exceptional case reveals a richer family of 'facewise absolutely monotonic maps'; see Section 9.2. We have also improved on all of the above results, by significantly relaxing the hypotheses required to obtain absolute monotonicity. Finally, and for completeness, we remark that Theorem 4.8 from our previous work [3], which is widely used herein, admits a generalization to all, possibly non-consecutive, integer powers, and again the bounds have closed form. This result is obtained through a careful analysis and novel results about Schur polynomials; we refer the reader to the recent paper by Khare and Tao [30] for more details. List of symbols. For the convenience of the reader, we list some of the symbols used in this paper. • Given a subset I ⊂ R, P k N (I) is the set of positive semidefinite N × N matrices with entries in I and of rank at most k. We let P N (I) := P N N (I) and P N := P N (R). • H + (I) denotes the set of positive semidefinite Hankel matrices of arbitrary dimension with entries in I. • H ++ n denotes the set of n × n totally non-negative Hankel matrices, and H ++ denotes the set of all totally non-negative Hankel matrices. • H (1) denotes the truncation of a possibly semi-infinite matrix H obtained by excising the first column. • F [H] is the result of applying F to each entry of the matrix H. • For K ⊂ R, we denote by Meas + (K) the set of admissible measures, i.e., nonnegative measures µ supported on K and admitting moments of all orders. • The kth moment of a measure µ is denoted by s k (µ); the corresponding moment sequence is s(µ) := (s k (µ)) k≥0 . The associated Hankel moment matrix H µ has (i, j) entry s i+j (µ). In particular, the moment sequence of µ is the leading row and column of H µ . • Given K ⊂ R, M(K) denotes the set of moment sequences associated to elements of Meas + (K). For any k ≥ 0, M k (K) denotes the corresponding set of truncated moment sequences: M k (K) := {(s 0 (µ), . . . , s k (µ)) : µ ∈ Meas + (K)}. • Given K ⊂ R and a scalar ρ with 0 < ρ ≤ ∞, M ρ (K) denotes the subset of M(K) with moments s j ∈ (−ρ, ρ) for all j ≥ 0, and, for any k ≥ 0, we let M ρ k (K) denote the subset of M k (K) with s j ∈ (−ρ, ρ) for j = 0, . . . , k. • Given ρ with 0 < ρ ≤ ∞, an integer k ≥ 0, and x ∈ [−1, 1), we let M ρ,+ ({1, x}) and M ρ,+ k ({1, x}) denote the subsets of M({1, x}) and M k ({1, x}), respectively, with total mass s 0 < ρ and such that 1 and x both have positive mass. • Given an integer m ≥ 1, a function F : R m → R acts on tuples of moment sequences of admissible measures in M(K 1 ) × · · · × M(K m ) as follows: F [s(µ 1 ), . . . , s(µ m )] := (F (s k (µ 1 ), . . . , s k (µ m ))) k≥0 . (1.1) • Given h > 0 and an integer n ≥ 0, ∆ n h F denotes the nth forward difference of the function F with step size h. • 1 m×n denotes the m × n matrix with all entries equal to 1. • C + := {z ∈ C : ℜz > 0} denotes the right open half-plane. 1.3. Organization. The plan of the article is as follows. Section 2 recalls notation and reviews previous work, while Section 3 lists our main results for classical positive Hankel matrices transformers, which, in particular, go beyond previous classical results. Sections 4,6,7, and 8 are devoted to proofs, arranged by the domains of the entries of the relevant Hankel matrices. For these proofs, we work with measures with restricted total mass, which is reflected in the domains of the test sets of matrices, and helps unify previously known results. Thus, we end up showing stronger results than in Section 2; these results were tabulated in a concise form in Section 1.1 above. An additional strengthening involves severely reducing the supports of the test measures, which translates to rank constraints on the test sets of Hankel matrices and hence stronger results. This technical point is not mentioned in the above tables, but is detailed in the aforementioned Sections 4, 6, 7, and 8 devoted to proofs. Section 5 contains the classifications of preservers of total non-negativity for several different sets of matrices, in the dimension-free setting. Section 9 deals with multivariable transformers of Hankel kernels. Section 10 makes the natural link with Laplace transforms and interpolation of entire functions. The appendix is devoted to algebraic properties of adjugate matrices. Preliminaries We collect in this section the basic concepts and notation necessary for accessing the rest of the article. Bibliographical indications will rely on classical texts. We are fortunate to be able to refer to a few very recent outstanding monographs, including [40,46]. Matrices of moments. Our raw material consists of structured matrices of moments and functions acting on them. In this subsection, we concentrate on the first. Henceforth N is a positive integer. Definition 2.1. Given a subset I ⊂ R, denote by P N (I) the set of positive semidefinite N × N matrices with entries in I, and let P N := P N (R). The set P N is a convex cone, closed in the Euclidean topology of R N ×N . Schur's product theorem asserts A•B ∈ P N whenever A, B ∈ P N ; here A•B = (a ij b ij ) denotes the entrywise product of two equidimensional matrices A = (a ij ) and B = (b ij ). For a proof it is sufficient to decompose B into a sum of rank-one positive matrices and follow the definition of matrix positivity. Recall that a matrix is said to be totally non-negative if all its minors are nonnegative. Totally non-negative matrices occur in a variety of areas; see [13] and the references therein. For instance, a well-known observation due to Schoenberg asserts that given vectors x 1 , x 2 , . . . , x N , in an inner-product space, the corresponding matrix (exp(− x j − x k 2 ) N j,k=1 is totally non-negative. Definition 2.2. For an integer n ≥ 1, let H ++ n denote the set of n × n totally nonnegative Hankel matrices, and let H ++ := n≥1 H ++ n denote the set of totally nonnegative Hankel matrices of arbitrary size. The moment problem, in the widely accepted meaning of the term, is arguably the quintessential inverse problem. It has a long history and continues to lead to unexpected impacts in pure and applied mathematics; see, for instance, [1,31,40,45]. Moments of positive measures are in general observables, with a physical or probabilistic interpretation. These observed real numbers are not free, but are subject to an array of semi-algebraic constraints, which are generally hard to deal with directly. A convenient and numerically friendly approach is to organize the moments into matrices with red[undant entries, the simplest case being associated to measures supported on subsets of the real line. We will start with this generic situation. Let µ be a non-negative measure on R, rapidly decreasing at infinity, that admits moments of all orders; let its moment data and associated Hankel matrix be denoted as follows: s k (µ) = s k := R x k dµ, s(µ) := (s k (µ)) k≥0 , H µ :=      s 0 s 1 s 2 · · · s 1 s 2 s 3 · · · s 2 s 3 s 4 · · · . . . . . . . . . . . .      . (2.1) All measures appearing in this paper are taken to be non-negative and are assumed to possess moments of all orders. We will henceforth call such measures admissible. Throughout this paper, we allow matrices to be semi-infinite in both coordinates. We also identify without further comment the space of real sequences (s 0 , s 1 , . . . ) and the corresponding Hankel matrices, as done in (2.1). To verify the positivity of the matrix H µ , it is sufficient to observe that 0 ≤ \| N j=0 c j x j \| 2 dµ = N j,k=0 H µ (j, k)c j c k . Definition 2.3. Given subsets I, K ⊂ R, let Meas + (K) denote the admissible measures supported on K, and let H + (I) denote the set of complex Hermitian positive semidefinite Hankel matrices with entries in I. We will henceforth use the adjective 'positive' to mean 'complex Hermitian positive semidefinite' when applied to matrices. The following theorem combines classical results of Hamburger, Stieltjes, and Hausdorff. Theorem 2.4. A sequence s = (s k ) ∞ k=0 is a moment sequence for an admissible measure on R if and only if the Hankel matrix with first column s is positive. In other words, the map Ψ : µ → (s k (µ)) ∞ k=0 is a surjection from Meas + (R) onto H + (R). Moreover, (1) restricted to Meas + ([0, ∞)), the map Ψ is a surjection onto the positive Hankel matrices with non-negative entries, such that removing the first column still yields a positive matrix; x 2n dµ. (2) restricted to Meas + ([−1, 1]), The first integral remains uniformly bounded as a function of n, while the second tends to infinity with n whenever the measure µ has positive mass on R \ [−1, 1]. Definition 2.5. In view of the above correspondence, we denote by M(K) the set of moment sequences associated to measures in Meas + (K). Equivalently, M(K) is the collection of first columns of Hankel matrices associated to admissible measures supported on K. We write H (1) to denote the truncation of a matrix H in which the first column is excised. For technical reasons which will become apparent from the proofs below, we introduce an additional parameter via the following definition. Definition 2.6. Given 0 < ρ ≤ ∞ and I ⊂ R, let M ρ (I) denote the set of moment sequences (s k (µ)) ∞ k=0 of admissible measures µ supported on I, with all moments in (−ρ, ρ). Also, for any n ≥ 0, let M ρ n (I) denote the corresponding set of truncated moment sequences (s k (µ)) n k=0 . Note that M ρ (I) = M(I) and M ρ n (I) = M n (I) when ρ = ∞. Moreover, for a nonnegative measure µ supported on [−1, 1], the mass s 0 (µ) dominates \|s k (µ)\| for all k ≥ 0. Studying moment sequences of admissible measures having mass s 0 < ρ is therefore equivalent to working with Hankel matrices with entries in a bounded interval (−ρ, ρ). This will be our approach in the remainder of the paper. A simple characterization of rank-one Hankel matrices is stated below. u j = u 1 (u 2 /u 1 ) j−1 if u 1 = 0, 0 if u 1 = 0 and 1 ≤ j < N. (2.2) Proof. This is immediate for N ≥ 2. For N > 2, each principal 3 × 3 block submatrix of uu T with successive rows and columns is of the form   u 2 j−1 u j−1 u j u j−1 u j+1 u j u j−1 u 2 j u j u j+1 u j+1 u j−1 u j+1 u j u 2 j+1   , whence u j−1 u j+1 = u 2 j for all j ≥ 2. Identity (2. 2) follows immediately. We invite the reader to find all positive measures on the real line which produce a rank-one Hankel matrix. In general, one can read off from a positive Hankel matrix whether the representing measure is unique, and estimate the shape of the support of the representing measure(s) (of utmost importance in polynomial optimization), and enter into the Lebesgue decomposition of the representing measure(s). We refer to [1,31,40] for aspects of such refined analysis pertaining to the moment problem and its current applications. In Section 9, we will treat multivariable moment problems. In that context, Hankel matrices are replaced by kernels with a Hankel-type property. The semigroup approach proves to be superior in the multivariablee setting; see [6] for more details. To conclude, we note that the study of Hankel matrices forms an important chapter of modern analysis, with ramifications for approximation theory, probability theory and control theory [35]. Absolutely monotonic functions. We turn now to operators on moments by identifying two relevant classes of functions. Central to our study is the class of absolutely monotonic entire functions. These are entire functions with non-negative Taylor coefficients at every point of (0, ∞). Equivalently, it is sufficient for such a function to have non-negative Taylor coefficients at zero. Their structure was unveiled in a fundamental memoir by Bernstein [7]; see also Widder's book [54] or the recent treatise [39]. One can restrict the absolute monotonicity definition to a finite interval, with the following outcome. Recall that a function is said to be completely monotonic on an interval (a, b) if the map x → f (−x) is absolutely monotonic on (−b, −a), i.e., if (−1) k f (k) (x) ≥ 0 for all x ∈ (a, b). Similarly, a function is completely monotonic on an interval I ⊂ R if it is continuous on I and is completely monotonic on the interior of I. Complete monotonicity can also be defined using finite differences. Let ∆ n h f denote the nth forward difference of f with step size h: ∆ n h f (x) := n k=0 (−1) n−k n k f (x + kh). Then f is completely monotonic on (a, b) if and only if (−1) n ∆ n h f (x) ≥ 0 for all nonnegative integers n and for all x, h such that a < x < x + h < · · · < x + nh < b. See [54, Chapter IV] for more details on completely monotonic functions. Such functions were also characterized in a celebrated result of Bernstein. Atomic measures are not excluded in Bernstein's theorem, hence series of exponentials and Dirichlet series are an integral part of the theory of absolutely or completely monotonic functions. One of the major advantages of absolute monotonicity is the analytic extension of the respective function to a complex domain. We will exploit this quality further on in the present work. 2.3. Matrix positivity transforms. The main theme of our work is permanence properties of moment matrices A under entrywise operations. From the very beginning we warn the reader that our framework is in contrast to the classical functional calculus A → f (A) which is the subject of Loewner's celebrated theorem: a real function f preserves matrix ordering (i.e., A ≤ B implies f (A) ≤ f (B)) among self-adjoint matrices if and only if f extends analytically to the upper-half plane and it has positive imaginary part there. For ample details and a dozen different proofs, see [12,46]. Entrywise operations on matrices and kernels also have a long and interesting history, see [4]. We will provide the outlines of a few significant results. Transformations which leave invariant Fourier transforms of various classes of measures on groups or homogeneous spaces were studied by many authors, including Schoenberg [42], Bochner [9], Helson, Kahane, Katznelson, and Rudin [23,28]. From the latter works, Rudin extracted [38] an analysis of maps which preserve moment sequences for admissible measures on the torus; equivalently, these are functions which, when applied entrywise, leave invariant the cone of positive semidefinite Toeplitz matrices. Rudin's result, originally proved by Schoenberg [42] under a continuity assumption, is as follows. The facts that (3) =⇒ (1) and (3) =⇒ (2) follow from the Schur product theorem [44]. However, the converse results are highly non-trivial. In the present paper, we consider moments of measures on the line rather than Fourier coefficients, so power moments rather than complex exponential moments. Hence we study functions F mapping moment sequences entrywise into themselves, i.e., such that for every admissible measure µ, there exists an admissible measure σ = σ µ satisfying F (s k (µ)) = s k (σ) for all k ≥ 0. Equivalently, by Theorem 2.4, we study entrywise endomorphisms of the cone of positive Hankel matrices with real entries. The following notion of entrywise calculus is central to this paper. for the matrix A = (a ij ). The function F also acts entrywise on moment sequences with all moments in D, so that F [s(µ)] k := F (s k (µ)) for all k ≥ 0, and similarly for truncated moment sequences. An observation on positivity preservers made by Loewner and developed by Horn [25] provides the following necessary condition for a function to preserve positivity on P N ((0, ∞)) when applied entrywise. Theorem 2.12 (Horn). If a continuous function F : (0, ∞) → R is such that F [−] : P N ((0, ∞)) → P N (R), then F ∈ C N −3 ((0, ∞)) and F (k) (x) ≥ 0 for all x > 0 and all 0 ≤ k ≤ N − 3. Moreover, if it is known that F ∈ C N −1 ((0, ∞)), then F (k) (x) ≥ 0 for all x > 0 and all 0 ≤ k ≤ N − 1. The main idea in the proof is to develop into Taylor series a perturbation determinant det[F (a + tu j u k )] N j,k=1 and isolate the first non-zero coefficient as a universal constant times the product F (a)F ′ (a) · · · F (N −3) (a). Our prior work in fixed dimension has amply exploited the symmetry and combinatorial flavor of similar determinants [3]. Main results in 1D We state in this brief section our main results, restricted to the one-variable case. The proofs will be given in subsequent sections with a gradual increase in technicality, which also applies the statements of these results. A leading thread is the isolation of minimal sets of matrices for the verification of preservers, without altering the conclusions. We remind the reader that all functions in this article act entry by entry on moment sequences and matrices. The following theorem, the first in a series to be established below, gives an idea of the type of positive Hankel-matrix preservers we seek. In particular, Theorem 3.1 strengthens the Schoenberg-Rudin Theorem 2.10, by relaxing the assumptions in [38,42] to require positivity preservation only for Hankel matrices. Theorem 3.1 is proved in Section 6 with three further strengthenings: we use test sets with at most three points (corresponding to Hankel test matrices of rank at most three), the measures are allowed to have a mass constraint, enabling us to classify functions F : (−ρ, ρ) → R, where 0 < ρ ≤ ∞, and we show that allowing functions to map entrywise into the co-domain M(R) does not enlarge the class of preservers. Our next result is a one-sided variant of the above characterizations, following Horn [25, Theorem 1.2]. Akin to Theorem 3.1, it arrives at the same conclusion under weaker assumptions than in [25]. In Section 4, we use results of Bernstein and Lorch-Newman to prove Theorem 3.2, and then provide a strengthening of it, Theorem 4.1, in the spirit described above after Theorem 3.1. Here, we can replace M([0, 1]) by test measures supported on at most two points. Next, we provide a classification of the preservers of M([0, ∞)), Theorem 3.3, which gives a Schoenberg-type characterization of functions preserving total non-negativity. It is akin to Theorem 3.2, and provides a connection between moment sequences, totally non-negative Hankel matrices, and their preservers; see Section 5 for the proof. (1) Applied entrywise, the function F preserves positive semidefiniteness on the set H ++ of all totally non-negative Hankel matrices. Our techniques lead to the following observation: the only non-constant maps which preserve the set of all totally non-negative matrices when applied entrywise are of the form F (x) = cx, where c > 0. See Theorem 5.7 for more details. Returning to moment sequences, in the present paper we also study preservers of M([−1, 0]), and show that these are classified as follows. Theorem 3.4. The following are equivalent for a function F : R → R. ( 1) Applied entrywise, F maps M([−1, 0]) into M((−∞, 0]). (2) There exists an absolutely monotonic entire function F such that F (x) =      F (x) if x ∈ (0, ∞), 0 if x = 0, − F (−x) if x ∈ (−∞, 0). It is striking to observe the possibility of a discontinuity at the origin, in both of the previous theorems. For the proof of this result, we refer the reader to Section 7. We also derive a similar description of the functions that transform M([−1, 0]) into M([0, ∞)); see Theorem 7.3. In this variant, we observe that F may also be discontinuous at 0. The arguments used to show Theorem 2.10 and its one-sided variant by Schoenberg, Rudin, and Horn do not carry over to our setting involving positive Hankel matrices. This is due to the fact that the hypotheses in Theorems 3.1 and 3.2 are significantly weaker. We show below how to further relax quite substantially the assumptions in Theorem 3.1 (Section 6), Theorem 3.2 (Section 4), and Theorem 3.4 (Section 7). In doing so, our goal is to understand the minimal amount of information that is equivalent to the requirement that a function preserves M([0, 1]) or M([−1, 1]) when applied entrywise. We will demonstrate that requiring a function to preserve moments for measures supported on at most three points, is equivalent to preserving moments for all measures. In particular, this shows that preserving positivity for positive Hankel matrices of rank at most three implies positivity preservation for all positive matrices. This latter point prompts a comparison to the case of Toeplitz matrices considered in [38]. Rudin proved that Theorem 2.10(3) holds if F preserves positivity on a twoparameter family of Toeplitz matrices with rank at most 3, namely {(a + b cos((i − j)θ)) i,j≥1 : a, b ≥ 0, a + b < 1}, (3.1) where θ is a fixed real number such that θ/π is irrational. Similarly, the present work shows that for power moments, it suffices to work with families of positive Hankel matrices of rank at most three. Theorem 6.1(1) contains the precise details. Moment transformers on [0, 1] Over the course of the next four sections, we will formulate and prove strengthened versions of the announced results. Here, we provide two proofs of Theorem 3.2. The first is natural from the point of view of moments and Hankel matrices. The proof proceeds by first deriving from positivity considerations some inequalities satisfied by all moments transformers. We then obtain the desired characterization by appealing to classical results on completely monotonic functions. This is in the spirit of Lorch and Newman [32], who in turn are very much indebted to the original Hausdorff approach to the moment problem via summation rules and higher-order finite differences. Using Theorem 2.9, we now provide our first proof of Theorem 3.2. µ ∈ Meas + ([0, 1]), there exists σ ∈ Meas + ([0, 1]) such that F (s k (µ)) = s k (σ) for all k ≥ 0. Let p(t) = a 0 t 0 + · · · + a d t d be a real polynomial such that p(t) ≥ 0 on [0, 1]. Then, 0 ≤ 1 0 p(t) dσ(t) = d k=0 a k s k (σ) = d k=0 a k F (s k (µ)). (4.1) Here and below, we employ (4.1) with a careful choice of measure µ and polynomial p to deduce additional information about the function F . In the present situation, fix finitely many scalars c j , t j > 0 and an integer n ≥ 0, and set p(t) = (1 − t) n and µ = j e −t j α c j δ e −t j h ,(4.2) where α > 0 and h > 0. Now let g(x) := j c j e −t j x , and apply (4.1) to see that the forward finite differences of F • g alternate in sign. That is, n k=0 (−1) k n k F   j c j e −t j α−t j kh   ≥ 0, so (−1) n ∆ n h (F • g)(α) ≥ 0. As this holds for all α, h > 0 and all n ≥ 0, it follows that F • g : (0, ∞) → (0, ∞) is completely monotonic for all µ as in (4.2). Using the weak density of such measures in Meas + ((0, ∞)), together with Bernstein's theorem (Theorem 2.9), it follows that F •g is completely monotonic on (0, ∞) for all completely monotonic functions g : (0, ∞) → (0, ∞). Finally, a theorem of Lorch and Newman [32, Theorem 5] now gives that F : (0, ∞) → (0, ∞) is absolutely monotonic. Our second proof of Theorem 3.2 involves a significant relaxation of its hypotheses. Our first observation is that, if F preserves positivity for 2 × 2 matrices, and sends M({1, u 0 }) to M(R) for a single u 0 ∈ (0, 1), then F is absolutely monotonic on (0, ∞). Further relaxation may be obtained by working with mass-constrained measures. Theorem 4.1. Fix scalars ρ and u 0 , with 0 < ρ ≤ ∞ and u 0 ∈ (0, 1). Given a function F : [0, ρ) → R, the following are equivalent. ( 1) The map F [−] sends M ρ ({1, u 0 }) ∪ M ρ ({0, 1}) into M(R), and F (a)F (b) ≥ F ( √ ab) 2 for all a, b ∈ [0, ρ). (2) The map F [−] sends M ρ ([0, 1]) into M([0, 1]). (3) The function F agrees on (0, ρ) with an absolutely monotonic entire function and 0 ≤ F (0) ≤ lim ǫ→0 + F (ǫ). If F is known to be continuous on (0, ρ), then the second hypothesis in (1) may be omitted. Note that assertion (1) is a priori significantly weaker than the requirement that F preserves M([0, 1]), at least when ρ = ∞, say. Moreover, hypothesis (3) here is the same as hypothesis (4) in Theorem 3.3, and Theorem 4.1 is used to prove that result in Section 5. We now turn to proving Theorem 4.1. This requires results on functions preserving positivity for matrices of a fixed dimension, which we now develop. As shown in [21, Theorem 4.1], the same result can be obtained by working only with a particular family of rank-two matrices, without the continuity assumption, and on any domain (0, ρ) as above. In the next theorem, Horn's hypotheses are relaxed even further by making appeal only to Hankel matrices. {A = a1 N ×N + buu T : a ∈ [0, ρ), b ∈ [0, ρ − a), 0 < a + b < ρ}. (4.3) Then F ∈ C N −3 (I), with F (k) (x) ≥ 0 for all x ∈ I (0 ≤ k ≤ N − 3), and F (N −3) is a convex non-decreasing function on I. If, further, F ∈ C N −1 (I), then F (k) (x) ≥ 0 for all x ∈ I and 0 ≤ k ≤ N − 1. Finally, if F is assumed to be continuous on I, then the assumption that F preserves positivity on P 2 (I) is not necessary. Remark 4.3. In fact, our proof of Theorem 4.2 reveals that these hypotheses may be relaxed slightly, by replacing the test set P 2 ((0, ρ)) with the collection of rank-one matrices P 1 2 ((0, ρ)) and all matrices of the form a b b b with a > b > 0. (4.4) The proof of Theorem 4.2 relies on Lemma 2.7. Proof of Theorem 4.2. If F ∈ C(I), then the result follows by repeating the argument in [25, Theorem 1.2], but with the vector α replaced by a vector u ∈ R N as in Lemma 2.7. Now suppose F is an arbitrary function, which is not identically zero on (0, ρ); we claim that F must be continuous. We first show that F (x) = 0 for all x ∈ (0, ρ). Indeed, suppose F (c) = 0 for some c ∈ (0, ρ). Given d ∈ (c, ρ), define a sufficiently long geometric progression u ′ 0 = c, . . . , u ′ n = d, such that u ′ n+1 ∈ (d, ρ) . By considering the matrices F [A j ], where A j := u ′ j u ′ j+1 u ′ j+1 u ′ j+2 , 0 ≤ j ≤ n − 1, we obtain that F (d) = 0 for all d ∈ (c, ρ). A similar argument applies to d ∈ (0, c), showing that F ≡ 0 on (0, ρ). Next, since F [−] preserves positivity on P 1 2 ((0, ρ)) and is positive on (0, ρ), it follows that g : x → log F (e x ) is midpoint convex on the interval (−∞, log ρ). Moreover, applying F [−] to matrices of the form (4.4) shows that F is non-decreasing. Hence, by [37,Theorem 71.C], the function g is necessarily continuous on (−∞, log ρ), and so F is continuous on (0, ρ). This proves the result in the general case. Using the above result, we can now prove Theorem 4.2. Proof of Theorem 4.2. If F ∈ C(I), then the result follows by repeating the argument in [25, Theorem 1.2], but with the vector α replaced by a vector u ∈ R N as in Lemma 2.7. Now suppose F is an arbitrary function, which is not identically zero on (0, ρ); we claim that F must be continuous. We first show that F (x) = 0 for all x ∈ (0, ρ). Indeed, suppose F (c) = 0 for some c ∈ (0, ρ). Given d ∈ (c, ρ), define a sufficiently long geometric progression u ′ 0 = c, . . . , u ′ n = d, such that u ′ n+1 ∈ (d, ρ). By considering the matrices F [A j ], where A j := u ′ j u ′ j+1 u ′ j+1 u ′ j+2 , 0 ≤ j ≤ n − 1, we obtain that F (d) = 0 for all d ∈ (c, ρ). A similar argument applies to d ∈ (0, c), showing that F ≡ 0 on (0, ρ). Next, since F [−] preserves positivity on P 1 2 ((0, ρ)) and is positive on (0, ρ), it follows that g : x → log F (e x ) is midpoint convex on the interval (−∞, log ρ). Moreover, applying F [−] to matrices of the form (4.4) shows that F is non-decreasing. Hence, by [37,Theorem 71.C], the function g is necessarily continuous on (−∞, log ρ), and so F is continuous on (0, ρ). This proves the result in the general case. Finally, we turn to the proof of Theorem 4.1, which provides a second proof of Theorem 3.2 which is more informative. We first observe that Theorem 4.2 can be reformulated in terms of moment sequences, using the fact that the matrices occurring in the statement of the theorem can be realized as truncations of positive Hankel matrices; see Definition 2.5. Theorem 4.4. Let F : I → R, where I = (0, ρ) and 0 < ρ ≤ ∞, and fix N ≥ 3. Suppose F [−] maps the moment sequences in M ρ 2N −2 ({1, u 0 }) with positive entries to {(s 0 (µ), . . . , s 2N −3 (µ), s 2N −2 (µ) + t) : µ ∈ Meas + (R), t ≥ 0} for some u 0 ∈ (0, 1), and the moment sequences in M ρ 2 ({0, 1}) ∪ M ρ 2 ({u}) with positive entries to M 2 (R) for all u ∈ (0, 1). Then F ∈ C N −3 (I), with F (k) (x) ≥ 0 for all x ∈ I (0 ≤ k ≤ N − 3), and F (N −3) is a convex non-decreasing function on I. If, further, it is known that F ∈ C N −1 (I), then F (k) (x) ≥ 0 for all x > 0 and 0 ≤ k ≤ N − 1. If F is continuous on I, then the assumption that F [−] maps elements of M ρ 2 ({u}) into M 2 (R) for all u ∈ (0, 1) may be omitted. Proof. In view of Hamburger's Theorem for truncated moment sequences, a Hankel matrix with entries in the first and last columns given by (2). s 0 , . . . , s N −1 and s N −1 , . . . , s 2N −2 is positive if and only if (s 0 , . . . , s 2N −3 ) ∈ M 2N −3 (R), and s 2N −2 ≥ x 2N −2 dµ,F [H µ ] (1) = n≥0 c n [H (1) µ ] •n , where F (x) = n≥0 c n xIt remains to show (1) =⇒ (3). It is immediate that mapping M ρ 2 ({0, 1}) into M 2 (R) is equivalent to mapping M ρ ({0, 1}) into M(R). Thus, by Theorem 4.4, it holds that F (k) (x) ≥ 0 for all x > 0 and all k ≥ 0. Theorem 2.8 now gives the result, apart from the assertion about F (0), but this is immediate. We conclude this part by explaining why Theorem 4.1 provides a minimal set of rankconstrained positive semidefinite matrices for which positivity preservation is equivalent to absolute monotonicity. Remark 4.6. A smaller set of rank-constrained matrices than that employed for The- orem 4.1 could not include a sequence of matrices in ∞ N =1 P 2 N ([0, ρ)) of unbounded dimension, hence would be contained in P ′ N := N n=1 P 2 n ([0, ρ)) ∪ ∞ n=1 P 1 n ([0, ρ)) for some N ≥ 1. However, as noted in the paragraphs preceding Proposition 4.11 below, the map x → x α preserves positivity on P ′ N for all α ≥ N − 2, and such a function may be non-analytic. Remark 4.7. The proof of Theorem 4.1 also strengthens a 1979 result of Vasudeva [52]. Vasudeva showed for I = (0, ∞) that if F : I → R preserves positivity entrywise on P N (I) for all N ≥ 1 then F is absolutely monotonic and so is represented by a convergent power series on I. The proof above shows that Vasudeva's result also holds if I is replaced by (0, ρ) for any ρ > 0 and, for every N , the set P N (I) is replaced by the subset of Hankel matrices within it of rank at most 2. 4.1. Hankel-matrix positivity preservers in fixed dimension. We conclude this section by addressing briefly the fixed-dimension case for powers and analytic functions, as studied by FitzGerald and Horn, and also in previous work by the authors [3,15,20]. Our first result shows that considerations of Hankel matrices may be used to strengthen the main result in [3]. (1) F [−] preserves positivity on P N (D(0, ρ)), where D(0, ρ) is the closed disc in the complex plane with center 0 and radius ρ. (2) The coefficients c j satisfy either c 0 , . . . , c N −1 , c ′ ≥ 0, or c 0 , . . . , c N −1 > 0 and c ′ ≥ −C(c; z M ; N, ρ) −1 , where C(c; z M ; N, ρ) := N −1 j=0 M j 2 M − j − 1 N − j − 1 2 ρ M −j c j . (3) F [−] preserves positivity on Hankel matrices in P 1 N ((0, ρ)). The strengthening here is the addition of the word 'Hankel' to hypothesis (3). {b n ρu(b)u(b) T , ρu(b n )u(b n ) T : n ≥ 1}, for any fixed b ∈ (0, 1), where u(ǫ) := (1 − ǫ, (1 − ǫ) 2 , . . . , (1 − ǫ) N ) T , for any ǫ ∈ (0, 1). (4.5) Note that u(ǫ)u(ǫ) T ∈ P 1 N (R) is Hankel, by Lemma 2.7. Thus, Remark 4.9 gives a notable reduction of the N -dimensional parameter space, P 1 N ((0, ρ)), to the countable subset of Hankel matrices required in (3 ′ ). If N > 1, this is indeed minimal information required to derive Theorem 4.8 (2), since the extreme critical value C(c; z M ; N, ρ) cannot be attained on any finite set of matrices in P 1 N ((0, ρ)). As a first step towards the proof of Theorem 4.8, we recall from [3, Lemma 2.4] that, under suitable differentiability assumptions, the conclusions of Theorem 4.2 still hold if one considers only rank-one matrices. We now formulate a slightly stronger version of this result. Proposition 4.10. Let F ∈ C ∞ ((−ρ, ρ)), where 0 < ρ ≤ ∞. Fix a vector u ∈ (0, √ ρ) N with distinct coordinates, and suppose F [b n uu T ] ∈ P N (R) for a positive real sequence b n → 0 + . Then the first N non-zero derivatives of F at 0 are strictly positive. The assumptions and conclusions of this result are similar to those of Theorem 4.2 above; a common generalization of both results can be found in [29]. Proof. For ease of exposition, we will assume F has at least N non-zero derivatives at 0, say of orders m 1 < · · · < m N , where m 1 ≥ 0. By results on generalized Vandermonde determinants [17, Chapter XIII, §8, Example 1], the vectors {u •m j : 1 ≤ j ≤ N } are linearly independent. Now, by Taylor's theorem, F [b n uu T ] = N j=1 F (m j ) (0) m j ! b m j n u •m j (u •m j ) T + o(b m N n ). (4.6) For each 1 ≤ k ≤ N , choose v k ∈ R N such that v T k u •m j = δ j,k m j !. Then, b −m k n v T k F [b n uu T ]v k = m k !F (m k ) (0) + o(b m N −m k n ) ≥ 0, and letting n → ∞ concludes the proof. We now use Proposition 4.10 to prove the theorem. In the latter case, to prove that c M ≥ −C(c; z M ; N, ρ) −1 , we use the sequence ρu(b n )u(b n ) T , where u(b n ) is defined as in (4.5). Let u n := √ ρu(b n ) for n ≥ 1. Then [3, Equation (3.11)] implies that 0 ≤ det \|c M \| −1 F [u n u T n ] , and so \|c M \| −1 ≥ N −1 j=0 s µ(M,N,j) (u n ) 2 c j ,\|c M \| −1 ≥ N −1 j=0 M j 2 M − j − 1 N − j − 1 2 ρ M −j c j = C(c; z M ; N, ρ). Thus (2) holds, and this concludes the proof. Finally, we consider the question of which real powers preserve positivity on N × N Hankel matrices. Recall that the Schur product theorem guarantees that integer powers x → x k preserve positivity on P N ((0, ∞)). It is natural to ask if any other real powers do so. In [15], FitzGerald and Horn solved this problem, and uncovered an intriguing transition. In their main result, they show that the power function x → x α preserves positivity entrywise on P N ((0, ∞)) if and only if α is a non-negative integer or α ≥ N − 2. The value N − 2 is known in the literature as the critical exponent for preserving positivity. As shown in [20], the critical exponent remains unchanged upon restricting the problem to preserving positivity on P k N ((0, ∞)) for any k ≥ 2. More precisely, for each non-integral α ∈ (0, N − 2), there exists a rank-two matrix A ∈ P 2 N ((0, ∞)) such that A •α ∈ P N ; see [20] for more details. As we now show, the result does not change when restricted to the set of positive semidefinite Hankel matrices. Proposition 4.11. Let 2 ≤ k ≤ N and let α ∈ R. The following are equivalent. (1) The power function x → x α preserves positivity when applied entrywise to Hankel matrices in P k N ((0, ∞)). (2) The power α is a non-negative integer or α ≥ N − 2. Moreover, there exists a Hankel matrix A ∈ P 2 N ((0, ∞)) such that A •α ∈ P N for all non-integral α ∈ (0, N − 2). Proof. By the main result in [27], for pairwise distinct real numbers Note that replacing (0, ∞) with (0, ρ) for some ρ with 0 < ρ < ∞ leads to the same classification of entrywise powers preserving positivity on the reduced test set. x 1 , . . . , x N > 0, the matrix ((1 + x i x j ) α ) N i, Totally non-negative matrices With a better understanding of the endomorphisms of moment sequences of positive measures, we turn next to the structure of preservers of total non-negativity, in both the fixed-dimension and dimension-free settings. Recall that a rectangular matrix is totally non-negative if every minor is a non-negative real number. We begin with the well-known fact that moment sequences of positive measures on [0, ∞) are in natural correspondence with totally non-negative Hankel matrices. Proof. The first claim is a consequence of well-known results in the theory of moments [18,45], as outlined in the introduction to [14]. For measures on [0, 1], the result now follows via Theorem 2.4(3). Lemma 5.1 also has a finite-dimensional version, which will be required in the proof of Theorem 3.3. s 0 = 0 =⇒ s 1 = 0 ⇐⇒ s 2 = 0 ⇐⇒ · · · ⇐⇒ s 2N −3 = 0 ⇐= s 2N −2 = 0. Consequently, if (4) holds and an entry of H is zero, then F [H] ∈ P N . Remark 5.3. While Theorem 3.3 is more natural to state for functions with domain [0, ∞), the proof goes through verbatim for F : [0, ρ) → R, where 0 < ρ < ∞. In this case, the test set H ++ in the first two assertions of Theorem 3.3 (but not the target set) must be replaced by its subset of matrices with entries in [0, ρ). Next we examine the class of polynomial maps that, when applied entrywise, preserve total non-negativity for Hankel matrices of a fixed dimension. First, note that the analogue of the Schur product theorem holds for totally non-negative Hankel matrices [14,Theorem 4.5]; this also follows from Lemma 5.2. Second, note that the Hankel matrix H ǫ := u(ǫ)u(ǫ) T is totally non-negative for all ǫ ∈ (0, 1), where u(ǫ) was defined in (4.5): u(ǫ) := (1 − ǫ, . . . , (1 − ǫ) N ) T . This holds because the elements of H ǫ are all positive, and the k × k minors of H ǫ vanish if k ≥ 2. As a consequence, Proposition 4.10 implies that if F is a polynomial which preserves positive semidefiniteness on H ++ N , then the first N non-zero coefficients of F must be positive. The following result shows that the next coefficient can be negative, with the same threshold as in Theorem 4.8. Thus the set of powers preserving total non-negativity for Hankel matrices coincides with the set of powers preserving positivity on P N ([0, ∞)), as identified by FitzGerald and Horn [15]. C(c; z M ; N, ρ) := N −1 j=0 M j 2 M − j − 1 N − j − 1 2 ρ M −j c j .(3 Remark 5.6. We note that Theorem 5.5 follows from a result of Jain [27, Theorem 1.1], since for x ∈ (0, 1), the semi-infinite Hankel matrix (1 + x i+j ) ∞ i,j=0 arises as the moment matrix of the measure δ 1 + δ x , and is therefore totally non-negative, by Lemma 5.1. We conclude this section by examining entrywise preservers of total non-negativity in the general setting, where the matrices are not assumed to have a Hankel structure, or to be symmetric or even square. By Theorem 3.3, every such preserver must be absolutely monotonic on (0, ∞). However, it is not immediately clear how to proceed further with non-symmetric matrices; the analogue of the Schur product theorem no longer holds in this situation, as noted in [14,Example 4.3]. Our next result shows that, when working with rectangular or symmetric matrices, the set of functions preserving total non-negativity is very rigid. Contrast this result, especially hypothesis (2), with Theorem 3.3. We defer the proof of Theorem 5.7 until we have more closely examined the case of entire maps. This will give what is needed to overcome the main technical difficulty in proving Theorem 5.7. Recall from [14, Section 5] that if A is a totally non-negative matrix which is 3 × 3, or symmetric and 4 × 4, then the Hadamard power A •α is totally non-negative for all α ≥ N − 2, where N is the number of rows of A. For larger matrices, very few entire functions preserve total non-negativity. Proof. First we consider the 4 × 4 case. Note that one implication is immediate, so suppose F [−] preserves total non-negativity and is not constant. Let A y := y Id 3 ⊕0 1×1 , where y ≥ 0 and Id k denotes the k × k identity matrix for k ≥ 1. Observing that F [A y ] is totally non-negative, it follows that F (y) ≥ F (0) ≥ 0 for all y ≥ 0. If, moreover, y > 0 is such that F (y) > F (0), then from the same observation we conclude that and let A(x) := 1 4×4 + xM . By the analysis in [14, Example 5.9], the matrix A(x) is totally non-negative for all x ≥ 0, while for every real α > 1 there exists δ α > 0 such that det A(x) •α < 0 for all x ∈ (0, δ α ). Fix z ∈ (0, δ m ), let t > 0, and note that F [tA(z)] = c m t m A(z) •m + t m+1 C(t, z) for some 4 × 4 matrix C(t, z). Since the matrix on the left-hand side is totally nonnegative, it follows that 0 ≤ t −4m det F [tA(z)] = c 4 m det A(z) •m + O(t) . Letting t → 0 + gives a contradiction. Hence c 1 = 0. Finally, note that F [t A(x)] = ∞ n=1 c n t n (1 4×4 + xM ) •n = ∞ n=1 c n t n n j=0 n j x j M •j = ∞ j=0 β j (t)x j M •j , where t ≥ 0 and β j (t) := ∞ n=j c n n j t n . Using a Laplace expansion, it is not hard to see that det F [t A(x)] = det M 4 (t) + O(x 5 ), where M 4 (t) := 4 j=0 β j (t)x j M •j . If R is a commutative unital ring containing x and α 1 , . . . , α 4 then Appendix A gives that det M 4 = −57168 α 0 α 2 1 α 2 x 4 + O(x 5 ), where M 4 := 4 j=0 α j x j M •j . (5.2) Taking R = R[t, x] and α j = β j (t), we have that M 4 equals M 4 (t). Since F [tA(x)] is totally non-negative for all x ≥ 0, dividing through by x 4 and letting x → 0 + , it follows that β 0 (t)β 1 (t) 2 β 2 (t) vanishes on an interval. Since β j (t) = F (j) (t)/j!, each β j is also entire; thus at least one β j ≡ 0, whence β 2 (t) ≡ 0. It follows that c n = 0 for all n ≥ 2, as claimed. That c 1 ≥ 0 now follows by considering F [Id 4 ]. This concludes the proof for 4 × 4 totally non-negative matrices. The proof for symmetric 5 × 5 matrices now follows, as [14, Example 5.10] gives a 5 × 5 symmetric totally non-negative matrix containing the matrix A(x) as a 4 × 4 minor. With this result in hand, we can now complete the outstanding proof in this section. Proof of Theorem 5.7. Clearly (3) =⇒ (1) =⇒ (2). Suppose (2) holds. Then, by Theorem 3.3, the function F is absolutely monotonic on (0, ∞), and F (0) ≥ 0. If F is not constant, then F (y) > F (0) for some y > 0. As F [y Id 3 ] is totally non-negative, looking at 2 × 2 minors now shows that F (0) = 0. To see that F is continuous at 0, note first that Recall that Schoenberg and Rudin's result, Theorem 2.10, characterizes positivity preservers for matrices with entries in (−1, 1). As a consequence of Theorem 6.1, we obtain the following generalization of Theorem 2.10 with a much reduced test set. Corollary 6.2. The hypotheses of Theorem 2.10 are equivalent to F [−] preserving positivity on Hankel matrices arising from moment sequences, with entries in (−1, 1) and rank at most 3. Furthemore, this theorem holds with (−1, 1) and (0, 1) replaced by (−ρ, ρ) and (0, ρ), respectively, whenever ρ ∈ (0, ∞]. The proof of Theorem 6.1 requires new ideas, as previous techniques to prove analogous results are not amenable to the more general Hankel setting; see Remark 6.7. As a first step, we obtain the following lemma; together with Theorem 2.4, it explains why assertion (1) in Theorem 6.1 can be relaxed to assertion (2). Recall the notion of truncated moment sequence from Definition 2.6. Proof. Akin to the proof of Theorem 4.2, the assumption implies that F is nondecreasing, whence locally bounded, on (0, ρ). Now let µ = aδ −1 for any a ∈ (0, ρ). By considering the leading principal 2 × 2 submatrix of F [H µ ], where H µ is the Hankel matrix (2.1) associated to the measure µ, it follows that \|F (−a)\| ≤ F (a). The next step is to use assertion (2) in Theorem 6.1 to establish the continuity of F on (−ρ, ρ). Proposition 6.4. Fix v 0 ∈ (0, 1) and suppose the function F : (−ρ, ρ) → R, where 0 < ρ ≤ ∞, maps entrywise M ρ 2 ({−1, 1}) ∪ M ρ 3 ({−1, v 0 }) ∪ u∈(0,1) M ρ 4 ({1, u}) into M 2 ([−1, 1]) ∪ M 3 ([−1, 1]) ∪ M 4 ([−1, 1]) . Then F is continuous on (−ρ, ρ). Proof. As F maps M ρ 2 ({−1, 1}) into M 2 ([−1, 1]), considering µ = a + b 2 δ 1 + a − b 2 δ −1 and ν = bδ 1 shows that F (a) ≥ F (b) ≥ 0 whenever 0 ≤ a ≤ b < ρ. It follows immediately that F maps M ρ 2 ({0, 1}) into M 2 (R). Then, by Theorem 4.4 for N = 3 and our assumptions, F is continuous, non-negative, and non-decreasing on (0, ρ). In particular, F has a right-hand limit at 0, and 0 ≤ F (0) ≤ lim F (a + b) − F (a + bv 2 0 ) ≥ ± F (−a + bv 0 ) − F (−a + bv 3 0 ) , or, equivalently, F (β + b + bv 0 ) − F (β + bv 0 + bv 2 0 ) ≥ F (−β) − F (−β − b(v 0 − v 3 0 ) ) . Letting b → 0 + and using the continuity of F on (0, ρ), we conclude that F is left continuous at −β. We proceed similarly to show right continuity of F at −β; let a := β + bv 3 0 and µ = aδ −1 + bδ v 0 , where b is such that 0 < b < (ρ − β)/(1 + v 3 0 ), and take b → 0 + as before. Remark 6.5. The integration trick (4.1) used in the proof of Proposition 6.4 shows that certain linear combinations of moments are non-negative. The integral it employs may also be expressed using the quadratic form given by the Hankel moment matrix for the ambient measure. To see this, suppose σ is a non-negative measure on [−1, 1] with moments of all orders, and let H σ := (s j+k (σ)) j,k≥0 be the associated Hankel moment matrix. If f : [−1, 1] → R + is continuous then so its radical √ f : [−1, 1] → R + , and the latter can be uniformly approximated on [−1, 1] by a sequence of polynomials p n (t) = dn j=0 c n,j t j . Thus 1 −1 f dσ = lim n→∞ 1 −1 p n (t) 2 dσ = lim n→∞ j,k≥0 c n,j c n,k 1 −1 t j+k dσ = lim n→∞ v T n H σ v n , where v n := (c n,0 , c n,1 , . . . , c n,dn , 0, 0, . . . ) T (n ≥ 1). Now, since the matrix H σ is positive, the limit on the right-hand side is non-negative and so 1 −1 f dσ ≥ 0. With continuity in hand, we can now complete the proof of Theorem 6.1. (1); that (1) =⇒ (2) follows from the remarks preceding Lemma 6.3. Now suppose (1) holds. By Proposition 6.4, the function F is continuous on (−ρ, ρ), so Theorem 4.1 gives that F agrees on (0, ρ) with a power series F having non-negative Maclaurin coefficients, which is convergent on the disc D(0, ρ). Proof of Theorem 6.1. Clearly (4) =⇒ (3) =⇒ (1) and (2) =⇒ Let µ := aδ −1 + e x δ e −h , where a ∈ (0, ρ), x < log(ρ − a), and h > 0, and let the polynomial p ±,n (t) := (1 ± t)(1 − t 2 ) n . Then p ±,n (t) is non-negative for all t ∈ [−1, 1] and all n ≥ 0. Applying (4.1) gives that Let H ±,a (x) := F (±a + e x ) and suppose F is smooth; dividing (6.2) by h n and taking h → 0 + , we see that \|H (n) +,a (x)\| ≥ \|H (n) −,a (x)\|. Since H +,a is real analytic, we conclude that the Taylor series for H −,a has a positive radius of convergence everywhere, so H −,a is real analytic on (−∞, log(ρ − a)). The change of variable x = log(y + a) has a convergent power-series expansion for \|y\| < a. It follows that y → F (y) is real analytic on (−ρ, ρ), hence is the restriction of F . When F is not necessarily smooth, we may use a mollifier argument. Fix 0 < ρ ′ < ρ and let G := F \| (−ρ ′ ,ρ ′ ) . For any δ ∈ (0, ρ − ρ ′ ), choose g δ ∈ C ∞ (R) such that g δ is non-negative, supported on (0, δ), and integrates to 1, and let F δ (x) := δ 0 G(x + t)g δ (t) dt for all x ∈ (−ρ ′ , ρ ′ ). As the function x → G(t + x) satisfies hypothesis (1) of the theorem with ρ replaced by ρ ′ , so does the smooth function F δ ; let F δ be an analytic function on the disc D(0, ρ ′ ) which is absolutely monotonic on (0, ρ ′ ) and agrees on (−ρ ′ , ρ ′ ) with F δ . Since \|F (x) − F δ (x)\| = δ 0 (G(x) − G(x + t))g δ (t) dt ≤ sup 0≤t≤δ \|G(x) − G(x + t)\| for all x ∈ (−ρ ′ , ρ ′ ), it follows that F δ converges to F locally uniformly on (−ρ ′ , ρ ′ ) as δ → 0 + . The function F δ is absolutely monotonic, so \| F δ (z)\| ≤ F δ (a) whenever \|z\| ≤ a < ρ ′ , and F δ (a) → F (a) as δ → 0 + . Hence { F δ (z) : δ > 0} is uniformly bounded on D(0, a), and therefore forms a normal family. Thus for some sequence δ n → 0 + , the functions F δn converge locally uniformly to a function F that is analytic on D(0, ρ ′ ), and F agrees with F on (−ρ ′ , ρ ′ ). As this argument holds for all ρ ′ ∈ (0, ρ), the proof is complete. Remark 6.6. The proof of Theorem 6.1 requires measures whose support contains the point 1, in order to be able to employ the mollifier argument to move from continuous to smooth functions. Remark 6.7. Recall that Rudin [38] showed that F must be analytic on (−1, 1) and absolutely monotonic on (0, 1) if F [−] preserves positivity for the two-parameter family of Toeplitz matrices defined in (3.1). A natural strategy to prove Theorem 6.1 would be to show that there exists θ ∈ R with θ/π irrational, such that the matrices (cos((i − j)θ)) n i,j=1 can be embedded into positive Hankel matrices, for all sufficiently large n. However, this is not possible: given 0 < m 1 < m 2 such that cos(m 1 θ) < 0 and cos(m 2 θ) < 0, if there were a measure µ ∈ Meas + ([−1, 1]) such that cos(m j θ) = s k j (µ) for j = 1 and j = 2, then, by the Toeplitz property, k 1 , k 2 , and k 1 + k 2 must all be odd, which is impossible. Moment transformers on [−1, 0] We now study the structure of endomorphisms of M([−1, 0]). The following result strengthens Theorem 3.4 and reveals that such functions may be discontinuous at the origin, in contrast to Theorem 6.1. (3) There exists an absolutely monotonic entire function F such that F (x) =      F (x) if x ∈ (0, ρ), 0 if x = 0, − F (−x) if x ∈ (−ρ, 0). In particular, the function F is odd, but may be discontinuous at 0. Proof. To show that (3) =⇒ (2), note first that if µ ∈ Meas + ([−1, 0]), so that µ = aδ 0 for some a, then F [H µ ] = H F (a)δ 0 , so we may assume µ is not of this form, whence the Hankel matrix H µ has no zero entries, and the moment sequence alternates in sign and is uniformly bounded, by Theorem 2.4. In particular, F (s 2k (µ)) = F (s 2k (µ)) and F (s 2k+1 (µ)) = − F (−s 2k+1 (µ)) (k ≥ 0). Recalling the form of the Hankel matrix H δ −1 , it follows that F [H µ ] = H δ −1 • F [H δ −1 • H µ ] (7.1) where • denotes the entrywise matrix product. This shows (2) We conclude by showing that F is odd. Let µ = aδ −1 for some a ∈ (0, ρ) and note that p n (t) = (−t) n (1 + t) is non-negative on [−1, 0] for any non-negative integer n. If Taking n = 0 and 1 gives that 0 ≤ F (a) + F (−a) ≤ 0, and the final claim follows. Theorem 7.1 has the following consequence. Corollary 7.2. Define a checkerboard matrix to be any real matrix A = (a ij ) such that (−1) i+j a ij > 0 for all i, j. Given a function F : R → R, the following are equivalent. (1) Applied entrywise, F maps the set of positive Hankel checkerboard matrices of all dimensions into itself. We conclude this section with an even analogue of Theorem 7.1. Theorem 7.3. Given u 0 ∈ (0, 1) and F : (−ρ, ρ) → R, where 0 < ρ ≤ ∞, the following are equivalent. ( 1) F [−] maps M ρ ({−1, −u 0 }) into M([0, ∞)) and M ρ 4 ({−1, 0}) ∪ u∈(0,1) M ρ 4 ({−u}) into M 4 ([0, ∞)). (2) F [−] sends M ρ ([−1, 0]) to M([0, 1]). (3) There exists an absolutely monotonic entire function F such that F (x) = F (x) if x ∈ (0, ρ), F (−x) if x ∈ (−ρ, 0). Moreover, 0 ≤ F (0) ≤ lim ǫ→0 F (ǫ). Proof. This is similar to the proof of Theorem 7.1; to show that (1) =⇒ (3), one may use the polynomials p n (t) = t n (1 − t). We omit further details. Transformers with compact domain The goal of this section is to show how results in the previous sections can be refined when the moments are contained in a compact domain. Indeed, when the domain of F is a compact interval I, the situation is more complex; absolute monotonicity, or even continuity of F , does not extend automatically from the interior of I to its end points. This was already observed by Rudin via specific counterexamples; see Remark (a) at the end of [38]. To the best of our knowledge, characterization results in this setting are not known. We now take a closer look at this phenomenon. We begin by characterizing the functions preserving positivity of Hankel matrices in P N (I) for all N , where I = [0, ρ] and 0 < ρ < ∞. The only Hankel matrix in P N +1 ([−ρ, ρ]) with an entry −ρ is the checkerboard matrix with (i, j)th entry (−1) i+j ρ. H a :=   a 0 a 0 a a a a 2a   , where a ∈ [0, ρ/2). As F [H a ] is positive, so 0 ≤ F (0) ≤ F + (0) := lim a→0 + F (a). Furthermore, 0 ≤ lim a→0 + det F [H a ] = −F + (0)(F (0) − F + (0)) 2 , whence F (0) = F + (0), To prove the claim, let the rows and columns of the positive Hankel matrix A be labelled by 0, . . . , N , and suppose a ij = −ρ. Then i + j is odd and a ll = a l+1,l+1 = ρ, where 2l + 1 = i + j. Repeatedly considering principal 2 × 2 minors shows that a pq = ρ if p + q is even. Now let m, n ∈ [0, N ] be odd, with m < n, and denote by C the principal 3 × 3 minor of A corresponding to the labels 0, m, and n. Writing C =   ρ a 0m ρ a 0m ρ a 0n ρ a 0n ρ   , we have that 0 ≤ det C = −ρ(a 0m − a 0n ) 2 , whence a 0m = a 0n . Taking m or n to equal i + j shows that these entries equal −ρ, which gives the claim. We end this section by considering functions preserving positivity for all matrices in N ≥1 P N ([−ρ, ρ]). Theorem 6.1 implies that every such function F is real analytic when restricted to (−ρ, ρ), and absolutely monotonic on (0, ρ). The following result provides a sufficient condition for F to preserve positivity, which is also necessary if the analytic restriction is odd or even. The inequality (8.2) says that any jump in F at −ρ is bounded above by the jump at ρ, which is non-negative. F (−ρ) − lim x→−ρ + F (x) ≤ F (ρ) − lim x→ρ − F (x),(8. Proof. Let g denote the continuous function on [−ρ, ρ] which agrees with F on (−ρ, ρ), and let the jumps ∆ ± := F (±ρ) − g(±ρ). Then (8.2) is equivalent to \|∆ − \| ≤ ∆ + . By the Schur product theorem and Proposition 8.1, F [−] preserves positivity on P N ((−ρ, ρ]) for all N . Now suppose A ∈ P N ([−ρ, ρ]) has some entry equal to −ρ, where N ≥ 1. Then the entries of A with modulus ρ form a block diagonal submatrix upon suitable relabelling of indices. This follows from the argument given in the proof of Proposition 8.1, applied to the ρ 2 -entries of A • A. Given this, and after further relabelling of indices, each block submatrix is of the form ρ1 n j ×n j −ρ1 n j ×m j −ρ1 m j ×n j ρ1 m j ×m j , by the main result in [24], where j = 1, . . . , r. Then F [A] = g[A] + B ′ , where B ′ = ⊕ k j=1 ∆ + · 1 n j ×n j ∆ − · 1 n j ×m j ∆ − · 1 m j ×n j ∆ + · 1 m j ×m j , and this is positive semidefinite, by (8.2). Thus F [−] preserves N ≥1 P N ([−ρ, ρ]). For the converse, we show that (8.2) holds if F [−] preserves positivity on just the set P 3 ([−ρ, ρ]) and F \| (−ρ,ρ) is odd or even. Note first that ∆ + ≥ 0, working with 2 × 2 matrices as above. Next, consider the positive matrix A :=   a 2 /ρ −a a −a ρ −ρ a −ρ ρ   , and note that 0 ≤ lim a→ρ − det F [A] = g(ρ) g(−ρ) g(ρ) g(−ρ) F (ρ) F (−ρ) g(ρ) F (−ρ) F (ρ) = ∆ + (g(ρ)F (ρ) − g(−ρ) 2 ) − g(ρ)∆ 2 − . It follows that ∆ 2 − g(ρ) ≤ ∆ 2 + g(ρ) if g(ρ 2 ) = g(−ρ) 2 , so if g = F \| (−ρ,ρ) is odd or even. This gives the result. Indeed, if g is an odd or even function which is continuous on [−ρ, ρ] and absolutely monotonic on (0, ρ), define F to be equal to g on (−ρ, ρ], and take F (−ρ) to be any element of (−F (ρ), F (ρ)]. Then F preserves positivity on all Hankel matrices with entries in [−ρ, ρ], but does not preserve positivity on N ≥1 P N ([−ρ, ρ]). Multivariable generalizations In this section we classify the preservers of moments arising from admissible measures in higher-dimensional Euclidean space, both in their totality and by considering their marginals. 9.1. Transformers of multivariable moment sequences. The initial generalization to higher dimensions of our characterization of moment-preserving functions raises no complications. However, the failure of Hamburger's theorem in higher dimensions, that is, the lack of a characterization of moment sequences by positivity of an associated Hankel-type kernel, means some extra work is required. Below, we isolate this additional challenge and provide a generalization of our main result. Let µ be a non-negative measure on R d , where d ≥ 1, which has moments of all orders; as before, such measures will be termed admissible. The multi-index notation x α = x α 1 1 . . . x α d d (x ∈ R d ) allows us to define the moment family s α (µ) = x α dµ(x) (α ∈ Z d + ), where Z + denotes the set {0, 1, 2, . . .} of non-negative integers. As before, we focus on measures with uniformly bounded moments, so that F (s α ( µ)) = s α ( σ) for all α ∈ Z d + , and a short calculation shows that F [s n (µ)] = s n (σ) for all n ∈ Z + , where σ is the pushforward of σ under the projection onto the first coordinate. Theorem 6.1 now gives that F is as claimed. sup α∈Z d + \|s α (µ)\| < ∞, To prove the converse, we need to explore the structure of the set M([−1, 1] d ). Denote the generators of the semigroup Z d + by setting (s α+β ), (s α+β − s α+β+21 j ), 1 ≤ j ≤ d, indexed over α, β ∈ Z d + are positive semidefinite [36]. Now suppose F is absolutely monotonic and entire; given a multisequence s α subject to these positivity constraints, we have to check that the multisequence F (s α ) satisfies the same conditions. As F is absolutely monotonic, Schoenberg's Theorem 2.10 gives that the kernels (α, β) → F (s α+β ) and (α, β) → F (s α+β+21 j ) are positive semidefinite. It remains to prove that the kernel (α, β) → F (s α+β ) − F (s α+β+21 j ) is positive semidefinite, for 1 ≤ j ≤ d. However, as F has the Taylor expansion F (x) = ∞ n=0 c n x n , with c n ≥ 0 for all n ∈ Z + , it is sufficient to check that the kernel (α, β) → (s α+β ) •n − (s α+β+21 j ) •n is positive semidefinite for any n ∈ Z + . This follows from a repeated application of the Schur product theorem: if matrices A and B are such that A ≥ B ≥ 0, then A •n ≥ A •(n−1) • B ≥ A •(n−2) • B •2 ≥ · · · ≥ B •n . This proof also shows that the transformers of M([−1, 1] d ) into M(R d ) are the same absolutely monotonic entire functions. On the other hand, we will see in Section 10 that, in general, a mapping F as in Theorem 9.1 does not preserve the semi-algebraic supports of the underlying measures. 9.2. Transformers of moment-sequence tuples: the positive orthant case. Our next objective is to characterize functions F : R m → R which map tuples of moments (s k (µ 1 ), . . . , s k (µ m )) arising from admissible measures on R, to a moment sequence s k (σ) for some admissible measure σ on R. This is a multivariable generalization of Schoenberg's theorem which we will demonstrate under significantly relaxed hypotheses. More precisely, we will study the preservers F : I m → R, where m ≥ 1 is a fixed integer, and I = (0, ρ) or [0, ρ) or (−ρ, ρ), where 0 < ρ ≤ ∞. By the Schur product theorem, every real analytic function F of the form F (x) = α∈Z m + c α x α (x ∈ I m ) (9.3) preserves positivity on P N (I) m if c α ≥ 0 for all α ∈ Z m + . The reverse implication was shown by FitzGerald, Micchelli, and Pinkus in [16] for ρ = ∞, and can be thought of as a multivariable version of Schoenberg's theorem. We now explain how results on several real and complex variables can be used to generalize the work in previous sections to this multivariable setting, including over bounded domains in the original spirit of Schoenberg and Rudin. Namely, we characterize functions mapping tuples of positive Hankel matrices into themselves. Of course, this is equivalent to mapping tuples of moment sequences of admissible measures into the same set. First we need some notation and terminology. Given I as in (9.1), suppose the sets K 1 , . . . , K m ⊂ R are such that all sequences in M ρ (K j ) have entries in I, for j = 1, . . . , m. A function F : I m → R acts on m-tuples of moment sequences of measures in M ρ (K 1 ) × · · · × M ρ (K m ) to produce real sequences, so that F [s(µ 1 ), . . . , s(µ m )] k := F (s k (µ 1 ), . . . , s k (µ m )) (k ∈ Z + ). (9.4) Given I ′ ⊂ R m , a function F : I ′ → R is absolutely monotonic if F is continuous on I ′ , and for any interior point x ∈ I ′ and α ∈ Z m + , the mixed partial derivative D α F (x) exists and is non-negative. As usual, for a tuple α = (α 1 , . . . , α m ) ∈ Z m + , we set D α F (x) := ∂ \|α\| ∂x α 1 1 · · · ∂x αm m F (x 1 , . . . , x m ), where \|α\| := α 1 + · · · + α m . The analogue of Bernstein's Theorem for the multivariable case is proved and put in its proper context in Bochner's book; see [10,Theorem 4 .2.2]. Our first observation is the connection between functions acting on tuples of moment sequences and on the corresponding Hankel matrices. Given admissible measures µ 1 , . . . , µ m and σ supported on the real line, it is clear that F [s(µ 1 ), . . . , s(µ m )] = s(σ) ⇐⇒ F [H µ 1 , . . . , H µm ] = H σ . In particular, equality holds at each finite truncation, that is, for the corresponding leading principal N ×N submatrices, for any N ≥ 1. We will henceforth switch between moment sequences and positive Hankel matrices without further comment. We begin by considering the case of matrices with positive entries, arising from tuples of sequences in M ρ ([0, 1]). To state and prove the main result in this subsection, we require a preliminary technical result. In other words, a facewise absolutely monotonic function is piecewise absolutely monotonic, with the pieces being the relative interiors of the faces of the truncated polyhedral cone [0, ρ) m . The following example illustrates this in the case m = 2. Example 9.4. Let F (x 1 , x 2 ) :=          x 2 1 + x 2 2 + 1 if x 1 , x 2 > 0, 2x 1 if x 1 > 0, x 2 = 0, x 2 2 + 1 if x 1 = 0, x 2 > 0, 0 if x 1 = x 2 = 0. Then F is facewise absolutely monotonic, with g ∅ = 0, g {1} (x 1 ) = 2x 1 , g {2} (x 2 ) = x 2 2 + 1, and g {1,2} (x 1 , x 2 ) = x 2 1 + x 2 2 + 1. In this example, and, in fact, for every facewise absolutely monotonic function, the function g J extends to an absolutely monotonic function on the closure [0, ρ) J of its domain, for all J ⊂ [m]. We denote this extension by g J . Furthermore, for Example 9.4, the functions g J satisfy a form of monotonicity that is compatible with the partial order on their labels: K ⊂ J ⊂ [m] =⇒ 0 ≤ g K ≤ g J on [0, ρ) K . (9.6) A word of caution: while g {1} (x 1 ) ≤ g {1,2} (x 1 , 0) for all x 1 ≥ 0, it is not true that the difference of these functions is absolutely monotonic on [0, ρ). With this definition and example in hand, together with Lemma 9.2, we can now characterize the preservers of tuples of moment sequences in M ρ ([0, 1]). Reformulating this result, as in the one-dimensional case above, it suffices to work only with Hankel matrices of rank at most two. Moreover, Theorem 4.1 is precisely Theorem 9.5 when m = 1. The proof builds on Theorem 4.1; however, the higher dimensionality introduces several additional challenges. A large part of Theorem 9.5 can be deduced from the following reformulation on the open cell in the positive orthant. If the function F : (0, ρ) m → R is such that F [−] preserves positivity on P 2 ((0, ρ)) m and on H + 1 (N ) × · · · × H + m (N ) for all N ≥ 1, then F is absolutely monotonic and is the restriction of an analytic function on D(0, ρ) m . Remark 9.7. As noted in Remark 4.3 for the one-variable case, the proof of Theorem 9.6 goes through under a weaker hypothesis, with the test sets replaced by the set of rank-one m-tuples P 1 2 ((0, ρ)) m and the set a 1 b 1 b 1 b 1 , . . . , a m b m b m b m : 0 < b l < a l < ρ, 1 ≤ l ≤ m . (9.7) The matrices in H + l (N ) and (9.7) are precisely the truncated moment matrices of admissible measures supported on {1, y l } and on {0, 1}, respectively. Proof of Theorem 9.6. We begin by recording a few basic properties of F . First, either F is identically zero, or it is everywhere positive on (0, ρ) m ; this may be shown similarly to the proof of Theorem 4.2. Moreover, using only tuples from P 1 2 ((0, ρ)) and (9.7), as well as the hypotheses, one can argue as in the proof of Theorem 4.2, and show that F is continuous on (0, ρ) m . Next, given c = (c 1 , . . . , c m ) T ∈ (0, ρ) m , the function g such that g(x) := F (x + c) for all x ∈ (0, ρ − c 1 ) × · · · × (0, ρ − c m ) satisfies the same hypotheses as F , but with ρ replaced by ρ − c l in each H + l (N ), and with P 1 2 ((0, ρ)) m replaced by P 1 2 ((0, ρ − c 1 )) × · · · × P 1 2 ((0, ρ − c m )). Therefore, as in the proof of [16, Theorem 2.1], a mollifier argument reduces the problem to considering only smooth F . We now follow the proof of [16,Proposition 2.5], but with suitable modifications imposed by the weaker hypotheses. Given r ≥ 0, we take N ≥ r+m m , and let y With this result in hand, we can now proceed. Proof of Theorem 9.5. Clearly, (2) =⇒ (1). We will show (1) =⇒ (3) by induction on m. As noted above, the case m = 1 is precisely Theorem 4.1. For the induction step, we first restrict F to the relative interior of any face of the truncated polyhedron [0, ρ) m , say (0, ρ) J for some J ⊂ [m]. The induction hypothesis and Theorem 9.6 give that F is facewise absolutely monotonic, so F ≡ g J on (0, ρ) J , with g J absolutely monotonic. To see that (9.6) holds, we claim that, for any pair of subsets L ⊂ K J ⊂ [m], g K (x) ≤ g J (x) whenever x ∈ (0, ρ) L ⊂ [0, ρ) m .g J (x 1 , x 2 , x 3 ) g K (x 1 , x 2 , 0) ≥ g K (x 1 , x 2 , 0) 2 , and taking limits as x 2 = x K\L → 0 + and x 3 = x J\K → 0 + , we have that g J (x 1 , 0, 0) g K (x 1 , 0, 0) ≥ g K (x 1 , 0, 0) 2 , and so (9.6) holds as required. For example, given a, b, c, d > 0, we have that F a b b c , d 0 0 0 , 0 0 0 0 = g {1,2} (a, d) g {1} (b) g {1} (b) g {1} (c) = (g {1,2} (a, d) − g {1} (a)) 1 0 0 0 + g {1} a b b c . The proof concludes by observing that both terms in the right-hand side of (9.8) are positive semidefinite, by the Schur product theorem and hypothesis (3): g J (a l,11 : l ∈ J) ≥ lim a l,11 →0 + ∀l∈J\K g J (a l,11 : l ∈ J) = g J (a l,11 : l ∈ K) ≥ g K (a l,11 : l ∈ K). As Theorem 9.5 shows, the notion of facewise absolutely monotone maps on [0, ρ) m is a refinement of absolute monotonicity, emerging from the study of positivity preservers of tuples of moment sequences, or, rather, of the Hankel matrices arising from them. If, instead, one studies maps preserving positivity on tuples of all positive semidefinite matrices, or even all Hankel matrices, then this richer class of maps does not arise. . . , c m ) ∈ I m \ (0, ρ) m . Note that at least one coordinate of c is zero. We choose u n = (u 1,n , . . . , u m,n ) ∈ (0, ρ) m such that u n → c, and we wish to show that F (u n ) = g(u n ) → F (c). Let H :=   1 0 1 0 1 1 1 1 2   and A l,n := u l,n 1 3×3 if c l > 0, u l,n H if c l = 0. Using (1) and the induction hypothesis for the (1, 2) and (2, 1) entries, it follows that lim n→∞ F [A 1,n , . . . , A m,n ] =   g(c) F (c) g(c) F (c) g(c) g(c) g(c) g(c) g(c)   ∈ P 3 . Computing the determinants of the leading principal minors gives g(c) ≥ 0, g(c) ≥ \|F (c)\|, and − g(c)(g(c) − F (c)) 2 ≥ 0. Hence F (c) = g(c), and the proof is complete. We can now state our final main result in this section. In particular, analogously to the one-variable case, Theorem 9.11 strengthens the multivariable analogue of Schoenberg's theorem in [16] by using only Hankel matrices arising from tuples of moment sequences. Moreover, akin to the m = 1 case, the proof reveals that one only requires Hankel matrices of rank at most 3. Given any v > 0, we let Corollary 9.12. The hypotheses in Theorem 9.11 are also equivalent to the following. (5) There exist ǫ > 0 and u 0 ∈ (0, 1) such that F [−] maps (M ρ [u 0 ] ) m ∪ v 1 ,...,vm∈(0,1+ǫ) M ρ v 1 × · · · × M ρ vm into the set of possibly truncated moment sequences of measures on R. As the reader will observe, hypothesis (5) is stronger, even in the one-dimensional case, than the corresponding hypothesis in Theorem 6.1. As the proof shows, these extra assumptions are required to obtain continuity on every orthant and on 'walls' between orthants, as well as real analyticity on one-parameter curves. Remark 9.13. Theorem 9.11 is the only instance when we deviate from Table 1.2 in the Introduction, but it should not come as a surprise that stronger conditions are required to guarantee real analyticity in several variables. Proof of Theorem 9.11 and Corollary 9.12. Clearly (4) =⇒ (3) =⇒ (2) =⇒ (1) =⇒ (5) by the Schur product theorem. Thus, we will assume (5) and obtain (4). By Theorem 9.6, the function F is absolutely monotonic on the open positive orthant (0, ρ) m , and equals the restriction to (0, ρ) m of an analytic function g : D(0, ρ) m → C. We now show that F ≡ g on all of (−ρ, ρ) m . The proof follows the m = 1 case in Section 6; for ease of exposition, we break it up into steps. Step 1. We first prove F is locally bounded. This follows by using M ρ 2 ({−1, 1}) m , as in the proof of Lemma 6.3. As above, this gives that Computing the moments of µ l,n gives the following: s 0 (µ l,n ) = \|c l \| + sgn(c l )u 0 + sgn(v l,n ) u 0 − u 3 0 v l,n , s 1 (µ l,n ) = c l , s 2 (µ l,n ) = \|c l \| + sgn(c l )u 0 + sgn(v l,n )u 2 0 u 0 − u 3 0 v l,n , s 3 (µ l,n ) = c l + v l,n . (9.12) As n → ∞, by the continuity of F in (0, ρ) m , the left-hand side of (9.11) goes to zero, whence so does the right-hand side, which is \|F (c 1 , . . . , c m )−F (c 1 +v 1,n , . . . , c m +v m,n )\|. This proves the continuity of F at (c 1 , . . . , c m ), so in every open orthant of (−ρ, ρ) m . To conclude this step, we show F is continuous on the boundary of the orthants, that is, on the union of the coordinate hyperplanes: Z := {(x 1 , . . . , x m ) ∈ (−ρ, ρ) m : x 1 · · · x m = 0}. The proof is by induction on m, with the case m = 1 shown in Proposition 6.4. For general m ≥ 2, by the induction hypothesis F is continuous when restricted to Z. It remains to prove F is continuous at a point c = (c 1 , . . . , c m ) ∈ Z when approached along a sequence {(c 1 + v 1,n , . . . , c m + v m,n ) : n ≥ 1} which lies in the interior of some orthant in (−ρ, ρ) m . Repeating the computations for (9.12), with the same sequences a l,n and µ l,n , and the polynomials p ± (t), we note that if c l = 0 then s 0 (µ l,n ) > 0 and s 2 (µ l,n ) > 0 for all sufficiently large n, while if c l = 0 then s 0 (µ l,n ) > 0 and s 2 (µ l,n ) > 0 for all n, since c l + v l,n = 0 by assumption. Therefore, in all cases, the left-hand side of (9.11) eventually equals F (u n ) − F (u ′ n ), with u n and u ′ n in the positive open orthant (0, ρ) m , and both converging to \|c\| := (\|c 1 \|, . . . , \|c m \|). Since F ≡ g on (0, ρ) m for some analytic function g on D(0, ρ) m , so (9.11) gives that lim n→∞ \|F (c) − F (c 1 + v 1,n , . . . , c m + v m,n )\| ≤ lim n→∞ F (u n ) − F (u ′ n ) = g(\|c\|) − g(\|c\|) = 0. It follows that F is continuous at all c ∈ Z, and hence on all of (−ρ, ρ) m , as claimed. Step 3. The next step in the proof is to show that it suffices to consider F to be smooth. This is achieved using a mollifier argument, exactly as in the one-variable situation. Step 4. Henceforth we assume F is smooth on (−ρ, ρ) m ; akin to the one-variable case, we will show that F is in fact real analytic. The proof extends across multiple steps below. The first step is encoded into the following technical lemma, for convenience. Lemma 9.14. Fix ρ ∈ (0, ∞] and a non-zero vector v ∈ R m . For any c ∈ (−ρ, ρ) m , let η v,c := e − v ∞ if ρ = ∞, e − v ∞ (ρ − c ∞ ) if ρ < ∞. (9.13) Then, for any w ∈ (−ρ, ρ) m , there exists c ∈ (−ρ, ρ) m such that w = c + η v,c 1, where the vector 1 := (1, . . . , 1). Proof. The assertion is immediate if ρ = ∞, so we suppose henceforth that ρ is finite. Let g(t) := w − t1 ∞ − (ρ − te v ∞ ) (t ≥ 0) . Clearly g(0) < 0 < g(ρ), so g has a root t 0 ∈ (0, ρ). Now the vector c := w − t 0 1 is as required (and t 0 = η v,c ). Step 5. We now claim that for every c ∈ (−ρ, ρ) m and every unit direction vector v = (v 1 , . . . , v m ) ∈ S m−1 , the function F is real analytic in the one-parameter space {c + η v,c e −xv : x ∈ (−1, 1)} ⊂ (−ρ, ρ) m , at the point x = 0, i.e., at w = c + η v,c 1. Here η v,c and 1 are as in Lemma 9.14, and we also use the notation e −xv := (e −xv 1 , . . . , e −xvm ). Notice moreover that the lth coordinate of c + η v,c e −xv is strictly bounded above in absolute value by c ∞ + η v,c e v ∞ , which is no more than ρ. To show the claim, we use the notation \|c\| := (\|c 1 \|, . . . , \|c m \|) and also fix a scalar x ∈ (−1, 1). We let p ±,n (t) := (1 ± t)(1 − t 2 ) n for n ≥ 0 and µ l,s := \|c l \|δ sgn(c l ) + η v,c e −xv l δ e −sv l whenever 0 < s < (1 − x)/(2n + 1), where 1 ≤ l ≤ m. As p ±,n (t) ≥ 0 for all t ∈ [−1, 1] and all n ≥ 0, applying (4.1) gives that v,c (x) whenever x ∈ (−1, 1). These estimates prove that the function F is real analytic at the point in the oneparameter space as claimed. Step 6. We now complete the proof. The real-analytic local diffeomorphism T : (u 1 , · · · , u m ) → (e u 1 − 1, e u 2 − 1, · · · , e um − 1) maps the origin to itself and, by the previous step, the function u → F (c + η v,c 1 + η v,c T (−u)) is smooth and real analytic in the unit ball along every straight line passing through the origin. Standard criteria for real analyticity (see [2,Theorem 5.5.33], for example) now give that F is real analytic at the point c + η v,c 1, hence at every point w ∈ (−ρ, ρ) m , by Lemma 9.14. Finally, recall that F agrees on (0, ρ) m with an analytic function g : D(0, ρ) m → C. As F : (−ρ, ρ) m → R is real analytic, so F = g\| (−ρ,ρ) m and the proof is complete. Remark 9.15. As Step 2 in the proof above shows, we may replace (M ρ [u 0 ] ) m in hypothesis (5) of Corollary 9.12 by M ρ [u 1 ] × · · · × M ρ [um] for any u 1 , . . . , u m ∈ (0, 1). Remark 9.16. Akin to the one-dimensional case, one may now show that Theorems 9.5 and 9.11 hold more generally for tuples of measures with bounded mass. More precisely, one should fix ρ 1 , . . . , ρ m ∈ (0, ∞) and work with tuples of admissible measures (µ 1 , . . . , µ m ) supported in [−1, 1] and such that s 0 (µ l ) < ρ l for l = 1, . . . , m, whence s k (µ l ) < ρ l for every k ≥ 0 and all such l. As discussed in the Introduction, this explains how our results unify and strengthen the Schoenberg-Rudin theorem and the FitzGerald-Micchelli-Pinkus result for positivity preservers. To prove Theorem 9.5 for F : I 1 × · · · × I m → R, where I l = [0, ρ l ), one should first define facewise absolutely monotonic maps on I 1 × · · · × I m using the relative interiors of the faces cut out by the same functionals as for [0, ρ) m . The existing proof for the case ρ 1 = · · · = ρ m goes through with minimal modifications, including to Theorem 9.6. The same is true for proving Theorem 9.11 with the domain (−ρ 1 , ρ 1 ) × · · · × (−ρ m , ρ m ) in place of (−ρ, ρ) m . Remark 9.17. There is a simple and potentially very useful conditioning operation which can assist with numerical or computational entrywise manipulation of Hankel matrices or Hankel kernels arising from moments. Namely, the moments s α = K x α dµ(x) (α ∈ Z m + ) of a positive measure with compact support K can be rescaled, s α → u α = t \|α\| s α , by a factor t > 0, so that u α are the moments of a positive measure supported by the unit cube, or even by its interior. Of course, a priori information on the size of the support K is essential for this step, but in this way some of the complications outlined in Theorem 9.11 and its proof can be avoided. Laplace-transform interpretations When speaking about completely monotonic or absolutely monotonic functions one cannot leave aside Laplace transforms. We briefly touch the subject below, in connection with our theme. Let F be an absolutely monotonic function on (0, ∞), and let µ and σ be admissible measures supported on [0, 1] such that Our assumption (10.1) becomes F (Lµ 1 (k)) = Lσ 1 (k) for all k ≥ 1, and a classical observation due to Carlson [11] implies that F (Lµ 1 (z)) = Lσ 1 (z) for all z ∈ C + . More precisely, Carlson's Theorem asserts that a bounded analytic function in the right half-plane is identically zero if it vanishes at all positive integers. The proof relies on the Phragmén-Lindelöf principle [34]; see also [8] or [51, §5.8] for more details. In this section, we will show some results from the interplay between the Laplace transform and functions which transform positive Hankel matrices. For point masses, the situation is rather straightforward. If µ = δ e −a for some point a ∈ [0, ∞), and F (x) = ∞ n=0 c n x n , then More generally, if µ has countable support, then the transform F [−] will yield a measure with countable support also. A strong converse to this is the following result. Proposition 10.1. Let a ∈ (0, 1) and suppose the function F : x → ∞ n=0 c n x n is absolutely monotonic on (0, ∞). The following are equivalent. (1) There exists an admissible measure µ on [0, 1] such that F (s k (µ)) = a k for all k ≥ 0. (2) F (x) = x N for some N ≥ 1, and µ = δ a 1/N . Proof. That Since c n ≥ 0, it follows that c n ν n (A) = 0 for all measurable sets A not containing λ, and all n ∈ Z + . Hence, either c n = 0, or ν n = δ λ . Moreover, ∞ n=0 c n = 1. Now, suppose c n = 0 for some n. By the above argument, we must have ν n = δ λ . Thus, Lν n (z) = ∞ 0 e −zt dν(t) n = e −λz for all z ∈ C + . Equivalently, ∞ 0 e −zt dν(t) = e −λz/n , and applying the uniqueness principle for the Laplace transform one more time gives that ν = δ λ/n . Hence c n = 0 for at most one n, say for n = N , so F (x) = x N and ν = δ λ/N . Finally, since ν = ψ * µ, we conclude that µ = δ a 1/N , as claimed. Appendix A. Two lemmas on adjugate matrices In this appendix we prove two lemmas. These allow us to establish Equation (5.2), which is key to our proof of Theorem 5.8, and they may be of independent interest. Let F denote an arbitrary field. Given a matrix M ∈ F N ×N , where N ≥ 1, and a function f : F → F, we let adj(M ) denote the adjugate matrix of M and f [M ] ∈ F N ×N the matrix obtained by applying f to each entry of M . Lemma A.1. Given a polynomial f (x) = α 0 + α 1 x + · · · + α n x n + · · · ∈ F[x] and a matrix M ∈ F N ×N , the polynomial det f [xM ] = α 0 α N −1 1 1 1×N adj(M )1 N ×1 x N −1 + O(x N ). Proof. Let M have columns m 1 , . . . , m N ; we write M = (m 1 \| · · · \|m N ) to denote this. Using the multi-linearity of the determinant, we see that det f [xM ] = ∞ i 1 ,...,i N =0 α i 1 · · · α i N x i 1 +···+i N det(m •i 1 1 \| · · · \| m •i N N ). (A.1) Observe that the only way to obtain a term where x has degree less than N − 1 is for at least two of the indices i l to be 0. The corresponding determinants are all 0 since they contain two columns equal to 1 N ×1 . For terms containing x N −1 , the only ones where the determinant does not contain two columns equal to 1 N ×1 sum to give α 0 α N −1 1 x N −1 N l=1 det(m 1 \| · · · \| m l−1 \| 1 N ×1 \| m l+1 \| · · · \| m N ). By Cramer's Rule, this sum is precisely 1 T N ×1 adj(M )1 N ×1 . We also require the following result, which we believe to be folklore. We include a proof for completeness. Note that these matrices are totally non-negative, and would be Hankel but for one entry. F [−] preserves positivity on:Domain I, ∪ N ≥1 P N (I) H + (I) µ ∈ M([0, 1]) or M([−1, 1]), ρ ∈ (0, ∞] s 0 (µ) ∈ I ∩ [0, ∞) F [−] preserves positivity on m-tuples of elements in: Domain I, ∪ N ≥1 P N (I) H + (I) µ ∈ M([0, 1]) or M([−1, 1]), ρ ∈ (0, ∞] s 0 (µ) ∈ I ∩ [0, ∞) Lemma 2. 7 . 7A rank-one N ×N matrix uu T , with entries in any field, is Hankel if and only if either the successive entries of u are in a geometric progression, or all entries but the last are 0. More precisely, the matrix uu T is Hankel if and only if IV, Theorem 3a]). If f is absolutely monotonic on [a, b), then it can be extended analytically to the complex disc centered at a and of radius b−a. Theorem 2 . 9 ( 29Bernstein [54, Chapter IV, Theorem 12a]). A function f : [0, ∞) → R is completely monotonic on 0 ≤ x < ∞ if and only if f (x) = ∞ 0 e −xt dµ(t)for some finite positive measure µ. Theorem 2 . 210 (Schoenberg, Rudin). Given a function F : (−1, 1) → R, the following are equivalent.(1) Applied entrywise, F preserves positivity on the space of positive matrices with entries in (−1, 1) of all dimensions.(2) Applied entrywise, F preserves positivity on the space of positive Toeplitz matrices with entries in (−1, 1) of all dimensions. (3) The function F is real analytic on (−1, 1) and absolutely monotonic on (0, 1). Definition 2 . 11 . 211Given a domain D ⊂ R and a function F : D → R, the function F [−] acts on the set of matrices with entries in D, by applying F entrywise: F [A] := (F (a ij )) Theorem 3.1. A function F : R → R maps M([−1, 1]) into itself when applied entrywise, if and only if F is the restriction to R of an absolutely monotonic entire function. Theorem 3.2. A function F : [0, ∞) → R maps M([0, 1]) into itself when applied entrywise, if and only if F is absolutely monotonic on (0, ∞), so non-decreasing, and 0 ≤ F (0) ≤ lim ǫ→0 + F (ǫ). Theorem 3. 3 . 3For a function F : [0, ∞) → R, the following are equivalent. ( 2 ) 2Applied entrywise, the function F preserves the set H ++ . (3) Applied entrywise, the function F sends M([0, ∞)) to itself. (4) The function F agrees on (0, ∞) with an absolutely monotonic entire function and 0 ≤ F (0) ≤ lim ǫ→0 + F (ǫ). . 2 . 2The 'if' part follows from two statements: (i) absolutely monotonic entire functions preserve positivity on all matrices of all orders, by the Schur product theorem; (ii) moment matrices from elements of M([0, 1]) have zero entries if and only if µ = aδ 0 for some a ≥ 0.Conversely, suppose the function F preserves M([0, 1]) when applied entrywise, i.e., given any Theorem 4 . 2 . 42Let F : I → R, where I := (0, ρ) and 0 < ρ ≤ ∞. Fix u 0 ∈ (0, 1) and an integer N ≥ 3, and let u := (1, u 0 , . . . , u N −1 0 ) T . Suppose F [−] preserves positivity on P 2 (I), and F [A] ∈ P N (R) for the family of Hankel matrices Definition 4 . 5 . 45For 1 ≤ k ≤ N , let P k N (I) denote the matrices in P N (I) of rank at most k. Theorem 4 . 8 . 48Fix ρ > 0 and integers N ≥ 1 and M ≥ 0, and let F (z) = N −1 j=0 c j z j + c ′ z M be a polynomial with real coefficients. The following are equivalent. Remark 4. 9 . 9As the following proof of Theorem 4.8 shows, assumption (3) can be relaxed further, by assuming F preserves positivity on a distinguished family of Hankel matrices. More precisely, it can be replaced by (3 ′ ) F [−] preserves positivity on two sequences of rank-one Hankel matrices, Proof of Theorem 4 . 8 . 48In view of Remark 4.9 and [3, Theorem 1.1], it suffices to show that (3 ′ ) =⇒ (2). Assume (3 ′ ) holds, and consider first the sequence b n ρu(b)u(b) T . If 0 ≤ M < N , then the result follows from Proposition 4.10, since the critical value is precisely C(c; z M ; N, ρ) = c −1 M . Now suppose M ≥ N . Again using Proposition 4.10, either c 0 , . . . , c N −1 and c ′ are all non-negative, or else we have that c 0 , . . . , c N −1 > 0 > c M . where µ(M, N, j) is the hook partition (M − N + 1, 1, . . . , 1, 0, . . . , 0), with N − j − 1 ones after the first entry and then j zeros, and s µ(M,N,j) is the corresponding Schur polynomial. As n → ∞, so u n → √ ρ(1, . . . , 1) T . The Weyl Character Formula in type A gives that s µ(M,N,j) (1, . . . , 1) = M j M − j − 1 N − j − 1 , and it follows that j=1 is positive semidefinite if and only if α is a non-negative integer or α ≥ N − 2. The result now follows immediately, by Lemma 2.7. Lemma 5. 1 . 1A real sequence (s k ) ∞ k=0 is the moment sequence of a positive measure on [0, ∞) if and only if the corresponding semi-infinite Hankel matrix H := (s i+j ) ∞ i,j=0 is totally non-negative. The measure is supported on [0, 1] if and only if the entries of H are uniformly bounded. Lemma 5.2 ([14, Corollary 3.5]). Let A be an N × N Hankel matrix. Then A is totally non-negative if and only if A and its truncation A (1) have non-negative principal minors. With Lemmas 5.1 and 5.2 in hand, we can now establish our characterization of positivity preservers on H ++ . Proof of Theorem 3.3. Suppose (1) holds, and let A ∈ H ++ . Then both F [A] and F [A] (1) = F [A (1) ] have non-negative principal minors, so F [A] ∈ H ++ , by Lemma 5.2. Thus (1) =⇒ (2). That (2) =⇒ (3) follows directly from Lemma 5.1. Next, suppose (3) holds and let a > 0 and b ≥ 0. Applying F [−] to the first few moments of the measure aδ √ b/ashows that F (a)F (b) ≥ F ( √ ab) 2 . By Theorem 4.1, we conclude that (4) holds. Finally, suppose (4) holds and let H ∈ H ++ N for some N ≥ 1. If every entry of H is non-zero, then F [H] is positive semidefinite, by the Schur product theorem. Otherwise, suppose H has a zero entry. Denote the entries in the first row and last column of H by s 0 , . . . , s N −1 and s N −1 , . . . , s 2N −2 , respectively. By considering 2 × 2 minors, it is easy to show that Theorem 5. 4 . 4Let ρ, N , M and F (z) = N −1 j=0 c j z j + c ′ z M be as in Theorem 4.8. The following are equivalent. (1) F [−] preserves total non-negativity for elements of H ++ N with entries in [0, ρ). (2) The coefficients c j satisfy either c 0 , . . . , c N −1 , c ′ ≥ 0, or c 0 , . . . , c N −1 > 0 and c ′ ≥ −C(c; z M ; N, ρ) −1 , where ) F [−] preserves positivity for rank-one elements of H ++ N with entries in (0, ρ). Proof. Clearly (1) =⇒ (3), and (3) =⇒ (3 ′ ), where the assertion (3 ′ ) is as in Remark 4.9. That (3 ′ ) =⇒ (2) follows from the proof of Theorem 4.8. To prove (2) =⇒ (1), first observe from Theorem 4.8 that F [−] preserves positivity on P N ([0, ρ]). Given any matrix A ∈ H ++ N with entries in [0, ρ), let B denote the N × N matrix obtained by deleting the first column and last row of A, and then adding a last row and column of zeros. Both A and B are positive semidefinite, and therefore so are F [A] and F [B]. Hence F [A] and F [A] (1) = F [A (1) ] have non-negative principal minors, since the principal minors of the latter are included in those of F [B]. Lemma 5.2 now gives the result.It is trivial that the Hadamard (entrywise) power H •α is totally non-negative for all H ∈ only if α ≥ 0. For higher dimensions, the situation is as follows. Theorem 5. 5 5([14, Theorem 5.11 and Example 5.5]). Let α ∈ R and N ≥ 2. The power function x α preserves H ++ N if and only if α is a non-negative integer or α ≥ N −2. Theorem 5. 7 . 7Suppose F : [0, ∞) → R.The following are equivalent. ( 1 ) 1Applied entrywise, the function F preserves total non-negativity on the set of all rectangular matrices of arbitrary size.(2) Applied entrywise, the function F preserves total non-negativity on the set of all real symmetric matrices of arbitrary size.(3) The function F is constant or linear. In other words, there exists c ≥ 0 such that either F (x) ≡ c, or else F (x) = cx for all x ≥ 0. Theorem 5. 8 . 8Let F (x) = ∞ n=0 c n x n be entire with real coefficients. The entrywise map F [−] preserves total non-negativity for 4 × 4 matrices if and only if F (x) ≡ c 0 with c 0 ≥ 0, or F (x) = c 1 x for all x ≥ 0 with c 1 ≥ 0. The same conclusion holds if F [−] preserves total non-negativity for symmetric 5 × 5 matrices. F ( 0 ) 14 , 014= c 0 = 0. Next, suppose for contradiction that F (x) = c m x m + O(x m+1 ), where m > 1 and c m = 0. We make use of the matrix studied in [Example non-negative. If L := lim x→0 + F (x), then det F [tA] → −L 3 ≥ 0 as t → 0 + , whence L = 0, as desired.Thus F has the form required to apply Theorem 5.8, so F (x) = c 1 x for all x ∈ [0, ∞), as required.6. Moment transformers on [−1, 1] Equipped with the one-sided result from Theorem 4.1, we now classify the functions which map the set M([−1, 1]) into M(R) when applied entrywise. The goal of this section is to prove the following strengthening of Theorem 3.1, in the spirit of Theorem 4.1. Theorem 6.1. Let F : (−ρ, ρ) → R, where 0 < ρ ≤ ∞. The following are equivalent.(1) F [−] maps the sequences u∈(0,1) M ρ ({−1, u, 1}) into M(R). (2) F [−] maps the sequences u∈(0,1) M ρ ({−1, u, 1}) into M([−1, 1]). (3) F [−] maps M ρ ([−1, 1]) into M(R).(4) F is the restriction to (−ρ, ρ) of an absolutely monotonic entire function. Lemma 6. 3 . 3If F : (−ρ, ρ) → R maps entrywise the sequences M ρ 2 ({−1, 1}) into M 2 (R), then F is locally bounded. IfF is known to be locally bounded on (0, ρ), then the set M ρ 2 ({−1, 1}) may be replaced by M ρ 2 ({−1}). p fix v 0 ∈ (0, 1) and use the truncated moment sequences in M ρ3 ({−1, v 0 }) to prove two-sided continuity of F at all points in (−ρ, 0]. Fix β ∈ [0, ρ), and for b such that 0 < b < (ρ − β)/(1 + v 0 ), let a := β + bv 0 and µ = aδ −1 + bδ v 0 .By assumption, we have thatF [−] : M ρ 3 ({−1, v 0 }) → M 3 ([−1, 1]), so there exists σ ∈ Meas + [−1, 1] such that (F (s 0 (µ)), . . . , F (s 3 (µ))) = (s 0 (σ), . . . , s 3 (σ)).If the polynomials p ± (t) := (1 ± t)(1 − t 2 ) then, ± (t) dσ ≥ 0, since p ± (t) are non-negative on [−1, 1]. Hence (4.1) gives that n−k F (−a + e x−(2k+1)h ) . (6.2) Theorem 7. 1 . 1Given u 0 ∈ (0, 1) and F : (−ρ, ρ) → R, where 0 < ρ ≤ ∞, the following are equivalent. ( 1 ) 1F [−] maps M ρ ({−1, −u 0 }) into M((−∞, 4 ((−∞, 0]). (2) F [−] maps M ρ ([−1, 0]) into M([−1, 0]). because F [−] is the composite of two operations: the map F [−], which sends M ρ ([0, 1]) into M([0, 1]), by Theorem 4.1, and entrywise multiplication by the matrix H δ −1 , which maps H µ for some measure µ to the Hankel matrix of the reflection of µ about the origin.That(2) =⇒ (1) is immediate. We now prove (1) =⇒ (3). Suppose (1) holds. Since F [H aδ 0 ] = (F (a) − F (0))H δ 0 + F (0)H δ 1 = H (F (a)−F (0))δ 0 +F (0)δ 1 ,the uniqueness in Theorem 2.4 gives that F (0) = 0. By considering only even rows and columns of Hankel matrices corresponding to moments in M ρ 4 ({−u}), M ρ 4 ({−1, 0}), and M ρ ({−1, −u 0 }), we have embeddings M ρ 2 ({u 2 }) ֒→ M ρ 4 ({−u}), M ρ 2 ({0, 1}) ֒→ M ρ 4 ({−1, 0}), and M ρ ({1, u 2 0 }) ֒→ M ρ ({−1, −u 0 }). Thus F [−] maps M ρ 2 ({u 2 }) into M 2 (R), M ρ 2 ({0, 1}) into M 2 (R), and M ρ ({1, u 2 0 }) into M(R). Theorem 4.4 now gives that F agrees with an absolutely monotonic entire function F on (0, ρ). Next, considering M ρ 2 ({−1}) gives that \|F (−a)\| ≤ F (a) for any a ∈ (0, ρ), whence F is locally bounded. In particular, F maps M ρ ({−1}) into M([−1, 0]), by Theorem 2.4. (t) dσ = (−1) n (F (s n (aδ −1 )) + F (s n+1 (aδ −1 )) = (−1) n (F ((−1) n a) + F ((−1) n+1 a)). ( 2 ) 2Applied entrywise, F maps the set of positive checkerboard matrices of all dimensions into itself. (3) F is odd and agrees on (0, ∞) with an absolutely monotonic entire function. Proposition 8 . 1 . 81Suppose F : I → R, where I = [0, ρ] and 0 < ρ < ∞. The following are equivalent. ( 1 ) 1F [−] preserves positivity on all positive Hankel matrices with entries in I. (2) F is absolutely monotonic on [0, ρ) and F (ρ) ≥ lim x→ρ − F (x). (3) F [−] preserves positivity on all positive matrices with entries in I. If, instead, I = [0, ρ) where 0 < ρ ≤ ∞, then the same equivalences hold, with(2)replaced by the requirement that F is absolutely monotonic on [0, ρ).Note the contrast with Theorem 4.1: if F [−] is required only to preserve positive Hankel matrices arising from moment sequences, then F may be discontinuous at 0, but this cannot occur here.Proof. Clearly (3) =⇒ (1). Next, suppose (1) holds and note that F is absolutely monotonic on (0, ρ), by Theorem 4.1. Consider the positive Hankel matrices  and F is right continuous at the origin. Finally, considering the first two leading principal minors of the Hankel matrix for the measure (ρ − a)δ 1 + aδ 0 , where a → ρ − , gives that F (ρ) ≥ lim a→ρ − F (a). Hence (1) =⇒(2).Finally, suppose (2) holds. We first claim that if A ∈ P N ((−∞, ρ]) then the entries of A equalling ρ form a (possibly empty) block diagonal submatrix, upon suitably relabelling the indices. = −ρ(ρ − a) 2 , so a = ρ.(8.1) Now let B A be the block-diagonal matrix with (i, j)th entry equal to 1 if a ij = ρ and 0 otherwise. If g is the continuous extension of F \| [0,ρ) to ρ, then F [A] = g[A] + (F (ρ) − g(ρ))B A ≥ 0, since both matrices are positive semidefinite. Hence (2) =⇒ (3). Finally, when I = [0, ρ), that (2) =⇒ (3) =⇒ (1) is immediate, and (1) =⇒ (2) is shown as above. Remark 8.2. A similar argument to Proposition 8.1 reveals that F [−] preserves positivity on the set {s(µ) ∈ M([0, 1]) : s 0 (µ) ∈ [0, ρ]} if and only if F is absolutely monotonic on (0, ρ) and such that 0 ≤ F (0) ≤ lim ǫ→0 + F (ǫ) and lim x→ρ − F (x) ≤ F (ρ). We next examine the case where the domain of F is a symmetric compact interval [−ρ, ρ]. The functions preserving positivity of Hankel matrices when applied entrywise are completely characterized as follows. Proposition 8 . 3 . 83Suppose F : I → R, where I = [−ρ, ρ] and 0 < ρ ≤ ∞. The following are equivalent. (1) F [−] preserves positivity on all positive Hankel matrices with entries in I. (2) F [−] preserves positivity on all positive Hankel matrices with entries in I that arise from moment sequences. (3) F is real analytic on (−ρ, ρ), absolutely monotonic on (0, ρ), and such that F (ρ) ≥ lim x→ρ − F (x) and \|F (−ρ)\| ≤ F (ρ). Proof. That (1) =⇒ (2) is immediate, while (2) =⇒ (3) follows from the extension of Theorem 3.1 given by Theorem 6.1, and the proofs of Proposition 8.1 and Lemma 6.3. Finally, if (3) holds, then (1) follows by Proposition 8.1, the Schur product theorem, and the following claim. Proposition 8. 4 . 4Given ρ ∈ (0, ∞), let I = [−ρ, ρ] and suppose F : I → R is real analytic on (−ρ, ρ), absolutely monotonic on (0, ρ), and such that the limits lim x→ρ − F (±x) both exist and are finite. If 2 ) 2then F [−] preserves positivity on the space of positive matrices with entries in I. The converse holds if F \| (−ρ,ρ) is either odd or even. Remark 8 . 5 . 85Propositions 8.3 and 8.4 indicate the existence of functions discontinuous at ±ρ which preserve positivity for Hankel matrices, but not all matrices, in contrast to the one-sided setting of Proposition 8.1. or, equivalently, supp(µ) ⊂ [−1, 1] d . In line with above, we let M(K) denote the set of all moment families of admissible measures supported on K ⊂ R d .Theorem 9.1. The map F [−] : R → R maps M([−1, 1] d ) into itself if and only if F is absolutely monotonic and entire. Proof. Any admissible measure µ on [−1, 1] pushes forward to an admissible measure µ on [−1, 1] d via the canonical embedding onto the first coordinate. If F maps M([−1, 1] d ) to itself then there exists an admissible measure σ on [−1, 1] d such that 1 j := (0, . . . , 0, 1, 0, . . . , 0), with 1 in the jth position. A multisequence of real numbers (s α ) α∈Z d + is the moment sequence of an admissible measure supported on [−1, 1] d if and only if the weighted Hankel-type kernels that F : I m → R acts entrywise on any m-tuple of N × N matrices with entries in I, so that F [−] : P N (I) m → R N ×N , F [A 1 , . . . , A m ] ij := F (a 1,ij , . . . , a m,ij ). (9.2) Lemma 9 . 2 . 92Given an integer m ≥ 1, let Y m denote the set of all y = (y 1 , . . . , y m ) T ∈ (0, 1) m such that the scalars for all α ∈ Z m + . Then the complement of Y m in (0, 1) m has zero mdimensional Lebesgue measure.Proof. LetX := {x = log y ∈ (−∞, 0) m : x ⊥ α for all α ∈ Z m \ {(0, . . . , 0)}}.The complement of X in (−∞, 0) m is a countable union of hyperplanes, and so has measure zero. The result now follows since Y m is the image of X under a smooth map.The new notion of a facewise absolutely monotonic function on [0, ρ) m plays an important role in our next result. In order to define it, recall that the truncated orthant [0, ρ) m is the truncation of a convex polyhedron, and as such, is the disjoint union of the relative interiors of its faces. These faces are in bijection with subsets of[m] := {1, . . . , m} via the mapping J → [0, ρ) J := {(x 1 , . . . , x m ) ∈ [0, ρ) m : x l = 0 for all l ∈ J}, (9.5) and this face has relative interior (0, ρ) J × {0} [m]\J . Definition 9.3. A function F : [0, ρ) m → R, where m ≥ 1 and 0 < ρ ≤ ∞, is facewise absolutely monotonic if, for each set of indices J ⊂ [m], the function F agrees on (0, ρ) J × {0} [m]\J with an absolutely monotonic function g J on (0, ρ) J . Here and henceforth, we identify without further comment (0, ρ) J and (0, ρ) J × {0} [m]\J . Theorem 9. 5 . 5Let F : [0, ρ) m → R,where m ≥ 1 and 0 < ρ ≤ ∞, and fix y = (y 1 , . . . , y m ) T ∈ Y m , as in Lemma 9.2. The following are equivalent. ( 1 ) 1F [−] maps M ρ ({1, y 1 }) × · · · × M ρ ({1, y m }) ∪ M ρ ({0, 1}) m into M(R), and F ((a 1 , . . . , a m ))F ((b 1 , . . . , b m )) ≥ F ( a 1 b 1 , . . . , a m b m ) 2for all a 1 , . . . , a m , b 1 , . . . , b m ∈ [0, ρ).(2) F [−] maps M ρ ([0, 1]) m into M([0, 1]). (3) F is facewise absolutely monotonic, and the functions {g J : J ⊂ [m]} satisfy the monotonicity condition (9.6). Theorem 9 . 6 . 96Fix ρ ∈ (0, ∞], an integer m ≥ 1 and a point y = (y 1 , . . . , y m ) T ∈ Y m , as in Lemma 9.2. For 1 ≤ l ≤ m and N ≥ 1, let u l,N := (1, y l , . . . , y N −1 l ) T , and H + l (N ) := {a1 N ×N + bu l,N u T l,N : a ∈ (0, ρ), b ∈ [0, ρ − a)}. ′ l := (1, y l , . . . , y N −1 l ) T for 1 ≤ l ≤ m. Fix some c ∈ (0, ρ) m , choose b l ∈ (0, ρ − c l ) for all l and letA l := c l 1 N ×N + b l y ′ l y ′ l T ∈ H + l (N ), so that F [A 1 , . . . , A m ] ∈ P N (R). We now use Lemma 9.2: since y ∈ Y m and N ≥ r+m m by assumption, for each β ∈ Z m+ with \|β\| ≤ r we can choose v β ∈ R N such that v β ⊥ (1, y α , y 2α , . . . , y (N −1)α ) T for all α ∈ Z m + \ {β} with \|α\| ≤ r,and (1, y β , . . . , y (N −1)β )v β = 1. An application of Taylor's theorem (similar to its use in Proposition 4.10 or [16, Proposition 2.5]) now gives that the derivative D β F (c) ≥ 0. Thus F is absolutely monotonic on (0, ρ) m , and Schoenberg's observation [41, Theorem 5.2] implies that F is the restriction to (0, ρ) m of an analytic function on D(0, ρ) m . For ease of exposition, we show this for an illustrative example; the general case follows with minimal modification. Suppose J = {1, 2, 3}, K = {1, 2}, and L = {1}. For any (x 1 , 0, 0) ∈ (0, ρ) L , we set (a 1 , a 2 , a 3 ) := (x 1 , x 2 , x 3 ) and (b 1 , b 2 , b 3 ) := (x 1 , x 2 , 0), where x 2 > 0 and x 3 > 0. By hypothesis (1), it follows that Finally, to show that (3) =⇒ (2), given positive Hankel matrices A 1 , . . . , A m arising from moment sequences in M ρ ([0, 1]), let J := {l ∈ [m] : a l,11 > 0} and K := {l ∈ [m] : a l,22 > 0}. Note that K ⊂ J ⊂ [m]. Recalling that the only Hankel matrices arising from M ρ ([0, 1]) and having zero entries are of the form H aδ 0 for some a ∈ [0, ρ), we may write F [A 1 , . . . , A m ] = (g J (a l,11 : l ∈ J) − g K (a l,11 : l ∈ K)) H δ 0 + g K [A l : l ∈ K]. (9.8) Proposition 9 . 8 . 98Suppose ρ ∈ (0, ∞] and F : I m → R, where I = [0, ρ). The following are equivalent. (1) F [−] preserves positivity on the space of m-tuples of positive Hankel matrices with entries in I. (2) F is absolutely monotonic on I m . (3) F [−] preserves positivity on the space of m-tuples of all positive matrices with entries in I. Proof. Clearly (2) =⇒ (3) =⇒ (1). Now suppose (1) holds. By Theorem 9.6, F is absolutely monotonic on the domain (0, ρ) m , and agrees there with an analytic function g : D(0, ρ) m → C. We now claim F ≡ g on I m . The proof is by induction on m, with the m = 1 case shown in Proposition 8.1. Suppose m > 1, and let c = (c 1 , . Definition 9. 9 . 9Given K ⊂ R and ρ ∈ (0, ∞], let M ρ (K) be the collection of possibly truncated moment sequences for all measures µ ∈ Meas + (K), where each sequence is truncated prior to the first moment of µ that lies outside (−ρ, ρ), or is not truncated if no such moment exists.Clearly, if ρ = ∞, then M ρ (K) = M ρ (K), while if ρ is finite and K ⊂ [−1, 1], then M ρ (K) = M ρ (K).Next is a more complex example, which occurs in the following theorem; see particularly Step 5 of its proof.Example 9.10. If K = {−1, v, 1} for v > 1, and ρ < ∞, then M ρ (K) consists of M ρ ({−1, 1}) together with truncated moment sequences of measures with positive mass at v. If µ = aδ −1 + bδ 1 + cδ v with a, b ≥ 0 and c > 0, then the moments of µ are unbounded, and M ρ (K) contains the truncated moment sequence (s 0 (µ), . . . , s n−1 (µ)), where n is the smallest positive integer such that \|a(−1) n + b + cv n \| ≥ ρ. Theorem 9 . 11 . 911Suppose F : I m → R, where m ≥ 1 and I = (−ρ, ρ) with 0 < ρ ≤ ∞. The following are equivalent. (1) For some δ > 0, the function F , when applied entrywise, maps M ρ ([−1, 1+δ]) m into the set of possibly truncated moment sequences of measures on R. (2) For every δ > 0, the function F , when applied entrywise, maps M ρ ([−1, 1+δ]) m into the set of possibly truncated moment sequences of measures on R. (3) Applied entrywise, the function F maps P N (I) m into P N (R) for any N ≥ 1. (4) The function F is absolutely monotonic on [0, ρ) m and agrees on I m with an analytic function. M ρ v := M ρ ({−1, v, 1}) and M ρ [v] := s 1 ∈{−1,0,1},s 2 ∈{−v,0,v} M ρ ({s 1 , s 2 }). Fp [−] : (M ρ [u 0 ] ) m ∪ M ρ v 1 × · · · × M ρ vm → M([−1, 1]) for all v 1 , . . . , v m ∈ (−1, 1). (9.9) Step 2. Next, we show that F is continuous on (−ρ, ρ) m . The first objective is to show continuity of F inside each open orthant of (−ρ, ρ) m . Given non-zero scalars c 1 , . . . , c m with \|c 1 \|, . . . , \|c m \| < ρ, and any sequence {(v 1,n , . . . , v m,n ) : n ≥ 1} ⊂ R m converging to the origin, let a l,n := \|c l \| + sgn(c l )u 0 u 0 − u 3 0 v l,n and µ l,n := a l,n δ sgn(c l ) + \|v l,n \| u 0 − u 3 0 δ − sgn(v l,n )u 0 (9.10) for l = 1, . . . , m. Note that, for all sufficiently large n, the sequence s(µ l,n ) ∈ M ρ [u 0 ] . We now follow the proof of Proposition 6.4. Suppose that F [H µ 1,n , . . . , H µm,n ] = H σn for some admissible measure σ n ∈ Meas + ([−1, 1]), for every n ≥ 1. The polynomials p ± (t) := (1 ± t)(1 − t 2 ) are non-negative on [± (t) dσ n ≥ 0, so F (s 0 (µ l,n ) m l=1 ) − F (s 2 (µ l,n ) m l=1 ) ≥ \|F (s 1 (µ l,n ) m l=1 ) − F (s 3 (µ l,n ) m l=1 )\| . (9.11) k F (\|c\| + η v,\|c\| e −(x+2ks)v ) n−k F (c + η v,c e −(x+(2k+1)s)v ) ,where we note that η v,c = η v,\|c\| , and that all arguments of F lie in (−ρ, ρ) m by the restriction on s. Note that we use the fact that our test set contains M ρ ({−1, v, 1}) for v ∈ (1, 1 + ǫ) here, and only here, in this proof. Now setting H v,c (x) := F (c + η v,c e −xv ), dividing both sides of this inequality by s n , and then taking s → 0 + , it follows that d n dx n H F (s k (µ)) = s k (σ) for all k ≥ 0.(10.1) By the change of variable x = e −t , we can push forward the restriction of the measure µ to (0, 1] to a measure µ 1 on [0, ∞), and similarly for σ. Thus, with the possible loss of is a complex-analytic function in the open half-plane C + := {z ∈ C : ℜz > 0}. F (Lµ(z)) = F (e −az ) = ∞ n=0 c n e −anz = Lσ 1 (δ e −an . ( 2 )e 2=⇒ (1) is clear. Now suppose (1) holds. Setting ψ(t) := − log t, −kt dν(t) = Lν(k) for all k ≥ 0,where ν := ψ * µ is the push-forward of µ under ψ. If a = e −λ for some λ > 0, then, by assumption, F (Lν(k)) = e −λk for all k ≥ 1.and, by Carlson's Theorem,F (Lν(z)) = e −λz for all z ∈ C + . (10.2)In view of Bernstein's theorem, Theorem 2.9, the function Lν is completely monotonic on [0, ∞). Now, since the composition of an absolutely monotonic function and a completely monotonic function is completely monotonic, so z → (Lν(z)) monotonic on [0, ∞) for all k ∈ Z + . Thus, by another application of Bernstein's theorem, there exists an admissible measure ν k on [0, ∞) such that(Lν(z)) k = ∞ 0 e −zt dν k (t)for all z ∈ C + .Using the above expression, we can rewrite (10.2) asF (Lν(z)) = ∞ n=0 c n (Lν n )(z) = L ∞ n=0 c n ν n (z) = e −λz = (Lδ λ )(z),and, by the uniqueness principle for Laplace transforms, we conclude that∞ n=0 c n ν n = δ λ . Now, let A be any measurable subset of [0, ∞) that does not contain λ. Then, ∞ n=0 c n ν n (A) = δ λ (A) = 0. Lemma A. 2 . 2Suppose M ∈ F N ×N has rank N − 1. If u spans the null space of M T , and v spans the null space of M , then A = adj M is a non-zero scalar multiple of vu T .Proof. That A = 0 follows by considering the rank of M . Since det M = 0, we have that AM = 0 and M A = 0. After taking the transpose, the first identity implies that the rows of A are multiples of u T ; the second identity implies immediately that the columns of A are multiples of v. This gives the result.We may now show that det M 4 = −57168 α 0 α 2 1 α 2 x 4 + O(x 5 Table 1 . 1 . 11The one-variable case. Table 1 . 2 . 12The multivariable case. 1.4. Acknowledgements. The authors extend their thanks to the International Centre for Mathematical Sciences, Edinburgh, where the major part of this work was carried out. D.G. is partially supported by a University of Delaware Research Foundation grant, by a Simons Foundation collaboration grant for mathematicians, and by a University of Delaware Research Foundation Strategic Initiative grant. A.K. is partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017, MATRICS grant MTR/2017/000295, and SwarnaJayanti Fellowship grants SB/SJF/2019-20/14 and DST/SJF/MS/2019/3 from SERB and DST (Govt. of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), by a Tata Trusts gravel grant, and by a Young Investigator Award from the Infosys Foundation. We are grateful to the referees for valuable comments and enriching bibliographical indications. the map Ψ is a bijection onto the positive Hankel matrices with uniformly bounded entries; (3) restricted to Meas + ([0, 1]), the map Ψ is a bijection onto the positive Hankel matrices with uniformly bounded entries, such that removing the first column still yields a positive matrix.Proof. The first assertion is classical; for example, see Akhiezer's book [1, Theorems 2.1.1, 2.6.4, and 2.6.5]. For the last two statements, we simply remark that for an admissible measure µ, s 2n (µ) = [−1,1] x 2n dµ + R\[−1,1] where µ is any non-negative measure with the first 2N − 2 moments equal to (s 0 , . . . , s 2N −3 ). (For details, see Akhiezer's book [1, Theorem 2.6.3].) Furthermore, in order to show continuity in Theorem 4.2 we only required 2 × 2 submatrices, of the form (4.4) or of rank one. Moreover, every matrix in P 2 (R) is a truncated moment matrix. These observations show that Theorem 4.4 is equivalent to Theorem 4.2. We now prove Theorem 4.1, with the help of Theorem 4.4. Meas + ([0, 1]) with s 0 (µ) < ρ. If µ = aδ 0 for some a ≥ 0 then (2) is immediate; henceforth we will assume H µ has no zero entries, where H µ is as defined in (2.1). Now, F [H µ ] is positive, by the Schur product theorem and the fact that the only moment matrices arising from elements of M ρ ([0, 1]) with zero entries come from M ρ ({0}). Clearly F [s(µ)] is uniformly bounded, hence comes from a unique measure σ supported on [−1, 1], by Theorem 2.4. Recalling Definition 2.5, we have thatProof of Theorem 4.1. Clearly (2) =⇒ (1). Next, assume (3) holds, and suppose µ ∈ n by the hypotheses and Theorem 2.8. Note that F [H µ ] (1) is positive, by the above computation and Theorem 2.4, since µ is supported on [0, 1]. By the same result, σ ∈ Meas + ([0, 1]), which gives 9.3. Transformers of moment-sequence tuples: the general case. Having resolved the characterization problem for functions defined on the positive orthant, we now work over the whole of R m . This requires us to consider admissible measures which may have support outside [−1, 1]. For such measures, the mass no longer dominates all moments, and so we include in our test sets truncations of the corresponding moment sequences, whereas for measures supported in [−1, 1], the full moment sequence lies in the test set. More precisely, we have the following definition. ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR (A. Belton) Department of Mathematics and Statistics, Lancaster University, Lan- By Lemma A.1, det M 4 has no constant, linear, or quadratic term. Moreover, since the matrix M has rank 3 and the vectors v = (6, −11, 6, −1) and u = (46, −59, 18, −1) span the null spaces of M and M T , respectively, Lemma A.2 gives that adj(M ) is equal to cvu T for some non-zero c ∈ R. The cubic term in det M 4 equals c 1 T v u T 1 α 0 α 3 1 x 3 , by Lemma A.1, and this vanishes becauseFinally, we compute the coefficient of the quartic term; we need to examine all the terms in (A.1) that arise from quadruples (i 1 , i 2 , i 3 , i 4 ) which sum to 4. Terms with indices of the form (4, 0, 0, 0), (3, 1, 0, 0), and (2, 2, 0, 0), and their permutations, are zero since the determinants contain two identical columns. We are therefore left with quadruples of the form (2, 1, 1, 0) and (1, 1, 1, 1). The term corresponding to (1, 1, 1, 1) is zero since det M = 0, as one can see as M does not have full rank. Thus the only non-zero quartic terms arise from one of the twelve permutations of the quadruple (2, 1, 1, 0). Therefore det M 4 = k α 0 α 2 1 α 2 x 4 + O(x 5 ), and to find the constant k, we compute all twelve determinants. The sum of these determinants is −57168, as claimed. The classical moment problem and some related questions in analysis. N I Akhiezer, Kemmer. Hafner Publishing CoNew YorkN.I. Akhiezer. The classical moment problem and some related questions in analysis. Translated by N. Kemmer. Hafner Publishing Co., New York, 1965. Real submanifolds in complex space and their mappings. 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[ "Darboux-Moutard transformations and Poincare-Steklov operators ", "Darboux-Moutard transformations and Poincare-Steklov operators ", "Darboux-Moutard transformations and Poincare-Steklov operators ", "Darboux-Moutard transformations and Poincare-Steklov operators " ]	[ "R G Novikov [email protected] ", "I A Taimanov [email protected] \nSobolev Institute of Mathematics\nNovosibirsk State University\n630090, 630090Novosibirsk, NovosibirskRussia, Russia\n", "\nCentre de Mathématiques Appliquées\nCentre de Mathématiques Appliquées ofÉcole Polytechnique. † CNRS (UMR 7641\nÉcole Polytechnique\n91128PalaiseauFrance\n", "R G Novikov [email protected] ", "I A Taimanov [email protected] \nSobolev Institute of Mathematics\nNovosibirsk State University\n630090, 630090Novosibirsk, NovosibirskRussia, Russia\n", "\nCentre de Mathématiques Appliquées\nCentre de Mathématiques Appliquées ofÉcole Polytechnique. † CNRS (UMR 7641\nÉcole Polytechnique\n91128PalaiseauFrance\n" ]	[ "Sobolev Institute of Mathematics\nNovosibirsk State University\n630090, 630090Novosibirsk, NovosibirskRussia, Russia", "Centre de Mathématiques Appliquées\nCentre de Mathématiques Appliquées ofÉcole Polytechnique. † CNRS (UMR 7641\nÉcole Polytechnique\n91128PalaiseauFrance", "Sobolev Institute of Mathematics\nNovosibirsk State University\n630090, 630090Novosibirsk, NovosibirskRussia, Russia", "Centre de Mathématiques Appliquées\nCentre de Mathématiques Appliquées ofÉcole Polytechnique. † CNRS (UMR 7641\nÉcole Polytechnique\n91128PalaiseauFrance" ]	[]	Formulas relating Poincare-Steklov operators for Schrödinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary. * The work was supported by the French-Russian grant (RFBR 17-51-150001 NCNI a/PRC 1545 CNRS/RFBR) and done during the visit of the second author (I.A.T.) to	10.1134/s0081543818060160	[ "https://export.arxiv.org/pdf/1808.03236v1.pdf" ]	119,168,752	1808.03236	dedd5841036a5419613fcd7ee55689120fec9de6	Darboux-Moutard transformations and Poincare-Steklov operators * 9 Aug 2018 R G Novikov [email protected] I A Taimanov [email protected] Sobolev Institute of Mathematics Novosibirsk State University 630090, 630090Novosibirsk, NovosibirskRussia, Russia Centre de Mathématiques Appliquées Centre de Mathématiques Appliquées ofÉcole Polytechnique. † CNRS (UMR 7641 École Polytechnique 91128PalaiseauFrance Darboux-Moutard transformations and Poincare-Steklov operators * 9 Aug 2018To S.P. Novikov on the 80th birthday Formulas relating Poincare-Steklov operators for Schrödinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary. * The work was supported by the French-Russian grant (RFBR 17-51-150001 NCNI a/PRC 1545 CNRS/RFBR) and done during the visit of the second author (I.A.T.) to Investigations of inverse problems for two-dimensional Schrödinger operators at a given energy level were initiated within the framework of the theory of solitons by S.P. Novikov and his scientific school. In particular, in [1] the spectral data were introduced for the two-dimensional periodic Schrödinger operator, in a magnetic field, which is finite-gap at one energy level, and the inverse problem of reconstructing the operator from these data was solved; in [2,3], in terms of these data, potential operators were singled out and there were derived the evolutionary equations (the Novikov-Veselov equations) which preserve this class of operators and their spectra at the given energy level, and theta-functional formulas for solving the equations were obtained; in [4] the first results on the inverse scattering problem at the negative energy level were obtained and for the first time in the theory of inverse problems the methods of the theory of generalized analytic functions were used. In this paper we develop an approach to direct and inverse problems, for the two-dimensional Schrödinger operator, based on the Moutard transformation. Since the Darboux transformation is a one-dimensional reduction of the Moutard transformation, our results on two-dmensional operators from §2 are extended to the case of one-dimensional Schrödinger operators (see §3). ∂ v ∂ w θ + uθ = 0 and is determined by a solution of the initial equation [5]. We consider its special reduction for which the variables v and w are complex-conjugate and after renormalizations the second order equation reduces to the Schrödinger equation Hϕ = −∆ϕ + uϕ = 0,(1) where ∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is the Laplace operator on the two-plane R 2 . Let us take a solution ω of (1) and construct by using this solution the new Schrödinger operator H = −∆ + u with the potential u = u − 2∆ log ω = −u + 2 ω 2 x + ω 2 y ω 2 .(2) Is is easy to show by straightforward computations that if a function ϕ satisfies (1), then the function θ, which is determined by the relations (ωθ) x = −ω 2 ϕ ω y , (ωθ) y = ω 2 ϕ ω x ,(3) satisfies the equation Hθ = −∆θ + uθ = 0,(4) obtained from (1) by the Moutard transformation determined by the initial solution ω. We note that the relations (3) determine θ up to summands of the form C ω , C = const, the function ω −1 = 1 ω satisfies the equation H 1 ω = 0 and determines the inverse Moutard transformation from H to H: u ω −→ u = u − 2∆ log ω 1/ω −→ u = u − 2∆ log 1 ω = u + 2∆ log ω. For a potential u = u(x) which depends on one variable the Moutard transformation reduces to the Darboux transformation [6]. Let H 1 be a one-dimensional Schrödinger operator H 1 = − d 2 dx 2 + u and ω 1 be its eigenfunction: H 1 ω 1 = Eω 1 , E 0 = κ 2 . The Moutard transformation, of the two-dimensional Schrödinger operator, determined by the solution ω(x, y) = e κy ω 1 (x) of (1), maps the potential of the operator into u(x) = u(x) − 2∆ log(e κy ω 1 (x)) = u − 2 d 2 dx 2 log ω 1 (x). Therewith the corresponding one-dimensional Schrödinger operator H 1 is transformed into the one-dimensional operator H 1 = − d 2 dx 2 + u. This transformation is called the Darboux transformation. Its action on eigenfunctions takes the following form. Let H 1 ϕ 1 = Eϕ 1 , ϕ 1 = ϕ 1 (x), E = µ 2 . We put ϕ = e µy ϕ 1 (x). It is clear that Hϕ = 0 and, by (3), we get the Moutard transformation of ϕ in the form θ = e µy θ 1 (x), were the second of relations (3) gives the Darboux transformation of eigenfunctions: θ 1 = 1 µ + κ d dx − d log ω 1 dx ϕ 1 ,(6)H 1 θ 1 = − d 2 θ 1 dx 2 + uθ 1 = Eθ 1 , E = µ 2 , and the first one gives its inversion ϕ 1 = 1 κ − µ d dx + d log ω 1 dx θ 1 .(7) In this case, since the image of the transformation is sought in the form (5), the transformation of eigenfunctions becomes single-valued, because the addition of summands of multiple 1 ω does not preserve the form (5) (for µ = κ). The Darboux transformation was repeatedly used and was often rediscovered (see, for example, [7]) for solving problems of mathematical physics and the spectral theory [8,9] (see also the review [10]). The Moutard transformation and its extension for solutions of the Novikov-Veselov equations [11,12] in recent years has been applied to the construction of the first examples of two-dimensional Schrödinger operators with fast decaying potential and with a nontrivial kernel [12] and blowing-up solutions of the Novikov-Veselov equation with regular initial data [13,14] (see the numerical analysis of the negative discrete spectrum and its dynamics for these examples in [15]), to the construction of explicit examples of two-dimensional potentials of Wigner-von Neumann type [16]. In [17], the action of the Moutard transformation on the Faddeev eigenfunctions at the zero energy level was described, and in [18], using this transformation, Faddeev's eigenfunctions at the zero energy level for multipoint delta-like potentials were found. In [19,20], it was established a relation of the generalized Moutard transformation for two-dimensional Dirac operators [21] to the conformal geometry of surfaces in three-and four-dimensional spaces, and with this were constructed blowing-up solutions of the modified Novikov-Veselov equation with regular initial data [22,23]. A generalization of the Moutard transformation to the case of generalized analytic functions, in particular, gave an approach to constructing the theory of generalized analytic functions with contour poles [24,25,26,27] and also allowed to construct a Moutard-type transformation for the conductivity equation [28]. Poincare-Steklov operators Let in the domain D with the boundary ∂D there is given an elliptic differential equation Lψ = Eψ.(8) We single out two boundary conditions, each of which, as a rule, completely determines the solution of the equation. Then the Poincare-Steklov operator, by definition, takes the value of one boundary condition into the value of another condition. Let us consider the most well-known particular cases of Poincare-Steklov operators. Let the elliptic equation (8) be given by a linear differential secondorder expression L. Then 1. if E is not an eigenvalue of the problem (8) with the Dirichlet condition ψ\| ∂D = 0, then the boundary data ψ\| ∂D determine the solution of (8) uniquely and we define the DN (Dirichlet-to-Neumann) operator, which takes the values of ψ on the boundary to the values of the derivatives of ψ along the exterior normal ν to the boundary (the data of the Neumann problem): DN : ψ\| ∂D −→ ∂ψ ∂ν \| ∂D ;(9) 2. if E is not an eigenvalue of the problem (8) with the Neumann condition ∂ψ ∂ν \| ∂D = 0, then the ND (Neumann-to-Dirichlet) operator is defined: N D : ∂ψ ∂ν \| ∂D −→ ψ\| ∂D ;(10) 3. the above operators are special cases of the RR (Robin-to-Robin) operator, which in the general case relates mixed boundary conditions (the Robin conditions). If E is not an eigenvalue of the problem (8) with the boundary condition cos α ψ − sin α ∂ψ ∂ν \| ∂D = 0, then the RR operator maps the boundary data Γ α ψ = cos α ψ − sin α ∂ψ ∂ν \| ∂D to the boundary data Γ α−π/2 ψ: RR : Γ α ψ −→ Γ α−π/2 ψ. As particular cases, we get the DN operator for α = 0 and the ND operator for α = π 2 . The action of the Moutard transformation on Poincare-Steklov operators We assume that equation (1) holds in a bounded simply-connected two-dimensional domain D ⊂ R 2 with a smooth boundary ∂D and that u is a regular function on D ∪ ∂D. Let H and u be the operator and the potential from (4), i.e., the Moutard transformations of H and u, determined by a solution ω of (1) via formulas (2). For equation (1) For solutions ϕ of (1) we consider also the following boundary data on ∂D: Γ τ ω ϕ = ω ϕ ω τ = ϕ τ − ω τ ω ϕ(11) and Γ ν ω ϕ = ω ϕ ω ν = ϕ ν − ω ν ω ϕ,(12) where ω is the fixed solution of (1) in D, ν is the outer normal to ∂D, τ is the path-length parameter on ∂D, which grows in the direction of ν ⊥ = (−ν 2 , ν 1 ) with ν = (ν 1 , ν 2 ); the lower indices τ and ν denote the derivations in τ and along the normal ν. For simplicity, we assume that ω has no zeroes on ∂D. The important observation is that relations (3) on ∂D can be rewritten as Γ ν ω −1 θ = −Γ τ ω ϕ,(13)Γ τ ω −1 θ = Γ ν ω ϕ.(14) Lemma 1 Assuming that 0 / ∈ σ D (H), the following formulas hold: Ker Γ τ ω = {cω : c ∈ C}, Im Γ τ ω = {f : ∂D ω −1 f dτ = 0}, where Γ τ ω is considered as an operator on solutions of (1). Lemma 1 follows from the definition of Γ τ ϕ by (11) and the fact that under our assumptions the Dirichlet problem for (1) is uniquely solvable. Lemma 2 Assuming that 0 / ∈ σ D ( H), we have Ker Γ ν ω = {cω : c ∈ C},(15)Im Γ ν ω = {f : ∂D ωf dτ = 0}, where Γ ν ω is considered as an operator on solutions of (1). Remark 1. The condition 0 / ∈ σ D ( H) we understand in the sense that the Dirichlet problem for (4) is uniquely solvable. That may be essential if ω has zeroes, on D, implying singularities of u. Lemma 2 follows from relation (14), the fact that relations (3) determine θ from ϕ up to summands of the form const · ω −1 and ϕ from θ up to summands of the form const · ω, and from Lemma 1 applied to equation (4). For equation (1) and boundary data (11) and (12) we consider the following Poincare-Steklov operators Λ u,ω and Λ −1 u,ω : Λ u,ω : Γ τ ω ϕ → Γ ν ω ϕ, 0 / ∈ σ D (H)(16) and Λ −1 u,ω : Γ ν ω ϕ → Γ τ ω ϕ, 0 / ∈ σ D ( H).(17) Proposition 1 Assuming that 0 / ∈ σ D (H), the operators Φ u and Λ u,ω are related by the following formulas: cases where ω is not a lower index), ω −1 , and ω ν denote the operators of multiplication by the corresponding functions on ∂D and Λ u,ω Γ τ ω ϕ = (Φ u ωIω −1 − ω ν Iω −1 )Γ τ ω ϕ, Φ u ϕ = (Λ u,ω Γ τ ω + ω ν ω −1 )ϕ, where ω (in theIω −1 f (τ ) = τ 0 ω −1 (t)f (t) dt for f ∈ Im Γ τ ω . Proposition 1 follows from the definitions given by formulas (9), (11), and (16), and from straightforward computation. Proposition 2 If 0 / ∈ σ N (H) ∪ σ D ( H) , the operators Φ −1 u and Λ −1 u,ω are related as follows: Λ −1 u,ω Γ ν ω ϕ = Γ τ ω Φ −1 u Id − ω ν ω Φ −1 u −1 Γ ν ω ϕ, where Id is the identity operator. Therewith, Ker Id − ω ν ω Φ −1 u = {cω ν : c ∈ C} and the inverse operator Id − ων ω Φ −1 u −1 is defined on Im Γ ν ω up to summands of the form cω ν , c ∈ C. Proposition 2 follows from formulas (10), (11), (12), (15), and (17), from the representation Γ ν ω ϕ = Id − ω ν ω Φ −1 u ϕ ν , and from straightforward computations. In the assumptions of this section on D and ω we have the following result. Theorem 1 Let 0 / ∈ σ D (H) ∪ σ D ( H) . Then the following formuals hold: (13) and (14) and from Lemmas 1 and 2 which describe the domains of definition of the operators Λ u,ω , Λ −1 u,ω , Λ u,ω −1 , and Λ −1 u,ω −1 under the assumption of Theorem 1. Propositions 1 and 2 and Theorem 1 provide ways to find the operators Φ u and Φ −1 u from Φ u and Φ −1 u and from the restrictions of ω and ω ν onto ∂D. In this case it is required only the invertibility of operators acting on functions on ∂D. In this sense these methods are essentially simpler than the direct reconstruction of Φ u and Φ −1 u from u, when it is necessary to invert an operator acting on functions defined on the whole domain D. This effect becomes quite obvious for the Darboux transformation that we are demonstrating in the next section. Λ u,ω −1 = −Λ −1 u,ω , Λ −1 u,ω −1 = −Λ u,ω . Theorem 1 follows from relations The action of the Darboux transformation on Poincare-Steklov operators Let us consider the Schrödinger equation H 1 ψ = − d 2 dx 2 + u ψ = Eψ, E = µ 2(18) on the interval D =]a, b[⊂ R and the Schrödinger equation H 1 ψ = − d 2 dx 2 + u ψ = E ψ,(19) which is obtained from (18) by using the Darboux transformation determined by the solution ω 1 of the equation H 1 ω 1 = κ 2 ω 1 .(20) We assume that u(x) is a regular function of the closed interval [a, b] = D ∪ ∂D. For an equation of type (18) we consider the operators Q u and Q −1 u such that Q u : ψ\| ∂D −→ dψ dx \| ∂D ,(21)Q −1 u : dψ dx \| ∂D −→ ψ\| ∂D ,(22) i.e., the DN and ND operators of the form (9) and (10), where, for simplicity, the derivation along the out normal ν is replaced by the derivation in x. Let σ D (H 1 ) and σ N (H 1 ) define the spectra of the operators defined by H 1 and the Dirichlet and Neumann conditions, respectively. For solutions ψ of (18) we also consider the following boundary condition on ∂D: Γ x ω1 ψ = ω 1 ψ ω 1 x = ψ x − ω 1,x ω 1 ψ, where ω 1 is the given solution of (20) which determines the Darboux transformation. We assume that ω 1 has no zeroes on ∂D. The relations (6) and (7) on ∂D can be rewritten as follows: 1 κ − µ Γ x ω −1 1 ψ = ψ,(23)ψ = 1 κ + µ Γ x ω1 ψ.(24) Lemma 3 Assuming that 0 / ∈ σ D ( H 1 ) and κ = ±µ, we have Ker Γ x ω1 = 0, Coker Γ x ω1 = 0, where Γ x ω1 is considered as an operator on solutions of (18). Remark 2. The condition 0 / ∈ σ D ( H 1 ) is understood in the sense similar to Remark 1. Lemma 3 follows from (7), (24), and the two-dimensionality of the space of functions on ∂D and the spaces of solutions of (18) and (19). Theorem 2 Assuming that 0 / ∈ σ N (H 1 ) ∪ σ D ( H 1 ) and κ = ±µ, the following formula is valid: Q u = − ω 1,x ω 1 + (κ 2 − µ 2 )Q −1 u Id − ω 1,x ω 1 Q −1 u −1 ,(25) where ω1,x ω1 is the operator of multiplication by the corresponding function on ∂D. The formula (25) explicitly specifies the transformation of the DN and ND operators due to the Darboux transformation. Such formulas can be used, in particular, for testing the algorithms for reconstructing the potential from measurements at the boundary. Proof of Theorem 2. We have the representation Γ x ω1 ψ = Id − ω 1,x ω 1 Q −1 u ψ x .(26) It follows from Lemma 3 and (26) that the operator Id− ω1,x ω1 Q −1 u is invertible. Further, using (21) and (22), we rewrite the relations (23) and (24) in the form Q u + ω 1,x ω 1 ψ = (κ − µ)ψ,(27)ψ = 1 κ + µ Id − ω 1,x ω 1 Q −1 u ψ x .(28) With the help of (21) and (28), we get ψ x = (κ + µ) Id − ω 1,x ω 1 Q −1 u −1 ψ, ψ = (κ + µ)Q −1 u Id − ω 1,x ω 1 Q −1 u −1 ψ.(29) Formula (25) follows from (27) and (29) after equating the expressions for ψ. Theorem 2 is proved. Example. Let us consider the operator H = − d 2 dx 2 on the interval ]a, b[ with 0 < a < b and its Darboux transformation determined by ω 1 = x. We have u = 0, u = 2 x 2 . Let us take µ = κ, where κ = 0. A general solution of the equation Hψ = −ψ ′′ = µ 2 ψ(30) has the form ψ = α cos µx + β sin µx. Therefore Q 2/x 2 = 1 µ −µ 2 + 1 x 1 x − Q 0 µ Q 0 − 1 x −1 = − 1 x − µ 2 Q 0 − 1 x −1 , which is a particular case of (25). we consider the operator Φ u = DN of the form (9) with E = 0 and the operator Φ −1 u = N D of the form (10) with E = 0, where u is the potential from (1). Let σ D (H) and σ N (H) denote the spectra of the operators defined by H = −∆ + u and the Dirichlet and Neumann boundary conditions, respectively. DN operator for the problem (30) takes the formQ 0 = BA −1 = µ sin µ(b − a) − cos µ(b − a) 1 −1 cos µ(b − a) .It is clear that sin µ(b − a) = 0 if and only if µ 2 does not belong to the spectrum of the Dirichlet problem for H = − d 2 dx 2 . The Darboux transformation for solutions takes the form The Moutard transformation constructs from solutions of a second order equation of the type ∂ v ∂ w ϕ + uϕ = 0 solutions ϑ of another second order equation of the same type1 Preliminary facts 1.1 Darboux-Moutard transformations The Schrödinger equation in a periodic field and Riemann surfaces. B A Dubrovin, I M Krichever, S P Novikov, Sov. Math. Dokl. 17Dubrovin, B.A., Krichever, I.M., and Novikov, S.P.: The Schrödinger equation in a periodic field and Riemann surfaces. Sov. Math. Dokl. 17 (1976), 947-952. 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[ "Stability of JSQ in queues with general server-job class compatibilities", "Stability of JSQ in queues with general server-job class compatibilities" ]	[ "James Cruise ", "Matthieu Jonckheere ", "Seva Shneer " ]	[]	[]	We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is rooted to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We provide a generic condition for stability of this model. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, e.g., complete graphs.	10.1007/s11134-020-09656-w	[ "https://arxiv.org/pdf/2001.09921v2.pdf" ]	212,647,280	2001.09921	cf26973581e701a8e7e3b5a1501beae7c87504c9	Stability of JSQ in queues with general server-job class compatibilities arXiv:2001.09921v2 [math.PR] 10 Mar 2020 March 11, 2020 James Cruise Matthieu Jonckheere Seva Shneer Stability of JSQ in queues with general server-job class compatibilities arXiv:2001.09921v2 [math.PR] 10 Mar 2020 March 11, 2020 We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is rooted to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We provide a generic condition for stability of this model. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, e.g., complete graphs. Introduction Load balancing schemes involving various types of load information have become absolutely essential for the quality of service in many important modern areas of applications: call centers, commercial server farms, scientific computing, vehicles systems, data centres, and others. In the last decades, many theoretical results have focused on the case of parallel servers (i.e. the complete graph model in our setting, see below). In particular, a lot of attention has been given to mean-field type results for complete graphs models, and schemes like join the shortest queue (JSQ) among n available or join the shortest of d among n queues (JSQ(d)), where n is large, started with the seminal work of [10,16] and complemented by several papers. (This last scheduling is also known as power-of-d). Transient functional law of large numbers and propagation of chaos for JSQ and JSQ(d) have been obtained for instance in [8,13] for FIFO scheduling. For general service time distributions, the results are scarcer. For service time distributions with decreasing hazard rate and FIFO service discipline, propagation of chaos properties and asymptotic behaviour of the number of occupied servers were obtained for the JSQ(d) policy in [3]. In [6], the convergence of the mean-field limit of the join-the-idle-queue (JIQ) policy in the stationary regime was proved under light traffic conditions. More recently, [15] obtains mean-field limit for the JIQ, and [11] computes the diffusive limit in the Halfin-Whitt regime for a class of policies of which JIQ and JSQ(d) policies are special cases. Interestingly they show that JIQ is optimal at this diffusive scale. For the JSQ policy, the large-server heavy-traffic limit was derived in [5]. Different scalings (asymptotic relations between number of servers, loads, buffer sizes) were considered for instance in [9]. We consider here a spatial generalization of these models where independent Poisson streams arrive to edges of a hypergraph and are routed (either statically or dynamically) to one of the vertices corresponding to the edge. The well-known case of JSQ(d) can be retrieved simply by considering a hypergraph where every set of d vertices forms an edge (i.e., a complete dhypergraph). It is also closely related to a model introduced for instance in [12]. In their context, each node has an associated arrival process and upon arrival a customer associated with vertex i considers the queue lengths of i and at all its neighbours. It is then allocated to be served by the server at a vertex with the smallest queue length from all those examined. This model is also a special case of the one considered here, as we can construct a hypergraph with n edges, each edge representing a node in the original graph and all the neighbours of that node in the original graph. Then, arrivals on that edge are equivalent to arrivals at the associated node in the original graph. Note that a further extension to this model is considered in [4], where a random subset of the neighbours is considered when making the decision about routing rather than the full neighbourhood. Again we can create an equivalent hypergraph model by introducing an edge for each random combination which might be considered. In the models of [12] and [4] stability is trivial as the state of the system is always dominated, at least in the sense of the maximal queue size, by that in a system where arrivals at a specific node have to be served by the server at that node. The authors are thus interested in the occupancy measures in some growing and/or random topologies. In our more general model stability is non-trivial and we thus study it, in fixed topologies. It is also worth noting that our model is equivalent to a bipartite graph between customers arrival processes and servers (i.e. customer arrival processes connected by edges to the servers that can serve the arriving jobs). These models have been referred to as skill-based systems and studied by several papers (see, e.g., [2], [1] and references therein. The load balancing studied in the aforementioned papers however is join the shortest workload (JSW), which has (for a specific state description) a product form stationary measure, and thus stability condition is easily derived. This is not the case for JSQ networks. Furthermore, even if the JSW discipline has been shown to have the largest stability region (and, in particular, larger than that of the JSQ discipline) in highly symmetric models, like multi-server queues or networks of these queues, (see [7,14]), it is not clear that it can be generalized to asymmetric networks topologies. The article is organized as follows. In Section 2, we precisely define our load balancing model on a hypergraph. In Section 3, we derive the stability conditions of static allocations and show that they do not necessarily correspond to a trivial rate conservation condition. In Section 4, we present the main contribution of our paper. Namely, we show that the JSQ load balancing on a hypergraph ensures stability if and only if there is a stable static allocation. Model Let G = (V, E) be a hypergraph with vertices V and edges E, where each e ∈ E is a subset of V. Associated with each edge e ∈ E is a class of customers who arrive as a Poisson process with rate λ e . Associated with every vertex v ∈ V is a single server, and we denote its queue size at time t by Q v (t). Each customer served by the server at vertex v, irrespective of its arrival class, requires an exponential service with rate µ v . Customers in the class associated with an edge e ∈ E can be served by any of the servers at vertices incident with the edge. In other words, customers in class in e ∈ E can be served by any server at a vertex v ∈ e. Upon arrival, a customer is allocated to a server and joins the relevant queue. The customers in each queue are served using the FIFO discipline. We now introduce static and dynamic allocation policies which we analyse. Static allocation Associated with each edge e ∈ E there are probabilities {p v,e } v∈e such that p v,e ≥ 0 for all v ∈ e and v∈e p v,e = 1. When a customer arrives upon an edge e ∈ E, it is allocated to a node v ∈ e with probability p v,e , independently of all other arrivals and services. Let P = {{p v,e } v∈e } e∈E refer to a given allocation for each edge. Therefore the total arrival process at node v is a Poisson process, independent of all other nodes, of rate λ v (P) = e∈E(v) p v,e λ e , where E(v) = {e ∈ E : v ∈ e} is the set of all edges containing node v. Dynamic allocation The dynamic allocation aims to load balance across the network by utilising join-the-shortestqueue dynamics. Upon an arrival of a customer on an edge e ∈ E, the queue sizes at all nodes v ∈ e are examined, and the customer is routed to the shortest of these. If there are more than one queues with the smallest size, the customer is routed to any one of them, at random with equal likelihoods. Note this is a natural definition of join-the-shortest-queue in this setting. Stability of static allocations For static allocations, the queues decouple and the stability condition is straightforward. Proposition 1. A static allocation is stable if and only if e∈E(v) p v,e λ e < µ v for all v ∈ V, or, alternatively, max v∈V 1 µ v e∈E(v) p v,e λ e < 1. Proof. For the static allocation we know that for a given allocation P the arrival process at each vertex is an independent Poisson process with arrival rate λ v (P) = e∈E(v) p v,e λ e . From this we have the stability condition associated with each node is e∈E(v) p v,e λ e < µ v , so that the stability condition for the whole system follows. ✷ While the previous result concerns a single possible allocation, we now consider the best possible allocation and the maximal stability region of the graph. The stability region for a given graph is maximized by minimizing over the possible allocations, as shown in the following. Proposition 2. There exists a stable static allocation if and only if min P   max v∈V 1 µ v e∈E(v) p v,e λ e   < 1. Particular case: symmetric system In this subsection we consider a particular case of our general setting where all customer classes have the same arrival intensity and all jobs require service times with the same distribution. More precisely, λ e = λ for all e ∈ E and µ v = µ for all v ∈ V. The general stability condition thus reduces to the requirement λ   min P   max v∈V e∈E(v) p v,e     < µ. Note that practically, the maximal arrival rate characterizing the optimal static stability condition in this case can be computed as λ * = µ z * where z * is the solution of the following linear program: min p,z z, e∈E(v) p v,e ≤ z, ∀v ∈ V, 0 ≤ p e,v , v∈e p e,v = 1, ∀e ∈ E. It is worth noting that if there exists an allocation P which equalizes the e∈E(v) p v,e over all vertices v, then there is no loss in stability region due to the restrictions imposed by the graph structure, i.e. the maximum possible total arrival rate into the network is equal to the total service rate of the network. Indeed, as all the values of e∈E(v) p v,e are equal, each of them is necessarily equal to \|V\|/\|E\|, where \|E\| is the number of edges and \|V\| is the number of vertices. The stability condition hence reads λ\|E\| < µ\|V\|, which is exactly the requirement that the total arrival rate is smaller than the total service rate. In this case we obtain complete resource pooling in the sense of stability (but possibly in a weaker sense than state space collapse). An interesting question thus arises: can we understand what properties of the graph enable us to find a balanced (i.e. maximal stable in terms of rate conservation) allocation, and when it is not possible? We partially answer this question in the next section. Addition of edges can lead to smaller stability region To better understand the question posed above, we provide three revealing examples: firstly two extreme cases where balance is always achievable and then an example where balance is not achievable and we do observe a loss of capacity. We focus on standard graphs in this section. Let us consider two extreme graphs on n vertices, the circle and the complete graph. In both cases the allocation of (1/2, 1/2) on every edge balances the loads and enables the maximum stability regions in these cases. For an example where balance can not be obtained, consider a graph containing 2k vertices for k > 2 and separate them into two groups of k vertices. The first k vertices form a clique. The remaining k vertices are then leaves in a graph connected to a single node in the clique and each node in the clique is connected to a single leaf. It is not difficult to see that the best allocation you can achieve here is to equalize across the clique and then on all leaf edges, to allocate all the traffic to the leaf vertex. This gives the following pair of stability constraints: λ < µ and k − 1 2 λ < µ. Note that for k > 3 the first condition is superfluous, so the maximum arrival rate per edge is 2µ k−1 . Since there are k(k + 1)/2 edges in this graph the maximum total stable arrival rate is k(k + 1) (k − 1) µ, which is substantially below the total service rate of 2kµ. Stability of dynamic allocations We now turn to our main result which characterizes the stability of dynamic allocations in terms of the maximal static stability condition. Theorem 1. The dynamic allocation is stable if and only if the maximal static allocation is stable, i.e., min P   max v∈V 1 µ v e∈E(v) p v,e λ e   < 1. Proof. Necessary condition. Suppose the dynamic allocation is stable. Then there exist stationary probabilities, π v,e say, for a customer arriving at edge e to be routed to vertex v ∈ e. Since the network is stationary, rate stability implies that λ e∈∈E(v) π v,e = µ v P (X v > 0). In particular: λ e∈∈E(v) π v,e < µ v . Hence the collection {{π v,e } v∈e } e∈E clearly forms a stable static allocation. Sufficient condition. Assume there exist a stable static allocation P: e∈E(v) p v,e λ e < µ v for all v ∈ V. As there is a finite number of vertices, fix ε > 0 such that e∈E(v) p v,e λ e − µ v < −ε (4.1) for all v ∈ V. Consider now the system with dynamic allocations (with arbitrary tie breaks) and consider the Lyapunov function L(x) = v∈V x 2 v . We know that at rate µ v there is a departure from node v, and at rate λ e there is an arrival at edge e, which will then go to the minimal adjacent queue. Therefore, conditioned on the current queue lengths being (x v ) v∈V , the drift of the Lyapunov function is equal to x v + c, thanks to (4.1). The drift is therefore smaller than −δ < 0 as long as v∈V x v > (c + δ)/(2ε), which is sufficient for stability. ✷ Conclusion We provided necessary and sufficient conditions of stability for a model of load balancing on fixed hypergraphs that generalize most previous models in the literature. Interesting and difficult challenges consist in characterizing these conditions for large classes of random graphs. ,e x v , as v∈e p v,e = 1. The drift can then be bounded from above by AcknowledgementsThe authors are grateful to the associate editor for their careful reading of the paper and useful comments and suggestions, especially for brining to our attention relevant work on skill-based routing. Fcfs parallel service systems and matching models. 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[ "Driver Behavior Recognition via Interwoven Deep Convolutional Neural Nets with Multi-stream Inputs", "Driver Behavior Recognition via Interwoven Deep Convolutional Neural Nets with Multi-stream Inputs" ]	[ "Chaoyun Zhang ", "Rui Li ", "Woojin Kim ", "Daesub Yoon ", "Paul Patras " ]	[]	[]	Recognizing driver behaviors is becoming vital for in-vehicle systems that seek to reduce the incidence of car accidents rooted in cognitive distraction. In this paper, we harness the exceptional feature extraction abilities of deep learning and propose a dedicated Interwoven Deep Convolutional Neural Network (InterCNN) architecture to tackle the accurate classification of driver behaviors in real-time. The proposed solution exploits information from multi-stream inputs, i.e., in-vehicle cameras with different fields of view and optical flows computed based on recorded images, and merges through multiple fusion layers abstract features that it extracts. This builds a tight ensembling system, which significantly improves the robustness of the model. We further introduce a temporal voting scheme based on historical inference instances, in order to enhance accuracy. Experiments conducted with a real world dataset that we collect in a mock-up car environment demonstrate that the proposed InterCNN with MobileNet convolutional blocks can classify 9 different behaviors with 73.97% accuracy, and 5 aggregated behaviors with 81.66% accuracy. Our architecture is highly computationally efficient, as it performs inferences within 15 ms, which satisfies the real-time constraints of intelligent cars. In addition, our InterCNN is robust to lossy input, as the classification remains accurate when two input streams are occluded.	10.1109/access.2020.3032344	[ "https://arxiv.org/pdf/1811.09128v1.pdf" ]	53,712,545	1811.09128	4e53aca7c732363f919a72f7e1924ea35ce2f42f	Driver Behavior Recognition via Interwoven Deep Convolutional Neural Nets with Multi-stream Inputs Chaoyun Zhang Rui Li Woojin Kim Daesub Yoon Paul Patras Driver Behavior Recognition via Interwoven Deep Convolutional Neural Nets with Multi-stream Inputs 1Index Terms-Driver behavior recognitiondeep learningconvolutional neural networks Recognizing driver behaviors is becoming vital for in-vehicle systems that seek to reduce the incidence of car accidents rooted in cognitive distraction. In this paper, we harness the exceptional feature extraction abilities of deep learning and propose a dedicated Interwoven Deep Convolutional Neural Network (InterCNN) architecture to tackle the accurate classification of driver behaviors in real-time. The proposed solution exploits information from multi-stream inputs, i.e., in-vehicle cameras with different fields of view and optical flows computed based on recorded images, and merges through multiple fusion layers abstract features that it extracts. This builds a tight ensembling system, which significantly improves the robustness of the model. We further introduce a temporal voting scheme based on historical inference instances, in order to enhance accuracy. Experiments conducted with a real world dataset that we collect in a mock-up car environment demonstrate that the proposed InterCNN with MobileNet convolutional blocks can classify 9 different behaviors with 73.97% accuracy, and 5 aggregated behaviors with 81.66% accuracy. Our architecture is highly computationally efficient, as it performs inferences within 15 ms, which satisfies the real-time constraints of intelligent cars. In addition, our InterCNN is robust to lossy input, as the classification remains accurate when two input streams are occluded. I. INTRODUCTION D RIVER's cognitive distraction is a major cause of unsafe driving, which leads to severe car accidents every year [1]. Actions that underlie careless driving include interacting with passengers, using a mobile phone (e.g., for text messaging, game playing, and web browsing), and consuming food or drinks [2]. Such behaviors contribute significantly to delays in driver's response to unexpected events, thereby increasing the risk of collisions. Identifying drivers' behaviors is therefore becoming increasingly important for car manufacturers, who aim to build in-car intelligence that can improve safety by notifying drivers in real-time of potential hazards. Further, although full car automation is years ahead, inferring driver's behaviour is essential for vehicles with partial ("hands off") and conditional ("eyes off") automation, which will dominate the market at least until 2030 [3]. This is because the driver must either be ready to take control at any time or intervene in situations where the vehicle cannot complete certain critical functions [4]. Modern driver behavior classification systems usually rely on videos acquired from in-vehicle cameras, which record the movements and the facial expressions of the driver [5]. The videos captured are routinely partitioned into sequences of image frames, which are then pre-processed for features selection [6]. Such features are fed to pre-trained classifiers to perform identification of different actions that the driver performs. Subsequently, the classifier may trigger an alarm system in manual driving cars or provide input to a driving mode selection agent in semi-autonomous vehicles. We illustrate this pipeline in Fig. 1. During this process, the accuracy of the classifier is directly related to the performance of the system. In addition, the system should perform such classification in real-time, so as to help the driver mitigate unsafe circumstances in a timely manner. Achieving high accuracy while maintaining runtime efficiency is however challenging, yet striking appropriate trade-offs between these aims is vital for intelligent and autonomous vehicles. Underpinned by recent advances in parallel computing, deep neural networks [7] have achieved remarkable results in various areas, including computer vision [8], control [9], and autonomous driving [10], [11], as they can automatically extract features from raw data without requiring expensive hand-crafted feature engineering. Graphics processing units (GPUs) allow to train deep neural networks rapidly and with great accuracy, and perform inferences fast. Moreover, System on Chip (SoC) designs optimized for mobile artificial intelligence (AI) applications are becoming more powerful and computationally efficient [12], and embedding deep learning in car systems increasingly affordable [13]. Therefore, potential exists to build high precision driver behavior classification systems without compromising runtime performance. In this paper, we design a driver behavior recognition system that uniquely combines different Convolutional Neural Network (CNN) structures, to accurately perform this task in real-time. As such, we make the following key contributions: 1) We build a mock-up environment to emulate self-driving car conditions and instrument a detailed user study for data collection purposes. Specifically, we deploy side and front facing cameras to record the body movements and facial expressions of 50 participant drivers, throughout a range of tasks they performed. This leads to a large driver behavior video dataset, which we use for model training and evaluation. 2) We architect original Interwoven Convolutional Neural Networks (InterCNNs) to perform feature extraction and fusions across multiple levels, based on multi-stream video inputs and optical flow information. Our design allows to plug in different lightweight CNN architectures (e.g. MobileNet [14], [15]) to improve the computation efficiency of in-vehicle systems. 3) We demonstrate that our InterCNNs with MobileNet blocks and a temporal voting scheme, which enhances accuracy by leveraging historical inferences, can classify 9 different behaviors with 73.97% accuracy, and 5 aggregated behaviors (i.e., grouping tasks that involve the use of a mobile device, and eating & drinking) with 81.66% accuracy. Our architecture can make inferences within 15 ms, which satisfies the timing constraints posed by real car systems. Importantly, our architecture is highly robust to lossy input, as it remains accurate when two streams of the input are occluded. The results obtained demonstrate the feasibility of accurate inference of driver's behavior in real-time, making important steps towards fulfilling the multi-trillion economic potential of the driverless car industry [16]. The rest of the paper is organized as follows. In Sec. II we discussed relevant related work. In Sec. III we present our data collection and pre-processing efforts, which underpin the design of our neural network solution that we detail in Sec. IV-A. We demonstrate the performance of the proposed InterCNNs by means of experiments reported in Sec. V. Sec. VI concludes our contributions. II. RELATED WORK The majority of the driver behavior classification systems are based on in-vehicle vision instruments (i.e., cameras or eye-tracking devices), which constantly monitor the movements of the driver [17]. The core of such systems is therefore tasked with a computer vision problem, whose objective is to classify actions performed by drivers, using sequences of images acquired in real-time. Existing research can be categorized into two main classes: non deep learning approaches and deep learning approaches. A. Non Deep Learning Based Driver Behavior Identification In [18], Liu et al. employ Laplacian Support Vector Machine (SVM) and extreme learning machine techniques to detect drivers' distraction, using labelled data that captures vehicle dynamic and drivers' eye and head movements. Experiments show that this semi-supervised approach can achieve up to 97.2% detection accuracy. Li et al. pursue distraction detection from a different angle. They exploit kinematic signals from the vehicle Controller Area Network (CAN) bus, to reduce the dependency on expensive vision sensors. Detection is then performed with an SVM, achieving 95% accuracy. Ragab et al. compare the prediction accuracy of different machine learning methods in driving distraction detection [19], showing that Random Forests perform best and require only 0.05 s per inference. Liao et al. consider drivers' distraction in two different scenarios, i.e., stop-controlled intersections and speed-limited highways [1]. They design an SVM classifier operating with Recursive Feature Elimination (RFE) to detect driving distraction. The evaluation results suggest that by fusing eye movements and driving performance information, classification accuracy can be improved in stop-controlled intersection settings. B. Deep Learning Based Driver behavior Identification Deep learning is becoming increasingly popular for identifying driver behaviors. In [20], a multiple scale Faster Region CNN is employed to detect whether a driver is using a mobile phone or their hands are on the steering wheel. The solution operates on images of the face, hands and steering wheel separately, and then performs classification on these regions of interest. Experimental results show that this model can discriminate behaviors with high accuracy in real-time. Majdi et al. design a dedicated CNN architecture called Drive-Net to identify 10 different behaviors of distracted driving [21]. Experiments suggest that applying Region of Interest (RoI) analysis on images of faces can significantly improve accuracy. Tran et al. build a driving simulator named Carnetsoft to collect driving data, and utilize 4 different CNN architectures to identify 10 distracted and non-distracted driving behaviors [22]. The authors observe that deeper architectures can improve the detection accuracy, but at the cost of increased inference times. Investigating the trade-off between accuracy and efficiency remains an open issue. Yuen et al. employ a CNN to perform head pose estimation during driving [23]. Evaluation results suggest that incorporating a Stacked Hourglass Network to estimate landmarks and refine the face detection can significantly improve the accuracy with different occlusion levels. In [24], Streiffer et al. investigate mixing different models, i.e., CNNs, recurrent neural networks (RNNs), and SVMs, to detect driver distraction. Their ensembling CNN + RNN approach significantly outperforms simple CNNs in terms of prediction accuracy. Recognizing driver's behavior with high accuracy, using inexpensive sensing infrastructure, and achieving this in realtime remains challenging, yet mandatory for intelligent vehicles that can improve safety and reduce the time during which the driver is fully engaged. To the best of our knowledge, existing work fails to meet all these requirements. III. DATA COLLECTION AND PRE-PROCESSING In this work, we propose an original driver behavior recognition system that can classify user actions accurately in realtime, using input from in-vehicle cameras. Before delving into our solution (Sec. IV-A), we discuss the data collection and pre-processing campaign that we conduct while mimicking an autonomous vehicle environment, in order to facilitate the design, training, and evaluation of our neural network model. A. Data Collection We set up the mock-up car cockpit environment illustrated in Fig. 2 and conduct a user behavior study, whereby we emulate automated driving conditions and record data from two cameras (one to the side of the driver, the other frontfacing) that capture driver's actions. We recruit a total of 50 participants, 72% male and 38% female, with different age, years of driving experience, first spoken language, and level of computer literacy. During the experiments, we use a 49-inch Ultra High Definition monitor, onto which each participant is shown 35-minute of 4K dashcam footage of both motorway and urban driving, while being asked to alternate between 'driving' and performing a range of tasks. The cameras record the behavior of the participants from different angles, capturing body movements and facial expressions with 640×360 perframe pixel resolution and frame rate ranging between 17.14 and 24.74 frames per second (FPS). We use the OpenCV vision library [25] together with Python, in order to label each of the frames of the approximately 60 hours of video recorded, distinguishing between the following actions: 1) Normal driving: The participant focuses on the road conditions shown on the screen and acts as if driving. 2) Texting: The participant uses a mobile phone to text messages to a friend. 3) Eating: The participant eats a snack. 4) Talking: The participant is engaged in a conversation with a passenger. 5) Searching: The participant is using a mobile phone to find information on the web through a search engine. 6) Drinking: The participant serves a soft drink. 7) Watching video: The participant uses a mobile phone to watch a video. 8) Gaming: The participant uses a mobile phone to play a video game. 9) Preparing: The participant gets ready to begin driving or finishes driving. Pr ep ar in g D ri vi ng Te xt in g Ea ti ng Ta lk in g Se ar ch in g D ri nk in g V id eo G am in g In each experiment, the participant was asked to perform actions (2)-(8) once, while we acknowledge that in real-life driving such behaviors can occur repeatedly. Fig. 3 summarizes the amount of data (number of video frames) collected for each type of action that the driver performed. B. Data Pre-processing Recall that the raw videos recorded have 640×360 resolution. Using high-resolution images inevitably introduces storage, computational, and data transmission overheads, which would complicate the model design. Therefore, we employ fixed bounding boxes to crop all videos, in order to remove the background, and subsequently re-size the videos to obtain lower resolution versions. Note that the shape and position of the bounding boxed adopted differ between the videos recorded with side and front cameras. We illustrate this process in Fig. 4. Adding Optical Flow (OF) [26] to the input of a model has proven effective in improving accuracy [27]. The OF is the instantaneous velocity of the moving objects under scene surface. It can reflect the relationship between the previous and current frames, by computing the changes of the pixel values between adjacent frames in a sequence. Therefore, OF can explicitly describe the short-term motion of the driver, without requiring the model to learn about it. The OF vector d (x,y) at point (x, y) can be decomposed into vertical and horizontal components, i.e., done pixel-by-pixel. We show an example of the OF in Fig. 4. Our classifier will use OF information jointly with labelled video frames as the input. The experiments we report in Sec. V confirm that indeed this improves the inference accuracy. Lastly, we downsample the videos collected, storing only every third frame and obtaining a dataset with 5.71-8.25 FPS, which reduces data redundancy. We will feed the designed model with 15 consecutive video frames and corresponding 14 OF vectors, spanning 1.82 to 2.62 seconds of recording. Such duration has been proven sufficient to capture entire actions, while obtaining satisfactory accuracy [28]. d (x,y) = {d (x,y) v , d (x,y) h }. It IV. MULTI-STREAM INTERWOVEN CNNS We design a deep neural network architecture, named Interwoven CNN (InterCNN) that uses multi-stream inputs (i.e., side video streams, side optical flows, front video streams, and front optical flows) to perform driver behavior recognition. We illustrate the overall architecture of our model in Fig. 5(a). For completeness, we also show two simpler architectures, namely (i) a plain CNN, which uses only the side video stream as input (see Fig. 5(b)); and a (ii) two-stream CNN (TS-CNN), which takes the side video stream and the side optical flow as input (see Fig. 5(c)). Both of these structures can be viewed as components of the InterCNN. A. The InterCNN Architecture Diving into Fig. 5(a), the InterCNN is a hierarchical architecture which embraces multiple types of blocks and modules. It takes four different streams as input, namely side video stream, side optical flow, front video stream and front optical flow. Note that these streams are all four-dimensional, i.e., (time, height, width, RGB channels) for each video frame, and (time, height, width, vertical and horizontal components) for OF frames. The raw data is individually processed in parallel by 7 stacks of 3D CNN blocks. A 3D CNN block is comprised of a 3D convolutional layer to extract spatiotemporal features [29], a Batch Normaliazation (BN) layer for training acceleration [30], and a Scaled Exponential Linear Unit (SELU) activation function to improve the model nonlinearity and representability [31]. Here, SELU(x) = λ x, if x > 0; αe x − α, if x ≤ 0, where the parameters λ = 1.0507 and α = 1.6733 are frequently used. We refer to these four streams of 3D CNNs as side spatial stream, side temporal stream, front spatial stream, and front temporal stream respectively, according to the type of input handled. Their outputs are passed to two temporal fusion layers to absorb the time dimension and perform the concatenation operation along the channel axis. Through these temporal fusion layers, intermediate outputs of spatial and temporal streams are merged and subsequently delivered to 25 stacks of Interweaving modules. We illustrate the construction of such modules in Fig. 6 and detial their operation next. B. Interweaving Modules The Interweaving module draws inspiration from ResNets [32] and can be viewed as a multi-stream version of deep residual learning. The two inputs of the module are processed by different CNN blocks individually, and subsequently delivered to a spatial fusion layer for feature aggregation. The spatial fusion layer comprises a concatenation operation and a 1×1 convolutional layer, which can reinforce and tighten the overall architecture, and improve the robustness of the model [33]. Experiments will demonstrate that this enables the model to maintain high accuracy even if the front camera is blocked completely. After the fusion, another two CNN blocks will decompose the merged features in parallel into two-stream outputs again. This maintains the information flow intact. Finally, the residual paths connect the inputs and the outputs of the final CNN blocks, which facilitates fast backpropagation of the gradients during model training. These paths also build ensembling structures with different depths, which have been proven effective in improving inference accuracy [34]. After processing by the Interweaving blocks, the intermediate outputs obtained are sent to a Multi-Layer Perceptron (MLP) to perform the final classification. C. CNN Blocks Employed The CNN blocks employed within the interweaving modules are key to performance, both in terms of accuracy and inference time. Our architecture is sufficiently flexible to allow different choices for these CNN blocks. In this work, we explore the vanilla CNN block, MobileNet [14], and MobileNet V2 [15] structures, and compare their performance. We show the architectures of these choices in Fig. 7. The vanilla CNN block embraces a standard 2D CNN layer, a BN layer and a Rectified Linear Unit (ReLU) activation function. This is a popular configuration and has been employed in many successful classification architectures, such as ResNet [32]. However, the operations performed in a CNN layer are complex and involve a large number of parameters. This may not satisfy the resource constraints imposed by vehicular systems. The MobileNet [14] decomposes the traditional CNN layer into a depthwise convolution and a pointwise convolution, which significantly reduces the number of parameters required. Specifically, depthwise convolution employs single a convolutional filter to perform computations on individual input channels. Thereby, this generates an intermediate output that has the same number of channels as the input. The outputs are subsequently passed to a pointwise convolution module, which applies a 1×1 filter to perform channel combination. MobileNet further employs a hyperparameter α to control the number of channels, and ρ to control the shape of the feature maps. We set both α and ρ to 1 in our design. In summary, the MobileNet block breaks the filtering and combining operations of a traditional CNN block into two layers. This significantly reduces the computational complexity and number of parameters required, while improving efficiency in resource-constrained devices. The MobileNet V2 structure [15] improves the MobileNet by introducing an inverted residual and linear bottleneck. The inverted residual incorporates the residual learning specific to ResNets. The input channels are expanded through the first 1×1 convolutional layer, and compressed through the depthwise layer. The expansion is controlled by a parameter t, which we set to 6 as default. To reduce information loss, the ReLU activation function in the last layer is removed. Compared to MobileNet, the second version has fewer parameters and higher memory efficiency. As we will see, this architecture may sometimes exhibit superior accuracy. Both MobileNet and MobileNet V2 are tailored to embedded devices, as they make inferences faster with smaller models. These makes them suitable for in-vehicle classification systems. V. EXPERIMENTS In this section, we first describe briefly the implementation of the proposed InterCNN for driver behavior recognition, then compare the prediction accuracy of different CNN blocks A. Implementation We implement the different neural network architectures studied in TensorFlow [35] with the TensorLayer library [36]. We train all models on a computing cluster equipped with 1-2 NVIDIA TITAN X and Tesla K40M GPUs for approximately 10 days and perform early-stop based on the validation error. For training, we use the Adam optimization algorithm [37], which is based on stochastic gradient descent. With this we seek to minimize the standard cross-entropy loss function between true labels and the outputs produced by the deep neural networks. To maintain consistency, we test all models using an NVIDIA TITAN X GPU when evaluating their computation efficiency. B. Accuracy Assessment We randomly partition the entire dataset into a training set (30 videos), a validation set (10 videos) and a test set (10 videos). We assessed the accuracy of our solution on two categories of behaviors. The first considers all the 9 different actions performed by the driver (see Sec. III). In the second, we aggregate the behaviors that are visually similar and carry similar cognitive status. In particular, [Texting, Searching, Watching Video, Gaming] are aggregated into a "Using phone" behavior, and [Eating, Drinking] are combined into a single "Eat & Drink" action. In Fig. 8 of "full behavior" recognition (top subfigure), the proposed In-terCNN with MobileNet blocks achieves the highest prediction accuracy, outperforming the plain CNN by 7.93%. Further, we can see that feeding the neural network with richer information (optical flows and front images) improves accuracy, as our two-stream CNN and InterCNN on average outperform the plain CNN by 2.26% and 3.81% respectively. This confirms that the OF and facial expressions provide useful descriptions of the behaviors, which our architectures effectively exploits. It is also worth noting that, although the performance gains over the plain CNN may appear relatively small, the amount of computational resource required by our architecture, inference times, and complexity are significantly smaller. We will detail these aspects in the following subsection. Turning attention to the aggregated behavior (bottom sub- figure), observe that the accuracy improves significantly compared to when considering all the different actions the driver might perform, as we expect. This is because some behaviors demonstrate similar patterns (e.g., texting and web searching), which makes discriminating among these extremely challenging. Overall, the InterCNN with MobileNet blocks obtains the best prediction performance when similar behaviors are aggregated, outperforming other architectures by up to 4.83%. In addition, our two-stream CNN and InterCNN consistently outperform the plain CNN. C. Model Complexity & Inference Time Next, we compare the model size (in terms of number or weights and biases to be configured in the model), inference time, and complexity (quantified through floating point operations -FLOPs) of InterCNNs with different CNN blocks. Lower model size will pose small storage and memory requirements on the in-vehicle system. The inference time refers to how long it takes to perform one instance of driver behavior recognition. This is essential, as such application are required to perform in real-time. Lastly, the number of FLOPs [38] is computed by counting the number of mathematical operation or assignments that involve floating-point numbers, and is routinely used to evaluate the complexity of a model. We illustrate this comparison in Fig. 9. Observe that Mo-bileNet and MobileNet V2 have similar numbers of parameters, and these are 4 times fewer than those of vanilla CNN blocks. This is consistent with the conclusions drawn in [14] and [15]. In addition, InterCNNs with MobileNet blocks can infer driver behavior within 15 ms per instance (center subfigure) with the help of a GPU, which satisfies the real-time constraints of intelligent vehicle systems. Runtime performance is indeed similar to that of an architecture employing CNN blocks, yet less complex, while an architecture with MobileNet blocks is 46.2% faster than with MobileNet V2 blocks. Lastly, the number of FLOPs required by an InterCNN with MobileNet blocks is approximately 4.5 and 6 times smaller than when employing CNN and respectively MobileNet V2 blocks. This requirement can be easily handled in real-time even by a modern CPU. D. Temporal Voting In the collected dataset, since the videos are recorded at high FPS rate, we observe that the behavior of a driver will not change very frequently over consecutive frames that span less than 3 seconds. Therefore, we may be able to reduce the likelihood of misclassification by considering the actions predicted over recent frames. To this end, we employ a temporal voting scheme, which constructs an opinion poll storing the inferred behaviors over the n most recent frames, and executes a "voting" procedure. Specifically, the driver's action is determined by the most frequent label in the poll. We illustrate the principle of this Temporal Voting (TV) procedure in Fig. 10. We set n = 15, by which the poll size bears the same temporal length as the inputs. We show the prediction accuracy before and after applying TV in Tab I and II. Observe that the TV scheme improves the classification accuracy of all architectures on both full and aggregated behavior sets. In particular, the accuracy on full behavior recognition increases by 1.99%, and that of aggregated behavior recognition by 1.80%. This demonstrates the effectiveness of the proposed TV scheme. E. Operation with Lossy Input In this subsection, we investigate the robustness of our InterCNN to losses in the input data streams, by blocking the front front video and the front OF inputs when performing inferences. Such circumstances can occur in real world settings, e.g., the camera may be intentionally or accidentally occluded by the driver. To cope with such conditions, we finetune our model by performing "drop-outs" over the inputs [39]. Specifically, we block the front video and the front OF streams with probability p = 0.5 during the training and testing of the InterCNN with MobileNet blocks. We summarize the obtained performance in Fig. 11. Note that by blocking the front video and the front OF streams, the input of the InterCNN is the same as that fed to the two-stream CNN, while the architecture remains unchanged. Observe that although the prediction accuracy of InterCNN drops slightly when the inputs are blocked, the occluded InterCNN remains better than the two-steam CNN in the full behavior recognition task. This suggests that out proposed architecture is highly robust to lossy input. This also confirms the effectiveness of the Interweaving modules, which we employ to improve the robustness of the model. We conclude that, by employing MobileNet blocks in Inter-CNNs, we achieve the highest accuracy in the driver behavior recognition task, as compared with any of the other candidate CNN block architectures. The InterCNN + MobileNet combo also demonstrates superior computational efficiency, as it requires the lowest number of parameters, exhibits the fastest inference times and the least FLOPs. Importantly, our design is robust to lossy inputs. The sum of these advantages make the proposed InterCNN with MobileNet blocks an excellent solution for accurate driver behavior recognition, easily pluggable in modern in-vehicle intelligent systems. F. Model Insights Lastly, we delve into the inner workings of the InterCNN by visualizing the output of the hidden layers of the model, aiming to better understand how the neural network "thinks" of the data provided as input and what knowledge it learns. T-distributed Stochastic Neighbor Embedding Vizualization: We first adopt the t-distributed Stochastic Neighbor Embedding (t-SNE) [40] to reduce the dimension of the last layer (the MLP layer in Fig. 5(a)), and plot the hidden representations of a testing video (35,711 frames) into a twodimensional plane, as shown in Fig. 12. In general, the t-SNE approach arranges data points that have a similar code nearby in the embedding. This typically reflects how the model "thinks" of the data points, as similar data representations will be clustered together. Interestingly, the embedding of eating and drinking remain close to each other, as both actions require to grasp a snack or drink and bring this close to the mouth. Furthermore, the embedding of actions that use a phone (i.e., web searching, texting, watching videos, and gaming) are grouped to the right side of the plane, as they are visually similar and difficult to differentiate. Moreover, as "Preparing" involves a combination of actions, including sitting down and talking to the experiment instructor, its representation appears scattered. These observations suggest that our model effectively learns the feature of different behaviors after training, as it projects similar data points onto nearby positions. Hidden Layer Output Visualization: We also investigate the knowledge learned by the model from a different perspectives, by visualizing the output of the hidden layers of the 3D CNN block before the temporal fusion layers. This will reflect the features extracted by each individual neural network stream. We show a snapshot of such visualization in Fig. 13. Observe that the spatial streams perform "edge detection", as the object edges in the raw inputs are outlined by the 3D CNN. On the other hand, the output of the hidden layers in the temporal steams, which process the optical flows, are too abstract to interpret. In general, the abstraction level of the features extracted will increase with the depth of the architecture; it is the sum of such increasingly abstract features that enables the neural network to perform the final classification of behaviors with high accuracy. VI. CONCLUSION In this paper, we proposed an original Interwoven Convolutional Neural Network (InterCNN) to perform driver behavior recognition. Our architecture can effectively extract information from multi-stream inputs that record the activities performed by drivers from different perspectives (i.e., side video, side optical flow, front video, and front optical flow), and fuse the features extracted to perform precise classification. We further introduced a temporal voting scheme to build an ensembling system, so as to reduce the likelihood of misclassification and improve accuracy. Experiments conducted on a real-world dataset that we collected with 50 participants demonstrate that our proposal can classify 9 types of driver behaviors with 73.97% accuracy, and 5 classes of aggregated behaviors with 81.66% accuracy. Our model makes such inferences within 15 ms per instance, which satisfies the real-time constraints of modern in-vehicle systems. The proposed InterCNN is further robust to lossy data, as inference accuracy is largely preserved when the front video and front optical flow inputs are occluded. Video frames Optical Flows Outputs of spatial streams Outputs of temporal streams p C. Zhang, R. Li, and P. Patras are with the Institute for Computing Systems Architecture (ICSA), School of Informatics, University of Edinburgh, UK. Emails: {chaoyun.zhang, rui.li, paul.patras}@ed.ac.uk. o W. Kim and D. Yoon are with the Electronics and Telecommunications Research Institute (ETRI), Daejon, South Korea. Emails: {wjinkim, eyetracker}@etri.re.kr. Fig. 1 : 1The typical pipeline of driver behavior classification and alarm/driving mode selection systems. Fig. 2 : 2Mock-up car cockpit environment (left) used for data collection and snapshots of the software interface employed for video capture (right). The curved screen shows a driving video to the participant, while the side and front cameras record their body movements and facial expressions, respectively. Fig. 3 : 3Summary of the total amount of data (video frames) collected for each driver behavior. ( c ) cThe architecture of a two-stream CNN (TS-CNN). Fig. 5 : 5Three different neural network architectures employed in this study. The plain CNN only uses the side video stream as input, the two-streaming CNN adds extra side OFs, while the InterCNN employs both side and front video streams and optical flows. Fig. 6 : 6The anatomic structure of an Interweaving module. Fig. 9 : 9Comparison of number of parameter (left), inference time (middle) and floating point operations (FLOPs) on Inter-CNNs with different CNN blocks. Fig. 10 : 10Illustration of the Temporal Voting (TV) scheme. Fig. 12 : 12Two-dimensional t-SNE embedding of the representations in the last hidden layer of the InterCNN with MobileNet blocks. Data generated using a full test video (35,711 frames). Fig. 13 : 13The input videos frames, optical flows, and hidden outputs of the 3D CNN blocks before two temporal fusion layers. Figures shown correspond to a single instance. Chaoyun Zhang is currently working towards his Ph.D. degree at the University of Edinburgh within the School of Informatics. He obtained an MSc Artificial Intelligence from the University of Edinburgh, with a focus on machine learning. He also obtained a BSc degree from the School of Electronic Information and Communications at Huazhong University of Science and Technology, China. His current research interests include the application of deep learning to problems in computer networking, including traffic analysis, resource allocation and network control. Rui Li is pursuing a Ph.D. in School of Informatics at the University of Edinburgh, UK, supervised by Dr. Paul Patras. Her research interests mainly focus on applied machine learning in the networking domain. Rui obtained MSc with Distinction in Embedded System and Control Engineering from University of Leicester, UK (2014), and BEng in Communications Engineering from Northwestern Polytechnical University, China (2013). has the same resolution as the original images, as the computation isFig. 4: The data pre-processing procedure (cropping and resizing) and optical flow quiver plots. The architecture of a plain CNN.Raw side image (640 × 360) Raw front image (640 × 360) Processed image (108 × 87) Processed image (54 × 58) Side optical flow (108 × 87) Front optical flow (54 × 58) Cropping + Resizing Cropping + Resizing z Front Spatial Stream Front Temporal Stream z 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block CNN Block CNN Block CNN Block Spatial Fusion CNN Block CNN Block CNN Block CNN Block Spatial Fusion CNN Block CNN Block CNN Block CNN Block Spatial Fusion CNN Block MLP Spatial Fusion ... ... ... ... ... ... Side Images Side Optical Flows Front Images Front Optical Flows Classification Temporal Fusion Temporal Fusion Interweaving Module Side Spatial Stream Side Temporal Stream (a) The architecture of the Interwoven CNNs (InterCNNs). 3D CNN Block 3D CNN Block CNN Block CNN Block MLP ... ... Side Images Classification Fusion (b) z 3D CNN Block 3D CNN Block 3D CNN Block 3D CNN Block CNN Block CNN Block MLP ... ... ... Side Images Side Optical Flows Classification Fusion Side Spatial Stream Side Temporal Stream , we show the prediction accuracy of the InterCNN architecture with all the CNN block types considered, as well as that of plain and two-stream CNN architectures, each employing the same three types of blocks. Observe that in the caseCNN Mobile- Net Mobile- Net V2 Block Type 0.0 0.2 0.4 0.6 0.8 Parameter 1e8 CNN Mobile- Net Mobile- Net V2 Block Type 0 5 10 15 20 25 Inference time [ms] CNN Mobile- Net Mobile- Net V2 Block Type TABLE I : IInference accuracy with different CNN blocks over full behaviors, before\after applying the TV scheme.Block Plain CNN TS-CNN InterCNN CNN 66.32%\69.74% 67.03%\68.27% 69.39%\68.40% MobileNet 62.68%\63.59% 68.41%\70.76% 70.61%\73.97% MobileNet V2 67.97%\69.35% 68.33%\70.89% 68.46%\70.61% TABLE II : IIInference accuracy with different CNN blocks over aggregated behaviors before\after applying the TV scheme.Block Plain CNN TS-CNN InterCNN CNN block 78.01%\79.70% 78.82%\80.77% 79.67%\81.25% MobileNet 75.29%\77.78% 79.37%\81.40% 80.62%\81.66% MobileNet V2 77.21%\79.14% 79.42%\80.97% 79.13%\80.55% Two-stream CNN InterCNN Occluded InterCNN Architecture 0 25 50 75 100 Accuracy [%] 70.76 73.97 71.09 81.4 81.66 81.3 Full Behavior Aggregated Behavior Fig. 11: Comparison of accuracy of two-stream CNN, Inter- CNN, and InterCNN with occluded inputs. All architectures employ MobileNet blocks. Woojin Kim received the B.S. degree in electrical engineering from Korea Advanced Institute of Technology, Daejeon, Korea, in 2008, and the Ph.D. degrees from the Seoul National University, Seoul, Korea. He is currently a Senior Researcher at Electronics and Telecommunication Research Institute, Daejeon, Korea. His research interests are cooperative control of multiple robots or humanmachine systems, and their applications. Daesub Yoon received the M.S. and Ph.D. degree in Computer Science & Software Engineering from Auburn University. From 2001 to 2005, he was a Research Assistant with the Intelligent & Interactive System Laboratory in Auburn University. He has joined at Electronics & Telecommunication Research institute, Korea since 2005. His research interests include assistive technology, eye tracking, attentive user interface, mental workload and human factors in automated driving vehicles and smart factories. Paul Patras [SM'18, M'11] received M.Sc. (2008) and Ph.D. (2011) degrees from Universidad Carlos III de Madrid (UC3M). He is a Lecturer and Chancellors Fellow in the School of Informatics at the University of Edinburgh, where he leads the Internet of Things Research Programme. His research interests include performance optimisation in wireless and mobile networks, applied machine learning, mobile traffic analytics, security and privacy, prototyping and test beds. ACKNOWLEDGMENT This work was partially supported by a grant (18TLRP-B131486-02) from the Transportation and Logistics R&D Program funded by Ministry of Land, Infrastructure and Transport of the Korean government. Detection of driver cognitive distraction: A comparison study of stop-controlled intersection and speed-limited highway. Y Liao, S E Li, W Wang, Y Wang, G Li, B Cheng, IEEE Transactions on Intelligent Transportation Systems. 176Y. Liao, S. E. Li, W. Wang, Y. Wang, G. Li, and B. 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[ "SPITZER IRAC IDENTIFICATION OF HERSCHEL-ATLAS SPIRE SOURCES", "SPITZER IRAC IDENTIFICATION OF HERSCHEL-ATLAS SPIRE SOURCES" ]	[ "Sam Kim \nDraft version\n\n", "Julie L Wardlow \nDraft version\n\n", "Asantha Cooray \nDraft version\n\n", "S Fleuren \nDraft version\n\n", "W Sutherland \nDraft version\n\n", "A A Khostovan \nDraft version\n\n", "R Auld \nDraft version\n\n", "M Baes \nDraft version\n\n", "R S Bussmann \nDraft version\n\n", "S Buttiglione \nDraft version\n\n", "A Cava \nDraft version\n\n", "D Clements \nDraft version\n\n", "A Dariush \nDraft version\n\n", "G De Zotti \nDraft version\n\n", "L Dunne \nDraft version\n\n", "S Dye \nDraft version\n\n", "S Eales \nDraft version\n\n", "J Fritz \nDraft version\n\n", "R Hopwood \nDraft version\n\n", "E Ibar \nDraft version\n\n", "R Ivison \nDraft version\n\n", "M Jarvis \nDraft version\n\n", "S Maddox \nDraft version\n\n", "M J Michałowski \nDraft version\n\n", "E Pascale \nDraft version\n\n", "M Pohlen \nDraft version\n\n", "E Rigby \nDraft version\n\n", "D Scott \nDraft version\n\n", "D J B Smith \nDraft version\n\n", "P Temi \nDraft version\n\n", "P Van Der Werf \nDraft version\n\n" ]	[ "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n", "Draft version\n" ]	[]	We use Spitzer-IRAC data to identify near-infrared counterparts to submillimeter galaxies detected with Herschel-SPIRE at 250 µm in the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). Using a likelihood ratio analysis we identify 146 reliable IRAC counterparts to 123 SPIRE sources out of the 159 in the survey area. We find that, compared to the field population, the SPIRE counterparts occupy a distinct region of 3.6 and 4.5 µm color-magnitude space, and we use this property to identify further 23 counterparts to 13 SPIRE sources. The IRAC identification rate of 86% is significantly higher than those that have been demonstrated with wide-field ground-based optical and near-IR imaging of Herschel fields. We estimate a false identification rate of 3.6%, corresponding to 4 to 5 sources. Among the 73 counterparts that are undetected in SDSS, 57 have both 3.6 and 4.5 µm coverage. Of these, 43 have [3.6] − [4.5] > 0 indicating that they are likely to be at z 1.4. Thus, ∼ 40% of identified SPIRE galaxies are likely to be high redshift (z 1.4) sources. We discuss the statistical properties of the IRAC-identified SPIRE galaxy sample including far-IR luminosities, dust temperatures, star-formation rates, and stellar masses. The majority of our detected galaxies have 10 10 to 10 11 L ⊙ total IR luminosities and are not intense starbursting galaxies as those found at z ∼ 2, but they have a factor of 2 to 3 above average specific star-formation rates compared to near-IR selected galaxy samples.	10.1088/0004-637x/756/1/28	[ "https://arxiv.org/pdf/1112.3653v3.pdf" ]	119,296,091	1112.3653	d5d35087c8694f9e22a743005ce01a723124a1b3	SPITZER IRAC IDENTIFICATION OF HERSCHEL-ATLAS SPIRE SOURCES 29 Jun 2012 DRAFT VERSION JULY 2, 2012 July 2, 2012 Sam Kim Draft version Julie L Wardlow Draft version Asantha Cooray Draft version S Fleuren Draft version W Sutherland Draft version A A Khostovan Draft version R Auld Draft version M Baes Draft version R S Bussmann Draft version S Buttiglione Draft version A Cava Draft version D Clements Draft version A Dariush Draft version G De Zotti Draft version L Dunne Draft version S Dye Draft version S Eales Draft version J Fritz Draft version R Hopwood Draft version E Ibar Draft version R Ivison Draft version M Jarvis Draft version S Maddox Draft version M J Michałowski Draft version E Pascale Draft version M Pohlen Draft version E Rigby Draft version D Scott Draft version D J B Smith Draft version P Temi Draft version P Van Der Werf Draft version SPITZER IRAC IDENTIFICATION OF HERSCHEL-ATLAS SPIRE SOURCES 29 Jun 2012 DRAFT VERSION JULY 2, 2012 July 2, 2012Preprint typeset using L A T E X style emulateapj v. 08/22/09Subject headings: galaxies: high-redshift -infrared: galaxies -galaxies:starburst We use Spitzer-IRAC data to identify near-infrared counterparts to submillimeter galaxies detected with Herschel-SPIRE at 250 µm in the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). Using a likelihood ratio analysis we identify 146 reliable IRAC counterparts to 123 SPIRE sources out of the 159 in the survey area. We find that, compared to the field population, the SPIRE counterparts occupy a distinct region of 3.6 and 4.5 µm color-magnitude space, and we use this property to identify further 23 counterparts to 13 SPIRE sources. The IRAC identification rate of 86% is significantly higher than those that have been demonstrated with wide-field ground-based optical and near-IR imaging of Herschel fields. We estimate a false identification rate of 3.6%, corresponding to 4 to 5 sources. Among the 73 counterparts that are undetected in SDSS, 57 have both 3.6 and 4.5 µm coverage. Of these, 43 have [3.6] − [4.5] > 0 indicating that they are likely to be at z 1.4. Thus, ∼ 40% of identified SPIRE galaxies are likely to be high redshift (z 1.4) sources. We discuss the statistical properties of the IRAC-identified SPIRE galaxy sample including far-IR luminosities, dust temperatures, star-formation rates, and stellar masses. The majority of our detected galaxies have 10 10 to 10 11 L ⊙ total IR luminosities and are not intense starbursting galaxies as those found at z ∼ 2, but they have a factor of 2 to 3 above average specific star-formation rates compared to near-IR selected galaxy samples. INTRODUCTION The extragalactic background at far-infrared (IR) and submillimeter (sub-mm) wavelengths is well-constrained from total intensity measurements (Puget et al. 1996;Fixsen et al. 1998;Dwek et al. 1998). However, the properties of the discrete galaxies that make up this background are still largely unknown. These sub-mm galaxies are expected to capture the dusty star-formation out to redshifts of 4 and beyond and are now understood to be an integral component of galaxy formation and evolution (Hughes et al. 1998;Eales et al. 1999;Blain et al. 2002;Chapman et al. 2003;Austermann et al. 2010). A challenge for studies of sub-mm galaxies is that widefield surveys at these wavelengths have poor spatial resolution, caused by the limited apertures of single dish submm survey telescopes (Smail, Ivison & Blain 1997;Scott et al. 2002;Coppin et al. 2006). Typical resolutions are of the order of tens of arcseconds, making the identification of multiwavelength counterparts challenging, particularly in the optical where sub-mm galaxies are usually faint due to high dust extinction. Furthermore, the surface density of faint optical galaxies is such that several potential counterparts may be situated within the positional error of each submm source. However, the identification of counterparts to sub-mm galaxies is critical for both photometric and spectroscopic studies, which yield redshifts, spectral energy distributions (SEDs) and morphological information. The measurements of these quantities for statistically significant samples of sub-mm galaxies are required to compare their properties with theoretical predictions, and to fully understand the role of the sub-mm bright phase in galaxy evolution. The most reliable way to pinpoint the positions of sub-mm galaxies is through high-resolution sub-mm interferometry, which directly traces the dust emission at wavelengths comparable to the original selection function (e.g. Downes et al. 1999;Gear et al. 2000;Iono et al. 2006;Younger et al. 2007). The sensitivity of early generations of sub-mm interferometric instrumentation limited anlayses to samples of no more than a few sources (e.g. Downes et al. 1999;Gear et al. 2000;Iono et al. 2006;Younger et al. 2007Younger et al. , 2008Younger et al. , 2009Ivison et al. 2008;Cowie et al. 2009;Aravena et al. 2010a;Hatsukade et al. 2010;Tamura et al. 2010;Chen et al. 2011 ), and although rapid progress is being made (e.g. Smolcic 2012a,b) such observations are still unfeasible for large surveys of sources. In addition, extended sources (e.g. Matsuda et al. 2007), or sources that are a blend of multiple components (e.g. Wang et al. 2011;Smolcic 2012b), can be difficult to detect. An alternative technique to identify sub-mm galaxies is to take advantage of the far-IR-radio correlation (e.g. Condon et al. 1998;Garrett 2002) and the relatively low surface density of radio sources by searching for counterparts in deep interferometric radio data (e.g. Ivison et al. 1998Ivison et al. , 2002Borys et al. 2004;Dye et al. 2009;Dunlop et al. 2010;Biggs et al. 2011). Similarly, the dusty, active sub-mm galaxies are typically bright at mid-IR wavelengths, a property that has also been utilized to identify counterparts (e.g. Ivison et al. 2004;Pope et al. 2006;Dye et al. 2008Dye et al. , 2009Clements et al. 2008;Dunlop et al. 2010;Biggs et al. 2011). Deep radio and mid-IR (typically Spitzer MIPS 24 µmbut also Spitzer IRAC 3.6 µm; Biggs et al. 2011) searches typically identify counterparts to 60-80% of sub-mm galaxies, and nearest-neighbor positional matching is then used to investigate the properties of these sources at other wavelengths. The remaining 20-40% of sources are thought to be at high-redshifts (z 3) or be dominated by cold dust (T D 25 K at z ∼ 2; Chapman et al. 2005). The Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS; Eales et al. 2010) is the largest open-time key project being undertaken by the Herschel Space Observatory (Pilbratt et al. 2010) 19 . The total planned survey area is 550 deg 2 within which it is expected to detect > 300,000 bright sub-mm galaxies. During the Science Demonstration Phase (SDP), H-ATLAS observed 14.4 deg 2 in the Galaxy And Mass Assembly (GAMA; Driver et al. 2011) 9 hour field, to 5σ depths of 35 to 132 mJy in five bands from 100 to 500 µm, using PACS (100 and 160µm; Poglitsch et al. 2010) and SPIRE (250, 350 and 500µm; Griffin et al. 2010) in parallel mode. The lack of deep radio and 24 µm data across the field restricts counterpart identification to optical (e.g. Sloan Digital Sky Survey [SDSS]; York et al. 2000) or near-IR data (e.g. VISTA Kilo-Degree Infrared Galaxy Survey [VIKING]; Sutherland et al. in prep, UKIRT Infrared Deep Sky Survey [UKIDSS]; Lawrence et al. 2007). Smith et al. (2011) identified SDSS and GAMA galaxies as the counterparts to 37% of the SPIRE sources in the H-ATLAS SDP field. In the wider area of the whole GAMA-9 hour field Fleuren et al. (2012) increased the fraction of SPIRE sources with reliable identifications to 51% by using K s -band imaging from the VIKING survey. In the GAMA-15 hour field 50% of the SPIRE sources have reliable counterparts identified in Wide-Field Infrared Survey Explorer (WISE) data at 3.4 µm (Bond et al. 2012). In this paper we cross-identify H-ATLAS SPIRE sources with Spitzer Infrared Array Camera (IRAC; Fazio et al. 2004a) galaxies, which are selected from 3.6 and 4.5-µm observations of 0.4 deg 2 of the H-ATLAS SDP field. The IRAC data are advantageous for counterpart identification because of their mid-IR wavelengths and depth (∼3 magnitudes deeper than WISE). Thus, counterpart identification is less bi-ased towards galaxies with relatively low redshifts or dust obscurations than the existing analyses. The paper is organized as follows: the data analysis and sample selection are presented in Section 2. In Section 3 we describe the counterpart identification method and in Section 4 we discuss our results, including the identification rate and the colors and properties of SPIRE counterparts. Our conclusions are presented in Section 5. A table of the identified counterparts, including magnitudes and fluxes is presented in the Appendix. We use J2000 coordinates and ΛCDM cosmology with Ω m = 0.27, Ω Λ = 0.73 and H 0 = 70 km s −1 Mpc −1 throughout. All photometry is on the AB magnitude system where 1 µJy= 23.9 mag; IRAC 3.6 and 4.5 µm AB magnitudes are designated [3.6] and [4.5], respectively. SAMPLE SELECTION AND DATA ANALYSIS The H-ATLAS fields were chosen to minimize contamination from Galactic cirrus and to maximize the overlap with existing and planned wide-area imaging and spectroscopic surveys. The SDP field, which we study here, spans ∼ 14 deg 2 in the GAMA-9 hour field and has existing SDSS, VIKING, and UKIDSS data. It is observed with PACS at 100 and 160 µm and SPIRE at 250, 350 and 500 µm. The H-ATLAS PACS and SPIRE map-making processes are described in Ibar et al. (2010) and Pascale et al. (2011), respectively, and details of the source extraction procedures are given in Rigby et al. (2011). A public catalog of SDP sources is available from the H-ATLAS website 20 . The data reach 5σ point-source depths of 132, 126, 32, 36 and 45 mJy, with beam sizes of 8.7, 13.1, 18.1, 25.2 and 36.9 ′′ (FWHM) at 100, 160, 250, 350, and 500 µm, respectively. Although PACS 100 and 160 µm data have significantly higher angular resolution than SPIRE data we do not use PACS data for counterpart identification because only five of the 159 SPIRE sources studied here are detected by PACS. In all five cases the position of the identified IRAC counterpart (table 2) is consistent with the PACS emission. Furthermore, the counterparts to all of these five sources are low-redshift late-type galaxies, which is consistent with expectations for galaxies that are detectable with PACS. Spitzer IRAC 3.6 and 4.5 µm staring mode observations of 0.4 deg 2 , in three regions of the H-ATLAS SDP field were carried out during the warm mission (Spitzer program 548; PI: A. Cooray). Two of the areas targeted contain bright SPIRE SDP sources that are now known to be gravitationally lensed (ID81 and ID130; Negrello et al. 2010;Hopwood et al. 2011) and the third area was chosen as a test field. In staring mode simultaneous observations at 3.6 and 4.5 µm are offset from each other by 6.8 ′ . Due to the offset, 0.22 deg 2 of the targeted area has imaging data at both 3.6 and 4.5 µm, and the remaining 0.18 deg 2 is split between 3.6 and 4.5 µm coverage. There are 159 SPIRE sources in the total IRAC footprint. Of these, 101 are observed at both 3.6 and 4.5 µm, and 30 (28) have only 3.6 µm (4.5 µm) data. The total exposure time is 1080 seconds per pixel in each filter. Data reduction and mosaicking is performed on the Corrected Basic Calibrated Data (cBCD) with MOPEX (MOsaicker and Point source EXtractor; Makovoz & Marleau 2005). The final mosaicked images have 0.6 ′′ × 0.6 ′′ pixels and the FWHM of the IRAC point spread function (PSF) at 3.6 and 4.5 µm is 1.66 and 1.72 ′′ , respectively. Our source detection and extraction procedure is designed to simultaneously select sources at 3.6 and 4.5 µm, and provide equivalent photometry in both bands. The source detection is performed on a mean combined 3.6 and 4.5 µm image. The 3.6 and 4.5 µm image combination is carried out with MONTAGE 21 , which models background flux by co-adding and re-projecting background corrected images into a mosaic. Source detection is performed with SEXTRACTOR (Bertin & Arnouts 1996) and detected sources are required to comprise at least three contiguous pixels with fluxes at least 1.5σ above the local background. There are 27170 unique 3.6 and 4.5 µm sources that meet this requirement and that are detected at ≥ 5σ. Photometry is measured in 3.8 ′′ diameter apertures with the APPHOT task in IRAF. The advantage of APPHOT is that the photometry is measured in fixed apertures at specified source positions. The measured aperture photometry is corrected to "total" fluxes, assuming point-source profiles, and using the calibration derived by the SWIRE team for IRAC data with multiplicative correction factors of 1.36 and 1.40 at 3.6 and 4.5 µm, respectively (Surace et al. 2005). The completeness of the IRAC catalog is established by inserting artificial galaxies with a maximum half-light radius of 1.5 ′′ in the IRAC images. Our source extraction and photometric procedure is repeated and a source is considered to be recovered if the extracted position and magnitude are within 1.5 ′′ and 5σ, respectively, of the input values. The 50% completeness limits are 22.5 mag (3.63 µJy) and 22.2 mag (4.79 µJy) at 3.6 and 4.5 µm respectively. The far-IR emission detected by SPIRE traces emission from cold dust. Therefore, point sources in blank-field surveys, such as H-ATLAS, are primarily expected to be external galaxies and not stars in the Milky Way. Indeed, of the ∼ 6600 SPIRE sources in ∼ 14 deg 2 in the H-ATLAS SDP field only 78 are Galactic stars, and two are candidate debris disks (Thompson et al. 2010). Since stars dominate the IRAC catalog at the brightest fluxes and have magnitude distributions that are different to galaxies, it is necessary to remove them prior to counterpart identification (e.g. Smith et al. 2011). Unfortunately the angular resolution of IRAC (FWHM ∼ 1.7 ′′ ) is insufficient to morphologically distinguish between stars and high-redshift galaxies. Furthermore, while stars can be identified in [3.6] − [4.5] color-magnitude space (e.g. Eisenhardt et al. 2004) we are limited to just single IRAC band data for ∼ 45% of the total IRAC catalog. Therefore, instead of using IRAC data alone to reliably identify stars, we make use of the stellar classification in SDSS. For this we match the IRAC catalog to SDSS DR7 (Abazajian, et al. 2009) using positional information. ∼ 40% of IRAC sources are matched with SDSS catalog within search radius 1.5 ′′ . Out of total 27170 IRAC sources, there are 4239 (16%) sources that are classified as stars by SDSS. Once these are removed, the final IRAC catalog has 22931 galaxies. This catalog is used for the SPIRE source identification and subsequent analysis. We note that Fazio et al (2004b) showed that faint stars are a only minor contribution to the IRAC population, with < 10 3 mag −1 deg −2 for magnitudes fainter than 16 mag at both 3.6 and 4.5 µm. Therefore, stars that are too faint to be detected as such by SDSS are not expected to affect our statistical analysis. In addition, SDSS QSO population statistics (Schneider et al. 2007) Fazio et al. 2004b) and EGS (Barmby et al. 2008). H-ATLAS galaxy counts are more consistent with SDWFS. The 50% IRAC completeness limits of our imaging at 22.5 and 22.2 mag at 3.6 and 4.5 µm, respectively, are indicated by vertical dash-dot-dot lines. 5σ detection limit of SDWFS with 3 ′′ aperture is 22.8 and 22.3 mag at 3.6 and 4.5 µm, respectively. centroids), and therefore the effect of QSOs on our statistics is also expected to be negligible. We conclude that the counterpart identification statistics are unlikely to be affected by unresolved source contamination in our input catalog. In figure 1 we compare the IRAC galaxy counts in H-ATLAS/Spitzer area with some of the previous IRAC deep and wide fields. H-ATLAS galaxy counts are more consistent with SDWFS (Ashby et al. 2009), although the variations between fields are likely to be due to cosmic aariance. 3. SPITZER IRAC IDENTIFICATION OF SPIRE SOURCES The size of the SPIRE beam is 18.1 ′′ (FWHM) at 250 µm, and 25.2 and 36.9 ′′ at 350 and 500 µm, respectively. The spatial density of IRAC-bright galaxies is high enough that multiple sources may lie within the SPIRE beam. Typically, this will include the true SPIRE counterpart and unassociated foreground and background galaxies. However, the surface density of 250-µm bright sources is sufficient that a single SPIRE source may be composed of emission from multiple galaxies. Therefore, choosing the nearest object as the counterpart of a given SPIRE source can be misleading. Instead, we perform a likelihood ratio (LR) analysis (Section 3.1; Sutherland & Saunders 1992;Ciliegi et al. 2003;Brusa et al. 2007;Smith et al. 2011;Fleuren et al. 2012), which uses positional information and the magnitude distribution of counterparts and background sources, to identify SPIRE sources in the IRAC catalog. Furthermore, the LR analysis shows that galaxies associated with SPIRE sources occupy a distinct region of IRAC color-flux space (figure 4), which we use in Section 3.4 to identify additional counterparts. We note that other statistical matching techniques have been used in astronomical implementations. For example, the corrected Poissonian probability (p-statistic; Downes et al. 1986) uses the surface density to calculate the probability of a source with the observed magnitude being detected within the radius investigated (e.g. Ivison et al. 2002Ivison et al. , 2007Pope et al. 2006;Chapin et al. 2009). However, this technique is most appropriate for catalogues in which the surface density is low (e.g. radio data) and favors counterparts that are brighter than the background population. Our IRAC data do not have lowsurface density, and we do not wish to assume that the Herschel counterparts are typically the brightest IRAC sources. Bayesian techniques (e.g. Budavari & Szalay 2008), which use a priori knowledge of the counterpart population to guide the identification process, could also be employed (e.g. Brand et al. 2006;Gorjian et al. 2008). However, we do not wish to bias our results by assuming a priori knowledge of the SPIRE population at IRAC wavelengths. The LR analysis is advantageous because it computes the intrinsic IRAC magnitude distribution of the SPIRE sources from the data provided (see e.g. Fleuren et al. 2012). The Likelihood Ratio method The LR, L, is the ratio of the probability that the IRAC source is the correct SPIRE counterpart with the equivalent probability for an unassociated background source. Following Sutherland & Saunders (1992), L is calculated as: L = q(m)f (r) n(m) ,(1) where q(m) and n(m) are the normalized magnitude distributions of counterparts and background sources, respectively. The radial probability distribution of the separation between the SPIRE 250 µm and IRAC positions is denoted by f (r). We estimate n(m), the normalized magnitude distribution of background sources, by averaging the source counts in areas of circle within 7.2 ′′ radius of 100 different realizations of 159 random positions. The radius is three times the positional uncertainty of the SPIRE catalog. We use σ pos = 2.4 ± 0.9 ′′ (Smith et al. 2011) calculated in the H-ATLAS SDP field from the positional offsets between SDSS DR7 galaxies and 6621 SPIRE sources. The center of each random circle is required to be at least 15 ′′ away from the nearest SPIRE centroid to minimize possible contamination of real association. The quantity q(m), the normalized magnitude distribution of true IRAC counterparts to SPIRE sources cannot be directly derived. We empirically estimate q(m) by first subtracting the magnitude distribution of background galaxies from the magnitude distribution of all IRAC sources within search radius. This results in an estimate of the magnitude distribution of sources that are in excess of the background, N excess (m), and assumed to be the true counterparts to SPIRE sources, Then FIG. 2.-The magnitude distribution of IRAC galaxies at 3.6 (top) and 4.5 µm (bottom) within the 7.2 ′′ counterpart search radius around SPIRE sources. We calculate the distribution of background galaxies and subtract these to reveal an excess around SPIRE sources which peaks at 19.0 mag at both 3.6 and 4.5 µm . These are the magnitude distributions of detected IRAC galaxies that are associated with SPIRE sources. The magnitude distribution of background sources is obtained by averaging the counts within 7.2 ′′ of 100 different realizations of 159 random positions that are each > 15 ′′ from the nearest SPIRE centroid. To reduce the Poisson fluctuations associated with the small sample of 159 SPIRE centroids, the magnitude distribution of IRAC counts within SPIRE error circles is averaged over two bins. The 50% Spitzer-IRAC completeness limits of 22.5 and 22.2 mag at 3.6 and 4.5 µm, respectively, are indicated by vertical dash-dot-dot lines. All of the excess galaxies that are associated with SPIRE sources are brighter than these detection limits indicating that the IRAC data are sufficiently deep to detect counterparts to all typical SPIRE sources. q(m) is calculated via: q(m) = N excess (m) m N excess (m) × Q 0 ,(2) where Q 0 is the fraction of true counterparts above the IRAC detection limit. The where r is the separation between the SPIRE 250 µm and IRAC centroids, and σ pos is the positional uncertainty of the SPIRE catalog. The median 250 µm signal-to-noise ratio of the SPIRE sources in our sample is 6.33 and only 43 (27%) have SNR> 8. Therefore, we ignore the improvement in σ pos for sources with high signal-to-noise ratio (Smith et al. 2011). Each H-ATLAS source may have several potential counterparts. Therefore, we calculate the reliability, R j , of each IRAC galaxy, j, in SPIRE counterpart search radius. R j is the probability that the galaxy, j, is the correct IRAC counterpart to the SPIRE source. Following Sutherland & Saunders (1992): R j = L j i L i + (1 − Q 0 ) ,(4) where i represents each IRAC source in the search radius. To accept a potential counterpart as reliable we require that R j ≥ 0.8, i.e. there is < 20% chance of a false association. This criterion was used by Smith et al. (2011) for the counterpart search with SDSS data and our results on the identification rate with IRAC data can be directly compared to their results based on optical imaging. We note that the choice of the exact limiting value of R does not strongly affect our conclusions. Of the 123 SPIRE sources with R ≥ 0.8 counterparts (section 3.3) 107 (87%) have R ≥ 0.9, and 101 (82%) have R ≥ 0.95. For a catalog in which identified sources are required to have R ≥ 0.8 the overall false detection rate is N (false) = Rj ≥0.8 (1 − R j ) .(5) 3.2. Q 0 calculation In order to apply the LR method to the SPIRE and IRAC catalogs we must first compute Q 0 . Smith et al. (2011) calculated Q 0 from the total number of SDSS galaxies in the error circles of SPIRE sources. However, this method requires some a priori chosen radius and is only valid if each SPIRE source is associated with only one SDSS galaxy. Instead, we make use of an alternative method, derived by Fleuren et al. (2012), which is radius independent and unaffected by the clustering of IRAC sources. Clustering can be a significant complication because SPIRE sources are expected to reside in overdense regions similar to environments of sub-mm galaxies (e.g. Aravena et al. 2010b;Hickox et al. 2012). For the purposes of this discussion we define a SPIRE error circle without a true IRAC counterpart as a "blank". Directly observed blanks arise due to counterparts below the IRAC detection limit or beyond the search radius, r. However, the total number of blanks must also be statistically corrected for contamination by unassociated foreground or background galaxies. Thus, the true number of blanks (S t ) is equal to the number of observed SPIRE blanks (S(r)) and the number of SPIRE sources that are falsely matched with random background galaxies. We define R o (r) as the number of random positions that contain a IRAC galaxy whilē R(r) = N − R o (r) is the number of random blanks. Then, S t =S(r) + [S t × R o (r) N ] ,(6) with R o (r) andR(r) calculated from 100 random realizations of N = 159 error circles, which are all located at least 15 ′′ away from SPIRE centroids. Q 0 is defined as the fraction of true counterparts above the IRAC detection limit. Thus, for an infinite search radius, the rate of true blanks,S t /N is simply equal to 1 − Q 0 . FIG. 3.-Quantities employed in the calculation of Q 0 .S(r)/R(r), the ratio of SPIRE blanks to random blanks, is fitted with 1 − Q 0 F (r) in order to calculate Q 0 . The best fit values are Q 0 = 0.93 ± 0.09 and 0.92 ± 0.08 at 3.6 and 4.5 µm, respectively. We also showR(r)/N andS(r)/N , the rate of random positions with at least one IRAC source and the rate of SPIRE blanks, respectively. Fleuren et al. (2012) demonstrate that equation 6 can be rearranged to show thatS t /N =S(r)/R(r), and thus 1 − Q 0 = S(r)/R(r). However, there is also the possibility that the true counterpart is outside of the examined area and this must be accounted for in the Q 0 estimate. The probability that the real SPIRE source is outside the search radius can be derived analytically from the normalized SPIRE source distribution, f (r) (equation 3). This probability, F (r), is F (r) = r 0 P (r ′ ) dr ′ = 1 − e − r 2 2σ 2 ,(7) where P (r) = 2πrf (r) (Fleuren et al. 2012). Assuming that the probability of a SPIRE source being a true blank (1 − Q 0 ) and the probability that the detected counterpart is outside of the search radius (1 − F (r)) are independent, it can be shown that the total probability that there is no counterpart is (see Fleuren et al. 2012) S t N = 1 − Q 0 F (r) .(8) However, we have already shown thatS t /N =S(r)/R(r). FIG. 4.-Top: IRAC color-magnitude diagram. All LR counterparts with R ≥ 0.8 and potential counterparts are highlighted. Potential counterparts are IRAC galaxies within 7.2 ′′ of SPIRE sources that are not otherwise identified with the LR method. The LR counterparts typically have [3.6] − [4.5] > − 0.4 and are brighter the background IRAC population. IRAC galaxies above the solid line and within 7.2 ′′ of SPIRE positions have a probability of being random associations of ≤ 20%. This discriminator is used to identify an additional 23 IRAC counterparts to 13 SPIRE sources ("color IDs"; section 3.4). Bottom: SPIRE color-flux diagram for all sources in the H-ATLAS SDP field. We highlight sources with identified IRAC counterparts using the LR and color-magnitude methods. Sources that have formal 500 µm fluxes below the 1σ detection limit are shown as lower limits and are excluded from the statistical analysis. A two dimensional KS test between the background population and the identified SPIRE sources yields the probability that the two datasets are drawn from the same parent population of p = 0.172. Thus, we conclude that the identified galaxies are not a strongly biased subsample of SPIRE sources, although there may be a small selection effect. Average error bars are shown at the top-left hand corner of both plots. Thus, using the observablesS(r) andR(r) one can calcu- Figure 3 shows S(r)/R(r),R(r)/N , andS(r)/N , observed for search radii of one to 10 ′′ . We calculate the constant, Q 0 , by fitting 1 − Q 0 F (R) toS(r)/R(r). χ 2 minimization yields Q 0 = 0.93 ± 0.09, 0.92 ± 0.08 at 3.6 and 4.5 µm, respectively. late Q 0 via 1 − Q 0 F (r) =S(r)/R(r). LR counterparts We apply the LR technique outlined in section 3.1 and 3.2 at 3.6 and 4.5 µm, to identify IRAC counterparts to 123 of 159 H-ATLAS SPIRE sources (Tables 1 and 2). q(m) and n(m) are calculated in a radius of 3σ pos = 7.2 ′′ (although Q 0 is radius independent; section 3.2) and only IRAC galaxies within 7.2 ′′ of SPIRE sources are considered to be po-tential counterparts. This limit includes 99.7% of true counterparts, whilst minimizing the potential contamination from unassociated sources. The analysis is performed separately at 3.6 and 4.5 µm and any IRAC galaxy with R ≥ 0.8 at either wavelength is considered to be a SPIRE counterpart. In addition, where both 3.6 and 4.5 µm data is available we combine the probabilities (R) from each wavelength, and include five counterparts that have a combined probability of being considered to be reliable identifications ≥ 0.8, but have R < 0.8 at 3.6 and 4.5 µm individually. All 123 counterparts are presented in table 2, including 81 galaxies with both 3.6 and 4.5 µm photometry. The false identification rate is 1.9% or approximately two sources (equation 5); in the case of counterparts that are identified at both 3.6 and 4.5 µm we use the highest of the two R j in this calculation. The identification rate of 77% is significantly higher than that from optical (37% in SDSS; Smith et al. 2011) or near-IR analyses (51% in VIKING; Fleuren et al. 2012) of H-ATLAS sources. This difference is likely to be primarily driven by the wavelengths of the study, which are less sensitive to Kcorrection and dust absorption than optical data. The counterpart identification rate is also significantly higher than the 50% obtained with WISE 3.4-µm data (Bond et al. 2012); this is true even if we only consider the 3.6 µm data, where 97 of the 129 SPIRE sources with 3.6 µm coverage (75%) are identified. WISE is shallower than our IRAC observations -19.7 mag at 3.4 µm compared to 22.5 mag at 3.6 µm in IRAC. However, 79% of the 3.6-µm identified counterparts have [3.6] ≤ 19.7 mag, suggesting that IRAC data to this depth would identify counterparts to ∼ 60% of SPIRE sources. The remaining difference in the IRAC and WISE identification rates is likely to be due to the resolutions of the instruments -the WISE 3.4 µm beam is ∼ 3 times larger than IRAC -although Cosmic Variance may also contribute. We note that if instead the data were limited to [3.6] < 20.5, corresponding to the 5σ point-source detection limit reached in a 120 sec integration with IRAC at 3.6 µm, the expected identification rate drops from 77% to ∼ 70%. The results presented here are insensitive to the exact value of Q 0 . When using a 1σ lower value for Q 0 (i.e. Q 0 = 0.84 at both 3.6 and 4.5 µm), and comparing to the results presented in section 3.3 and table 2, the counterparts to 116 sources (95%) are unchanged. In this case there are four (3%) previously identified sources that no longer have reliable counterparts, and three (2%) previously unidentified SPIRE sources that now have R ≥ 0.8 counterparts. Counterparts identified in IRAC color-magnitude space While we reliably identify 77% of the SPIRE sources with the LR method, the values of Q 0 indicate that ∼ 90% of SPIRE counterparts are detected in IRAC. Figure 4 shows the IRAC color-magnitude diagram for galaxies in the H-ATLAS SDP field, with SPIRE counterparts identified with the LR method highlighted. These galaxies occupy a distinct region of IRAC color-magnitude space -they typically have [3.6] − [4.5] − 0.4 and are brighter than the background population. We use this property to identify counterparts to the SPIRE sources that have both 3.6 and 4.5-µm IRAC coverage but no R ≥ 0.8 counterparts from the LR method. A similar analysis was performed by Biggs et al. (2011) at 3.6 and 5.8-µm to identify to LABOCA 870-µm sources. We begin by determining the region of color-magnitude space in which there is a minimal chance of contamination by background sources (the ID region). A 20% contamination limit is used because this equivalent to R ≥ 0.8 for the LR method. We demarcate the ID region with the simplest reasonable function -a single diagonal line. The placement of the line is determined by calculating the gradients and intercepts that would yield 20% contamination from background sources within 7.2 ′′ radius of the LR counterparts. The number of LR counterparts returned is maximized to yield a bestfit gradient of 0.515 for a line that intercepts [3.6] = 16.0 mag at [3.6] − [4.5] = −2.142. This division is shown as a solid line in figure 4. IRAC galaxies that lie above this line and are within 7.2 ′′ of SPIRE sources that are otherwise unidentified, have > 80% probability of being physically associated with the SPIRE source, i.e., < 20% chance of being an unrelated foreground or background galaxies, and are considered counterparts. There are 36 SPIRE sources that do not have LR counterparts. Of these 19 have both 3.6 and 4.5 µm data, and 23 IRAC counterparts to 13 of these sources are identified with the color-magnitude method (seven have multiple counterparts). These sources are presented in table 2. The contamination rate for this method is 12.6% (approximately two IRAC galaxies), calculated by summing the probabilities of finding a background galaxy within 7.2 ′′ of a SPIRE source occupying a distinct region of color-magnitude space. In figure 4 we show the S 250 /S 500 color-flux plot for H-ATLAS SDP sources and consider whether sources with identified IRAC counterparts are representative of the whole SPIRE population. A 2D Kolmogorov-Smirnov (KS) test between the background H-ATLAS SPIRE population and the sources with identified counterparts has p = 0.172, suggesting that the two samples are drawn from similar, but not necessarily identical, parent populations. Considering the S 250 /S 500 color and 250 µm fluxes separately yields p = 0.710 and p = 0.289, respectively. Thus we conclude that the sources with identified IRAC counterparts are very similar to the whole SPIRE population, but may have a slightly different distribution of 250 µm flux. Astrometric offset To check whether there is any noticeable astrometric shift among the input catalogs for LR analysis, in figure 5 we plot the positional differences between the SPIRE sources and the corresponding IRAC counterparts. The median separation between SPIRE and IRAC sources is 0.10 ′′ . We also investigate the overall astrometric reference frame difference between IRAC and SDSS catalogs, using ∼ 40% of the IRAC sample that is also identified with SDSS galaxies within in 2 ′′ . The positional difference between IRAC and SDSS is 0.19 ± 0.17 ′′ and we found no systematic offset either in RA or Dec. Our IRAC data are complemented with VIKING Z, Y, J, H and K s -band photometry, which is measured in 2 ′′ diameter apertures (Sutherland et al. in prep). We cross-match FIG. 5.-The distribution of ∆RA and ∆Dec offsets between SPIRE sources and the corresponding IRAC counterparts identified in this work. The astrometric median separation between SPIRE sources and IRAC is 0.10 ′′ which is consistent with the overall IRAC astrometric uncertainty tied to the SDSS astrometry. The three concentric circles have radii of 1, 2 and 3σpos (7.2 ′′ ) from the SPIRE centroid. The overall offsets in RA and Dec show that there is no systematic offset between input SPIRE and IRAC sources. the IRAC (FWHM ∼ 1.6 ′′ ) to the nearest neighbor VIKING (FWHM ∼ 0.9 ′′ ) source within 2 ′′ search radius. Indeed, the mean offset between the IRAC and VIKING positions is 0.09 ± 0.50 ′′ and 0.28 ± 0.28 ′′ in RA and Dec, respectively. We find no statistically significant astrometric error between two catalogs. 4. DISCUSSION As described in sections 3.3 and 3.4 we identified IRAC counterparts to 136 (86%) of the SPIRE sources. 123 of these are identified with the LR method and 23 counterparts to 13 SPIRE sources are identified with the color-magnitude method ( Table 1). The contamination rate of 1.9% for the LR method is smaller than that from the color-magnitude method (12.6%) but corresponds to a similar number of IRAC galaxies -two to three. Combining the results from the two identification methods, we expect that the total false identification rate of the catalog presented in Table 2 is 3.6%, such that five to six of the IRAC counterparts presented in table 2 are false. . From left-to-right we show a three-color SDSS image (g, r, and i bands), the 3.6 µm, and the 4.5 µm data. The galaxies identified as the SDSS (SDP.2244) and IRAC (H-ATLAS ID-2244) counterparts to the SPIRE source are marked with an 'X' and a small circle, respectively. The large circle has 7.2 ′′ radius and encompasses the SPIRE 3σpos area in which counterparts are identified. Contours show the SPIRE 250 µm emission at 5, 7, 9, 11σ levels. In the absence of high-resolution sub-mm imaging we cannot determine whether this SPIRE source is the results of blended emission from the two identified galaxies, or whether one of those counterparts is a chance association. Statistics of Identified Sources FIG. 7.-H-ATLAS ID-891: an example of counterpart identification using the color-magnitude method. Symbols are as figure 6. There is no LR counterpart in the SDSS r-band (Smith et al. 2011) or with IRAC at 3.6 or 4.5 µm. However, there are two galaxies that are both identified as SPIRE counterparts on the basis of their IRAC colors and magnitudes. These two galaxies have R = 0.41 and 0.47 at 3.6 µm, and R = 0.44 and 0.50 at 4.5 µm and neither is detected in SDSS. It is probable that the SPIRE source is a blend of the emission from these two IRAC galaxies although, in the absence of spectroscopic redshifts it is unclear whether the configuration is caused by a line-of-sight alignment or an interaction between the two IRAC galaxies. FIG. 8.-H-ATLAS ID-827 has four counterparts identified with the IRAC color-magnitude method and is the most complex system in our sample. The westernmost and easternmost galaxies are detected by SDSS (but are not identified as counterparts in these data; Smith et al. 2011), and have photometric redshifts of z = 0.07 ± 0.01 and z = 0.20 ± 0.03, respectively. The two fainter counterparts are not detected in SDSS but have [3.6] − [4.5] > 0, indicating that they may be at z 1.4 (Papovich 2008). We conclude that the SPIRE source is most likely to be comprised of blended emission from the four galaxies. Symbols are as figure 6. Papovich 2008). We expect that more than half of the sub-mm bright sources reside at z 1.4. For reference the color-redshift tracks of Arp 220 (Silva et al. 1998), Mrk231 (Berta 2006) and the average sub-mm galaxy SED (Wardlow et al. 2011) are shown. are different galaxies to the SDSS counterparts. On the basis of the false-identification rates of the two studies (4.2% for SDSS and 3.6% for IRAC) approximately four counterparts are expected to disagree between the two surveys, which is consistent with that observed here. The two sources with different counterparts identified in IRAC and SDSS are H-ATLAS ID-2244 and 6962. We show one of these sources, H-ATLAS ID-2244, as an example in figure 6. In this case the SDSS (SDP.2244) and IRAC (H-ATLAS ID-2244) counterparts are separated by 3.2 ′′ and both have a ≤ 3% probability of being false identifications: R = 0.97 in SDSS and R = 0.99 at both 3.6 and 4.5 µm. In the absence of additional data, such as sub-mm or radio interferometry, the true nature of this source is unclear. It is possible that one of the counterparts is a chance association, although we cannot say which one. It is also possible that the SPIRE source is comprised of a blend of emission from both counterparts in a single SPIRE beam. Blending is also an important consideration for counterparts that are identified with the color-magnitude method. Seven of the 13 SPIRE sources identified with this method have multiple IRAC counterparts; conversely the LR method has none. This apparent disparity is not surprising because the LR method implicitly assumes that each SPIRE source has a single IRAC counterpart. Where there are N multiple counterparts contributing equally to the sub-mm flux the average R cannot exceed 1/N , and will typically be ∼ 1/N . Due to the large SPIRE beam (FWHM = 18.1 ′′ at 250 µm) it is likely than in at least a few cases a single SPIRE source may be composed of blended emission from multiple galaxies or from galaxy interactions. The color-magnitude method does not consider the presence of other IRAC sources and is therefore not biased against multiple counterparts. In addition, beyond requiring that counterparts to lie within the SPIRE counterpart search radius, the color-magnitude method does not consider the separation between the SPIRE and IRAC centroids and therefore it is not biased against wide-separation counterparts. However, it does require both 3.6 and 4.5 µm fluxes and it assumes that all SPIRE sources have a similar color-magnitude distribution at 3.6 and 4.5 µm. The seven SPIRE sources in the sample that have multiple IRAC counterparts are likely to be a combination of blended SPIRE sources and individual sub-mm galaxies associated with multiple interacting IRAC sources. H-ATLAS ID-891 is shown as an example in figure 7. In this case all the detected SDSS galaxies are 3.5 ′′ away from the SPIRE centroid and all have R < 0.8 (Smith et al. 2011). There are two potential counterparts in the IRAC data that are not detected in SDSS. Using the LR method alone these two nearby sources have reliability of R = 0.41 and 0.44 at 3.6 µm and R = 0.47 and 0.50 at 4.5 µm. However, both galaxies have [3.6] and [4.5] that place them in the region of color-magnitude space with < 20% probability of being random associations with the SPIRE emission and as such they are both identified as counterparts. The two galaxies have similar colors, with ([3.6] − [4.5]) = 0.38 and 0.43 mag, consistent with a redshift of z 1.4 (Papovich 2008). The two IRAC sources are separated by 2.9 ′′ (24.9 kpc at z = 2, 23.0 kpc at z = 3. Furthermore, the SPIRE source is bright (S 500 = 84 ± 9mJy) and red in the sub-mm bands, with a rising spectrum from 250 to 500 µm, indicating a sub-mm photometric redshift of z 3 (e.g. Lapi et al. 2011), although we caution that the dust temperature and redshift are degenerate in the sub-mm bands. It is likely that H-ATLAS ID-891 is a blend of emission from the two identified counterparts and it is possible that the sub-mm emission is the result of a merger or interaction of two gas-rich galaxies (e.g. Aravena et al. 2010b). Spectroscopic data are required to verify this scenario. and we identify ID-891 as a target for additional follow-up, especially to identify the exact nature of the sub-mm emission. H-ATLAS ID-827, shown in figure 8, is the most complex system in our sample. In this case there are four IRAC counterparts identified on the basis of their colors and magnitudes. The four galaxies have a wide range of [3.6] with values from 17.8 mag to 20.2 mag. The counterparts do not appeared to be clustered, as may be expected for gravitational lensing or a multi-component interaction, and each is separated from its nearest neighbor by ∼ 4 ′′ . The two brightest galaxies are detected by SDSS, but were not identified as SPIRE counterparts by Smith et al. (2011); they have photometric redshifts of z = 0.07 ± 0.01 and z = 0.20 ± 0.03 for the westernmost and easternmost, respectively. The two fainter counterparts are not detected in SDSS and both have [3.6] − [4.5] > 0, indicating that they are likely to be at z 1.4 (Papovich 2008). The morphology, astrometry and available redshift information suggests that gravitational lensing in unlikely and that H-ATLAS ID-827 is most likely to be a blended source, which is comprised of two low-redshift and two high-redshift components. Redshifts of Identified Sources Knowledge of the redshift distribution of SPIRE sources is central to understanding their role in the Universe. However, only 50% of the IRAC counterparts are detected in SDSS and 74% have VIKING photometry. Therefore, any redshift distribution derived from the optical data will be biased to low redshifts where the detection rates in these surveys are high. Instead, in figure 9 (Silva et al. 1998), Mrk 231 (Berta 2006) and the average 870-µm selected sub-mm galaxies (Wardlow et al. 2011) are shown, and an A V = 1 mag reddening vector and the median error bar are plotted in the lower lefthand corner. The SPIRE counterparts are slightly bluer than both these tracks suggesting that they may have less dust reddening and A V values that are 0.5 to 1 magnitudes smaller than the SED templates. 3.6 and 4.5 µm coverage, 47 of which have an optical redshift. All of the optical redshift are z < 0.8; 32 (68%) of these galaxies have ( Papovich 2008). Thus, up to ∼ 40% of the identified SPIRE population may lie at high redshifts. Deep spectroscopy and photometry, particularly at near-IR wavelengths, is required to determine more precise redshift information. We also investigate the redshift distribution of the SPIRE sources by calculating sub-mm photometric redshifts from the fluxes of the three SPIRE bands. We employ a template fitting method which uses χ 2 minimization, comparing the observed fluxes to the SEDs of Arp 220 (Silva et al. 1998), Mrk 231 (Berta 2006). We caution that the sub-mm photometric redshifts are limited by the absence of longer wavelength data, and the degeneracy between redshift and dust temperature. Furthermore, flux boosting is an important consideration and will add significant errors to the fluxes of the faintest sources. Due to these uncertainties, we do not consider the sub-mm FIG. 11.-Top: Dust temperature as a function of the infrared luminosity L IR (8 -1100 µm) for H-ATLAS/IRAC galaxies with SDSS redshifts. L IR is obtained by fitting the SPIRE flux with β = 1.5 in isothermal modified blackbody model (equation 9). Error bars are median 1σ standard deviation from the best-fit models with dust temperature (T d ) and total IR luminosities L IR as free parameters. Bottom: Dust temperature as a function of the redshift for H-ATLAS/IRAC galaxies. See text for the comparison galaxy samples plotted here. photometric redshifts on a source-by-source basis, but instead examine the overall distribution. This sub-mm photometric redshift distribution peaks at z ∼ 2 with a tail out to z ∼ 4, and suggests that the majority of the IRAC counterparts without SDSS redshifts lie at z ∼ 2. This result is consistent with the photometric redshifts of the whole SPIRE population (e.g. Amblard et al. 2010;Lapi et al. 2011) and the redshift distribution of 850 µm sources (e.g. Chapman et al. 2005;Wardlow et al. 2011). Now that the counterparts to a significant majority of SPIRE sources are known, follow-up spectroscopic campaigns to establish the redshifts of all the sources are feasible. Specific star formation rate (sSFR) for our IRACidentified SPIRE galaxies as a function of the redshift. Filled circles and triangles represent H-ATLAS galaxies in stellar mass bins 10 9.8 -10 10.8 M ⊙ and 10 10.8 -10 11.5 M ⊙ , respectively. Our data are compared to Pérez-Gozález et al. (2008) and Damen et al. (2009). Upper (lower) two lines are for the smaller (larger) mass bin. The error bar on the top left shows the average 68% confidence range on log(sSFR). Bottom: Star formation rate as a function of the stellar mass. We plot results from a previous study for comparison: PACS-COSMOS and PACS-GOODS South (Rodighiero et al. 2011). We also show the mean SFR and stellar mass correlation at z = 2 (Daddi et al. 2007), 1 and 0 (Elbaz et al. 2007). The majority of the H-ATLAS/IRAC galaxies fall between z = 0 ∼ 2 in SFR-M⋆ relation showing a slight excess of star formation rate compared with populations at similar redshifts, but selected at optical and near-IR wavelengths. Properties of Identified Sources Near-IR Colors Mrk 231 (Berta 2006) and the average 870-µm selected submm galaxy (Wardlow et al. 2011). The SPIRE counterparts are slightly bluer than both these tracks suggesting that they may have less dust reddening and A V values that are 0.5 to 1 magnitudes smaller than the SED templates. IR Luminosities and Dust Temperatures: To understand properties of the identified H-ATLAS/IRAC galaxies associated with SDSS redshifts, we first consider a SED analysis of the SPIRE data making use of the redshifts based on the IRAC identification. First we consider the far-IR/sub-mm portion of the SED with 3-band SPIRE fluxes at 250, 350, and 500 µm and reproduce the analysis of Amblard et al. (2010) where the dust temperature (T d ) and IR luminosities L IR were studied. We make use of a modified black-body with isothermal dust temperature to describe the dust emission, in which flux can be written as f µ ∝ µ 3+β e (hµ/kT d −1) .(9) To be consistent with previous other estimates from the literature we fix β = 1.5 and fit to the SPIRE fluxes to establish the dust temperature and the overall normalization to the SED. The IR luminosity is estimated from the best-fit SED over the wavelength range of 8 to 1100 µm. Since we only fit to the dust emission at wavelengths greater than 250 µm our luminosities are likely low by a factor of 2 to 3 if there are any warm dust, heated by AGNs, present in these galaxies. However, our overall uncertainty on the IR luminosity estimates is at least a factor of 5. For the sample as a whole we find an average dust temperature of 26 ± 9 K, which compares well with the average dust temperature of 27 ± 8 K for z < 0.1 SPIRE galaxies based on SDSS identifications only (Amblard et al. 2010). We summarize our results related to the FIR SED analysis in figure 11 where we show the L IR -T d relation and the L IR as a function of the redshift, determined from the reliable H-ATLAS IDs with either SDSS spectroscopic or photometric redshift. We also compare the H-ATLAS dust temperatures and IR luminosities with sub-mm bright galaxy samples in the literature. Samples include the sources in BLAST with COMBO-17 (Wolf et al. 2004) In figure 11 we find that the SPIRE-selected galaxies in H-ATLAS with IRAC-based identifications for redshifts span the luminosity range of 10 10 to 10 12 L ⊙ from sub-LIRG luminosities to ULIRG conditions. These luminosities and the ranges are comparable to BLAST-detected sources, while they span lower than the local IRAS-selected galaxies, which tend to be at lower redshifts z < 0.05 and have higher luminosities; such a difference is understandable since IRAS is an all-sky shallow survey and detects primarily the rare, bright galaxies in the near-by universe. Stellar Masses and Star-Formation Rates: With the crossidentification we can expand the SED analysis over 3 orders of magnitude in wavelength from optical to sub-mm. For optical and near-IR data we make use of SDSS, UKIDSS (Lawrence et al. 2007), VIKING (Sutherland et al. in prep), and IRAC 3.6 and 4.5 µm fluxes. The analysis of full optical to submm SEDs is similar in spirit to Dunne et al. (2011) where the SEDs of SPIRE sources with reliable identifications using SDSS data were analyzed. We make use of the MAGPHYS SED modelling code (da Cunha, Charlot & Elbaz 2008) for this work. The SED modelling in the code is done in a energybalance manner such that the absorbed light in shorter UV and optical wavelengths is accounted for by the thermal reradiation at far-infrared and sub-mm wavelengths. For the H-ATLAS/IRAC galaxy sample we have 5 optical fluxes from SDSS (u, g, r, i, z), 5 near-IR fluxes from UKIDSS+VIKING (Z, Y, J, H, K s ), and 2 IRAC channels (3.6, 4.5 µm), in addition to 3 SPIRE bands. Redshift information of the sources are from SDSS spectroscopic observations (z =0 -0.8). For comparative work with other sub-mm galaxy samples we focus on results related to stellar mass M ⋆ and starformation rate (SFR), as given by MAGPHYS best-fit models. In figure 12 (top figure) we plot the specific star-formation rate (sSFR) given as SFR/M ⋆ vs. redshift with our sample sub-divided to two stellar mass bins. We find that the sSFR of IRAC IDs marginally show an anticorrelation with galaxy stellar mass. Such a behavior may be related to the downsiz-ing scenario (Cowie et al. 1996). However, when compared to Damen et al. (2009) andPérez-Gozález et al. (2008), IRAC IDs shows 0.4 ∼ 0.5 dex higher sSFRs. The Damen et al. (2009) galaxy sample is an IRAC-selected sample of galaxies out to z ∼ 1.8 in ECDFS with photometry supplemented with optical, near-IR, and MIPS 24 µm fluxes. The Pérez-Gozález et al. (2008) sample is selected with IRAC at 3.6 and 4.5 µm in a variety of fields. While our SPIRE-selected sample has IRAC identifications, they are not likely to be the majority of the galaxies in both these studies. This is also clear from figure 4 where we show that most of the identified SPIRE sources are distributed differently from the background population in the color-magnitude space. They occupy the top-end of the starformation in galaxies selected at near-IR wavelengths. In Fig. 12 (bottom) we plot the SFR vs. the stellar mass of our sample. The SFR of normal local star-forming galaxies at z ∼ 1 is known to correlate strongly with the stellar mass and form the so-called star-formation main-sequence (Brinchmann et al. 2004;Salim et al. 2007;Peng et al. 2010). Such a correlation is also observed at higher redshifts, albeit with a different normalization (e.g. Daddi et al. 2007;Elbaz et al. 2007;Noeske et al. 2007;Pannella et al. 2009;Daddi et al. 2009;Gonzalez et al. 2010;Rodighiero et al. 2010;Karim et al. 2011). However, the typical dispersion of correlation is known to be ∼ 0.3 dex over a wide range of redshifts z = 0 ∼ 2. Local (U)LIRGs and high-redshift SMGs have SFRs in excess of the main-sequence and are defined as starburst galaxies (e.g. Elbaz et al. 2007Elbaz et al. , 2011. These galaxies generally occupy a region that is ∼ ×10 above the mainsequence. However, there is also evidence that some highredshift (U)LIRGs and SMGs do not have enhanced starformation rates relative to their stellar mass and that the starburst fraction may decrease at high redshifts (e.g. Tacconi et al. 2008;Rodighiero et al. 2011). This suggests changes to the mode of star-formation in (U)LIRGs between z = 0 and z = 2 away from the starburst mechanism (e.g. Daddi et al. 2010b;Genzel et al. 2010;Elbaz et al. 2011;Krumholz et al. 2012;Melbourne et al. 2012). In our work, ∼ 50% of the subsample of IRAC-identified SPIRE galaxies (z < 1) lie near the starburst region. Thus, it is possible that half of the IRAC-identified SPIRE galaxies with SDSS redshifts are in fact starbursts that lie off the main-sequence. SUMMARY We have identified Spitzer-IRAC counterparts to sources selected with Herschel-SPIRE at 250 µm in the H-ATLAS survey. Among 159 SPIRE centroids, we found 123 reliable IRAC counterparts using a likelihood ratio analysis. The identified SPIRE sources are distributed differently in IRAC colormagnitude space compared to the field population. Therefore, we made an additional selection based on the IRAC ID locality to yield a further 23 counterparts to 13 SPIRE sources. Seven SPIRE sources have multiple IRAC counterparts. These are likely to be due to blended emission in the SPIRE beam. In total 146 reliable IRAC counterparts to 136 SPIRE sources were identified, including seven that have more than one IRAC counterpart. The identification rate of 86% is higher than that of wide-field ground-based optical and near-IR imaging of Herschel fields. The galaxies with unknown redshifts and that are not detected in SDSS and VIKING imaging data have SPIRE colors indicative of high redshift sources and [3.6] − [4.5] > 0, suggesting that they are likely to be at z 1.4. We estimate the ∼ 40% of SPIRE sources lie at high redshifts, although the exact redshift distribution of SPIRE sources remains elusive. The counterparts presented here can now be pursued for followup data to further investigate the nature of SPIRE galaxies. The majority of our detected galaxies with sub-LIRG to LIRG luminosities are not intense starbursting galaxies in the local universe, though they have above average specific star-formation rates. Note. -⋆ ID with color-magnitude method. a Coordinates are the position of IRAC counterparts. b Photometry from the VIKING survey (Z, Y, J, H, Ks) is extracted in 2 ′′ diameter apertures (Sutherland et al. in prep). c The reported SPIRE 350 and 500 µm fluxes are measured at the postions of 250 µm sources; no SNR cut is applied. d R is reliability of the counterparts calculated separately at 3.6 and 4.5 µm from the LR method (section 3.1). e Redshifts are from the SDSS catalogue for sources detected in that survey. f The separation between the centroid of the SPIRE emission and the IRAC counterpart. Note. -⋆ ID with color-magnitude method. a Coordinates are the position of IRAC counterparts. b Photometry from the VIKING survey (Z, Y, J, H, Ks) is extracted in 2 ′′ diameter apertures (Sutherland et al. in prep). c The reported SPIRE 350 and 500 µm fluxes are measured at the postions of 250 µm sources; no SNR cut is applied. d R is reliability of the counterparts calculated separately at 3.6 and 4.5 µm from the LR method (section 3.1). e Redshifts are from the SDSS catalogue for sources detected in that survey. f The separation between the centroid of the SPIRE emission and the IRAC counterpart. Herschel-ATLAS is a project which uses IAU ID H-ATLAS RA a Dec a Z b Y b J b H b Ks b 3.6 µm 4.5 µm S 250 c S 350 c S 500 c R d R d z e Separation f ID (J2000) (J2000) (mag) (mag) (mag) (mag) (mag) (mag) (mag)( [ 3 . 6 ] 36and [4.5] distributions of n(m) and N excess (m) are shown in figure 2.The radial probability distribution, f (r), is given by We next compare the counterparts identified in the IRAC data with results based on SDSS (Smith et al. 2011). Of the 146 IRAC counterparts, 102 (70%) are undetected in SDSS, 52 (36%) agree with the SDSS identification, and two (1%) FIG. 6.-H-ATLAS ID-2244: an example of source that is identified using the LR method, but where the IRAC counterpart (R = 0.99 at both 3.6 and 4.5 µm) is different from the SDSS counterpart (SDP.2244, R = 0.97; Smith et al. 2011) FIG. 9 . 9-The distribution of IRAC [3.6] − [4.5] color for SPIRE counterparts with photometry at both wavelengths. For the 47 sources with SDSS spectroscopic or photometric redshifts we also plot the [3.6] − [4.5] color against redshift. The 57 SPIRE-IRAC identifications without reliable redshifts are either faint or undetected in SDSS and 43 of them have [3.6] − [4.5] > 0, an indicator of high redshift source (z 1.4; [3. 6 ] 6− [4.5]) < 0, while 15 (32%) have ([3.6]−[4.5]) > 0. The 57 identifications without reliable redshifts from the optical are undetected in SDSS and 43 of them have ([3.6] − [4.5]) > 0, a crude indicator that z 1.4 (e.g. : In figure 10 we show IRAC and VIKING color-color diagrams, highlighting identified SPIRE counterparts in comparison with background sources that are within 7.2 ′′ of the SPIRE centroids. As we show in figure 4 the SPIRE counterparts have distinct [3.6] − [4.5] color compared to the background sources, but the two populations are indistinguishable in [K s ] − [3.6] and [z] − [K s ]. We also compare to the redshift tracks of Arp 220 (Silva et al. 1998), FIG. 12.-Top: or a SWIRE photometric redshift(Rowan-Robinson et al. 2008) and Spitzer-MIPS 70 and 160 µm fluxes(Dye et al. 2009), local ULIRGS observed with SCUBA at 450 and 850 µm(Clements et al. 2010), and local IRAS-selected galaxies with 60 and 100 µm fluxes along with SCUBA 850 µm(Dunne et al. 2000). indicate that only 2-3 QSOs are expected in the IRAC area (< 1 in the counterpart search area around SPIRE 21 http://montage.ipac.caltech.edu FIG. 1.-Galaxy number counts from H-ATLAS/Spitzer area compared with previous IRAC deep and wide fields: SDWFS (Ashby et al. 2009), First Look Survey (FLS; TABLE 1 SUMMARY 1OF IRAC COUNTERPARTS TO SPIRE SOURCES Method NSPIRE a NID b ID rate c NIRAC d N multiple NOTE. -a Number of SPIRE sources considered. b Number of SPIRE sources with at least one counterpart. c Fraction of SPIRE sources with at least one identified counterpart. d Number of IRAC counterparts identified. e Number of SPIRE sources with multiple IRAC counterparts.e Contamination rate we plot the [3.6] − [4.5] color distribution, and [3.6] − [4.5] against redshift for the counterparts with spectroscopic or photometric redshift from SDSS. This plot includes the 104 IRAC counterparts that have both FIG. 10.-IRAC and VIKING color-color diagrams. Top: [Ks], [3.6] and [4.5]. Bottom: [z ′ ], [Ks] and [3.6]. We highlight SPIRE counterparts and background sources that are within 7.2 ′′ of SPIRE centroids. Tracks for Arp 220 Herschel, an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. The H-ATLAS website is http://www.h-atlas.org/. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. US participants in H-ATLAS also acknowledge support from the NASA Herschel Science Center through a contract from JPL/Caltech.Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.This work used data from VISTA at the ESO Paranal Observatory, programme 179.A-2004. VISTA is an ESO nearinfrared survey telescope in Chile, conceived and developed by a consortium of universities in the United Kingdom, led by Queen Mary University, London.We thank Giulia Rodighiero for providing us electronic tables of COSMOS and GOODS SED fitting results. The UCI group acknowledges support from NSF CAREER AST-0645427, NASA NNX10AD42G, and an award from Caltech/JPL for US participation in Herschel-ATLAS. COMBINED PHOTOMETRY FOR IRAC IDS.14 APPENDIX TABLE 2 TABLE 2 - 2continued. +00 • 25 ′ 57. ′′ 6 18.83 ± 18.83 18.59 ± 0.01 18.20 ± 0.01 17.80 ± 0.01 17.53 ± 0.01 17.76 ± 0.01 17.92 ± 0.01 41.50 ± 6.41 19.09 ± 7.30 −12.+00 • 19 ′ 40. ′′ 5 18.93 ± 18.93 18.84 ± 0.01 18.48 ± 0.01 18.20 ± 0.02 17.94 ± 0.01 18.22 ± 0.01 +00 • 46 ′ 23. ′′ 8 19.83 ± 19.83 19.59 ± 0.03 19.33 ± 0.02 18.95 ± 0.03 18.66 ± 0.02 ... 03 h 17. s 75 −00 • 46 ′ 16. ′′ 1 19.94 ± 19.94 19.69 ± 0.03 19.61 ± 0.03 19.30 ± 0.04 18.98 ± 0.03 ... 19.24 ± 0.01 41.18 ± 6.55 34.52 ± 7.14 20.07 ± 9.18 +00 • 31 ′ 08. ′′ 1 21.55 ± 21.55 21.01 ± 0.08 20.54 ± 0.06 19.82 ± 0.07 19.17 ± 0.04 18.69 ± 0.01 19.05 ± 0.01 41.09 ± 6.50 20.53 ± 7.20 −1.+00 • 31 ′ 54. ′′ 8 20.47 ± 20.47 20.17 ± 0.04 19.86 ± 0.03 19.27 ± 0.04 18.95 ± 0.03 19.09 ± 0.01 19.04 ± 0.01 39.63 ± 6.46 15.19 ± 7.J091001.4+004024 3792 09 m 08 h 01. s 44 +00 • 40 ′ 21. ′′ 8 21.29 ± 21.29 21.16 ± 0.07 20.61 ± 0.06 20.22 ± 0.09 19.73 ± 0.07 ... 19.56 ± 0.04 39.95 ± 6.38 28.10 ± 7.18 10.01 ± 9.26 ... J090903.5+002031 3825 09 m 09 h 03. s 34 +00 • 20 ′ 30. ′′ 8 18.27 ± 18.27 18.09 ± 0.01 17.80 ± 0.01 17.52 ± 0.01 17.48 ± 0.01 17.43 ± 0.01 17.76 ± 0.01 38.19 ± 6.48 12.71 ± 7J090821.6+002700 4344 09 m 08 h 21. s 24 +00 • 27 ′ 00. ′′ 0 21.66 ± 21.66 21.42 ± 0.12 21.41 ± 0.13 > 20.99 20.65 ± 0.14 19.73 ± 0.01 19.93 ± 0.02 38.39 ± 6.40 30.19 ± 7.18 20.33 ± 8+00 • 19 ′ 35. ′′ 2 21.32 ± 21.32 21.16 ± 0.09 20.56 ± 0.06 20.36 ± 0.11 19.84 ± 0.07 19.27 ± 0.01 19.63 ± 0.04 34.85 ± 6.33 45.90 ± 7.21 32.04. s 37 −00 • 51 ′ 42. ′′ 1 20.22 ± 20.22 19.94 ± 0.04 19.55 ± 0.03 19.08 ± 0.04 18.67 ± 0.02 18.73 ± 0.01 18.95 ± 0.01 37.35 ± 6.50 12.10 ± 7.18 Note. -⋆ ID with color-magnitude method. a Coordinates are the position of IRAC counterparts. b Photometry from the VIKING survey (Z, Y, J, H, Ks) is extracted in 2 ′′ diameter apertures (Sutherland et al. in prep). c The reported SPIRE 350 and 500 µm fluxes are measured at the postions of 250 µm sources; no SNR cut is applied. d R is reliability of the counterparts calculated separately at 3.6 and 4.5 µm from the LR method (section 3.1). e Redshifts are from the SDSS catalogue for sources detected in that survey. f The separation between the centroid of the SPIRE emission and the IRAC counterpart. +00 • 30 ′ 50. ′′ 7 20.16 ± 20.16 19.88 ± 0.03 19.69 ± 0.03 19.29 ± 0.04 18.96 ± 0.03 18.96 ± 0.01 19.19 ± 0.01 37.35 ± 6.56 17.17 ± 7.17 6.+00 • 18 ′ 18. ′′ 2 19.72 ± 19.72 19.66 ± 0.03 19.36 ± 0.02 19.01 ± 0.03 18.69 ± 0.02 18.84 ± 0.01 +00 • 17 ′ 24. ′′ 8 19.27 ± 19.27 19.19 ± 0.02 18.82 ± 0.01 18.54 ± 0.02 18.29 ± 0.02 18.58 ± 0.01 +00 • 20 ′ 34. ′′ 9 21.73 ± 21.73 21.39 ± 0.12 21.12 ± 0.10 20.86 ± 0.18 20.32 ± 0.10 19.49 ± 0.01 J090314.0+004235 5422 09 m 03 h 13. s 98 +00 • 42 ′ 31. ′′ 2 19.61 ± 19.61 19.39 ± 0.02 19.20 ± 0.02 19.00 ± 0.03 18.78 ± 0.03 18.97 ± 0.01 18.94 ± 0.01 33.19 ± 6.55 32.95 ± 7.19 4.J090914.8+002041 5450 09 m 09 h 14. s 82 +00 • 20 ′ 39. ′′ 5 21.62 ± 21.62 21.43 ± 0.09 21.07 ± 0.09 20.62 ± 0.13 20.47 ± 0.13 20.46 ± 0.03 20.85 ± 0.05 35.94 ± 6.42 24.72 ± 7.23 16.42 ± 9.J090837.1+005002 5530 09 m 08 h 37. s 04 +00 • 50 ′ 02. ′′ 2 20.91 ± 20.91 20.67 ± 0.06 20.10 ± 0.04 19.72 ± 0.06 19.21 ± 0.04 ... 18.17 ± 0.01 36.26 ± 6.56 16.39 ± 7.27 −0.12 ± 9.02 J090825.8+004217 5538 09 m 08 h 25. s 83 +00 • 42 ′ 13. ′′ 7 22.80 ± 22.80 22.16 ± 0.23 21.76 ± 0.18 21.21 ± 0.24 20.68 ± 0.14 20.32 ± 0.03 20.77 ± 0.03 35.38 ± 6.41 17.99 ± 7.32 −5.24 ± 9.06 0.736 0.661 ... 3.46 J090821.9+002445 5564 09 m 08 h 22. s 14 +00 • 24 ′ 47. ′′ 3 > 22.37 22.06 ± 0.21 21.45 ± 0.13 20.77 ± 0.16 20.13 ± 0.09 19.30 ± 0.01 19.18 ± 0.01 35.10 ± 6.62 19.57 ± 7.16 7.32 ± 8.97 0.992 0.993 ... 3.29 J090910.9+003517 5621 09 m 09 h 11. s 35 +00 • 35 ′ 17. ′′ 4 20.81 ± 20.81 20.71 ± 0.05 20.34 ± 0.05 20.13 ± 0.08 19.66 ± 0.06 18.97 ± 0.01 19.22 ± 0.01 33.80 ± 6.56 25.47 ± 7.15 26.89 ± 8.96 0.973 0.956 0.71 ± 0.11 5.72 J090848.1+002626 5691 09 m 08 h 48. s 13 +00 • 26 ′ 22. ′′ 1 20.10 ± 20.10 19.89 ± 0.03 19.57 ± 0.03 19.19 ± 0.04 18.86 ± 0.03 19.04 ± 0.01 19.29 ± 0.01 35.24 ± 6.38 18.50 ± 7.16 16.50 ± 8.90 0.885 0.890 0.39 ± 0.05 4.15 J090812.1+002430 5735 09 m 08 h 12. s 26 +00 • 24 ′ 30. ′′ 7 19.21 ± 19.21 18.78 ± 0.01 18.66 ± 0.01 18.45 ± 0.02 18.16 ± 0.02 18.31 ± 0.01 18.21 ± 0.01 34.91 ± 6.51 15.87 ± 7.25 3.13 ± 8.99 0.998 0.997 0.16 ± 0.02 1.00 J090852.9+004106 5774 09 m 08 h 52. s 90 +00 • 41 ′ 07. ′′ 1 > 21.85 > 21.40 > 20.94 > 19.61 > 19.20 19.82 ± 0.02 19.58 ± 0.01 33.83 ± 6.45 38.19 ± 7.17 22.59 ± 9.26 0.996 0.996 ... 0.79 J090912.3+002129 6012 09 m 09 h 12. s 65 +00 • 21 ′ 28. ′′ 0 19.82 ± 19.82 19.59 ± 0.02 19.27 ± 0.02 18.90 ± 0.03 18.63 ± 0.03 18.38 ± 0.01 18.81 ± 0.01 35.13 ± 6.65 24.35 ± 7.22 6.36 ± 9J090908.3+002545 6100 ⋆ 09 m 09 h 08. s 64 +00 • 25 ′ 48. ′′ 4 22.45 ± 22.45 21.84 ± 0.13 21.55 ± 0.14 21.16 ± 0.21 20.48 ± 0.13 19.52 ± 0.01 19.35 ± 0.01 33.08 ± 6.38 44.67 ± 7.17 42.48 ± 8.86 0.244 0.540 J090902.6+004737 6146 09 m 09 h 02. s 42 +00 • 47 ′ 39. ′′ 5 21.63 ± 21.63 21.56 ± 0.13 20.92 ± 0.08 20.48 ± 0.12 19.89 ± 0.07 ... 19.58 ± 0.01 32.24 ± 6.37 20.07 ± 7.18 2.26 ± 8.98 J090754.6+003345 6220 ⋆ 09 m 07 h 54. s 82 +00 • 33 ′ 42. ′′ 2 19.71 ± 19.71 19.41 ± 0.02 19.14 ± 0.02 18.79 ± 0.03 18.48 ± 0.02 18.68 ± 0.01 18.77 ± 0.01 34.05 ± 6.51 12.37 ± 7.23 8.23 ± 9.14 J090946.4+003847 6224 09 m 09 h 46. s 39 +00 • 38 ′ 47. ′′ 1 20.48 ± 20.48 20.19 ± 0.03 19.83 ± 0.03 19.25 ± 0.04 18.81 ± 0.03 18.65 ± 0.02 18.76 ± 0.01 34.77 ± 6.59 14.14 ± 7.24 4.J090832.0+002749 6324 09 m 08 h 31. s 94 +00 • 27 ′ 52. ′′ 4 22.04 ± 22.04 21.85 ± 0.17 21.59 ± 0.15 20.77 ± 0.16 20.22 ± 0.10 19.18 ± 0.01 19.21 ± 0.01 33.98 ± 6.35 23.50 ± 7.19 10.73 ± 9.12 0.989 0.972 ... 2.90 J090956.0+003739 6348 09 m 09 h 56. s 03 +00 • 37 ′ 37. ′′ 2 19.46 ± 19.46 19.18 ± 0.02 18.77 ± 0.01 18.31 ± 0.02 17.86 ± 0.01 18.22 ± 0.01 18.18 ± 0.01 33.90 ± 6.45 12.66 ± 7.24 −3.81 ± 9.38 0.999 0.999 0.28 ± 0.06 2.12 J090840.1+001928 6480 09 m 08 h 40. s 02 +00 • 19 ′ 30. ′′ 1 19.88 ± 19.88 19.80 ± 0.03 19.50 ± 0.02 19.37 ± 0.05 19.03 ± 0.03 19.25 ± 0.01 ... 33.32 ± 6.55 25.13 ± 7.23 27.36 ± 9.19 0.995 ... J090847.1+001512 6792 09 m 08 h 47. s 28 +00 • 15 ′ 14. ′′ 1 18.37 ± 18.37 18.16 ± 0.01 17.79 ± 0.01 17.45 ± 0.01 17.14 ± 0.01 17.55 ± 0.01 ... 33.60 ± 6.38 6.58 ± 7.14 −1.24 ± 8.87 0.999 ... J090324.0+003954 6893 09 m 03 h 23. s 95 +00 • 39 ′ 52. ′′ 1 21.76 ± 21.76 21.35 ± 0.11 20.98 ± 0.09 20.48 ± 0.13 19.86 ± 0.07 19.17 ± 0.01 19.32 ± 0.01 32.93 ± 6.49 16.27 ± 7.32 −0.91 ± 9.29 0.946 0.955 0.49 ± 0.19 2.54 J090808.2+002115 6962 09 m 08 h 08. s 14 +00 • 21 ′ 16. ′′ 5 22.38 ± 22.38 21.56 ± 0.13 21.07 ± 0.10 21.30 ± 0.27 20.14 ± 0.09 19.12 ± 0.01 ... 32.57 ± 6.48 17.35 ± 7.23 −4.52 ± 9.05 0.997 ... 36 ± 0.03 20.03 ± 0.02 30.39 ± 6.44 38.53 ± 7.14 27.37 ± 8.94 0.987 0.986 ... 2.47 J090915.4+003450 7107 09 m 09 h 15. s 06 +00 • 34 ′ 48. ′′ 8 20.68 ± 20.68 20.31 ± 0.04 19.99 ± 0.03 19.41 ± 0.04 19.00 ± 0.03 18.62 ± 0.01 18.81 ± 0.01 32.15 ± 6.41 8.28 ± 7.23 J090855.7+002121 7216 09 m 08 h 55. s 64 +00 • 21 ′ 23. ′′ 1 21.87 ± 21.87 21.44 ± 0.12 21.04 ± 0.09 20.82 ± 0.17 20.38 ± 0.11 19.34 ± 0.01 18.88 ± 0.01 32.34 ± 6.39 37.62 ± 7.16 33.56 ± 8.89 0.996 0.998 J090836.5+002513 7826 09 m 08 h 36. s 54 +00 • 25 ′ 09. ′′ 5 22.11 ± 22.11 21.54 ± 0.13 21.13 ± 0.10 21.16 ± 0.23 20.33 ± 0.11 19.41 ± 0.01 19.28 ± 0.01 30.94 ± 6.33 35.84 ± 7.13 27.94 ± 9.06 0.991 0.992 J090820.3+002808 11918 09 m 08 h 19. s 84 +00 • 28 ′ 08. ′′ 6 22.34 ± 22.34 21.49 ± 0.13 21.16 ± 0.10 20.69 ± 0.15 20.53 ± 0.13 19.95 ± 0.03 19.99 ± 0.03 26.08 ± 6.45 37.76 ± 7.28 16.30 ± 9.11 0.764 0.787IAU ID H-ATLAS RA a Dec a Z b Y J b H b Ks b 3.6 µm 4.5 µm S 250 c S 350 c S 500 c R d R d z e Separation f ID (J2000) (J2000) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mJy) (mJy) (mJy) (3.6µm) (4.5µm) (arcsec) J090925.2+003224 2281 09 m 09 h 25. s 11 +00 • 32 ′ 26. ′′ 8 > 23.06 > 22.35 21.43 ± 0.12 20.87 ± 0.16 20.32 ± 0.11 19.39 ± 0.01 19.04 ± 0.01 47.80 ± 6.50 41.33 ± 7.27 14.29 ± 9.27 0.992 0.993 ... 3.35 J091257.8-005508 2343 09 m 09 h 57. s 82 −00 • 55 ′ 02. ′′ 7 18.46 ± 18.46 18.13 ± 0.01 17.98 ± 0.01 17.91 ± 0.01 18.08 ± 0.02 18.22 ± 0.01 18.28 ± 0.01 42.07 ± 6.55 24.56 ± 7.29 10.37 ± 9.30 0.927 0.949 0.38 ± 0.07 5.75 J090841.3+002005 2432 09 m 08 h 41. s 38 +00 • 20 ′ 06. ′′ 6 20.27 ± 20.27 19.97 ± 0.03 19.79 ± 0.03 19.25 ± 0.04 18.80 ± 0.03 18.70 ± 0.01 ... 46.36 ± 6.45 27.64 ± 7.15 30.30 ± 9.12 0.998 ... 0.39 ± 0.07 1.44 J090752.3+002100 2437 09 m 07 h 52. s 22 +00 • 21 ′ 00. ′′ 3 ... ... ... ... ... 20.05 ± 0.02 ... 46.26 ± 6.46 34.17 ± 7.25 20.42 ± 9.32 0.989 ... ... 1.92 J090850.0+004309 2459 ⋆ 09 m 08 h 49. s 79 +00 • 43 ′ 07. ′′ 7 20.62 ± 20.62 20.11 ± 0.04 20.02 ± 0.04 19.56 ± 0.05 19.16 ± 0.04 18.80 ± 0.01 18.81 ± 0.01 46.74 ± 6.55 30.97 ± 7.16 6.70 ± 8.99 0.354 0.399 ... 4.45 J090850.0+004309 2459 ⋆ 09 m 08 h 50. s 16 +00 • 43 ′ 09. ′′ 7 > 22.37 21.98 ± 0.20 22.17 ± 0.26 21.09 ± 0.22 20.61 ± 0.14 19.76 ± 0.03 19.65 ± 0.01 46.74 ± 6.55 30.97 ± 7.16 6.70 ± 8.99 0.643 0.598 ... 1.50 J090803.8+002250 2549 09 m 08 h 03. s 57 +00 • 22 ′ 51. ′′ 6 19.52 ± 19.52 18.90 ± 0.02 18.62 ± 0.01 18.18 ± 0.02 17.83 ± 0.01 18.12 ± 0.01 ... 45.96 ± 6.55 23.21 ± 7.15 2.63 ± 9.01 0.995 ... 0.29 ± 0.08 4.67 J090855.5+002808 2565 09 m 08 h 55. s 59 +00 • 28 ′ 06. ′′ 8 > 21.79 > 21.40 > 20.94 > −99.00 > 19.20 21.25 ± 0.06 20.83 ± 0.04 45.75 ± 6.51 38.65 ± 7.14 32.42 ± 8.92 0.896 0.977 ... 1.85 J090930.2+002755 2680 09 m 09 h 30. s 22 +00 • 27 ′ 55. ′′ 0 22.07 ± 22.07 21.55 ± 0.10 20.99 ± 0.08 20.49 ± 0.11 20.08 ± 0.09 19.27 ± 0.01 19.21 ± 0.01 45.12 ± 6.36 37.32 ± 7.15 15.07 ± 8.93 0.953 0.966 ... 0.51 J090803.7+002921 2715 09 m 08 h 03. s 82 +00 • 29 ′ 22. ′′ 6 > 21.91 > 21.40 > 20.94 > 19.61 > 19.20 20.85 ± 0.04 20.39 ± 0.03 46.08 ± 6.52 28.66 ± 7.30 29.26 ± 9.02 0.985 0.990 ... 1.14 J090905.3+001525 2773 09 m 09 h 05. s 49 +00 • 15 ′ 23. ′′ 6 > 22.54 > 21.73 > 21.05 > 19.74 > 19.20 20.06 ± 0.02 ... 44.96 ± 6.53 38.06 ± 7.18 25.49 ± 8.90 0.836 ... ... 2.61 J090846.0+004339 2793 ⋆ 09 m 08 h 46. s 01 +00 • 43 ′ 36. ′′ 8 ... ... ... ... ... 20.41 ± 0.06 20.21 ± 0.03 45.21 ± 6.42 39.54 ± 7.23 18.88 ± 8.97 0.613 0.478 ... 2.57 J090846.0+004339 2793 ⋆ 09 m 08 h 45. s 88 +00 • 43 ′ 39. ′′ 0 ... ... ... ... ... 20.56 ± 0.07 20.08 ± 0.02 45.21 ± 6.42 39.54 ± 7.23 18.88 ± 8.97 0.378 0.515 ... 2.40 J090943.0+004322 2796 09 m 09 h 43. s 23 +00 • 43 ′ 22. ′′ 2 > 23.06 > 22.35 19.70 ± 0.03 19.38 ± 0.04 19.09 ± 0.04 ... 18.66 ± 0.01 39.85 ± 6.38 21.33 ± 7.15 11.52 ± 9.10 ... 0.998 0.16 ± 0.17 2.24 J090819.6+003259 2866 09 m 08 h 20. s 00 +00 • 33 ′ 02. ′′ 6 > 22.43 > 22.02 21.94 ± 0.21 > 20.99 20.84 ± 0.17 19.95 ± 0.02 19.63 ± 0.01 44.81 ± 6.48 44.94 ± 7.15 34.21 ± 9.12 0.882 0.881 ... 6.31 J091302.7-004618 2986 09 m 08 h 02. s 92 −00 • 46 ′ 19. ′′ 7 19.06 ± 19.06 18.85 ± 0.02 18.62 ± 0.01 18.39 ± 0.02 18.11 ± 0.02 ... 18.22 ± 0.01 44.02 ± 6.53 21.20 ± 7.40 9.62 ± 8.99 ... 0.928 0.25 ± 0.03 3.57 J090922.4+002715 3043 09 m 09 h 22. s 19 +00 • 27 ′ 15. ′′ 3 21.11 ± 21.11 20.80 ± 0.05 20.49 ± 0.05 20.15 ± 0.08 19.54 ± 0.06 18.87 ± 0.01 19.20 ± 0.01 42.74 ± 6.48 17.48 ± 7.14 14.55 ± 8.99 0.922 0.919 0.68 ± 0.12 3.57 J090333.3+004746 3056 09 m 03 h 33. s 53 +00 • 47 ′ 48. ′′ 8 ... ... ... ... ... ... 19.44 ± 0.03 43.97 ± 6.43 48.32 ± 7.15 25.88 ± 8.83 ... 0.891 ... 3.12 J090800.5+002457 3084 09 m 08 h 00. s 51 +00 • 24 ′ 57. ′′ 3 > 21.85 > 21.40 > 20.94 > 19.61 > 19.20 21.21 ± 0.05 ... 42.94 ± 6.51 61.39 ± 7.18 46.14 ± 8.76 0.918 ... ... 0.57 J090839.4+004107 3113 09 m 08 h 39. s 11 +00 • 41 ′ 06. ′′ 9 18.76 ± 18.76 18.39 ± 0.01 17.98 ± 0.01 17.57 ± 0.01 17.22 ± 0.01 17.62 ± 0.01 17.68 ± 0.01 44.18 ± 6.41 19.63 ± 7.23 −0.42 ± 9.33 0.959 0.948 0.25 ± 0.03 5.73 J090746.7+002148 3161 09 m 07 h 46. s 38 +00 • 21 ′ 47. ′′ 2 19.63 ± 19.63 19.24 ± 0.02 19.05 ± 0.02 18.64 ± 0.02 18.32 ± 0.02 18.52 ± 0.01 ... 44.71 ± 6.44 18.68 ± 7.27 9.49 ± 9.12 0.984 ... 0.35 ± 0.03 5.16 J090832.7+002406 3205 09 m 08 h 32. s 85 +00 • 24 ′ 07. ′′ 1 21.17 ± 21.17 20.64 ± 0.06 20.19 ± 0.04 19.64 ± 0.06 19.11 ± 0.04 18.65 ± 0.01 18.94 ± 0.01 40.89 ± 6.65 30.78 ± 7.31 14.69 ± 8.98 0.998 0.998 0.63 ± 0.10 2.13 J090829.1+001556 3242 09 m 08 h 28. s 96 +00 • 15 ′ 58. ′′ 8 > 22.37 > 22.02 21.90 ± 0.20 > 20.99 20.49 ± 0.12 19.29 ± 0.02 ... 42.99 ± 6.59 43.26 ± 7.16 15.88 ± 8.91 0.987 ... ... 4.12 J090843.7+001437 3251 09 m 08 h 43. s 61 +00 • 14 ′ 38. ′′ 4 19.81 ± 19.81 19.59 ± 0.03 19.28 ± 0.02 18.82 ± 0.03 18.46 ± 0.02 18.20 ± 0.01 ... 38.04 ± 6.61 28.16 ± 7.14 5.02 ± 8.95 0.961 ... 0.51 ± 0.12 2.68 J091303.6-004855 3285 09 m 08 h 03. s 34 −00 • 48 ′ 59. ′′ 1 18.82 ± 18.82 18.58 ± 0.01 18.24 ± 0.01 17.92 ± 0.01 17.67 ± 0.01 ... 18.20 ± 0.01 42.06 ± 6.40 11.54 ± 7.16 9.00 ± 9.00 ... 0.975 0.11 ± 0.02 6.28 J090836.4+002948 3304 09 m 08 h 36. s 62 +00 • 29 ′ 48. ′′ 3 21.90 ± 21.90 21.38 ± 0.12 21.04 ± 0.09 20.94 ± 0.19 20.43 ± 0.11 19.22 ± 0.01 19.15 ± 0.01 39.87 ± 6.41 45.74 ± 7.28 31.52 ± 8.86 0.996 0.996 ... 2.10 J090940.3+002939 3318 09 m 09 h 40. s 28 +00 • 29 ′ 42. ′′ 4 ... ... ... ... ... 20.56 ± 0.04 20.33 ± 0.03 42.01 ± 6.55 32.10 ± 7.21 20.21 ± 8.92 0.932 0.985 ... 2.56 J090921.9+002556 3457 09 m 09 h 21. s 78 39 ± 9.13 0.997 0.996 0.12 ± 0.02 2.33 J090832.2+001938 3573 09 m 08 h 32. s 24 ... 41.25 ± 6.53 22.19 ± 7.17 1.67 ± 9.10 0.967 ... 0.21 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3.67 .92 0.890 0.889 ... 6.24 J090935.1+002224 4352 09 m 09 h 35. s 26 +00 • 22 ′ 22. ′′ 3 ... ... ... ... ... 19.48 ± 0.03 19.47 ± 0.04 37.71 ± 6.61 26.04 ± 7.17 24.95 ± 9.08 0.983 0.986 ... 2.79 J090918.3+003409 4366 ⋆ 09 m 09 h 18. s 30 +00 • 34 ′ 09. ′′ 2 ... ... ... ... ... 19.96 ± 0.02 19.73 ± 0.02 38.79 ± 6.55 31.27 ± 7.24 14.04 ± 8.90 0.545 0.545 ... 0.64 J090918.3+003409 4366 ⋆ 09 m 09 h 18. s 44 +00 • 34 ′ 09. ′′ 7 20.82 ± 20.82 20.58 ± 0.04 20.50 ± 0.05 20.25 ± 0.09 19.99 ± 0.08 19.91 ± 0.02 19.87 ± 0.02 38.79 ± 6.55 31.27 ± 7.24 14.04 ± 8.90 0.452 0.452 0.46 ± 0.15 1.60 J091306.9-004719 4404 09 m 09 h 07. s 08 −00 • 47 ′ 20. ′′ 9 22.65 ± 22.65 22.23 ± 0.26 21.50 ± 0.16 20.84 ± 0.18 20.28 ± 0.10 ... 19.11 ± 0.01 37.63 ± 6.52 32.91 ± 7.19 29.28 ± 9.11 ... 0.996 ... 2.37 J091318.1-005409 4520 09 m 09 h 18. s 11 −00 • 54 ′ 16. ′′ 1 21.98 ± 21.98 21.85 ± 0.18 21.56 ± 0.17 20.99 ± 0.20 20.74 ± 0.16 20.36 ± 0.04 20.83 ± 0.07 39.29 ± 6.62 20.79 ± 7.25 4.80 ± 9.07 0.764 0.590 ... 6.51 J090744.7+002005 4524 09 m 07 h 44. s 70 +00 • 20 ′ 06. ′′ 0 > 21.86 > 21.40 > 20.94 > 19.61 > 19.20 21.32 ± 0.10 ... 37.82 ± 6.46 33.80 ± 7.11 19.53 ± 9.10 0.892 ... ... 1.98 J090902.0+001936 4587 ⋆ 09 m 09 h 02. s 05 +00 • 19 ′ 34. ′′ 2 ... ... ... ... ... 21.36 ± 0.08 20.72 ± 0.10 34.85 ± 6.33 45.90 ± 7.21 32.27 ± 8.92 0.075 0.326 ... 2.35 J090902.0+001936 4587 ⋆ 09 m 09 h 02. s 39 27 ± 8.92 0.357 0.219 0.33 ± 0.15 5.15 J090902.0+001936 4587 ⋆ 09 m 09 h 01. s 94 +00 • 19 ′ 39. ′′ 3 ... ... ... ... ... 20.27 ± 0.03 20.12 ± 0.05 34.85 ± 6.33 45.90 ± 7.21 32.27 ± 8.92 0.557 0.445 ... 3.19 J091304.1-005141 4658 09 m 09 h 8.14 ± 9.29 0.996 0.997 0.41 ± 0.04 3.12 J090851.6+003823 4693 09 m 08 h 51. s 66 +00 • 38 ′ 22. ′′ 3 ... ... ... ... ... 20.21 ± 0.03 19.95 ± 0.03 38.05 ± 6.62 23.97 ± 7.16 3.68 ± 9.22 0.950 0.926 ... 1.25 Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from 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[ "Adaptive Leader-Following Consensus for Uncertain Euler-Lagrange Systems under Directed Switching Networks", "Adaptive Leader-Following Consensus for Uncertain Euler-Lagrange Systems under Directed Switching Networks" ]	[ "Tao Liu ", "Jie Huang " ]	[]	[]	The leader-following consensus problem for multiple Euler-Lagrange systems was studied recently by the adaptive distributed observer approach under the assumptions that the leader system is neurally stable and the communication network is jointly connected and undirected. In this paper, we will study the same problem without assuming that the leader system is neutrally stable, and the communication network is undirected. The effectiveness of this new result will be illustrated by an example.I. INTRODUCTIONConsensus, as a fundamental problem of cooperative control, has received significant attention over the past decadeThere are two types of consensus problems, i.e., leaderless consensus and leader-following consensus. The leaderless consensus problem aims to design a distributed control law to make the states/outputs of all agents synchronize to each other, while the leader-following consensus problem attempts to drive the states/outputs of all agents to a prescribed trajectory generated by a leader system. Euler-Lagrange (EL) systems is an important class of nonlinear systems, that models a large class of mechanical systems including robotic manipulators and rigid bodies [5],[6]. The consensus problem for multiple EL systems has been extensively investigated.The leader-following consensus problem for multiple EL systems was first considered in[7]assuming that all followers have access to the leader. The same problem was further studied in[8]under the assumption that the communication network of the multiple EL systems is static, undirected and connected, and in [9], [10] under the assumption that the communication network of the multiple EL systems is static and connected.More recently, the leader-following consensus problem for multiple EL systems subject to jointly connected switching communication network was studied [11],[12]. Specifically, by employing a distributed observer, a distributed adaptive state feedback control law was synthesized to solve the leader-following consensus problem for multiple EL systems under a set of standard assumptions in[11]. A drawback of the distributed observer in [11] is that the system matrix of the leader has to be used by all followers, which may not be realistic in some applications. This drawback was overcome in [12] by replacing the distributed observer with a so-called adaptive distributed observer, which is capable of providing the estimated system matrix of the leader to all followers. Thus the control law in[12]does not require the system matrix of the leader be used by all the followers. Nevertheless, the success of[12]was obtained at two other costs. First, it required that the leader system be neurally stable, which precludes the frequently used ramp signal. Second, it assumed that the communication network was undirected, which also limited the scope of the applications of the result in[12].In this paper, we will offer two improvements over the main result in[12]. That is, we will obtain the same result as in[12]using the adaptive distributed observer approach but without assuming that the	10.1109/chicc.2016.7554658	[ "https://arxiv.org/pdf/1604.07261v1.pdf" ]	63,891,294	1604.07261	57d74f83fb5cf00e8996b2095f6f73304fa29cc7	Adaptive Leader-Following Consensus for Uncertain Euler-Lagrange Systems under Directed Switching Networks 25 Apr 2016 Tao Liu Jie Huang Adaptive Leader-Following Consensus for Uncertain Euler-Lagrange Systems under Directed Switching Networks 25 Apr 2016arXiv:1604.07261v1 [math.OC] 1 The leader-following consensus problem for multiple Euler-Lagrange systems was studied recently by the adaptive distributed observer approach under the assumptions that the leader system is neurally stable and the communication network is jointly connected and undirected. In this paper, we will study the same problem without assuming that the leader system is neutrally stable, and the communication network is undirected. The effectiveness of this new result will be illustrated by an example.I. INTRODUCTIONConsensus, as a fundamental problem of cooperative control, has received significant attention over the past decadeThere are two types of consensus problems, i.e., leaderless consensus and leader-following consensus. The leaderless consensus problem aims to design a distributed control law to make the states/outputs of all agents synchronize to each other, while the leader-following consensus problem attempts to drive the states/outputs of all agents to a prescribed trajectory generated by a leader system. Euler-Lagrange (EL) systems is an important class of nonlinear systems, that models a large class of mechanical systems including robotic manipulators and rigid bodies [5],[6]. The consensus problem for multiple EL systems has been extensively investigated.The leader-following consensus problem for multiple EL systems was first considered in[7]assuming that all followers have access to the leader. The same problem was further studied in[8]under the assumption that the communication network of the multiple EL systems is static, undirected and connected, and in [9], [10] under the assumption that the communication network of the multiple EL systems is static and connected.More recently, the leader-following consensus problem for multiple EL systems subject to jointly connected switching communication network was studied [11],[12]. Specifically, by employing a distributed observer, a distributed adaptive state feedback control law was synthesized to solve the leader-following consensus problem for multiple EL systems under a set of standard assumptions in[11]. A drawback of the distributed observer in [11] is that the system matrix of the leader has to be used by all followers, which may not be realistic in some applications. This drawback was overcome in [12] by replacing the distributed observer with a so-called adaptive distributed observer, which is capable of providing the estimated system matrix of the leader to all followers. Thus the control law in[12]does not require the system matrix of the leader be used by all the followers. Nevertheless, the success of[12]was obtained at two other costs. First, it required that the leader system be neurally stable, which precludes the frequently used ramp signal. Second, it assumed that the communication network was undirected, which also limited the scope of the applications of the result in[12].In this paper, we will offer two improvements over the main result in[12]. That is, we will obtain the same result as in[12]using the adaptive distributed observer approach but without assuming that the leader system is neutrally stable and the communication network is undirected. For this purpose, we need to first strengthen the result on the adaptive distributed observer [12] so that it applies to unbounded leader's signal in polynomial form. Then we will establish our main result using this strengthened version of the adaptive distributed observer. In what follows, we will adopt the following notation. 1N denotes an N dimensional column vector whose components are all 1. ⊗ denotes the Kronecker product of matrices. x denotes the Euclidean norm of a vector x and A denotes the induced norm of a matrix A by the Euclidean norm. λmax(A) and λmin(A) denote the maximum and the minimum eigenvalues of a matrix A, respectively. For Xi ∈ R n i ×p , i = 1, . . . , m, col(X1, . . . , Xm) = X T 1 , . . . , X T m T . We call a time function σ : [0, +∞) → P = {1, 2, . . . , n0} a piecewise constant switching signal if there exists a sequence {ti, i = 0, 1, 2, . . .} satisfying t0 = 0, ti+1 − ti ≥ τ0 for some positive constant τ0, such that, for all t ∈ [ti, ti+1), σ(t) = p for some p ∈ P. n0 is some positive integer. P is called the switching index set; ti is called the switching instant and τ0 is called the dwell time. II. PROBLEM FORMULATION AND ASSUMPTIONS Consider N EL systems described by the following dynamic equations: Mi(qi)qi + Ci(qi,qi)qi + Gi(qi) = τi, i = 1, . . . , N (1) where qi,qi ∈ R n are the generalized position and velocity vectors, respectively; Mi(qi) ∈ R n×n is the positive definite inertia matrix; Ci(qi,qi)qi ∈ R n is the Coriolis and centripetal forces vector; Gi(qi) ∈ R n is the gravity vector, and τi ∈ R n is the generalized forces vector. It is well known that the EL systems have the following two properties: Property 1:Ṁi(qi) − 2Ci(qi,qi) is skew symmetric. Property 2: For all x, y ∈ R n , Mi(qi)x + Ci(qi,qi)y + Gi(qi) = Yi(qi,qi, x, y)Θi where Yi(qi,qi, x, y) ∈ R n×p is a known regression matrix and Θi ∈ R p is a constant vector consisting of the uncertain parameters of (1). Like in [11], [12], let q0 ∈ R n denote the desired generalized position vector, which is assumed to be generated by the following exosystem:v = Sv, q0 = Cv(2) where v ∈ R m and S ∈ R m×m , C ∈ R n×m are constant matrices. Without loss of generality, we assume the pair (C, S) is observable. We view the system composed of (1) and (2) as a multi-agent system of (N + 1) agents with (2) as the leader and N subsystems of (1) as followers. Given systems (1), (2) and a piecewise constant switching signal σ(t), we can define a switching digraphḠ σ(t) = (V,Ē σ(t) ) 1 withV = {0, 1, . . . , N } andĒ σ(t) ⊆V ×V for all t ≥ 0. Here, node 0 is associated with the leader system (2) and node i, i = 1, . . . , N , is associated with the ith subsystem of (1). For i = 0, 1, . . . , N, j = 1, . . . , N , (i, j) ∈Ē σ(t) if and only if τj can use the state of agent i for control at time instant t. As a result, our control law has to satisfy the communication constraint described by the digraphḠ σ(t) . Such a control law is called a distributed control law. Our problem is described as follows. Problem Description: Given systems (1), (2) and a switching digraphḠ σ(t) , find a distributed state feedback control law of the following form: τi = fi qi,qi, ϕi, ϕj − ϕi, j ∈Ni(t) φi = gi ϕi, ϕj − ϕi, j ∈Ni(t) , i = 1, . . . , N(3) whereNi(t) denotes the neighbor set of agent i at time t, such that, for i = 1, . . . , N , and for any initial conditions v(0), qi(0) andqi(0), qi(t) andqi(t) exist for all t ≥ 0 and satisfy lim t→+∞ (qi(t) − q0(t)) = 0, lim t→+∞ (qi(t) −q0(t)) = 0. (4) Some assumptions for the solvability of the above problem are listed below. j=i kḠ σ(t j ) . Remark 1. Assumption 1 allows the generalized position vector q0 of the leader system (2) to be a polynomial in t and thus is much more general than the assumption that the leader system is neutrally stable required in [12]. Assumption 2 is more restrictive than Assumption 1. However, it still allows the generalized position vector q0 of the leader system (2) to be a ramp function, which is not allowed in [12]. [1] and is perhaps the mildest condition on a switching network since it allows the network to be disconnected at any time instant. Remark 2. Assumption 4 is called the jointly connected condition III. MAIN RESULTS Let us first recall the adaptive distributed observer introduced in [12]. For this purpose, letĀ σ(t) = [aij (t)] N i,j=0 denote the weighted adjacency matrix ofḠ σ(t) . Then, for each agent of (1), we define a dynamic compensator as follows: Si = µ1 N j=0 aij(t)(Sj − Si) ηi = Siηi + µ2 N j=0 aij (t)(ηj − ηi), i = 1, . . . , N(5) where Si ∈ R m×m , S0 = S, ηi ∈ R m , η0 = v, µ1 and µ2 are any positive constants. Furthermore, let G σ(t) = (V, E σ(t) ) denote the subgraph ofḠ σ(t) , where V = {1, . . . , N } and E σ(t) ⊆ V × V is obtained fromĒ σ(t) by removing all the edges between node 0 and the nodes in V. Let L σ(t) be the Laplacian of G σ(t) . Then, putting η = col(η1, . . . , ηN ), η = η − 1N ⊗ v,Ŝi = Si − S,Ŝ = col(Ŝ1, . . . ,ŜN ) and S d = block diag Ŝ 1, . . . ,ŜN , we can write (5) into the following compact form: S = −µ1 H σ(t) ⊗ Im Ŝ η = IN ⊗ S − µ2(H σ(t) ⊗ Im) η +Ŝ d η(6) where H σ(t) = L σ(t) + diag {a10(t), . . . , aN0(t)}. Now, let us establish the following result. Lemma 1. Under Assumptions 1 and 4, for any µ1, µ2 > 0, and for any initial conditionsŜ(0) andη(0), we have lim t→+∞Ŝ (t) = 0 (7) exponentially, and lim t→+∞η (t) = 0 (8) asymptotically. Proof: By Corollary 4 of [13], for any µ1 > 0, the origin of theŜ-subsystem of (6) is exponentially stable. That is to say, limt→+∞Ŝ(t) = 0, exponentially. Thus, we only need to prove (8). Denote A(t) = IN ⊗ S − µ2(H σ(t) ⊗ Im) and F (t) = S d (t)(1N ⊗ v). Then, the second equation of (6) is equivalent tȯ η = A(t)η +Ŝ d (t)η + F (t).(9) SinceŜ d (t) converges to zero exponentially, there exist α1 > 0 and λ1 > 0 such that Ŝ d (t) ≤ α1 Ŝ d (0) e −λ 1 t .(10) Note that (1N ⊗ v) ≤ (IN ⊗ e St ) (1N ⊗ v(0)) .(11) Under Assumption 1, there exists a polynomial p(t) such that (IN ⊗ e St ) ≤ p(t).(12) Then, F (t) ≤ Ŝ d (t) (1N ⊗ v) ≤ α1 Ŝ d (0) (1N ⊗ v(0)) p(t)e −λ 1 t ≤ α2 Ŝ d (0) (1N ⊗ v(0)) e −λ 2 t(13) for some α2 > 0 and λ1 > λ2 > 0. Thus, F (t) also converges to zero exponentially. By Lemma 2 of [13], under Assumptions 1 and 4, for any µ2 > 0, the origin of the linear switched systeṁ η = A(t)η(14) is exponentially stable. Let Φ(τ, t)η be the solution of (14) that starts at (t,η). Define P (t) = +∞ t Φ(τ, t) T QΦ(τ, t)dτ(15) where Q is some constant positive definite matrix. Clearly, P (t) is continuous for all t ≥ 0. Since the equilibrium pointη = 0 of (14) is exponentially stable, we have Φ(τ, t) ≤ α3e −λ 3 (τ −t) , ∀τ ≥ t ≥ 0(16) for some α3 > 0 and λ3 > 0. It can be easily verified that c1 η 2 ≤ η T P (t)η ≤ c2 η 2 for some positive constants c1 and c2. Hence P (t) is positive definite and bounded. Thus, we can assume that P (t) ≤ c3 for any t ≥ 0 with c3 being some positive constant. On the other hand, since A(t) is continuous on intervals [ti, ti+1), i = 0, 1, 2, . . ., we have, for t ∈ [ti, ti+1), i = 0, 1, 2, . . ., ∂ ∂t Φ(τ, t) = −Φ(τ, t)A(t), Φ(t, t) = Im.(17) Then we havė P (t) = +∞ t Φ(τ, t) T Q ∂ ∂t Φ(τ, t) dτ + +∞ t ∂ ∂t Φ(τ, t) T QΦ(τ, t)dτ − Q = − +∞ t Φ(τ, t) T QΦ(τ, t)dτ A(t) − A(t) T +∞ t Φ(τ, t) T QΦ(τ, t)dτ − Q = −P (t)A(t) − A(t) T P (t) − Q. (18) Let U (t) =η T (t)P (t)η(t). Then, along the trajectory of (9), for any t ∈ [ti, ti+1) with i = 0, 1, 2, . . ., we havė U (t) =η T Ṗ (t) + A(t) T P (t) + P (t)A(t) η + 2η T P (t)Ŝ d (t)η + 2η T P (t)F (t) = −η T Qη + 2η T P (t)Ŝ d (t)η + 2η T P (t)F (t) ≤ −η T Qη + 2c3 Ŝ d (t) η 2 + 2η T P (t)F (t) ≤ −λmin(Q) η 2 + 2c3 Ŝ d (t) η 2 + P (t) 2 ε η 2 + ε F (t) 2 ≤ − λmin(Q) − 2c3 Ŝ d (t) − c 2 3 ε η 2 + ε F (t) 2 .(19) Choose ε = 2c 2 3 λ min (Q) . Then, sinceŜ d (t) converges to zero exponentially, there exists some positive integer l, such that λmin(Q) − 2c3 Ŝ d (t) − c 2 3 ε > 0, ∀t ≥ t l .(20) Thus, we haveU (t) ≤ ε F (t) 2 , ∀t ≥ t l(21) which implies U (t) ≤ U (t l ) + ε t t l F (τ ) 2 dτ, ∀t ≥ t l .(22) Since F (t) converges to zero exponentially, limt→+∞ U (t) exists and is finite. Thus, we conclude that U (t) is bounded over t ≥ 0 and hence the solutionη(t) of (9) is also bounded over t ≥ 0. In addition, for any t ∈ [ti, ti+1), i = 0, 1, 2, . . ., we haveÜ (t) is bounded over [0, +∞) sinceη,η, P (t),Ṗ (t),Ŝ d (t),Ṡ d (t) F (t) andḞ (t), are all bounded over [0, +∞). Thus, U (t) satisfies the three conditions of Lemma 1 of [14]. As a result,U (t) → 0 as t → +∞, which in turn implies that the solutionη(t) of (9) converges to zero asymptotically. Hence the proof is completed. Remark 3. SinceU is only piecewise continuous over [0, +∞), instead of using Barbala's lemma, we have to use Lemma 1 of [14] to concludeU (t) → 0 as t → +∞. lim t→+∞ (Si(t) − S) = 0 (23) lim t→+∞ (ηi(t) − v(t)) = 0.(24) That is why (5) is called the adaptive distributed observer of the leader system (2). Moreover, let η di = µ2 N j=0 aij (t)(ηj − ηi). Then, (24) implies lim t→∞ η di (t) = 0. (25) Sinceη i −v = Siηi + η di − Sv = Si(ηi − v) +Ŝiv + η di , we have lim t→∞ (ηi −v) = 0, i = 1, . . . , N.(26) Remark 5. The adaptive distributed observer for the leader system (2) was first developed in Lemma 2 of [12] under the assumptions that all the eigenvalues of the matrix S are semi-simple with zero real parts and the digraph G σ(t) is undirected. Lemma 2 of [12] was strengthened recently by Lemma 4.1 of [15], which removed the assumption that the digraph G σ(t) is undirected. Here, Lemma 1 further replaced the neutral stability assumption on the matrix S required in [12] and [15] with Assumption 1. As a result, we can handle signals in polynomial form. Next, like in [12], we will synthesize an adaptive distributed control law utilizing the adaptive distributed observer as follows. Let ξi = Cηi andq ri = CSiηi − α(qi − ξi)(27) where α is a positive constant. Then, qri = C Ṡ iηi + Siηi − α(qi −ξi).(28) By Property 2, there exists a known matrix Yi = Yi(qi,qi,qri,qri) and an unknown constant vector Θi such that YiΘi = Mi(qi)qri + Ci(qi,qi)qri + Gi(qi). Let si =qi −qri. Then, we define our control law as follows: τi = −Kisi + YiΘi (31) Θi = −Λ −1 i Y T i si (32) Si = µ1 N j=0 aij(t)(Sj − Si) (33) ηi = Siηi + µ2 N j=0 aij (t)(ηj − ηi), i = 1, . . . , N(34) whereΘi ∈ R p , Ki and Λi are positive definite matrices. Now, we are ready to present our main result. Theorem 1. Given systems (1), (2) and a switching digraphḠ σ(t) , under Assumptions 2 to 4, the problem is solvable by a distributed state feedback control law composed of (31)-(34). Proof: First note that, under Assumption 2, the leader system also satisfies Assumption 1. Next, from (27) and (30), we havė qi + α(qi − ξi) = si + CSiηi(35) where CSiηi = C(ηi − η di ) =ξi − Cη di . Subtractingξi on both sides of (35) gives (qi −ξi) + α(qi − ξi) = ui(36) where ui = si − Cη di . Since α > 0, (36) is a stable first order linear system in (qi − ξi) with input ui. If ui decays to zero as t tends to infinity, then both (qi − ξi) and (qi −ξi) decay to zero as t tends to infinity. As a result, by (24), (26) and the following identities qi(t) − q0(t) = (qi(t) − ξi(t)) + C(ηi(t) − v(t)) qi(t) −q0(t) = (qi(t) −ξi(t)) + C(ηi(t) −v(t))(37) the proof is completed. By (25), under Assumptions 2 and 4, η di (t) → 0 as t → +∞. We only need to show si(t) → 0 as t → +∞. To this end, substituting (31) into (1) gives Mi(qi)qi + Ci(qi,qi)qi + Gi(qi) = −Kisi + YiΘi(38) and subtracting YiΘi on both sides of (38) gives CN (qN ,qN )} . Mi(qi)qi + Ci(qi,qi)qi − Mi(qi)qri − Ci(qi,qi)qri = −Kisi + YiΘi(Define V = 1 2 s T M (q)s +Θ T ΛΘ .(43) By (28) and (30), s(t) is differentiable on each interval [ti, ti+1), i = 0, 1, 2, . . ., so isV (t). Noticing thatṀi(qi) − 2Ci(qi,qi) is skew symmetric giveṡ V = s T M (q)ṡ + 1 2 s TṀ (q)s +Θ T ΛΘ = s T −C(q,q)s − Ks + YΘ + 1 2 s TṀ (q)s +Θ T ΛΘ = −s T Ks + s T YΘ −Θ T ΛΛ −1 Y T s = −s T Ks ≤ 0.(44) Since V (t) andV (t) are piecewise continuous over [0, +∞), we cannot use Barbala's lemma to concludeV (t) → 0 as t → +∞. We need to use Corollary 1 of [14] to conclude limt→+∞V (t) = 0, which implies limt→+∞ s(t) = 0. For this purpose, we need to show that there exists a positive number γ such that sup t i ≤t≤t i+1 , i=0,1,2,... \|V (t)\| ≤ γ.(45) SinceV (t) = −2s T Kṡ, it suffices to show that both s andṡ are bounded. Now note that V (t) is continuous, and M (q) and Λ are positive definite, (44) implies that s andΘ are bounded. Thus, the input ui in (36) is bounded. ¿From (41), to showṡ is bounded, we need to show C(q,q) and YΘ are bounded. We first note that (36) implies both both (qi − ξi) and (qi −ξi) are bounded since ui is bounded. By (26),ξi = Cηi is bounded sincė q0 = Cv is bounded. Thusqi is bounded, which implies Ci(qi,qi) is bounded under Assumption 3. ¿From (29), YΘ is bounded if bothqri andqri are bounded. Since we have already shown that si andqi are bounded, we haveqri is bounded by (30). We now showqri is bounded using (28). In fact, CṠiηi = CṠiηi = CṠiv + CṠi(ηi − v)(46)CSiηi = CŜiηi + CSηi = CŜiv + CŜi(ηi −v) + CSv + CS(ηi −v) = CŜiv + CŜi(ηi −v) +q0 + CS(ηi −v).(47) Thus,qri is bounded since, by Remark 4, under Assumptions 2 and 4, every term on the right hand side of (28) is bounded. Thus, (45) is satisfied. The proof is completed by invoking Corollary 1 of [14]. Remark 6. If we strengthen Assumption 1 to the one that the leader system is neutrally stable as assumed in [12], then the generalized position vector q0 as well as its derivative of any degree is bounded. In this case, Assumption 2 is satisfied automatically. Furthermore, ξi,ξi are bounded from (24) and (26), which implies that qi,qi are bounded. Thus, Assumption 3 is also satisfied automatically. It is worth mentioning that even in this case, we have extended the result of [12] from undirected communication networks to directed communication networks. IV. AN EXAMPLE In this section, we consider a group of four EL systems, each of which describes a two-link manipulator whose motion equation is taken from [5]: Mi(qi)qi + Ci(qi,qi)qi + Gi(qi) = τi, i = 1, 2, 3, 4, where qi = col(θi1, θi2) and Mi(qi) = ai1 + ai2 + 2ai3 cos θi2 ai2 + ai3 cos θi2 ai2 + ai3 cos θi2 ai2 Ci(qi,qi) = −ai3(sin θi2)θi2 −ai3(sin θi2)(θi1 +θi2) ai3(sin θi2)θi1 0 Gi(qi) = ai4g cos θi1 + ai5g cos(θi1 + θi2) ai5g cos(θi1 + θi2) with Θi = col(ai1, ai2, ai3, ai4, ai5). Then, Assumption 3 is satisfied. Let the leader's signal be as follows: q0(t) = 1 + t + cos t + sin t 1 + t + cos t − sin t . Then this leader's signal can be produced by the following leader system:v with initial condition v(0) = 14. It can be verified that the pair (C, S) is observable and Assumption 2 is satisfied. Let the switching digraphḠ σ(t) be dictated by the following switching signal: = Sv =     0 1 0 0 0 0 0 0 0 0 0 1 0 0 −1 0     v q0 = Cv = 1 0 1 0 1 0 0 1 v (a)Ḡ 1 (b)Ḡ 2 (c)Ḡ 3 (d)Ḡ 4σ(t) =            1, if sT0 ≤ t < (s + 1 4 )T0 2, if (s + 1 4 )T0 ≤ t < (s + 1 2 )T0 3, if (s + 1 2 )T0 ≤ t < (s + 3 4 )T0 4, if (s + 3 4 )T0 ≤ t < (s + 1)T0(48) where T0 = 2, and s = 0, 1, 2, . . .. The four digraphsḠi, i = 1, 2, 3, 4, are described by Figure 1 where node 0 is associated with the leader and the other nodes are associated with the followers. It can be seen that Assumption 4 is satisfied even thoughḠ σ(t) is disconnected at any time t ≥ 0. According to Theorem 1, we can design a control law in the form described by (31)-(34) with the following design parameters: µ1 = µ2 = 10, α = 10, Ki = 20I2, Λi = 0.2I5, for i = 1, 2, 3, 4. We let aij(t) = 1, i, j = 0, 1, 2, 3, 4, whenever (j, i) ∈Ē σ(t) . The actual values of Θi are given as follows: Simulation is conducted with randomly chosen initial conditions. The trajectories of qi andqi, i = 1, 2, 3, 4, are shown in Figure 2 and Figure 3, respectively. V. CONCLUSION In this paper, we have studied the leader-following consensus problem for multiple uncertain Euler-Lagrange systems under the jointly connected switching network. Due to the employment of the adaptive distributed observer in a strengthened version, we have removed the assumptions that the leader system is neutrally stable and the communication network is undirected. (i, j), and node i is called a neighbor of node j. Let Ni = {j\|(j, i) ∈ E }, which is called the neighbor set of node i. The edge (i, j) is called undirected if (i, j) ∈ E implies (j, i) ∈ E . The digraph G is undirected if every edge in E is undirected. If the digraph contains a sequence of edges of the form (i1, i2), (i2, i3), . . ., (i k , i k+1 ), then the set {(i1, i2), (i2, i3), . . . , (i k , i k+1 )} is called a directed path of G from node i1 to node i k+1 and node i k+1 is said to be reachable from node i1. A digraph Gs = (Vs, Es) is called a subgraph of G = (V, E ) if Vs ⊆ V and Es ⊆ E (Vs × Vs). Given a set of n0 digraphs {Gi = (V, Ei), i = 1, . . . , n0}, the digraph G = (V, E ) where E = n 0 i=1 Ei is called the union of the digraphs Gi, denoted by G = n 0 i=1 Gi. The weighted adjacency matrix of a digraph G is a nonnegative matrix A = [aij] ∈ R N×N , where aii = 0 and aij > 0 if and only if (j, i) ∈ E , i, j = 1, . . . , N . On the other hand, given a matrix A = [aij ] ∈ R N×N satisfying aii = 0 and aij ≥ 0 for i = j, we can always define a digraph G whose weighted adjacency matrix is A. The Laplacian of G is then defined as L = [lij ] ∈ R N×N , where lii = N j=1 aij , lij = −aij for i = j. Given a piecewise constant switching signal σ : [0, +∞) → P = {1, 2, . . . , n0}, and a set of n0 digraphs Gi = (V, Ei), i = 1, . . . , n0, with the corresponding weighted adjacency matrices being denoted by Ai, i = 1, . . . , n0, we call the time-varying graph G σ(t) = (V, E σ(t) ) a switching digraph, and denote the weighted adjacency matrix and the Laplacian of G σ(t) by A σ(t) and L σ(t) , respectively. APPENDIX A digraph G = (V, E ) Assumption 1 . 1None of the eigenvalues of S have positive real parts. Assumption 2.q0 is bounded. Assumption 3 . 3There exist positive constants km, km, kc, kg, such that, for i = 1, . . . , N , kmIn ≤ Mi(qi) ≤ k m In, Ci(qi,qi) ≤ kc qi , and Gi(qi) ≤ kg. Assumption 4 . 4There exists a subsequence {i k }, k = 0, 1, 2, . . ., of {i : i = 0, 1, 2, . . .} with ti k+1 − ti k < ǫ for some positive ǫ such that every node i, i = 1, . . . , N , is reachable from node 0 in the union digraph i k+1 −1 Remark 4 . 4As a result of Lemma 1, under Assumptions 1 and 4, for any µ1, µ2 > 0, and i = 1, . . . , N , 39)whereΘi =Θi − Θi. Then, by (30), we haveMi(qi)ṡi + Ci(qi,qi)si = −Kisi + YiΘi.(40)Let x = col(x1, . . . , xN ) for x = q,q, s,ṡ,Θ, and X = block diag{X1, . . . , XN } for X = K, Y, Λ −1 . Then (40) and (32) can be written asM (q)ṡ = −C(q,q)s − Ks + YΘ (41) Θ = −Λ −1 Y T s(42)where M (q) = block diag {M1(q1), . . . , MN (qN )} C(q,q) = block diag {C1(q1,q1), . . . , Fig. 1 . 1Switching topologyḠ σ(t) with P = {1, 2, 3, 4} Fig. 3 . 3consists of a finite set of nodes V = {1, . . . , N } and an edge set E ⊆ V ×V. An edge of E from node i to Generalized velocity of each agent node j is denoted by See Appendix for a summary on digraph. 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[ "Nanoscale rectification at the LaAlO 3 /SrTiO 3 interface", "Nanoscale rectification at the LaAlO 3 /SrTiO 3 interface" ]	[ "Daniela F Bogorin \nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvaniaUSA\n", "Chung Wung Bark \nDepartment of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n", "Ho Won Jang \nDepartment of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n", "Cheng Cen \nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvaniaUSA\n", "Chad M Folkman \nDepartment of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n", "Chang-Beom Eom \nDepartment of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n", "Jeremy Levy [email protected] \nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvaniaUSA\n" ]	[ "Department of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvaniaUSA", "Department of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA", "Department of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA", "Department of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvaniaUSA", "Department of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA", "Department of Materials Science and Engineering\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA", "Department of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPennsylvaniaUSA" ]	[]	Control over electron transport at scales that are comparable to the Fermi wavelength or mean-free path can lead to new families of electronic devices. Here we report electrical rectification in nanowires formed by nanoscale control of the metal-insulator transition at the interface between LaAlO 3 and SrTiO 3 . Controlled in-plane asymmetry in the confinement potential produces electrical rectification in the nanowire, analogous to what occurs naturally for Schottky diodes or by design in structures with engineered structural inversion asymmetry. Nanostructures produced in this manner may be useful in developing a variety of nanoelectronic, electro-optic and spintronic devices.	10.1063/1.3459138	[ "https://arxiv.org/pdf/0912.3714v2.pdf" ]	118,434,807	0912.3714	4047e17e281ba728eebd2e852a6fd6d81cce78aa	Nanoscale rectification at the LaAlO 3 /SrTiO 3 interface Daniela F Bogorin Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPennsylvaniaUSA Chung Wung Bark Department of Materials Science and Engineering University of Wisconsin-Madison 53706MadisonWisconsinUSA Ho Won Jang Department of Materials Science and Engineering University of Wisconsin-Madison 53706MadisonWisconsinUSA Cheng Cen Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPennsylvaniaUSA Chad M Folkman Department of Materials Science and Engineering University of Wisconsin-Madison 53706MadisonWisconsinUSA Chang-Beom Eom Department of Materials Science and Engineering University of Wisconsin-Madison 53706MadisonWisconsinUSA Jeremy Levy [email protected] Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPennsylvaniaUSA Nanoscale rectification at the LaAlO 3 /SrTiO 3 interface (Received1 Control over electron transport at scales that are comparable to the Fermi wavelength or mean-free path can lead to new families of electronic devices. Here we report electrical rectification in nanowires formed by nanoscale control of the metal-insulator transition at the interface between LaAlO 3 and SrTiO 3 . Controlled in-plane asymmetry in the confinement potential produces electrical rectification in the nanowire, analogous to what occurs naturally for Schottky diodes or by design in structures with engineered structural inversion asymmetry. Nanostructures produced in this manner may be useful in developing a variety of nanoelectronic, electro-optic and spintronic devices. The discovery of a high-mobility two-dimensional electron gas (2DEG) at the interface between LaAlO 3 and SrTiO 3 1, 2 has opened exciting new opportunities for electric-field controlled phenomena 3-10 and devices 11,12 . The interface between thin films of LaAlO 3 and TiO 2 -terminated SrTiO 3 exhibits an abrupt insulator-to-metal transition with increasing LaAlO 3 thickness. The 2DEG is n-type and strongly localized at the interface 13 . Films grown at a critical thickness of 3 unit cells (3uc-LAO/STO) exhibit a reversible and hysteretic interfacial metal-insulator phase transition that can be programmed by voltages V bg~+ /-100 V applied to the back SrTiO 3 substrate 3 . Nanoscale control of the interfacial metal-insulator transition in 3uc-LAO/STO can be achieved using a conducting AFM probe 11 . A positive voltage applied to the AFM tip with respect to the interface locally switches the interface into a conducting state, while a negative voltage locally restores the interface to an insulating state. Conducting nanostructures are created by scanning the AFM tip over the 3uc-LAO/STO surface along a trajectory (x(t),y(t)) while a voltage V tip (t) is applied (Fig. 1 17 , and hybrid organic/inorganic semiconductors 18 . In this letter, we report the creation of nanoscale rectifying junctions along nanowires In an initial experiment, a nanowire is created with a spatially uniform positive voltage Fig. 2(a)). The resulting surface charge is intentionally uniform (Fig. 2(b)) and V tip =+10 V ( produces a nanowire along the x-direction with highly linear current-voltage (I-V) characteristics ( Fig. 2(d)). The lateral width of this nanowire is determined by cutting it: a negative bias V tip =-10 V is applied to the AFM tip as it moves across the wire (along the y-direction), and the conductance of the nanowire is monitored using a lock-in amplifier. Analysis of the conductance profile as the nanowire is cut yields a width w=2.6 nm. The nanowire is subsequently erased and replaced by a structure created with an asymmetric sawtooth-shaped voltage pulse ( Fig. 2 (e)) described by   , V x y  , which is defined below:     0 2 1 / , 0 ,/ 1, otherwise dd x x x x V x y V            , where 0 10 V V  and x d =40 nm are the sawtooth amplitude, asymmetry direction and width, respectively. During the writing process, the AFM tip is scanned at a speed v x =400 nm/s. After writing, the measured I-V curve (Fig. 2(h)) becomes highly non-reciprocal and rectifying, 5 allowing substantial current flow only for positive bias. There is a small leakage current for the reverse bias, and an onset for reverse-field "breakdown" for 4.5 V V  . The nanostructure is subsequently erased and a third one is written with an asymmetric voltage in the opposite direction   , V x y  , using parameters 0 10 V V  and x d =40 nm (Fig. 2(i)). Again, the I-V curve shows rectifying behavior but with an opposite polarity (Fig. 2(l)). A comparison of the two diode structures (Fig. 2(h,l)) shows a ~20% variation in the reverse breakdown voltage; these variations are typical of those seen experimentally for a given set of parameters. Schematic energy diagrams for nanostructures written under uniform and non-uniform tip voltage profiles are presented in Fig. 2(c,g,k). The asymmetric conduction-band profile E c (x) (Fig. 2(g,k)) and I-V characteristics mimic that of a metal-semiconductor Schottky junction. Under forward bias ( Fig. 2(g)), substantial current flow is observed above a threshold voltage V>V th . This threshold voltage allows one to estimate the strength of the electrostatic field produced by the potential gradient 6 2.5 10 V cm th d E V x    . Under a reverse bias (Fig. 2(k)), current flow is suppressed by the sharp potential barrier. The rectifying properties of the nanostructures described above depend upon the modulation-doping profile along the nanowire. As demonstrated in Fig. 2, these profiles can be created by spatial modulation of the writing voltage V tip . Here we demonstrate a second method for producing non-reciprocal nanostructures. In this approach, spatial variations in the conduction-band profile are created by a precise sequence of erasure steps. In a first experiment, a conducting nanowire is created using V tip =+10 V. The initial I-V curve ( Fig. 3(a), green curve) is highly linear and reciprocal. This nanowire is then cut by scanning the AFM tip across the nanowire at a speed v y =100 nm/s using V tip =-2 mV at a fixed location (x=20 nm) along the length of the nanowire. This erasure process increases the conduction-band minimum E c (x) locally by an amount that scales monotonically with the number of passes N cut (Fig. 3(a) inset); the resulting nanostructure exhibits a crossover from conducting to activated to tunneling behavior 12 . Here we focus on the symmetry of the full I-V curve. As N cut increases, the transport becomes increasingly nonlinear; however, the I-V curve remains highly reciprocal. The canvas is subsequently erased and a uniform conducting nanowire is written in a similar fashion as before (V tip =+10 V, v x =400 nm/s). A similar erasure sequence is performed; however, instead of cutting the nanowire at a single x coordinate, a sequence of cuts is performed at nine adjacent x coordinates along the nanowire (separated by x=5 nm). The number of cuts at each location along the nanowire N cut (x) increases monotonically with x, resulting in a conduction band profile E c (x) that is asymmetric by design ( Fig. 3(b), inset). The resulting I-V curve for the nanostructure evolves from being highly linear and reciprocal before writing ( Fig. 3(b), green curve) to highly nonlinear and non-reciprocal ( Fig. 3(b), red curve). Nanoscale control over asymmetric potential profiles at the interface between LaAlO 3 and SrTiO 3 can have many potential applications in nanoelectronics and spintronics. Working as straightforward diodes, these junctions can be used to create half-wave and full-wave rectifiers for AC-DC conversion or for RF detection and conversion to DC. By cascading two or more such junctions, with a third gate for tuning the density in the intermediate regime could form the basis for low-leakage transistor devices. Generally speaking, the ability to control the potential () Vx along a nanowire could be used to create wires with built-in polarizations similar to those created in heterostructures that lack inversion symmetry 21 . Nanoscale control over inversion 7 symmetry breaking could in principle be used to produce nonlinear optical frequency conversion (i.e., second-harmonic generation or difference frequency mixing), thus providing a means for the generation of local sources of light or THz radiation. A straightforward modification of this idea involves the creation of potential profile asymmetries that are in-plane and transverse to the nanowire direction (i.e., along the y direction in Fig. 1(a)). Such asymmetries could give rise to significant Rashba spin-orbit interactions 22,23 . The resulting effective magnetic fields could allow control over spin precession along two orthogonal axes 24 , and thus exert full threedimensional control over electron spin 25 in a nanowire. Intermediate I-V curves are shown after every alternate cut. As the wire is cut, the potential barrier increases (inset), and the zero-bias conductance decreases; however, the overall I-V curve remains highly reciprocal. (b) I-V plots for a nanowire subject to a sequence of cuts N cut (x) at nine locations spaced 5 nm apart along the nanowire (see Table I). Green curve indicates I-V curve before the first cut. The asymmetry in N cut (x) results in a non-reciprocal I-V curve. Fig. 3(b). formed within 3uc-LAO/STO heterostructures. Thin films of LaAlO 3 are grown on TiO 2terminated (001) SrTiO 3 substrates 19 by pulsed laser deposition with in situ high pressure reflection high energy electron diffraction (RHEED)20 . The films are grown at a substrate temperature of 550C under oxygen pressure of 10 -3 mbar and cooled down to room temperature at a 10 -3 mbar. After growth, electrical contacts to the interface are defined by optical lithography using a combination of ion milling and Au/Ti deposition. Within a 3535 m 2 "canvas" defined by the electrode edges, nanostructures are "written" and "erased" at the interface using conducting AFM lithography 11, 12(Fig. 1). The nanoscale writing and subsequent transport experiments are performed at room temperature (295 K) under atmospheric conditions (35-50% relative humidity). During and after the writing process, the transport properties of these nanostructures are monitored by applying a small voltage V s =0.1V to an electrode connected to one end of the nanowire. The resulting current is measured at a second electrode connected to the other end of the nanowire which is held at virtual ground(Fig. 1).The series of three experiments described below demonstrates how the profile created by the AFM voltage can lead to linear or rectifying behavior. These experiments were performed on the same canvas, whereby a nanowire is first written, followed by transport experiments, followed by an erasure of the canvas (raster-scanning the entire area with V tip =-10 V) so that a new nanostructure can be created. Experiments were performed on multiple devices written onto two different 3uc-LAO/STO heterostructures grown under nominally identical conditions. For the purpose of comparison, the results shown here are for devices and structures written in a single experimental session. FIGURE LEGENDS Figure 1 . LEGENDS1(Color online) Schematic illustrating of the nanowriting process at the LaAlO 3 /SrTiO 3 interface. Au electrodes (shown in yellow) are electrically contacted to the LaAlO 3 /SrTiO 3 interface. The AFM tip with an applied voltage is scanned once between the two electrodes with a voltage applied V tip (x(t), y(t)). Positive voltages locally switch the interface to a conducting state, while negative voltages locally restore the insulating state. Here, a conducting nanowire (shown in green) is being written. The conductance between the two electrodes is monitored by applying a small voltage bias on one of the two gold electrodes (V s ) and reading the current at the second electrode (I D ). Figure 2 . 2(Color online) Structure and electrical properties of non-reciprocal nanoscale devices at the LaAlO 3 /SrTiO 3 interface. (a) Tip voltage profile during writing procedure. (b) Cross-sectional view illustrating the surface modulation-doping of the LaAlO 3 /SrTiO 3 interface, resulting in a spatially uniform nanowire. (c) Schematic energy-band diagram for the conduction band minimum E c and Fermi energy E F for the uniform nanowire. (d) Current-voltage (I-V) plot for the uniform nanowire. (e) Tip voltage profile V tip , (f) crosssectional view, (g) schematic energy-band diagram and (h) I-V characteristics for a positive sawtooth potential V tip (x,y)= V 0 V + (x,y), where V 0 =10 V. (i) Tip voltage profile V tip , (j) cross-sectional view, (k) schematic energy-band diagram and (l) I-V characteristics for a negative sawtooth potential V tip (x,y)= V 0 V -(x,y), where V 0 =10 V. Figure 3 . 3(Color online) (a) I-V plots for a nanowire cut at the same location multiple times with an AFM tip bias V tip =-2 mV. Green curve indicates I-V curve before the first cut. ). Writing with a smaller V tip or faster scan rate generally produces a narrower, less highly conducting nanostructure. Nanostructures are stable for ~1 day in atmospheric conditions at room temperature, and indefinitely under modest vacuum 12 . It is believed that the AFM writing procedure charges the top LaAlO 3 surface and modulation-dopes the interface with near-atomic spatial precision. Using this writing procedure, a variety of quasizero-dimensional and quasi-one-dimensional nanostructures have been created 11 . Nanoscale tunnel junctions and transistors with features as small as 2 nm have also been demonstrated 12 .heterostructures, and Schottky barriers are just a few examples of non-reciprocal devices that play an essential role in modern electronics . Non-reciprocal devices down to nanoscaleAsymmetries in the electronic confining profile generally lead to non-reciprocal behavior in transport (i.e.  I(V)  I(V)). 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[ "FILLING THE MASS GAP: HOW KILONOVA OBSERVATIONS CAN UNVEIL THE NATURE OF THE COMPACT OBJECT MERGING WITH THE NEUTRON STAR", "FILLING THE MASS GAP: HOW KILONOVA OBSERVATIONS CAN UNVEIL THE NATURE OF THE COMPACT OBJECT MERGING WITH THE NEUTRON STAR" ]	[ "C Barbieri ", "O S Salafia ", "M Colpi ", "G Ghirlanda ", "A Perego ", "A Colombo " ]	[]	[]	In this letter we focus on the peculiar case of a coalescing compact-object binary whose chirp mass is compatible both with a neutron star-neutron star and black hole-neutron star system, with the black hole in the ∼ 3 − 5 M range defined as the mass gap". Some models of core-collapse supernovae predict the formation of such low-mass black holes and a recent observation seems to confirm their existence. Here we show that the nature of the companion to the neutron star can be inferred from the properties of the kilonova emission once we know the chirp mass, which is the best constrained parameter inferred from the gravitational signal in low-latency searches. In particular, we find that the kilonova in the black hole-neutron star case is far more luminous than in the neutron star-neutron star case, even when the black hole is non spinning. The difference in the kilonovae brightness arises primarily from the mass ejected during the merger. Indeed, in the considered interval of chirp masses, the mass ejection in double neutron star mergers is at its worst as the system promptly forms a black hole. Instead mass ejection for black hole-neutron star case is at its best as the neutron stars have low mass/large deformability. The kilonovae from black hole-neutron star systems can differ by two to three magnitudes. The outcome is only marginally dependent on the equation of state. The difference is above the systematics in the modeling.	10.3847/2041-8213/ab5c1e	[ "https://arxiv.org/pdf/1912.03894v1.pdf" ]	208,910,359	1912.03894	2cc23ef2e49c9563713311abe39cfd270770dc9c	FILLING THE MASS GAP: HOW KILONOVA OBSERVATIONS CAN UNVEIL THE NATURE OF THE COMPACT OBJECT MERGING WITH THE NEUTRON STAR December 10, 2019 Draft version December 10, 2019 C Barbieri O S Salafia M Colpi G Ghirlanda A Perego A Colombo FILLING THE MASS GAP: HOW KILONOVA OBSERVATIONS CAN UNVEIL THE NATURE OF THE COMPACT OBJECT MERGING WITH THE NEUTRON STAR December 10, 2019 Draft version December 10, 2019Draft version Preprint typeset using L A T E X style emulateapj v. 01/23/15stars:neutronstars: black holesbinaries: generalgamma-ray burst: generalgravitational waves In this letter we focus on the peculiar case of a coalescing compact-object binary whose chirp mass is compatible both with a neutron star-neutron star and black hole-neutron star system, with the black hole in the ∼ 3 − 5 M range defined as the mass gap". Some models of core-collapse supernovae predict the formation of such low-mass black holes and a recent observation seems to confirm their existence. Here we show that the nature of the companion to the neutron star can be inferred from the properties of the kilonova emission once we know the chirp mass, which is the best constrained parameter inferred from the gravitational signal in low-latency searches. In particular, we find that the kilonova in the black hole-neutron star case is far more luminous than in the neutron star-neutron star case, even when the black hole is non spinning. The difference in the kilonovae brightness arises primarily from the mass ejected during the merger. Indeed, in the considered interval of chirp masses, the mass ejection in double neutron star mergers is at its worst as the system promptly forms a black hole. Instead mass ejection for black hole-neutron star case is at its best as the neutron stars have low mass/large deformability. The kilonovae from black hole-neutron star systems can differ by two to three magnitudes. The outcome is only marginally dependent on the equation of state. The difference is above the systematics in the modeling. INTRODUCTION During the O1 and O2 observing runs, the LIGO Scientific Collaboration and Virgo Collaboration (LVC) detected gravitational wave (GW) signals from ten coalescing stellar-mass black hole binaries (BHBH) and a neutron star binary system (NSNS), the latter accompanied by a multi-wavelength electromagnetic (EM) counterpart (Abbott et al. 2017a,b; The LIGO Scientific Collaboration & the Virgo Collaboration 2018). At the time of writing, as the third observing run (O3) is in progres, the LVC reported the detection of two probable black hole-neutron star (BHNS) binary merger candidates (S190814bv -The LIGO Scientific Collaboration & the Virgo Collaboration 2019a, and S190910d -The LIGO Scientific Collaboration & the Virgo Collaboration 2019b), plus candidates with a lower probability of being actual astrophysical events 5 . Before the beginning of O3, the estimated BHNS detection rate for this run was in the range 0.04 − 12 yr −1 . At the time of writing, there are no indications of observed EM counterparts associated with these candidates ; for S190814bv see e.g. Srivastav 2019; Soares-Santos 2019; Klotz 2019, for S190910d see e.g. On a theoretical point of view, BHNS mergers can be accompanied by an EM counterpart as in the NSNS case. This occurs when the NS is (at least partially) tidally disrupted before crossing the BH event horizon (Shibata & Taniguchi 2011). Tidal disruption is favoured in binaries with low mass ratio q = M 1 /M 2 and large NS tidal deformability Λ NS (corresponding to a small NS mass and/or to a "stiff" equation of state). A high black hole spin 6 , which brings the last stable circular orbit of the binary closer to the BH horizon, also greatly enhances the tidal disruption (Shibata & Taniguchi 2011;Kawaguchi et al. 2016;Foucart et al. 2018;Barbieri et al. 2019b,a). The unbound NS material ("ejecta") is thought to produce kilonova emission (Lattimer & Schramm 1974;Li & Paczyński 1998;Metzger 2017). Moreover, Shapiro (2017); Paschalidis (2017); Ruiz et al. (2018) showed that after a BHNS merger a relativistic jet can be launched, powering a short gamma-ray burst (sGRB) (Eichler et al. 1989;Narayan et al. 1992) and GRB afterglow emission (Sari et al. 1998;D'Avanzo et al. 2018;Ghirlanda et al. 2018;Salafia et al. 2019 (Zhang et al. 2019), and the NS with the highest and best estimated mass is the radio pulsar J0740+6620 with M NS = 2.14 +0.10 −0.09 M in a lowmass binary (Cromartie et al. 2019). Thus, observations appear to indicate a discontinuity between the observed mass distributions of NSs and stellar BHs, called mass gap", located approximately between ≈ 3 M (the maximum NS mass inferred from causality arguments) and ∼ 5 M (Lattimer & Prakash 2001). However recently Thompson et al. (2019) reported the discovery of a BH with mass 3.3 +2.8 −0.7 M in a non-interacting binary system with a red giant. The mass spectrum of compact objects depends sensitively on the mass of the carbon-oxygen core at the end of stellar evolution, on the compactness of the collapsing core at bounce and on the supernova (SN) explosion engine. Belczynski et al. (2012) and Fryer et al. (2012) showed that, in the presence of a significant amount of fallback, explosions happening over a large interval of post-bounce times lead to a continuous range in remnant masses. By contrast explosions happening predominantly within a few hundreds of ms after bounce, characterized by negligible amounts of fallback material, produce more easily the mass gap. Interestingly, at the time of writing, the LVC reported event candidates with binaries having at least one component in the mass gap (The LIGO Scientific Collaboration & the Virgo Collaboration 2019c,d). It is known that the binary chirp mass M c , a combination of the masses of the two components, is one of the best measured parameters encoded in the GW signal. It is the prime parameter used to identify in low-latency searches the nature of the binary -whether the system hosts two NSs, two stellar BHs or a BH and a NS. Interestingly, we note that if the NS and BH mass spectra join to form a continuum, i.e. there is no "mass gap" between BH and NS mass distributions (as Thompson et al. 2019 seem to indicate), there exists a range of values of the chirp mass M c where the nature of the binary cannot be identified uniquely based on the chirp mass only (see also Mandel et al. 2015). In particular, hereafter we call "ambiguous" the chirp masses whose values are compatible with either a NSNS or a light BHNS system (see Fig. 1). In this Letter, we aim at answering the following question: can EM observations of coalescing binaries in this "ambiguous" chirp mass interval help to disentangle their nature and narrow down the uncertainties on the existence of a mass gap? To this purpose, we study the properties of the kilonova emission of NSNS and BHNS systems which fall in this "ambiguous" chirp mass interval, using the semi-analytical model presented in Barbieri et al. (2019b,a). "AMBIGUOUS" CHIRP MASSES The binary chirp mass is defined as M c = (M 1 M 2 ) 3/5 (M 1 + M 2 ) 1/5 ,(1) where M 1 and M 2 are the masses of the two component stars (we take M 1 ≥ M 2 ). LVC public alerts follow a classification scheme to communicate probabilistic estimates of the nature of the merging system to the community. The scheme classifies as "BNS" any system with both masses M 1 and M 2 smaller than 3 M ; as "BBH" any system with both M 1 and M 2 larger than 5 M ; as "NSBH" any system with M 1 > 5 M and M 2 < 3 M , and as "MassGap" any system with at least one component carrying a mass between 3 and 5 M . An additional "Terrestrial" category is defined to represent triggers that are not of astrophysical origin (i.e. false alarms). In this work we follow a slightly different classification. We assume the SFHo equation of state (EoS), for which the maximum mass of a non-rotating NS is M max NS = 2.058 M (Steiner et al. 2013). We also fix the minimum NS mass to M min NS = 1 M (∼ 10% lower than the value found in Suwa et al. 2018). We thus classify as "NSNS" those systems with both M 1 and M 2 between 1 and 2.058 M (yellow region in Fig. 1); "BHNS" those with M 1 > 5 M and M 2 < 2.058 M (green region); "BHBH" those with both masses above 5 M as in the LVC classification (purple region). Considering that compact objects populating the mass gap have masses larger than M max NS , we assume these to be stellar-origin BHs. In Fig. 1 we divide the "MassGap" region in three sub-regions: "BH+gap" for those systems with a BH above 5 M and a BH in the gap; "gap+gap" for those with two BHs in the gap; "gap+NS" for those with a BH in the gap and a NS. Two limiting values of the chirp mass can be identified: are NSNS mergers. Similarly, M NSNS c,max = 1.792 M is the chirp mass corresponding to a NSNS binary with both NSs having the maximum allowed mass (blue line). Events with chirp mass above M NSNS c,max cannot be produced by a NSNS merger. Events with chirp mass between M gapNS c,min and M NSNS c,max can be either NSNS or gap+NS mergers (green-orange lines), i.e. they are "am-biguous". COMPUTATION OF EJECTA PROPERTIES FROM BHNS AND NSNS MERGERS During the final phase of a NSNS merger, tidal forces lead to a partial disruption of the stars, producing an outflow of neutron-rich material. When the crusts of the two NS impact each other, compression, shock heating and potentially neutrino ablation cause an additional outflow (Hotokezaka et al. 2013;Bauswein et al. 2013;Radice et al. 2016;Dietrich et al. 2017;Beloborodov et al. 2018). The released NS material can be divided into two components: the dynamical ejecta, gravitationally unbound, that leave the merger region, and a bound component, which forms an accretion disc around the merger remnant. On longer timescales, other outflows originate from the disc: faster ejecta produced by magnetic pressure and neutrino-matter interaction during the initial neutrinocooling-dominated accretion phase (we call these "wind ejecta"), and slower but more massive ejecta produced by viscous processes in the disc, especially during the advection-dominated phase (Dessart et al. 2009;Metzger & Fernández 2014;Perego et al. 2014;Just et al. 2015; Siegel & Metzger 2017 -we call these "secular ejecta"). Substantial differences in the ejecta properties arise depending on the post-merger scenario (see i.e. Kawaguchi et al. 2019). In order to calculate dynamical ejecta and disc mass from a NSNS merger we adopt the fitting formulae reported in Radice et al. (2018), which are calibrated on a large suite of high-resolution GRHD simulations 7 . Both quantities depend on the NS masses and tidal deformabilities. We also adopt their formula for the dynamical ejecta mass-weighted average asymptotic velocity v dyn . The NS tidal disruption can occur also in BHNS mergers. If the NS is disrupted outside the innermost stable circular orbit, then the released material remains outside the BH in the form of a crescent (e.g. Kawaguchi et al. 2016), otherwise the NS plunges directly onto the BH. We adopt the fitting formula from Foucart et al. (2018) to calculate the total mass remaining outside the BH, M out . This quantity depends on the BH mass and spin, and on the NS mass, tidal deformability and baryonic mass M b NS . We adopt the formulae in Kawaguchi et al. (2016) to calculate the dynamical ejecta mass and average velocity v dyn in this case. M dyn depends on the BH mass and spin, the NS mass, baryonic mass and compactness C NS , and on the angle ι tilt between the binary total angular momentum and the BH spin. We assume ι tilt = 0. v dyn depends only on the mass ratio q = M BH /M NS . We proceed as in Barbieri et al. (2019b) to calculate C NS from Λ NS and M b NS from M NS and C NS . We then calculate the disc mass as M disc = max[M out − M dyn , 0]. As in Barbieri et al. (2019b) we assume that M dyn cannot exceed 30%M out , considering recent BHNS simulations presented in Foucart et al. (2019). 7 We note that Kiuchi et al. (2019) showed that the predictions from these formulae might underestimate the produced disc mass in binaries with large mass ratios. However they consider the case with M NS,1 = 1.55 M and M NS,2 = 1.2 M , thus lowmass/largely deformable NSs. Instead, as can be seen in Fig. 2 In what follows, we conservatively assume the BH to be non-spinning (χ BH = 0), corresponding to the worst condition for ejecta production 8 . EJECTA PROPERTIES FOR "AMBIGUOUS" CHIRP MASSES In Fig. 2 we show the dynamical ejecta and disc masses on the (M 1 , M 2 ) plane along lines of constant M c . We limit the y axis to M max NS as we focus on systems that contain at least one NS. It is apparent that NSNS configurations compatible with "ambiguous" chirp masses do 8 among the co-rotating configurations. Indeed the counterrotating cases (χ BH < 0) are the worst conditions in absolute, more often leading to a direct plunge. However counter-rotating configurations are not expected for field binaries but for the dynamically formed ones, that represent a negligible contribution to the merger rate (Ye et al. 2019). not produce dynamical ejecta (upper panel of Fig. 2). In this parameter region, the fits from Radice et al. (2018) predict the absence of this kind of ejecta, due to the prompt collapse of the remnant to a BH. Conversely, BHNS configurations can more easily produce dynamical ejecta. These systems have small mass ratio q < 5 and low-mass (large Λ NS ) NSs, which is the optimal condition to produce massive ejecta in BHNS mergers (as shown in Barbieri et al. 2019b,a). The same arguments hold for disc masses (bottom panel in Fig. 2). Note that the value of M disc predicted by the fitting formula for NSNS systems in the considered range is set by the lower limit indicated in Radice et al. (2018), which is M disc = 10 −3 . For BHNS configurations, instead, discs with masses up to ∼ 7 × 10 −2 M are produced. It is important to note that when the differences of the ejecta mass for the BHNS and NSNS cases are substantial, they are larger than the systematic errors. The uncertainties on the fitting formulae for NSNS are ∆M (Foucart et al. 2018). Therefore, being M BHNS disc = M out − M BHNS dyn , we assume its uncertainty to be ∆M BHNS disc = (∆M out ) 2 + (∆M BHNS dyn ) 2 . We define σ dyn = (∆M NSNS dyn ) 2 + (∆M BHNS dyn ) 2 and σ disc = (∆M NSNS disc ) 2 + (∆M BHNS disc ) 2 . We indicate as pink shadowed area in Fig. 2 the regions where the differences in the mass of dynamical ejecta and disc for the BHNS and NSNS cases are greater than or equal to σ dyn and σ disc , respectively. In these regions the ejecta mass differences are larger than the systematic errors. Figure 3 summarizes the differences between two representative NSNS and BHNS systems with "ambiguous" chirp masses (cases II and III in the Figure), and also a "GW170817-like" NSNS case, for comparison. For the latter we consider a NSNS system with masses 1.46 M and 1.27 M . Merger (I) produces relatively low-mass dynamical ejecta at all latitudes, with a preferentially equatorial angular distribution ∝ sin 2 θ, where θ is the polar angle (Perego et al. 2017). The accretion disc is massive and ∼ 20% of its mass is unbound in the form of secular ejecta, with a similar angular distribution as for the dynamical ones, while ∼ 5% of its mass goes into the wind ejecta, mostly confined in the polar region (θ < π/3 rad - Perego et al. 2017). After the merger, an intermediate state with a hyper-massive NS could exist before collapsing to a BH (gray central object represented in Fig. 3-I). The strong neutrino winds produced in this state interact with the ejecta, increasing the electron fraction Y e or, equivalently, lowering the opacity. We consider a system with 2 M and 1.6 M stars (II) as our representative NSNS merger in the "ambiguous" chirp mass range. As explained above, in this case we expect no dynamical ejecta and a low-mass accretion disc, resulting in low-mass wind and secular ejecta. The merger remnant collapses promptly to a BH. The absence of an intermediate hyper-massive NS state implies little neutrino wind, giving a low Y e in the ejecta (Kawaguchi et al. 2019). Finally, as BHNS merger in the "ambiguous" chirp mass range we consider a system with M BH = 3 M and M NS = 1.1 M (III). In BHNS mergers, the dynamical ejecta have a crescent-like shape, extending into half of the equatorial plane and limited to the region with θ < 0.3 rad (Kawaguchi et al. 2016). In the considered system, the dynamical ejecta and accretion disc are massive. Due to the lack of a neutrino wind, the fraction of accretion disc flowing into wind ejecta is lower than in the NSNS case (we assume ∼ 1%). The disc fraction that goes into secular ejecta is the same as in the NSNS case. As a consequence, the secular ejecta are massive, while the wind ejecta have low mass. The ejecta Y e is lower than the NSNS case. Therefore, being the ejecta properties substantially different for the NSNS and BHNS cases in the "ambiguous" chirp mass range, we expect the kilonova light curves to present important differences as well. KILONOVA MODEL The neutron-rich material ejected in NSNS and BHNS mergers is an ideal site for r-process nucleosynthesis, which produces the heaviest elements in the Universe (Lattimer & Schramm 1974;Eichler et al. 1989;Korobkin et al. 2012;Wanajo et al. 2014). The synthesized nuclei are unstable and they decay radioactively, powering the kilonova emission (Li & Paczyński 1998;Metzger et al. 2010;Kasen et al. 2013). We compute the kilonova light curves using the composite semi-analytical model presented in Barbieri et al. (2019b,a) (in part based on Perego et al. 2017;Martin et al. 2015;Grossman et al. 2014). For the NSNS cases we assume the model parameters (ejecta geometry, opacity and the fractions of M disc that go into wind and secular ejecta) as in Perego et al. (2017). The model has been tested on GW170817: using the parameters inferred for this event (Perego et al. 2017;Abbott et al. 2017), we obtain multi-wavelength light curves consistent with the observed ones (Villar et al. 2017, paper in preparation). For BHNS systems we assume the same model parameters as in Barbieri et al. (2019b,a) (based on Kawaguchi et al. 2016;Fernández et al. 2017;Just et al. 2015). The kilonova light curves are highly degenerate with respect to binary parameters. Thus, it is impossible to infer the system properties from the kilonova light curve alone. This degeneracy can be (at least partially) broken using information from GW analysis. In particular, the measurement of the binary chirp mass reduces the number of parameters by one. Leaving i.e. M 1 as a free parameter, M 2 is constrained by the measured M c . KILONOVAE FOR "AMBIGUOUS" CHIRP MASSES In Fig. 4 we show the envelope of the kilonova light curves expected from NSNS and BHNS mergers, for four selected values of the chirp mass. We consider emission in the g (509 nm) and K (2143 nm) bands and the figure shows the absolute magnitude as a function of time. For all the "ambiguous" chirp masses the fitting formulae from Radice et al. (2018) in NSNS mergers predict no dynamical ejecta and a minimum allowed disc mass M disc = 10 −3 M . Thus we have a single light curve for NSNS mergers, and we can expect that these events would not produce kilonovae brighter than shown in Fig. 4. For BHNS mergers there exists a range of light curves for each M c , arising from the different combinations of the component masses, producing different ejecta properties. For M c = 1.45 M (panels 1a-1b) all kilonovae from BHNS mergers are much brighter at every time than that from NSNS mergers. Therefore a single observation in one of these bands would allow in principle to distinguish the nature of the merging system. At higher values of the chirp mass, there is only a small overlap between the BHNS and NSNS cases, at the bottom of the BHNS envelope. Therefore, except for observed light curves at low absolute magnitudes, it should be always possible to distinguish the nature of the merging system by the sole kilonova brightness. We note that the disentangling of the nature of the binary is optimal when M c = 1.45 M (panels 1a-1b). In this case, as shown in Fig. 2, the mass interval of the ejecta from BHNS mergers is the narrowest, and this in turn leads to the narrowest spread in the kilonova light curves. The prediction of BHNS kilonovae as bright or brighter than NSNS ones is presented in Kawaguchi et al. (2019). They find that BHNS kilonovae are brighter in the nearinfrared K-band, due to the smaller electron fraction Y e in the ejecta owing to the lack of strong neutrino irradiation from the central remnant. In the i band, Kawaguchi et al. (2019) find that NSNS configurations ending with the formation of a supermassive NS leads to brighter kilonovae than the BHNS case. This is due to the strong neutrino emission produced in this case, that increases Y e in the ejecta. However, in their study they compare sundry BHNS and NSNS configurations not selected on the bases of the chirp mass. By contrast, in our work, we compare BHNS and NSNS mergers at fixed chirp mass. This requirement restricts the NSNS binary configurations to cases producing no dynamical ejecta and very low mass discs. Therefore, whatever the value of Y e in the ejecta from NSNS merger is, the mass propelled in the merger is so low that almost all the BHNS kilonovae are brighter, at all wavelengths. For other comparisons between NSNS and BHNS merger outcomes and studies on distinguishing the nature of merging compact binaries see Hinderer et al. (2019); , who considered an unconventional BH companion with mass of ∼ 1.4 M , thus below the maximum NS mass. As a visual comparison we also show the kilonovae for BHNS binaries having a NS with a representative" mass of 1.4 M (aqua/magenta lines). For M c = 1.45 M such a binary does not exist, while for M c = 1.75 M it is fated to a direct plunge, thus there is no kilonova. We remark that the kilonova light curves from BHNS are inferred assuming non-spinning BHs (χ BH = 0). As explained in Barbieri et al. (2019b,a), increasing the BH spin (fixing all the other parameters) leads to more massive ejecta and, consequently, more luminous kilonovae. Therefore, if the BHs have a non-zero spin, our argument would be even stronger. As an example, for χ BH 0.5 all light curves from BHNS kilonovae would be brighter than those from NSNS binaries in each band and at any time, in this critical range of ambiguous" chirp masses. CONCLUSION The detection of a BHNS coalescence could be the next ground-breaking discovery in multi-messenger astronomy. At the time of writing, there are promising GW candidates detected by the LVC during the observation run O3. The associated detection of an electromagnetic signal from these new GW sources might contribute to our understanding of the physical processes that power the multi-wavelength EM emission (Gompertz et al. 2018;Rossi et al. 2019, and references therein). From the GW signal, one of the best constrained parameters in low latency is the binary chirp mass, a combination of the masses of the two components. This parameter is currently used to classify the binary, whether the system hosts two NSs, two stellar BHs or a BH and a NS. In the present Letter, we point out that in absence of a "mass gap" between the NS and BH mass distributions (as Thompson et al. 2019 seem to indicate), there exists a range of M c (as shown in Fig. 1) for which it is not possible to distinguish the nature of the binary on the basis of the chirp mass measurement alone 9 . For the SFHo EoS adopted in this analysis, we find that the values of the chirp mass between 1.233 M and 1.792 M are compatible either with NSNS and BHNS systems. In this Letter we show that the observation of the kilonova emission from these systems can break the degeneracy in the "ambiguous" chirp mass range, and constrain the nature of the merging system. We find that kilonova emission shows substantial differences in the lu-minosity and temporal evolution in NSNS and BHNS systems (see Fig. 4). In particular, the BHNS case is far more luminous than the NSNS case, even when the BH is non-spinning. This happens because in in this "ambiguous" M c range the NSNS configurations represent the worst cases for ejecta production, while the BHNS configurations represent the best ones. It is important to note that, when the differences of the ejecta mass for the BHNS and NSNS cases are substantial, they are larger than the systematic errors in the modeling. Therefore, observing the kilonova associated with such an event is of fundamental importance to break the degeneracy on the nature of the merging system. Furthermore, if the observed kilonova is compatible with a BHNS merger, this would provide evidence in support of the existence of low-mass BHs, filling the "mass gap". This work illustrates the potential of multi-messenger observations of compact binary mergers, and the impor-tance of an efficient exchange of information between the GW and EM communities (Biscoveanu et al. 2019;Margalit & Metzger 2019). * Crisp 2019 ; 2019Pereyra 2019). Figure 1 . 1M 1 − M 2 compact binaries having the same chirp mass Mc. Different lines indicate different values for Mc. We assume the SFHo EoS: the maximum NS mass is M max NS = 2.058 M , and M min NS is set equal to 1 M . Yellow, green and violet regions of the parameter space indicate, respectively, NSNS, BHNS and BHBH binaries. Gray-hatched areas indicate binary configurations with at least one component in the mass gap. Red and blue lines represent M gapNS c,min and M NSNS c,max , respectively (see text for definition). M gapNS c,min = 1.233 M is the chirp mass corresponding to a gap+NS binary with M NS = M min NS and M BH = M max NS (red line). All GW events with chirp mass below M gapNS c,min Figure 2 . 2Dynamical ejecta (top panel) and accretion disc (bottom panel) masses for different values of the binary chirp mass Mc. We assume the SFHo EoS (M max NS = 2.058 M ) and non-spinning BHs (χ BH = 0). The vertical red line separates NSNS configurations (left) and BHNS ones (right). Each line corresponds to a Mc (reported on it). The pink shadowed area is the region where differences of the ejecta mass for the BHNS and NSNS cases are larger than systematic errors. The yellow stars indicate the BHNS systems with the NS having a representative" mass of 1.4 M . Figure 3 . 3Cartoon of the ejecta and disc produced in different systems: (I) a NSNS merger with ∼ 1.46 M and ∼ 1.27 M stars, close to the masses in GW170817; (II) a NSNS merger with two massive stars of ∼ 2 M and ∼ 1.6 M ; and (III) a BHNS merger with a light BH of ∼ 3 M and a NS of ∼ 1.1 M . NSNS and BHNS configurations II and III correspond to the same "ambiguous" chirp mass Mc = 1.55 M . Red, orange, light blue and purple represent dynamical ejecta, accretion disc, wind ejecta and secular ejecta, respectively. Filled areas correspond to massive components, while hatched areas correspond to low mass components. disc +5×10 −4 M (Radice et al. 2018). The uncertainties on the fitting formulae for BHNS are ∆M BHNS dyn = 0.2 M dyn (Kawaguchi et al. 2016) and ∆M out = 0.1 M out Figure 4 . 4Light curve ranges for kilonovae produced by NSNS (dashed lines) and BHNS (filled areas) binaries with a chirp mass Mc = 1.45 M (1), 1.55 M (2), 1.65 M (3) and 1.75 M (4). The a" and b" panels show light curve ranges in the g (509 nm) and K (2143 nm) band, respectively. Aqua (magenta) lines represent the kilonova in the g (K) band for the BHNS systems with a NS having a representative mass of 1.4 M . [email protected] We defer to the LIGO/Virgo O3 Public Alerts webpage https: //gracedb.ligo.org/superevents/public/O3/ for a complete list of current candidates.1 Universit degli Studi di Milano-Bicocca, Dipartimento di Fisica "G. Occhialini", Piazza della Scienza 3, I-20126 Milano, Italy 2 INAF -Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate, Italy 3 INFN -Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy 4 Università degli Studi di Trento, Dipartimento di Fisica, via Sommarive 14, I-38123 Trento, Italy 5 ). The BH mass distribution observed so far in coalescing binaries is broad (The LIGO Scientific Collaboration et al. 2018), extending up to 50 +16.6 −10.2 M , with the lightest BH carrying a mass 7.6 +1.3 −2.1 M , close to the mean BH mass in observed Galactic X-ray binaries of ∼ 7.8 ± 1.2 M (Özel et al. 2010). Double NS systems observed so far carry masses in the interval 1.165 M − 1.590 M , we consider systems with M NS,1 > 1.65 M and M NS,2 > 1.35 M . Therefore the NSs in our systems are less deformable and we expect that the underestimation reported in Kiuchi et al. (2019) is less significant. We use the term "spin" to indicate the dimensionless spin parameter, χ BH = cJ/GM 2 BH , where J is the BH angular momentum. Offline GW signal analysis could provide more precise information. This would lead to stronger constraints on the component masses that could allow to distinguish the nature of the merging system using GW alone, in some cases. We thank F. Zappa and S. Bernuzzi for sharing EoS tables. The authors acknowledge support from INFN, under the Virgo-Prometeo initiative. O. S. and G.G. acknowledges the Italian Ministry for University and Research (MIUR) for funding through project grant 1.05.06.13. . B P Abbott, R Abbott, T D Abbott, 10.3847/2041-8213/aa9478ApJL. 85039Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, ApJL, 850, L39 . B P Abbott, R Abbott, T D Abbott, 10.1103/PhysRevLett.119.161101Phys. Rev. Lett. 119161101Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, Phys. Rev. Lett., 119, 161101 . B P Abbott, 10.3847/2041-8213/aa91c9Astrophys. J. 84812Abbott, B. P., et al. 2017b, Astrophys. J., 848, L12 . 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[ "Opto-Mechanical Design of ShaneAO: the Adaptive Optics System for the 3-meter Shane Telescope", "Opto-Mechanical Design of ShaneAO: the Adaptive Optics System for the 3-meter Shane Telescope" ]	[ "C Ratliff \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "J Cabak \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "D Gavel \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "R Kupke \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "D Dillon \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "E Gates \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "W Deich \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "J Ward \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "D Cowley \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "T Pfister \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n", "M Saylor \nUCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA\n" ]	[ "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA", "UCO Lick Observatories\nUniversity of California\n1156 High St. Santa Cruz95064Santa CruzCAUSA" ]	[]	A Cassegrain mounted adaptive optics instrument presents unique challenges for opto-mechanical design. The flexure and temperature tolerances for stability are tighter than those of seeing limited instruments. This criteria requires particular attention to material properties and mounting techniques. This paper addresses the mechanical designs developed to meet the optical functional requirements. One of the key considerations was to have gravitational deformations, which vary with telescope orientation, stay within the optical error budget, or ensure that we can compensate with a steering mirror by maintaining predictable elastic behavior. Here we look at several cases where deformation is predicted with finite element analysis and Hertzian deformation analysis and also tested. Techniques used to address thermal deformation compensation without the use of low CTE materials will also be discussed.	10.1117/12.2057064	[ "https://arxiv.org/pdf/1407.8209v1.pdf" ]	119,231,072	1407.8209	24d261c8f0e98ef5e50d481e7dc652816a9db739	Opto-Mechanical Design of ShaneAO: the Adaptive Optics System for the 3-meter Shane Telescope C Ratliff UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA J Cabak UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA D Gavel UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA R Kupke UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA D Dillon UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA E Gates UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA W Deich UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA J Ward UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA D Cowley UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA T Pfister UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA M Saylor UCO Lick Observatories University of California 1156 High St. Santa Cruz95064Santa CruzCAUSA Opto-Mechanical Design of ShaneAO: the Adaptive Optics System for the 3-meter Shane Telescope adaptive optics, opto-mechanical design A Cassegrain mounted adaptive optics instrument presents unique challenges for opto-mechanical design. The flexure and temperature tolerances for stability are tighter than those of seeing limited instruments. This criteria requires particular attention to material properties and mounting techniques. This paper addresses the mechanical designs developed to meet the optical functional requirements. One of the key considerations was to have gravitational deformations, which vary with telescope orientation, stay within the optical error budget, or ensure that we can compensate with a steering mirror by maintaining predictable elastic behavior. Here we look at several cases where deformation is predicted with finite element analysis and Hertzian deformation analysis and also tested. Techniques used to address thermal deformation compensation without the use of low CTE materials will also be discussed. The effects of flexure become especially problematic during long camera exposures. All structural systems experience some deformation with a change in applied load and a Cassegrain mounted telescope instrument will experience a continual change in gravitational load direction while tracking a celestial object. Figure 1 shows the Shane telescope fork axis (dashed line), which the telescope and instruments rotate about as a celestial object is tracked during an exposure. Since the Cassegrain instrument mount, located under the primary mirror, also rotates, there are two axes of motion that affect gravity induced flexure in the AO system. For most of the AO stages and mounts, we designed for a worst-case change in gravitational load which would correspond to a 3 hour camera exposure; a change in gravity load by roughly 45 degrees. II. DESIGN APPROACH Several designs will be presented to show our approach to designing positioning stages and structural support systems to meet our functional requirements. These designs include: 1. A 5 axis stage for the 32x32 MEMs deformable mirror, required to align the deformable mirror with the wavefront sensor; 2. Custom designed x-y flexure stage for the wave front sensor; 3. AO bench support structure. The location for the MEMs and wavefront sensor are shown in Figure 2, the center ray of the telescope light is represented by the light blue line. The flexure error budget for each subassembly is outlined in Table 1. MEMS STAGE The MEMS 5 axis positioning stage design is based on a Newport 9082 5 axis aligner mounted on a custom aluminum base. It consists of a plate supported by 6 spheres and springs that load the plate against the spheres. The position of each sphere is controlled by a fine pitch screw/wedge/motor mechanism that controls the position of each sphere in a direction that is normal (or nearly so) to the plate. This stage was chosen given the small footprint allowance in the optical prescription and large number of degrees of freedom to control. There was a problem with the off the shelf design because it allowed too much deformation under varying gravity loads, measured to be 5x higher than requirements. We decided to try to improve the stock design by making it stiffer. The first thing to investigate was more preload on the kinematic interfaces. Based on some deformation analysis, it was estimated the spring preloading force would need to be roughly 5x higher. With this higher preload, there would be more stress on the material. If this stress were to cause plastic deformation/brinneling, positioning issues and possibly hysteresis in the positioning mechanism would result. We therefore decided to investigate replacing the stock 6061 T6 aluminum material with several candidate materials which included M2 tool steel and sapphire pucks. Figure 5 shows the stress in the aluminum for a 30 N load and 9.5mm diameter spheres. The stress exceeds the 6061 T6 aluminum yield stress which is 40,000 psi (276 MPa). M2 tool steel turned out to have some margin of safety with respect to yielding for the same load and although less stiff than sapphire, it is much less expensive and quicker to fabricate. After fabrication we tested our design with the setup shown in Figure 6. It consists of an alt-azimuth index table, linear variable differential transformer displacement transducer with 0.1 micron resolution and the modified stage. In summary, we were able to transform a stock 5-axis kinematic stage into a much stiffer one. M2 tool steel was selected based on its cost, high modulus of elasticity, high strength, and ease of machining. Both the plate supported by spheres and translating wedges were replaced with this high strength steel. All mating surfaces were machined to a surface finish of approximately 10 microinches RMS. Preload spring forces were increased by a factor of 3-6 depending on orientation. The angular flexure was reduced to less than 10% of the stock stage. Experiments showed a much stiffer stage with no plastic deformation. The original stage with the stock aluminum and brass components exhibited less stiffness and considerable plastic deformation. WAVEFRONT SENSOR FLEXURE STAGE The Shane adaptive optics system utilizes a Shack-Hartmann wavefront sensor with selectable configurations. Depending on light levels, the user may choose an 8x8 or 16x16 array across-the-pupil subaperture arrangement, with an upgrade path to 32 subapertures across available in the future. The wavefront sensor consists of a fast 160x160, 21 m pixel Lincoln Laboratories CCD detector in a SciMeasure camera head. To control fine positioning on the sub pixel level, we mounted it on a custom x-y flexure stage. In front of the camera is the selector stage that allows choosing between optics that provide 8x or 16x Hartmann lenslet sampling across the beam. There is an open slot that may be occupied by a 32x lenslet barrel in the future. The x-y flexure stage on the camera head allows fine alignment to the lenslet barrel selected. The AO system will be used with both natural guidestars (NGS) and laser guidestars (LGS), therefore a requirement is that the entire wavefront sensor (lenslet optics and camera) ride on a focus stage to accommodate the shift in focus between NGS and LGS. Each item is indicated in Figure 7. The two other stages, the lenslet selector and focus stages, provide closed loop control with sub 100nm resolution and sub 2 micron resolution. One thing that had to be considered in this design was the Hertzian deformation of the Picomotor spherical tip against the flexure stage contact area. We found out that the spherical tip pushing against a flat contact would dominate the overall deformation for a 7.5 degree change in the gravity vector relative to the reference frame of the flexure stage. We were able to dramatically reduce the Hertzian deformation by having the Picomotor tip interface with a spherical receiving socket. A comparison of the Hertzian deformation for several cases is shown in Table 3. A load change of 1.7 N corresponds to a worst case change in gravity load by 7.5 degrees. A section view of the final design is shown in Figure 8. This improvement reduced the deformation by 95%. Worth noting is the Picomotor was near its axial force limit with the amount of preload used. In hindsight an actuator with substantially more axial force capability than the required preload would have been preferred. Another important aspect in designing this flexure stage is determining thicknesses. Our error budget in Z (as shown in Figure 8) is 1 micron, 10x that in the x and y, but still an important requirement to meet. Finite element analysis was used to determine if the as-designed thickness, labeled Z in Figure 8, is appropriate. Again the change in load was a 7.5 degree change in the gravity vector, rotated about the X axis. The change in the Z deformation for this case was small, about .04 microns, and well within specifications. As it turned out, the gravity load changes were not the most important factor in designing this aspect of the flexure stage. External cable forces, which are highly dependent on stress relief implementation, drove the decision to keep the Z value the same as that shown in the FEA rather than make the flexure stage thinner. SUPPORT STRUCTURE The support structure consists of several systems: the AO opto-mechanical support assemblies, the optical table, and the optical table support system. The AO opto-mechanical support assemblies include anything that interfaces with the optical beam or calibration of the beam. It consists of many subassemblies, two of which are discussed above. In addition to the deformable mirrors the assemblies include 30 axes of precision linear and rotary motion. The optical table is a custom sandwich structure that consists of 400 series steel plates bonded and welded to a trussed steel honeycomb interior. The table is all steel rather than a low CTE material like Invar. This table is the base support for all of the AO opto-mechanical support assemblies. The optical table support structure is shown in Figures 10 and 11. Six struts support the table that anchor into several welded and bolted steel frames that interface with the telescope. Figure 10 shows the optical path thru the system. Figure 11 shows the six struts connecting to the rear surface of the optical table. The six support struts form three bipods. One of these bipods is shown in Figure 12. The axes of the bipod struts intersect at the mounting surface. The concept of intersecting at the surface with spherical bearing support at the end of each strut eliminates transmission of moment loads to the optical table. This design reduces both bending stress imposed on the table and flexure which could cause misalignment between optical components. The goal is to support the table as a simple 6-degree-of-freedom rigid body and avoid introducing moments, which could add unwanted bench deformation. Each strut that makes up the bipods is adjustable in length along the strut axis with a fine thread turnbuckle design. This adjustability allows for kinematic positioning of the optical table in all 6 degrees of freedom and allowed for relatively easy alignment of the first steering mirror to the center ray of the telescope. Bipods Another aspect of the design was to ensure we maintained elastic deformation for all load conditions. One such scenario was earthquake loading. We looked at the response spectrum to a 2 G load with a forcing function between 0.5Hz and 100 Hz. The stress results (psi) are shown in Figure 13 for one direction of load excitation. The resonant frequencies for all of the significant modes are shown in Table 4. The results show that the stress is less than 10,000 psi for key structural components. Elastic deformation ensures there is no need to realign after exposure to these types of loads, and that any residual flexure motion is repeatable. This allows the AO system to use active compensation using steering mirrors to account for drift in the optical axis as the gravity vector changes. The thermal behavior of the support structure is no less important than the gravitational flexure. In our design, one objective was to ensure that we are insensitive to typical temperature swings that occur at the observatory while minimizing the use of more costly low coefficient of thermal expansion materials like Invar and glass ceramics. Again, the thermal behavior is designed to be repeatable so that the AO system can actively compensate beam wander as a function of temperature. The material coefficients of thermal expansion (CTE) for the welded frame (1020 steel), struts (stainless steel), optical bench (carbon steel and 400 stainless), and first steering mirror (stainless steel) were all selected to be close to that of the telescope itself (1020 steel). As a result, the system expands and contracts together. For a 15 C change in temperature, the struts supporting the table change length by over 60 microns. However the table and first fold mirror move less than 5 microns relative to the center beam of the telescope. We also used the strategy of matched CTE's for corresponding input sensors and actuators for the AO assemblies. The stages and support structure of the wave front sensor and MEMs have matched CTE's. Both are made primarily from 6061 T6 Aluminum. Although they move more than other parts of the system that are made of steel, Invar, or Zerodur, they move together to maintain a 1:1 relationship between each wave front sensor quad cell and MEMs actuator group. The same is true of the tip-tilt sensor and tip-tilt deformable mirror. Using aluminum allows lower cost mounts, better selection of commercial translation stages, and less weight. III. CONCLUSION: We have touched on some of the many design aspects to consider when building the opto-mechanical structure for a high precision astronomical instrument. AO systems improve angular resolution over seeing-limited instruments and this maps directly to putting more demanding precision and stiffness requirements on the opto-mechanical mounts and stages. The key to a successful design is to meet the functional requirements, which include motion range and precision, ease of use, and spatial constraints, while at the same time addressing the alignment problems introduced by load variations, vibration, or changes in temperature. Figure 1 : 1Gravity Vector and Fork Mounted Rotation Axis Figure 2 : 2MEMs and WFS in AO System Figure 3 :Figure 4 : 34MEMs Close Up View of Kinematic Translation Stage: Wedges Omitted, Base Plate is Shown as Transparent Figure 5 . 5Aluminum: 30 N Spring Preload Yields Material: Stress Exceeds 40,000 psi Figure 6 . 6Flexure Measurement Figure 7 : 7Wave Front SensorA fairly demanding stability requirement, 0.1 micron linear deformation for a 30 minute exposure, for this wave front sensor requires attention to certain aspects of the design. We decided on a custom flexure stage since space was limited and it needed to mate with the Scimeasure CCD camera head mount geometry. Flexure stages can provide high precision positioning with a high stiffness in axes other than the direction of travel. For driving the x-y flexure stage we chose an off the shelf product from Newport, a Picomotor™, which provides open loop control with 30 nm resolution. 3 :Figure 8 : 38Comparison of Change in Deformation for Load Change of 1.7 N Section View Camera Flexure Stage Design Figure 9 : 9FEA Results, 7.5 Degree Change in Gravity Vector Figure 10 :Figure 11 : 1011Front Side of AO System, Telescope Light Highlighted Blue Backside View Figure 12 : 12Optical Bench Support Bipod, Highlighted in Blue Figure 13 : 13Resulting Stress for Vibration Mode of Maximum Displacement (in psi) Figure 14 : 14Shane AO on Telescope, Covers Removed Table 1 : 1Opto-Mechanical Stability RequirementsSplits light into wavefront sensor (WFS). Dichroic changer will select one of three dichroics depending on science wavelength and LGS/NGS mode.Name Description Table 2 : 2Measured Deformation ResultsCase1 Stock Kinematic Stage Case2 M2 Tool Steel and 6 to 11 lb preload Case3 M2 Tool Steel and 15 to 20 lb preload Maximum Deflection ( μm) within 22.5° of Zenith 3.7 1.4 0.8 Maximum Angular Deflection (arcsec) within 22.5° of Zenith 7.3 2.8 1.6 Maximum Deflection ( μm) within 45° of Zenith 4.6 2.2 1.5 Angular (arcsec) Hysteresis 5.9 2.2 0.5 Table Table 4 : 4Resonant Vibration Frequencies Under 100 HzMode Frequency (Hz) 1 26.643 2 29.256 3 30.698 4 46.885 5 53.978 6 63.328 7 78.720 ShaneAO: an enhanced adaptive optics and IR imaging system for the Lick Observatory 3 meter telescope. R Kupke, D Gavel, Proceedings of SPIE. 8447Kupke, R., Gavel, D., et al. "ShaneAO: an enhanced adaptive optics and IR imaging system for the Lick Observatory 3 meter telescope," Proceedings of SPIE Vol. 8447, (2012). Development of an enhanced adaptive optics system for the Lick Observatory Shane 3 meter Telescope. Donald T Gavel, Proc. SPIE. SPIE7931793103Gavel, Donald T., "Development of an enhanced adaptive optics system for the Lick Observatory Shane 3 meter Telescope," Proc. SPIE 7931, 793103 (2011). Atmospheric Limitations to High Angular Resolution Imaging. F Roddier, Proc. Of ESO Conf. on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths. Of ESO Conf. on Scientific Importance of High Angular Resolution at IR and Optical WavelengthsRoddier F., "Atmospheric Limitations to High Angular Resolution Imaging." Proc. Of ESO Conf. on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths (1981) Formulas for Stress and Strain" 5 th edition. R Roark, W Young, McGraw Hill Book Co516Roark, R., Young, W., "Formulas for Stress and Strain" 5 th edition, McGraw Hill Book Co. (1975) p. 516. . John W Hardy, Oxford University PressNew YorkAdaptive Optics for Astronomical TelescopesHardy, John W. [Adaptive Optics for Astronomical Telescopes]. New York: Oxford University Press, 1998.	[]
[ "Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation", "Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation" ]	[ "Lorenzo Piroli \nSISSA and INFN\nvia Bonomea 26534136TriesteItaly\n", "Pasquale Calabrese \nSISSA and INFN\nvia Bonomea 26534136TriesteItaly\n" ]	[ "SISSA and INFN\nvia Bonomea 26534136TriesteItaly", "SISSA and INFN\nvia Bonomea 26534136TriesteItaly" ]	[]	We consider the ground-state properties of an extended one-dimensional Bose gas with pointwise attractive interactions. We take the limit where the interaction strength goes to zero as the system size increases at fixed particle density. In this limit the gas exhibits a quantum phase transition. We compute local correlation functions at zero temperature, both at finite and infinite size. We provide analytic formulas for the experimentally relevant one-point functions g2, g3 and analyze their finitesize corrections. Our results are compared to the mean-field approach based on the Gross-Pitaevskii equation which yields the exact results in the infinite system size limit, but not for finite systems.	10.1103/physreva.94.053620	[ "https://arxiv.org/pdf/1609.03854v2.pdf" ]	118,605,049	1609.03854	683ed60efe36fb1f7fe055bbae5d501093699946	Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation Lorenzo Piroli SISSA and INFN via Bonomea 26534136TriesteItaly Pasquale Calabrese SISSA and INFN via Bonomea 26534136TriesteItaly Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation We consider the ground-state properties of an extended one-dimensional Bose gas with pointwise attractive interactions. We take the limit where the interaction strength goes to zero as the system size increases at fixed particle density. In this limit the gas exhibits a quantum phase transition. We compute local correlation functions at zero temperature, both at finite and infinite size. We provide analytic formulas for the experimentally relevant one-point functions g2, g3 and analyze their finitesize corrections. Our results are compared to the mean-field approach based on the Gross-Pitaevskii equation which yields the exact results in the infinite system size limit, but not for finite systems. I. INTRODUCTION The study of one-dimensional quantum integrable models has produced many remarkable results over the past fifty years. Among its greatest successes, is the derivation of thermodynamical properties of extended systems from the underlying microscopic quantum theory. A unified theoretical framework of integrability is now well established as reported in many excellent textbooks [1][2][3][4][5][6]. Until recently, however, the interest for these studies has been mainly academic, due to the lack of experimental applications. The situation has completely changed during the past two decades, due to the new experimental possibilities coming from the physics of ultra-cold atoms. Indeed, optical and magnetic traps can nowadays be employed to effectively confine atoms in one spatial dimension where almost ideal Hamiltonians are engineered with a high degree of isolation and control over the experimental parameters [7,8]. Thus the results of exact calculations in integrable models can be tested in cold atomic laboratories, offering a playground where theory and experiments can be compared directly and without ambiguity. One of the prototypical examples of integrable models is the one-dimensional Lieb-Liniger gas, describing a system of bosons with pointwise interactions. This model has a long history [9][10][11] and has been intensively studied in the literature, but the exact computation of correlation functions still represents a remarkable theoretical challenge. At the same time, this problem is of paramount importance for a comparison with cold atomic realizations of confined bosons, where quantum correlations are routinely measured in experiments [12][13][14][15][16][17][18][19][20][21][22][23][24]. In the case of repulsive interactions, a significant amount of theoretical work has already been devoted to the computation of correlation functions . Over the years this problem has inspired the development and application of sophisticated techniques based, for example, on the Bethe ansatz method [26,27,32,38,39,41,46,47] or on field theoretical approaches [36,40]. It is worth mentioning that while the focus has been traditionally on ground states and thermal states, the past few years have also witnessed an increasing interest in the computation of correlation functions in arbitrary excited states of integrable systems [50][51][52][53][54][55][56][57], also in connection with its relevance in the study of non-equilibrium dynamics of one-dimensional Bose gases [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73]. Attractive interactions have been less studied in the literature [10,[74][75][76][77][78][79][80][81][82][83][84][85]. In this case, the traditional thermodynamic limit of the model is ill-defined, with divergences in the ground state energy and in local correlation functions [4,10]. These divergences reflect the physical property that strong attractive interactions lead to instabilities in a gas containing a large number of bosons. A stable, non-thermal stationary state can nevertheless be obtained in the thermodynamic limit as a result of an interaction quench, as for the super Tonks-Girardeau gas [20,[86][87][88][89][90][91] or in a quench from the non-interacting model [71,72]. In spite of these problems, there are two interesting regimes where the attractive Bose gas can be studied in thermal equilibrium both at zero or finite temperature. The first is the zero density limit (see e.g. [81]), where the system size is sent to infinity, keeping the number of particles finite. The second regime is the one investigated in this work, i.e. the infinite system size limit taken with fixed density of particles but with the attractive interaction sent to zero as system size increases. We will refer to this as a weakly interacting thermodynamic limit. Importantly, no divergences arise in this regime because an extended gas of attractive bosons is stable for sufficiently small attractive interactions. Furthermore, in this case the system exhibits interesting properties that are absent in the zero density limit such as a quantum phase transition with varying the (rescaled) interaction strength [74,76,85]. Here we compute local correlation functions at zero temperature in the weakly interacting thermodynamic limit. We consider the one-point functions g 2 and g 3 , which are accessible in cold atomic experiments, exploiting their relation to photoassociation and three-body recombination rates [16,17]. Besides the per se interest for a comparison with experimental implementations of the attractive 1D Bose gas [12,13], our results might be a starting point for the challenging task of computing correlation functions in arbitrary excited states of the attractive Lieb-Liniger model. In the first part of this paper we address the exact com-arXiv:1609.03854v2 [cond-mat.quant-gas] 19 Nov 2016 putation of these correlation functions using the Bethe ansatz method building upon the results of some recent works [41,85]. In the second part, our findings are compared to the mean-field approach based on the Gross-Pitaevskii equation [92,93]. While for one-dimensional systems it is known that the validity of the latter breaks down for finite interactions [9,94], it is expected to give accurate results in the small interaction regime. Our calculations show explicitly that the results for local correlations obtained by means of the Gross-Pitaevskii equation become exact in the limit of infinite system size and vanishing interaction, but they are incorrect for finite systems. This unveils a direct link between the descriptions of the system in terms of the Bethe ansatz method and of the Gross-Pitaevskii equation. The rest of this manuscript is organized as follows. In section II we briefly introduce the Lieb-Liniger gas and its exact solution. We then discuss in section III the weakly interacting thermodynamic limit and review some recent results [85] regarding the Bethe ansatz characterization of the ground state in the attractive regime. Section IV is devoted to the computation of one-point correlation functions g 2 and g 3 , both at finite size and in the infinite system size limit. The Gross-Pitaevskii equation is then introduced in section V, where we compare the mean-field results to those obtained by means of the Bethe ansatz method. Finally, conclusions are presented in section VI. Some technical aspects of our work are provided in the appendixes. II. THE LIEB-LINIGER MODEL We consider the Lieb-Liniger model [9] of N bosons with pointwise interactions on a ring of length L with Hamiltonian H = − 2 2m N j=1 ∂ 2 ∂x 2 j + 2c j<k δ(x j − x k ).(1) The interaction strength is related to the one dimensional scattering length a 1D through c = − 2 /ma 1D [95] and can be varied via Feshbach resonances [96] to take either positive or negative values. In the following we set = 2m = 1 and focus on the attractive regime c = −c < 0 .(2) The equivalent second quantization Hamiltonian is H = L 0 dx ∂ x Ψ † (x)∂ x Ψ(x) + cΨ † (x)Ψ † (x)Ψ(x)Ψ(x) ,(3) where Ψ † and Ψ are bosonic creation and annihilation operators satisfying [Ψ(x), Ψ † (y)] = δ(x − y). The Hamiltonian (1) can be diagonalized by means of the Bethe ansatz [9]. The N -body eigenfunctions are ψ N (x 1 , . . . , x N ) = P >k 1 + ic sgn(x − x k ) λ P − λ P k × N j=1 e iλ P j xj ,(4) where the sum is over the N ! permutations P of the rapidities {λ j } N j=1 . The latter are complex numbers which parametrize the different eigenstates of the Hamiltonian, and satisfy the quantization conditions (Bethe equations) e −iλj L = N k =j λ k − λ j − ic λ k − λ j + ic , j = 1, . . . , N .(5) The momentum (K) and energy (E) of a given eigenstate are expressed in terms of the rapidities λ j as K {λ j } N j=1 = N j=1 λ j ,(6)E {λ j } N j=1 = N j=1 λ 2 j .(7) Introducing the density of particles D and the dimensionless interaction γ [9] D = N L , γ = −c D ,(8) the standard thermodynamic limit is defined as N, L → ∞ with D, γ fixed. As we already mentioned in the introduction, this is ill-defined in the attractive regime because it gives rise to divergences in the ground state energy and local correlation functions [10]. It is possible to overcome this problem introducing the rescaled interaction κ = −γN 2 ,(9) and defining the weakly interacting thermodynamic limit as N, L → ∞ , D, κ fixed .(10) As we will show in the following, despite the interaction strength goes to zero as N increases, this limit is nontrivial and the physics of the model depends only on κ. III. THE GROUND-STATE RAPIDITY DISTRIBUTION FUNCTION In the attractive regime, for any number of particles N the rapidities corresponding to the ground state of the model are always aligned along the imaginary axis and centered around λ = 0 [4,10]. In the zero density limit, the rapidities satisfy the well-known string hypothesis [4], according to which they display a uniform spacingc between one another. This is no longer the case in the limit (10), where the rapidities arrange themselves along the imaginary axis according to a non-trivial distribution function. The latter has been recently derived for arbitrary κ [85] as reviewed in this section. We mention that partial results where also presented in previous works [80,97], while numerical studies of the groundstate rapidities are reported in [78,79]. The ground-state rapidities correspond to the unique set (up to permutations) of purely imaginary solutions {λ j } N j=1 of (5). They are pictorially displayed in Fig. 1 and in the limit (10) they shrink to the point λ = 0. It is then convenient to define the following rescaled ground-state rapidities (which have a different normalisation compared to [85]): x j = −iλ j L .(11) Plugging (11) into the Bethe equations (5) and taking the logarithm one obtains the following system of equations for the rescaled rapidities x j = N l=1 l =j log x j − x l + κ/N x j − x l − κ/N .(12) In the limit (10) the rescaled rapidities x j arrange themselves according to a non-trivial distribution function ρ κ (x), characterized by the property that for any function f (x) one can write N j=1 f (x j ) = N dxρ κ (x)f (x) + O(1) .(13) It has been found [80,85] that a critical value κ * of the interaction exists such that ρ κ (x) is qualitatively different for κ > κ * and κ < κ * , namely κ * = π 2 .(14) For 0 < κ ≤ κ * the rapidity distribution function is determined as the solution of the integral equation [85] x = 2κ− xmax xmin dy ρ(y) x − y . Here we introduced the principal value integral [98] − xmax xmin f (x) x − y dx ≡ lim ε→0 y−ε xmin f (x) x − y dx + xmax y+ε f (x) x − y dx ,(16) while x min , x max are chosen consistently with the normalization condition xmax xmin ρ κ (x) = 1 .(17) These equations share some similarities with the large-N limit of the Bethe equations in the Richardson pairing model [97,99]. The solution of (15) under the condition (17) is [85] ρ κ (x) = 1 κπ κ − x 2 4 , x ∈ [−2 √ κ, 2 √ κ] , 0 \|x\| > 2 √ κ .(18) An important constraint on the ground-state rapidities is \|x j − x k \| > κ/N [85], resulting in the condition ρ κ (x) ≤ 1 κ .(19) For κ < κ * , ρ κ (x) in (18) always satisfies (19). The critical point κ * = π 2 is identified with the value of the interaction such that ρ κ * (x) in (18) has a maximum (in x = 0) equal to 1/κ. The form of the ground-state rapidity distribution changes qualitatively for κ > κ * and it reads [85] ρ k (x) =    1/κ x ∈ [−bκ, bκ] , ρ κ (x) x ∈ [−aκ, −bκ] ∪ [bκ, aκ] , 0 \|x\| > aκ .(20) The parameters a and b are defined as the solution of the non-linear system    z = b 2 /a 2 , 4K(z) [2E(z) − (1 − z)K(z)] = κ , aκ = 4K(z) ,(21) while the functionρ κ (x) is determined by the singular integral equation x = 2 log x + κb x − κb + 2κ− Ω dyρ κ (y) x − y ,(22) where the principal value integral is over the domain Ω = [−aκ, −bκ] ∪ [bκ, aκ] .(23)0 0.1 0.2 −6 −4 −2 0 2 4 6 ρ κ (x) x κ = 2 κ = 5 κ = 10.5 κ = 12 FIG. 2. Rescaled rapidity distribution ρκ(x) for different val- ues of κ. The values κ = 2, 5 < κ * correspond to one quantum phase, while κ = 10.5, 12 > κ * correspond to the other. For κ > κ * there is a plateau ρκ(x) = 1/κ centered around x = 0. The solution of (22) can be found explicitly to be [85,100] ρ (21), (24) are the elliptic integrals of the first, second and third kind: (x) = 2 πa\|x\|κ 2 (a 2 κ 2 − x 2 )(x 2 − b 2 κ 2 ) × Π b 2 κ 2 x 2 , b 2 a 2 . (24) The functions K(x), E(x) and Π(x, y) appearing inK(z) = π/2 0 dϑ 1 1 − z sin 2 ϑ ,(25)E(z) = π/2 0 dϑ 1 − z sin 2 ϑ ,(26)Π(x, y) = π/2 0 dϑ 1 (1 − x sin 2 ϑ) 1 − y sin 2 ϑ .(27) We report in Fig. 2 the rapidity distribution functions corresponding to different values of κ in the two different regimes κ ≤ κ * and κ > κ * . The qualitative difference in the behavior of ρ κ (x) for κ ≤ κ * and κ > κ * is a signal of a quantum phase transition. We will return to the nature of the latter in section V. A qualitative change of the distribution of the Bethe rapidities in correspondence of a quantum phase transition has been observed also in other integrable models [101,102]. Using the above results, the ground state energy per particle can be computed as (29). The vertical dashed line corresponds to the critical value of the interaction κ * = π 2 , for which e0(κ) exhibits a discontinuity in its second order derivative. gs (κ) = 1 N N j=1 λ 2 j = − 1 N L 2 N j=1 x 2 j = − D 2 N 2 e 0 (κ) + O(1/N 3 ) , (28) where e 0 (κ) = dxρ κ (x)x 2 .(29) From (28) we have that gs (κ) → 0 as N → ∞ according to the limit (10): hence the ground state energy coincides with that of the non-interacting state. However, we will see that the ground-state local correlation functions in the limit (10) are qualitatively different from those of the free case. It is worth to discuss the relation of the limit (10) with other regimes studied in the literature. Consider the large N limit N → ∞ , L, κ fixed ,(30) where κ is as usual given by (9). In this case the value of the density grows indefinitely. In this regime, the ground state energy is non-vanishing and given bỹ gs (κ) = − 1 L 2 e 0 (κ) + O(1/N ) ,(31) where e 0 (κ) is the same as in (29). The limit (30) has been studied previously in a number of works [74-76, 82, 83] and it is known that the system undergoes a quantum phase transition. In particular e 0 (κ) [as well as˜ gs (κ), cf. (31)] exhibits a discontinuity in the second order derivative for κ = κ * , cf. Fig. 3. Conversely, all the physical quantities depending only on κ (such as the local correlations g K ) will have the same value in both limits (10) and (30). IV. LOCAL CORRELATION FUNCTIONS We address now the computation of the ground-state one-point correlation functions. In particular, we consider the K-body functions g K = GS\|(Ψ † (0)) K Ψ K (0)\|GS D K ,(32) where Ψ, Ψ † are the bosonic field operators in the second quantization formalism, D is the particle density (8), and \|GS the ground state. We focus on the cases K = 2, K = 3 which are directly relevant for experimental coldatomic realizations of bosons confined in one dimension. In particular, g 2 is the so called local pair correlation function which can be determined by measures of photoassociation rates [17]. Analogously, g 3 is proportional to the three-body recombination rate [16]. Intuitively, g K gives information about the probability of finding K bosons in the same position. A. Finite-size correlators The knowledge of the exact (normalized) ground state wave function ψ GS allows the computation of any correlation function. For example, g 2 can be expressed as g 2 = N (N − 1) D 2 L 0 dx N −2 ψ * GS (0, 0, x 1 , . . . , x N −2 ) × ψ GS (0, 0, x 1 , . . . , x N −2 ) . (33) However, the representation (33) involves the evaluation of ∼ (N !) 2 multiple integrals, because of the form of the wave function (4). Hence, the r.h.s. of (33) can be in practice evaluated only for very small values of N . A remarkable simplification of the problem was obtained by Balázs Pozsgay [41], who derived the following alternative representation for g K by means of algebraic Bethe ansatz methods g K = (K!) 2 D K {λ + }∪{λ − } \|{λ + }\|=K   j>l λ + j − λ + l (λ + j − λ + l ) 2 + c 2 )   × det H det G ,(34) where H jl = (λ j ) l−1 for l = 1, . . . , K , G jl for l = K + 1, . . . , N ,(35) and G jl being the Gaudin matrix G jl = δ jl L + N r=1 ϕ(λ j − λ r ) − ϕ(λ j − λ l ) ,(36) with ϕ(u) = 2c/(u 2 + c 2 ). The sum in (34) is over all the partitions of the set of rapidities {λ j } N j=1 into two disjoint sets {λ + j } K j=1 and {λ − j } N −K j=1 . Furthermore, the order of the rapidities in both H and G in each term of the sum is understood to be given by the ordered set (34) was obtained in [41], where only the repulsive regime was considered, but it holds also in the attractive case (2), because its derivation is purely algebraic. As an additional check, for small N and negative values of the interaction, we numerically verified that (34) agrees with the result obtained by direct integration of the ground-state wave function (33). {λ + j } K j=1 ∪ {λ − j } N −K j=1 . The result Despite Eq. (34) being a great simplification with respect to multiple integral representations of the form (33), it is still not completely satisfying from the computational point of view when large numbers of particles are considered. Furthermore, it is not suitable for the analysis of the thermodynamic limit N → ∞. In fact, it is possible to derive a more efficient representation by direct manipulation of (34). This requires a sequence of technical steps which are illustrated in appendix A, while here we report only the final result. The results obtained in appendix A can be written as g 2 (κ, N ) = 2 N κ N j=1 x 2 j − x j w (1) j ,(37)g 3 (κ, N ) = 1 N κ 2 N j=1 3x 2 j w (2) j − 4x 3 j w (1) j + x 4 j − 2 N κ N j=1 w (1) j x j + 1 N 3 N j=1 x j w (1) j − x j . (38) Here x j are the rescaled rapidities (11) while the parameters w w (l) m + 1 N N j=1 2κ w (l) m − w (l) j (x m − x j ) 2 − κ 2 /N 2 = x l m .(39) These formulas allow the exact computation of g 2 and g 3 for very large number of particles (we use up to N 2000 in Sec. IV C). In Fig. 4 we report g 2 and g 3 calculated for several N with this method. Obviously, no singularity occurs in the behavior of local correlations for finite systems, but for κ ∼ κ * a discontinuity in the first derivatives of both g 2 and g 3 emerges while increasing N , as we will analytically show in the next subsection. Finally, it is worth mentioning that by direct evaluation of (37) and (38), one has lim κ→0 g 2 (κ, N ) = 1 − 1 N ,(40)lim κ→0 g 3 (κ, N ) = 1 − 1 N 1 − 2 N ,(41) namely for κ → 0 we recover the ground-state correlators of the free system (i.e. the limit κ → 0 and the weakly interacting thermodynamic limit commute). B. Large-N limit We now address the computation of the one-point correlation functions in the weakly attractive thermodynamic limit (10). Our starting point is given by the formulas (37) and (38) for finite N . The calculation is rather cumbersome, but the final results can be easily written down. Thus, we anticipate the final results and their discussion, reporting the derivation soon after. The full dependence on κ of one-point local correlators in the large-N limit is g 2 (κ) κ N = 50 N = 100 N = 200 N = 500 N = ∞ g 3 (κ) κ N = 50 N = 100 N = 200 N = 500 N = ∞g K (κ) = 1 , 0 ≤ κ ≤ κ * , g s K (κ) κ > κ * ,(42) where g s K (κ) is for K = 2 g s 2 (κ) = 1 48 κ 16(a 2 + b 2 ) − (a 2 − b 2 ) 2 κ ,(43) and for K = 3 g s 3 (κ) = 1 240 κ 2 23a 4 + 82a 2 b 2 + 23b 4 − 2(a 2 − b 2 ) 2 (a 2 + b 2 )κ . (44) The parameters a and b are the solution of the system (21) and are easily evaluated numerically for any κ. Equations (43) and (44) give immediately the value of g 2 and g 3 in the thermodynamic limit. As it is evident from Fig. 4, the functions g 2 (κ) and g 3 (κ) are not smooth at the critical point κ * , where their derivative is discontinuous. For κ < κ * the local correlators coincide with those of a non-interacting systems but they rapidly increase for κ > κ * . This is expected: as the attractive interaction is increased the bosons tend to cluster and have a higher probability of being found in the same position. Note that the opposite behavior is observed for positive values of the coupling γ [28,29,40,41] where the repulsive nature of the interaction is responsible for a decrease in the one-point functions g 2 and g 3 . Equations (43) and (44) also allow for the analysis of the one-point functions in the two limits κ ∼ κ * and κ → ∞. The derivation presents no difficulty and it is sketched in appendix C. Setting κ = κ * + δ , κ > κ * ,(45) in the limit δ → 0 + one finds g 2 (δ) = 1 + 4δ π 2 + O(δ 2 ) ,(46)g 3 (δ) = 1 + 12δ π 2 + O(δ 2 ) .(47) Analogously in the limit κ → ∞ g 2 (κ) = κ 6 + O(1) ,(48)g 3 (κ) = κ 2 30 + O(κ) .(49) These asymptotic behaviors are displayed in Fig. 5, together with the numerical evaluation of (42). Derivation of the large-N results In the large-N limit, the parameters w (l) j are replaced by a continuous function of the rapidities w (l) (x) such that w (l) j → w (l) (x j ). From (37), (38) one readily obtains g 2 (κ) = 2 κ dxρ κ (x) x 2 − xw (1) (x) ,(50)g 3 (κ) = 1 κ 2 dxρ κ (x) 3x 2 w (2) (x) − 4x 3 w (1) (x) + x 4 − 2κxw (1) (x) ,(51) where the integrals are over the support of the rapidity distribution function ρ κ (x). The problem is then reduced to determining the auxiliary functions w (l) (x). (42). Logarithmic scales are used on both axes and the vertical dashed line corresponds to κ * = π 2 . Dotted black lines show the asymptotic behavior for large κ as given by (48), (49), while red dashed lines correspond to the first order expansion in κ − κ * as in (46), (47). The idea is to transform the discrete system (39) into a linear integral equation for w (l) (x), analogously to what was done in [41] for the repulsive case. Note, however, that in the case considered here one immediately faces the technical issue of dealing with singular integral kernels of the form K(x, y) = 1 (x − y) 2 .(52) Furthermore, when x m+1 x m + κ/N the denominator appearing in the l.h.s. of (39) vanishes and near contributions to the sum (corresponding to the terms \|j − m\| N ) might be important. The continuum limit of (39) is then non-trivial and has to be performed separately for κ < κ * and κ > κ * . The analysis of the large-N limit of the Bethe equations (12) (cf. also [85]), suggests that for κ < κ * the near contributions \|j − m\| N can be neglected in the sum of equation (39); the large-N limit of the latter can then be cast in the form w (l) (x) + 2κ= dyρ κ (y) w (l) (x) − w (l) (y) (x − y) 2 = x l .(53) Here we introduced the Hadamard principal value integral defined as [103,104] = xmax xmin f (x) (x − y) 2 dx ≡ lim ε→0 y−ε xmin f (x) (x − y) 2 dx + xmax y+ε f (x) (x − y) 2 dx − 2f (y) ε .(54) Equation (53) can be explicitly solved for l = 1, 2: one can explicitly verify, making use of (15), that the follow-ing functions are a solution of (53) w (1) (u) = 1 2 u ,(55)w (2) (u) = 1 3 u 2 + 2 3 κ .(56) Using now the explicit form of ρ κ (x) (18) and equations (50), (51) one obtains g 2 (κ) = g 3 (κ) = 1 , 0 ≤ κ ≤ κ * ,(57) namely for κ < κ * one-point functions are the same as a non-interacting system. In the regime κ > κ * , the computation of the auxiliary functions w (l) (x) is much more involved. From section III, we know that in the interval (−κb, κb) [where b is defined in (21)] the rescaled rapidities x j arrange themselves in such a way that for large N they display an equal spacing κ/N between one another and then one can use the parametrization x j+1 = x j + κ N + δ j N ,(58) where δ j vanishes in the thermodynamic limit. Then the corresponding term in the sum (39) apparently diverges as 1/δ j , but this divergence is canceled if w (l) j is approx- imately constant in (−κb, κb) , namely w (l) j+1 = w (l) j +δ j N ,(59) whereδ j is also vanishing for N → ∞. Hence, we make the following ansatz for the functions w (l) (x) w (l) (x) = C (l) x ∈ (−κb, κb), w (l) (x) x ∈ Ω ,(60) where Ω is defined in (23) while C (l) ,w (l) (x) are respectively a constant and a non-trivial function to be determined. This ansatz is well supported by numerical evidence, which provides a posteriori justification for (60). We now complete the task of explicitly computing the functionsw (1) (x),w (2) (x). First, note that w (1) (x) is odd with respect to x = 0. Hence, it has to be C (1) = 0. Next, we assume that in the region Ω defined in (23), near contributions to the sum in (39) can be neglected, so that one can plug the ansatz (60) directly into (53). As a result, we find that the functionw(x) is determined bỹ w (1) (x) 1 + 4κb x 2 − b 2 κ 2 +2κ= Ω dyρ κ (y)w (1) (x) −w (1) (y) (x − y) 2 = x ,(61) for x ∈ Ω. Making use of the identity [104] = Ω dy ρ κ (y) (x − y) 2 = − d dx − Ω dy ρ κ (y) (x − y) ,(62) and of (22), Eq. (61) is easily rewritten as 2κ= Ω dyρ κ (y)w (1) (y) (x − y) 2 = −x .(63) Rescaling the variables as ζ = y aκ , ξ = x aκ ,(64) we are left with the simple equation = −r −1 dζ + = 1 r dζ f (1) (ζ) (ζ − ξ) 2 = − a 2 κ 2 ξ ,(65) where r = b/a and where we introduced f (l) (ζ) = ρ κ (ζ)w (l) (ζ) .(66) Assuming the continuity of the function w (1) (x), we have that f (1) (ζ) satisfies the following conditions f (1) (±r) = f (1) (±1) = 0 .(67) Equation (65) belongs to the general family of integral equations with hypersingular kernel 1 π = −r −1 dζ + = 1 r dζ f (ζ) (ζ − ξ) 2 = χ(ξ) ,(68) which admits an explicit solution for an arbitrary regular function χ(x) [105] which is f (ζ) = 1 π − ζ −1 1 R(u) (B + Φ(u))du ζ ∈ (−1, −r) , 1 π − 1 ζ 1 R(u) (B + Φ(u))du ζ ∈ (r, 1) ,(69)where R(u) = (1 − u 2 )(u 2 − r 2 ) 1/2 ,(70)Φ(u) = − −r −1 dv χ(v)R(v) u − v − − 1 r dv χ(v)R(v) u − v ,(71) and where as usual we used the symbol of dashed integral for the principal value integral (16). The constant B is defined as B = P F ,(72) where P = 1 r du R(u) − 1 r tR(t) u 2 − t 2 (χ(t) + χ(−t)) dt , (73) F = 1 r dt R(t) .(74) In the special case of (65), from (72) and (73) we have B = 0, since χ(ξ) is an odd function. The remaining integrals can be performed analytically and after long but straightforward calculations one obtains from whichw (1) (ζ) follows directly from (66). One has now all the ingredients to explicitly compute g 2 (κ) for κ > κ * . From (50), using (24) and (75) and after straightforward integration one gets (43). The computation ofw (2) (x) can be performed analogously. However, the technical steps are now more involved and its derivation is reported in appendix B, together with that of the final result (44),. f (1) (ζ) = a 2 κ (1 − ζ 2 ) (ζ 2 − r 2 ) 4π ,(75) C. Finite-size corrections We now investigate the finite size corrections for g 2 and g 3 . Away from the critical point, finite size corrections are expected to exhibit an analytical behavior in 1/N . We evaluated numerically the formulas (37) and (38) for large system sizes up to N 1000 finding that indeed the leading correction is in 1/N . For κ < κ * , one could even try to tackle this problem analytically, generalizing the techniques of [106] where the Bethe equations in the isotropic spin-1/2 Heisenberg chain are studied and the leading corrections in the system size computed. Remarkably, the Bethe equations studied in [106] share a formal analogy with (15). However, the study of onepoint functions also requires inspection of finite-size corrections to the auxiliary equation (53). In any case, these techniques cannot be applied directly at the critical point where a more sophisticated treatment is required. At the critical point κ * = π 2 , finite-size corrections are more severe as it is clear from Fig. 4. To understand their behavior we consider the quantities ∆ K (N ) = g K (κ * , N ) − 1 ,(76) satisfying lim N →∞ ∆ K (N ) = 0. For several values of N we computed ∆ 2 (N ) and ∆ 3 (N ) from (37), (38), and reported our results in Fig. 6. As expected, the dependence on N is not consistent with an analytic behavior in 1/N . Accordingly, for large N we fit the numerical values of ∆ K (N ) using the function K (N ) = A K N α K + B K N .(77) For numbers of particles up to N 2000, the best fit for the exponents α K are α 2 = 0.667 ,(78)α 3 = 0.665 .(79) while the coefficients A K and B K are A 2 = 2.09 , B 2 = −1.5 ,(80)A 3 = 6.06 , B 3 = −3.1 .(81) The numerical estimates (78), (79) suggest the exact value for the exponents to be 2/3. The fitting function (77) is displayed in Fig. 6, showing excellent agreement with the numerical data. In particular, the exponents (78), (79) justify the slow approach of g K (κ * , N ) to the asymptotic value g K (κ * ) = 1 displayed in Fig. 4. V. THE GROSS-PITAEVSKII EQUATION In the previous section we considered the computation of one-point functions by means of the Bethe ansatz method. In this section we address the interesting comparison between these exact results and the mean-field approach based on the Gross-Pitaevskii equation [92,93]. While in one dimension the mean-field approximation breaks down for sufficiently strong interaction [9,94], it is expected to give accurate results in regimes of small coupling [107][108][109][110][111]. In the case of one-dimensional attractive bosons, this was investigated in [107] in the zero density limit showing that mean-field results for the ground-state energy and reduced one-body density matrix are exact to the leading order in N , when N → ∞. It is then of interest to test the mean-field approach also in the weakly attractive thermodynamic limit considered here. This is especially true for the higher-body one-point functions g 2 , g 3 which were not considered in previous studies and for which the question of the accuracy of mean-field calculations is non-trivial. In the mean-field description, the ground state is approximated by the product of single-particle wave functions as ψ GS (x 1 , . . . , x N ) = N j=1 φ(x j ) .(82) The optimal wave function φ(x) is obtained by minimization of the functional E[φ] = ψ GS \|H\|ψ GS ,(83) where H is the Hamiltonian (1). Following this prescription and using standard techniques, one is directly led to the time-independent Gross-Pitaevskii equation − ∂ 2 ∂x 2 + 2cN 1 − 1 N \|φ(x)\| 2 φ(x) = µ N φ(x) ,(84) where 0 ≤ x ≤ L, and where µ is the chemical potential, which is introduced to ensure the normalization condition L 0 dx\|φ(x)\| 2 = 1 .(85) The ground-state wave function then corresponds to the solution of (84) with the smallest energy (83). Solutions of (84) are known explicitly, and have been studied both in the repulsive [112] and attractive regime [74]. The exact solution is written in terms of the rescaled variable ξ = 2π L x ∈ [0, 2π],(86) together with the rescaled single-particle wave functioñ φ(ξ) = L 2π 1/2 φ(x(ξ)) .(87) Setting c = −κ/(N L), it is straightforward to obtain from the previous relations − ∂ 2 ∂ξ 2 − 2πν N (κ)\|φ(ξ)\| 2 φ (ξ) =μ Nφ (ξ) ,(88) where we introduced ν N (κ) = κ 2π 2 1 − 1 N ,μ N = µ 4π 2 D L ,(89) and where the normalization condition now reads 2π 0 dξ\|φ(ξ)\| 2 = 1 . The solution of minimal energy of (88) under the periodicity conditionφ(0) =φ(2π) can be found in [74,76]: where ν * = 1/2, K(x) and E(x) are the complete elliptic integrals in (25), (26) while dn(x\|m) is the Jacobian elliptic function [112]. The real parameter ζ ∈ [0, 2π] can be chosen arbitrarily while the other real parameter m N is the solution of the non-linear equation φ ζ (ξ) = 1/ √ 2π 0 ≤ ν N ≤ ν * , K(m N ) 2πE(m N ) dn K(m N ) π (ξ − ζ) ν N > ν * ,(91)K(m N )E(m N ) = π 2 ν N 2 .(92) We plot in Fig. 7 the wave function (91) for ζ = 0 and different values of ν N . Increasing ν N it displays a more distinct peak around ζ, corresponding to the emergence of a bright soliton [74,76]. Note also that (91) apparently breaks translational invariance, but this is not the case because the correct ground-state is recovered from (91) after averaging with respect to the peak position ζ. This is the same as the Bethe ansatz wave function (4) in which (given that the rapidities are purely imaginary) the ground state corresponds to the superposition of a family of many-body wave functions localized around the translated centers of mass of the bosons. Exploiting the factorized form (82), within the meanfield approach one-point functions can be simply obtained from the integral representations such as (33). Averaging with respect to ζ after performing the integration and expressing everything in terms of the rescaled wave function (87), one obtains the mean-field result g MF 2 = 2π 1 − 1 N 2π 0 dξ\|φ 0 (ξ)\| 4 ,(93)g MF 3 = (2π) 2 1 − 1 N 1 − 2 N 2π 0 dξ\|φ 0 (ξ)\| 6 ,(94) whereφ 0 (ξ) is given by (91) (with ζ = 0). The integrals (93), (94) can be performed analytically to yield g MF K (κ, N ) = g w K , 0 ≤ ν N (κ) ≤ ν * , g s K ν N (κ) > ν * ,(95)whereg w 2 = 1 − 1 N , g w 3 = 1 − 1 N 1 − 2 N ,(96)andg s 2 = K(m N ) 2 (m N − 1) 3E(m N ) 2 − 2(m N − 2)K(m N ) 3E(m N ) ,(97)g s 3 = − 8 m 2 N − 3m N + 2 K(m N ) 3 30E(m N ) 3 + 2 8m 2 N − 23m N + 23 K(m N ) 2 30E(m N ) 2 .(98) These expressions provide the mean-field results for the one-point functions at finite N . They are displayed in Fig. 8, where the comparison with the exact values of the previous section (also reported in the figure) show that they are close to the correct results, but show quantitative and qualitative differences. First, also for finite N the Gross-Pitaevskii equation predicts a critical point where the derivatives of g 2 and g 3 are discontinuous, which is of course only an artifact of the mean-field approach. Second, for κ ≤ κ * the mean-field finite-size corrections have the opposite sign compared to the exact ones. However for N → ∞, the Gross-Pitaevskii equation yields the exact results as we are going to show. From (89) we have ν ∞ (κ) = κ 2π 2 ,(99) so that we simply obtain g MF K (κ) = 1 , 0 ≤ κ ≤ κ * , g s K (κ) κ > κ * ,(100) whereg s K (κ) are still given by (97), (98), with the replacement m N → m ∞ ,(101) which satisfies K(m ∞ )E(m ∞ ) = κ 4 .(102) Remarkably, (100) coincides with the Bethe ansatz result (42), as it is explicitly shown in appendix D. VI. CONCLUSIONS We considered ground-state properties of the attractive Lieb-Liniger gas in the limit of large system size and weak interactions (10). We addressed the calculations of the physically relevant one-point functions g 2 and g 3 . We provided formulas valid at finite size and for arbitrary values of the interaction; furthermore, we showed that in the large-N limit they can be expressed in a simple analytical form. We analyzed numerically finite size corrections at the critical point. Finally we compared our calculations to the mean-field approach based on the Gross-Pitaevskii equation: while the latter provides approximate results for finite systems, it exactly predicts the correct values of one-point functions in the large-N limit. This result was not at all expected: while in the limit (10) the interaction γ vanishes with the system size L, one would have expected that many-body effects beyond mean-field would contribute to higher-body correlations such as g 2 and g 3 . Our explicit calculations show that this is not the case, providing a direct link between the Bethe ansatz and the Gross-Pitaevskii equation. The ground-state calculations performed here might be a useful starting point for the more challenging computation of correlation functions in highly excited states of the model. This would allow us to extend our results to the case of finite temperature and to provide an essential ingredient to characterize steady states in quantum quenches. Having an accurate description of the entire spectrum in the weakly interacting limit might be also useful to characterise the height distribution function in Kardar-Parisi-Zhang growth processes [113]. Indeed, through the replica trick this problem is related to the attractive Lieb-Liniger model [114]. This correspondence has been successfully used to describe the time evolution of the height distribution function in the thermodynamic limit for several experimentally relevant situations [115][116][117][118]. Exact calculations for finite systems are still challenging [119], but the results in this paper represent a promising starting point. VII. ACKNOWLEDGMENTS We thank Robert Konik, Austen Lamacraft and Andrea Trombettoni for useful discussions surrounding this work. P.C. acknowledges the financial support by the ERC under Starting Grant 279391 EDEQS. G jl = δ jl 1 + 1 N N r=1 2κ (x j − x r ) 2 − κ 2 /N 2 − 1 N 2κ (x j − x l ) 2 − κ 2 /N 2 .(A3) The determinant can be rewritten as det N G −1 H = det K W ,(A4) where W jk = G −1 jm x k−1 σ(m) .(A5) Here, σ is the permutation that maps the ordered set {x j } N j=1 into {x σ(j) } N j=1 = {x + j } K j=1 ∪{x − j } N −K j=1 Multiplying (A5) by G nj and summing over j, it is straightforward to show that W jk = w (k−1) σ(j) ,(A6) where w (k) j are the unique solution of the system (39). Next, we wish to simplify the sum over partitions, analogously to what was done in [41], where (34) was studied in the case of repulsive interactions. First, we consider the case K = 2. From (A1) we have g 2 = 2 N 2 N i,j=1 x i − x j (x i − x j ) 2 − κ 2 /N 2 w (1) i − w (1) j = 2 N κ N j=1 x j x j − w (1) j ,(A7) where in the first equality we used w (0) j = 1 while in last equality we used (39). The computation for g 3 is more involved. We define Λ ij = x i − x j (x i − x j ) 2 − κ 2 /N 2 ,(A8)Γ ij = 1 (x i − x j ) 2 − κ 2 /N 2 ,(A9) and present the following identity derived in [41] Λ ij Λ ik Λ jk = 1 3 Λ ij Γ jk + Λ jk Γ ki + Λ ki Γ ij −Λ ji Γ ik − Λ ik Γ kj − Λ kj Γ ji . (A10) Noting now det 3    1 w (1) k w (2) k 1 w (1) j w (2) j 1 w (1) i w (2) i    = − w (1) j − w (1) i w (2) k − w (2) j + w (1) k − w (1) j w (2) j − w (2) i ,(A11) and exploiting the properties of the determinant, g 3 can be rewritten as g 3 = 12 N 3 N i,j,k=1 Λ ij Γ jk − w (1) j − w (1) i w (2) k − w (2) j + w (1) k − w (1) j w (2) j − w (2) i . (A12) Summing now over the index k and using (39) we obtain g 3 = 6 κN 2 N i,j=1 x i − x j (x i − x j ) 2 − κ 2 /N 2 Ω ij , (A13) where Ω ij = w (1) j − w (1) i x 2 j − w (2) j − x j − w (1) j w (2) j − w (2) i .(A14) Using (39) it easy to show that N j,k=1 x k (x j − x k ) 2 − κ 2 /N 2 Ω jk = 0 ,(A15) so that from (A13) we are left to compute g 3 = 6 κN 2 N j,k=1 x j Ω jk (x j − x k ) 2 − κ 2 /N 2 = 3 κN 2 N j,k=1 Γ jk x j x k (x j − x k )(w (1) j − w (1) k ) − x j w (2) j (w (1) k − w (1) j ) + x j w (1) j (w (2) k − w (2) j ) − x k w (2) k (w (1) j − w (1) k ) + x k w (1) k (w (2) j − w (2) k ) , where the last equality comes from symmetrization of the numerator and simple rearrangements. The terms in the last two lines in the above expression can be simplified by summing over k and j respectively and using (39). This yields g 3 = 3 κN 2 N j,k=1 Γ jk x j x k (x j − x k )(w (1) j − w (1) k ) + 3 κ 2 N N j=1 (x 2 j w (2) j − x 3 j w (1) j ) .(A16) Finally, we wish to get rid of the double sum. From the simple identities x j x k (x j − x k ) = 1 3 (x k − x j ) 3 − x 3 k 3 + x 3 j 3 (A17) (x k − x j ) 3 = (x k − x j ) (x j − x k ) 2 − κ 2 N 2 + κ 2 N 2 (x k − x j ) ,(A18) and making once again use of (39), we can rewrite 3 κN 2 N j,k=1 Γ jk x j x k (x j − x k )(w (1) j − w (1) k ) = 1 κN 2 N j,k=1 (w (1) j − w (1) k )(x k − x j ) + 1 N 3 N k=1 x k × (w (1) k − x k ) + 1 κ 2 N N k=1 x 3 k (x k − w (1) k ) .(A19) As a last step, we exploit the symmetry of the sets {w (1) j } N j=1 , {x j } N j=1 to write N j,k=1 (w (1) j − w (1) k )(x k − x j ) = −2N N j=1 x k w (1) k . (A20) Putting everything together, we finally arrive at (38). Appendix B: Computation of w (2) (x) In this appendix we discuss the derivation of the function w (2) (x) in (60) and the proof of the expression for g 3 as given in (44). The treatment is analogous to the one presented in the main text for w (1) (x) and g 2 , even though technically more involved. First, note that in this case the constant C (2) in (60) is non-vanishing and has to be determined independently. Assuming as for w (1) (x) that near contributions to the sum (39) can be neglected in the region Ω [defined in (23)], we can plug the ansatz (60) into (53), yielding w (2) (x) 1 + 4κb x 2 − b 2 κ 2 − C (2) 4κb x 2 − b 2 κ 2 +2κ= Ω dyρ(y)w (2) (x) −w (2) (y) (x − y) 2 = x 2 . (B1) This equation admits a solution for arbitrary C (2) . Indeed, consider the translated functioñ ω (2) (x) =w (2) (x) − C (2) ,(B2) so thatf (2) (x) = ρ κ (x)ω (2) (x) (B3) satisfies again the condition (67). Note thatf (2) (x) is simply related to f (2) (x) in (66) through f (2) (x) = f (2) (x) − C (2) ρ κ (x) .(B4) Making use of the identity (62), Eq. (B1) can be written as 2κ= Ω dyf (2) (y) (x − y) 2 = −x 2 + C (2) ,(B5) and after performing the rescaling (64), we simply obtain = −r −1 dζ + = 1 r dζ f (2) (ζ) (ζ − ξ) 2 = − a 3 κ 2 2 ξ 2 − C (2) a 2 κ 2 , (B6) where r = b/a as usual. This equation is precisely of the form (68) and thus its general solution is given by (69). In order to fix the value of the constant C (2) , we observe that (39) where a, b are given in (21), while Ω is defined in (23). Equation (B9) uniquely fixes the value of C (2) . In particular, the following prescription can be used to numerically obtaining C (2) . One starts with the initial value C (2) = 0 and considers the corresponding solution of (B6) as given explicitly by (69). From this, one computes the l.h.s. of (B9), which yields a positive real number. Increasing the value of C (2) and repeating these steps, the l.h.s. of (B9) decreases monotonically: the correct value of C (2) is then simply obtained when the l.h.s. of (B9) vanishes. Once C (2) and hence f (2) (x) are known, the function w (2) (x) is immediately obtained by (66). Finally, g 3 can be numerically obtained from (51) after integration. In principle, the integrals involved in the exact solution (69) of (B6) can be performed analytically, as it was the case for the function f (1) (x) derived in the main text, cf. (75). However, now the constant C (2) is non-vanishing and the form of the solution is more involved: the analytical expressions arising in this case are unwieldy, making the full analytical derivation of (44) extremely tedious. On the other hand, as we described above the numerical value for C (2) and f (2) (x) can be obtained easily and g 3 computed accordingly. Then, for arbitrary values of κ one can numerically verify that this gives the same value as (44) to arbitrary numerical precision. The analytical expression (44), which is in this way proven numerically, is instead more easily obtained by comparison with the mean-field result, as discussed in appendix D. Appendix C: Asymptotics of correlators In this appendix we consider the asymptotic behavior of the one-point functions. First, we consider the limit δ = κ − κ * → 0 + . In this limit, z → 0, where z is the solution of the system (21). By considering the known series expansions of the functions K(z), E(z), we have from the second equation of the system (21) z = 2δ π 2 − 7δ 2 4π 4 + O(δ 3 ) .(C1) From the third equation of (21), using again the series expansion of K(z) we get a = 2 π − δ π 3 + 5δ 2 4π 5 + O(δ 3 ) ,(C2) while from the first equation b = 2 √ 2δ 1/2 π 2 − 15δ 3/2 4 √ 2π 4 .(C3) Plugging (C2), (C3) into (43) and (44) we obtain (46) and (47). Next, we consider the limit κ → ∞. In this limit z → 1 in such a way that (1 − z)K(z) → 0 .(C4) Then, from the second equation of the system (21) and using E(1) = 1 one obtains K(z) ∼ κ 8 .(C5) Making use of (C5), from the first and third equations of (21) it is then straightforward to obtain a, b ∼ 1 2 .(C6) Finally, plugging (C6) into (43), (44) we finally arrive at equations (48), (49). FIG. 1 . 1Pictorial representation of the ground-state rapidities in the weakly attractive thermodynamic limit. FIG. 3 . 3Rescaled energy per particle e0(κ) as defined in . . . , N ) are auxiliary variables determined as the solution of the equations FIG. 4 . 4One-point correlators g2 and g3 as a function of the interaction κ in (9) near the critical point κ * . The vertical dashed lines are a guide for the eye corresponding to the critical value κ * = π 2 . The results for increasing numbers of particles N are displayed, showing that large finite-size corrections are observed at the critical point κ * . FIG. 5 . 5One-point correlation functions in the limit (10), as evaluated from FIG. 6 . 6Finite size corrections ∆K (N ) [as defined in(76)] at the critical point κ * in log scales. The dots are the exact numerical values computed using formulas(37),(38), while lines correspond to the fit function(77). FIG. 7 . 7Squared absolute value of the ground-state Gross-Pitaevskii wave function (91) with ζ = 0. FIG. 8 . 8One-point correlators g2 and g3 as a function of the interaction κ near the critical point κ * . The vertical dashed lines correspond to the critical value κ * = π 2 . The exact numerical values of the correlators (solid lines) are displayed, together with the results obtained by means of the Gross-Pitaevskii equation (dashed lines) which become exact in the limit N → ∞. ( 2 ) 2(x) − 2bC (2) = 0 , (B9) Appendix A: Finite-size formulasIn this appendix we derive the finite size formulas for the one-point correlation functions(37),(38)reported in section IV.We start by rewriting(34)in terms of the rescaled rapidities(11). Exploiting the properties of the determinant we havewhere the matrices G and H are written in terms of the rescaled rapidities asandAppendix D: Equivalence of infinite-N limitsIn this appendix we show the equivalence of the Bethe ansatz and mean-field results(42)and(100)for the onepoint functions in the limit(10).Introducing the parameterone has the following identities[120]where we definedm ∞ = 1 − m ∞ . Using (D2), (D3) one can easily show from (102) that z ∞ satisfiesso that from the second equation of (21) we have z ∞ = z. Furthermore, exploiting again (21) one can showPlugging now (D5), (D6) and (D7) into (100) and after rearrangements one finally obtains(42). Exactly Solvable Models in Statistical Mechanics. 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[]	[ "Jane ", "Seth Sullivant " ]	[]	[]	We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a combinatorial interpretation for the coefficients in the numerator of the generating functions for OEIS A050446 in terms of the swap statistic on alternating permutations.	null	[ "https://arxiv.org/pdf/1901.07443v2.pdf" ]	119,164,555	1901.07443	bab45857f8e52aa69b10b5539d89e2e82afdf845	22 Jan 2019 Jane Seth Sullivant 22 Jan 2019THE h * -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET Date: January 23, 2019. h * -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET 5 We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a combinatorial interpretation for the coefficients in the numerator of the generating functions for OEIS A050446 in terms of the swap statistic on alternating permutations. Introduction and Preliminaries The zig-zag poset Z n on ground set {z 1 , . . . , z n } is the poset with exactly the cover relations z 1 < z 2 > z 3 < z 4 > . . . . That is, this partial order satisfies z 2i−1 < z 2i and z 2i > z 2i+1 for all i between 1 and ⌊ n−1 2 ⌋. The order polytope of Z n , denoted O(Z n ) is the set of all n-tuples (x 1 , . . . , x n ) ∈ R n that satisfy 0 ≤ x i ≤ 1 for all i and x i ≤ x j whenever z i < z j in Z n . We aim to understand the Ehrhart series of O(Z n ). The Ehrhart function of a polytope P , denoted i P (m) is the function that counts the number of lattice points in the m-th dilate of P for any positive integer m. That is, i P (m) = #(Z n ∩ mP ) where mP = {mv \| v ∈ P }. The Ehrhart series of P is the formal power series Ehr P (t) = ∞ m=0 i P (m)t m . The Ehrhart series of a polytope with integer vertices is a rational function of the form Ehr P (t) = h * P (t) (1 − t) d+1 where d is the dimension of P . The numerator polynomial in this rational expression is called the h * -polynomial of P . Our goal in this paper is to understand the h * -polynomials of the O(Z n ). For n = 1, 2, 3, 4, 5, 6, these have the following form 1, 1, 1 + t, 1 + 3t + t 2 , 1 + 7t + 7t 2 + t 3 , 1 + 14t + 31t 2 + 14t 3 + t 4 and appear in the Online Encyclopedia of Integer Sequences with reference number A205497 [4]. We began studying this problem because the Ehrhart polynomial of O(Z n ) is equal to that of the CFN-MC polytope of any rooted binary tree on n + 1 leaves [3]. Therefore, it is also equal to the Hilbert series of the toric ideal of phylogenetic invariants of the CFN-MC model on such a tree. To understand the h * polynomial of O(Z n ), we will interpret its coefficients in terms of a permutation statistic on alternating permutations. An alternating permutation on n letters is a permutation a 1 a 2 . . . a n such that a 1 < a 2 > a 3 < a 4 > . . . . Notice that alternating permutations coincide with bijective labelings of Z n with the numbers 1, . . . , n that agree with the partial order on Z n . We define the swap permutation statistic on an alternating permutation σ to be the number of integers i such that σ −1 (i) < σ −1 (i + 1) and swapping i and i + 1 in σ yields another alternating permutation. In other words, i to the left of i+1 and there is at least one other character between them. The goal of this paper is to prove the following theorem relating the h * -polynomial of O(Z n ) and the swap statistic. (Z n ) is h * O(Zn) (t) = σ t swap(σ) where this sum ranges over all alternating permutations of length n. In Section 2, we provide further background information on zig-zag posets and their order polytopes. We relate these to the theory of alternating permutations. Then we discuss the necessary background on Ehrhart theory. In Section 3, we prove our main result, Theorem 1.1, by giving a shelling of the canonical triangulation of the order polytope of the zig-zag poset. In Section 4, we give an alternate proof of Theorem 1.1 by counting chains in the lattice of order ideals of the zig-zag poset. This proof makes use of the theory of Jordan-Hölder sets of general posets developed in Chapter 3 of [7]. Preliminaries The zig-zag poset Z n on ground set {z 1 , . . . , z n } is the poset with the cover relations z 2i−1 < z 2i and z 2i > z 2i+1 for 1 ≤ i ≤ ⌊ n−1 2 ⌋. Linear extensions of the zig-zag poset are in bijection with alternating permutations of length n; that is, permutations a 1 . . . a n for which the a i 's satisfy a 2i−1 < a 2i and a 2i > a 2i+1 for 1 ≤ i ≤ ⌊ n−1 2 ⌋. The number of alternating permutations of length n is the nth Euler zig-zag number E n . The sequence of Euler zig-zag numbers starting with E 0 begins 1, 1, 1, 2, 5, 16, 61, 272, . . . . This sequence can be found in the Online Encyclopedia of Integer Sequences with identification number A000111 [4]. The exponential generating function for the Euler zig-zag numbers satisfies n≥0 E n x n n! = tan x + sec x. Furthermore, the Euler zig-zag numbers satisfy the recurrence 2E n+1 = n k=0 n k E k E n−k for n ≥ 1 with initial values E 0 = E 1 = 1. A thorough background on the combinatorics of alternating permutations can be found in [6]. The following new permutation statistic on alternating permutations is central to our results. Definition 2.1. Let σ be an alternating permutation. The permutation statistic swap(σ) is the number of i < n such that σ −1 (i) < σ −1 (i + 1) − 1. Equivalently, this is the number of i < n such that i is to the left of i + 1 and swapping i and i + 1 in σ yields another alternating permutation. The swap-set Swap(σ) is the set of all i < n for which we can perform this operation. We say that σ swaps to τ if τ can be obtained from σ by performing this operation a single time. For example, the alternating permutation 15342 has swap(15342) = 1 and Swap(15342) = {1}. Hence, 15342 swaps to 25341 and to no other alternating permutation. To every finite poset on n elements one can associate a polytope in R n by viewing the cover relations on the poset as inequalities on Euclidean space. Definition 2.2. The order polytope O(P ) of any poset P on ground set p 1 , . . . , p n is the set of all v ∈ R n that satisfy 0 ≤ v i ≤ 1 for all i and v i ≤ v j if p i < p j is a cover relation in P . Order polytopes for arbitrary posets have been the object of considerable study, and are discussed in detail in [5]. In the case of O(Z n ), the facet defining inequalities are those of the form −v i ≤ 0 for i ≤ n odd v i ≤ 1 for i ≤ n even v i − v i+1 ≤ 0 for i ≤ n − 1 odd, and −v i + v i+1 ≤ 0 for i ≤ n − 1 even. Note that the inequalities of the form −v i ≤ 0 for i even and v i ≤ 1 for i odd are redundant. The order polytope of Z n is also the convex hull of all (v 1 , . . . , v n ) ∈ {0, 1} n that correspond to labelings of Z n that are weakly consistent with the partial order on {p 1 , . . . , p n }. In [5], Stanley gives the following canonical unimodular triangulation of the order polytope of any poset P on ground set {p 1 , . . . , p n }. We turn our attention to the study of Ehrhart functions and series of lattice polytopes. Let P ⊂ R n be any polytope with integer vertices. Recall that the Ehrhart function, i P (m), counts the integer points in dilates of P ; that is, Chapter 3]. We further define the Ehrhart series of P to be the generating function −v 1 ≤ 0 −v 3 ≤ 0 v 1 − v 2 ≤ 0 v 3 − v 4 ≤ 0. v 2 ≤ 1 v 4 ≤ 1 −v 2 + v 3 ≤ 0i P (m) = #(Z n ∩ mP ), where mP = {mv \| v ∈ P } denotes the mth dilate of P . The Ehrhart function is, in fact, a polynomial in m [1,Ehr P (t) = m≥0 i P (m)t m . The Ehrhart series is of the form Ehr P (t) = h * P (t) (1 − t) d+1 , where d is the dimension of P and h * P (t) is a polynomial in t of degree at most n. Often we just write h * (t) when the particular polytope is clear. The coefficients of h * (t) have an interpretation in terms of a shelling of a unimodular triangulation of P , if such a triangulation exists. Definition 2.4. Let T be the collection of maximal dimensional simplices in a pure simplicial complex of dimension d with #T = s. An ordering ∆ 1 , ∆ 2 , . . . , ∆ s on the simplices in T is a shelling order if for all 1 < r ≤ s, r−1 i=1 ∆ i ∩ ∆ r is a union of facets of ∆ r . Equivalently, the order ∆ 1 , ∆ 2 , . . . , ∆ s is a shelling order if and only if for all r ≤ s and k < r, there exists an i < r such that ∆ k ∩ ∆ r ⊂ ∆ i ∩ ∆ r and ∆ i ∩ ∆ r is a facet of ∆ r . This means that when we build our simplicial complex by adding facets in the order prescribed by the shelling order, we add each simplex along its highest dimensional facets. Keeping track of the number of facets that each simplex is added along gives the following relationship between shellings of a triangulation of an integer polytope and the Ehrhart series of the polytope, which is proved in [1, Chapter 3]. Theorem 2.5. Let P be an integer polytope. Let ∆ 1 , . . . , ∆ s be a unimodular triangulation of P using no new vertices. Denote by h * j the coefficient of t j in the h * polynomial of P . If ∆ 1 , . . . , ∆ s is a shelling order, then h * j is the number of ∆ i that are added along j of their facets in this shelling. Equivalently, h * (t) = s i=1 t a i , where a i = #{k < i \| ∆ k ∩ ∆ i is a facet of ∆ i }. Example 2.6. Consider the order polytope O(Z 4 ) with its canonical triangulation by alternating permutations ∆ 3412 , ∆ 2413 , ∆ 2314 , ∆ 1423 , ∆ 1324 . This particular ordering of the facets in the canonical triangulation is a special case of the shelling order that will be established and proved in the next section. The fact that this is a shelling order can be checked directly in this example, for instance: ∆ 2314 ∩ (∆ 3412 ∪ ∆ 2413 ) = conv     1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0     which is a facet of ∆ 2314 . Since the intersection consists of a single facet, it will contribute a 1 to the coefficient of t in h * O(Z 4 )) (t) = 1 + 3t + t 2 . Shelling the Canonical Triangulation of the Order Polytope In this section we describe a family of shelling orders on the simplices of the canonical triangulation of O(Z n ). Let σ be an alternating permutation. We will denote by vert(σ) the set of all vertices of the simplex ∆ σ . Note that this is the set of all 0/1 vectors v of length n that have v i ≤ v j whenever σ(i) < σ(j). i with 1 ≤ i ≤ n−1 such that vert(σ)−{v σ i } = vert(τ )−{v τ i }. By definition of each v σ j and v τ j , this occurs if and only if σ −1 (j) = τ −1 (j) for all j = i, i + 1 and e σ −1 (i) + e σ −1 (i+1) = e τ −1 (i) + e τ −1 (i+1) . This is true if and only if swapping the positions of i and i + 1 in σ yields τ , as needed. Denote by inv(σ) the number of inversions of a permutation σ; that is, inv(σ) is the number of pairs i < j such that σ(i) > σ(j). We similarly define a non-inversion to be a pair i < j with σ(i) < σ(j). We call an inversion or non-inversion (i, j) relevant if i < j − 1; that is, if it is not required by the structure of an alternating permutation. Note that performing a swap on an alternating permutation always decreases its inversion number by exactly one. Theorem 1.1 follows as a corollary of Proposition 3.1 and the following theorem. Theorem 3.2. Let σ 1 , . . . , σ En be an order on the alternating permutations such that • if i < j then inv(σ i ) ≥ inv(σ j ) and • if σ j swaps to σ i then i < j. Then the order ∆ σ 1 , . . . , ∆ σ En on the simplices of the canonical triangulation of O(Z n ) is a shelling order. For any alternating permutation σ, define the exclusion set of σ, excl(σ) to be the set of all v σ k ∈ vert(σ) such that k is a swap in σ. In other words, excl(σ) = {v \| v ∈ ∆ σ − ∆ τ for some τ such that σ swaps to τ }. Proposition 3.1 implies that in order to prove Theorem 3.2, it suffices to check that if inv(σ) ≤ inv(τ ), then excl(σ) ⊂ vert(τ ). This fact will follow from the next two propositions. Proof. Consider a vertex v σ k ∈ vert(σ). Note that we may read all of the non-inversions (i, j) with σ(i) ≤ k < σ(j) from v σ k since these correspond to pairs of positions in v σ k with a 0 in the first position and a 1 in the second. That is to say, we have v σ k (i) = 0, v σ k (j) = 1, and i < j. We claim that every relevant non-inversion of σ can be read from an element of excl(σ) in this way. To prove this, it suffices to show that if (i, j) is a relevant non-inversion of σ, then there exists a k with σ(i) ≤ k < σ(j) such that k is a swap in σ. We will prove this by induction on σ(j) − σ(i). If σ(j) − σ(i) = 1, then since (i, j) is a relevant non-inversion, σ(i) is a swap in σ. Let σ(j) − σ(i) > 1. Consider the position of σ(i) + 1 in σ. If σ −1 (σ(i)+1) < j −1, then (σ −1 (σ(i)+1), j) is a relevant non-inversion, and we are done by induction. If σ −1 (σ(i) + 1) > j, then σ(i) is a swap in σ. If σ −1 (σ(i) + 1) = j − 1, then note that i < σ −1 (σ(i) + 1) − 1 since otherwise, σ(i), σ(i) + 1, σ(j) would be an adjacent increasing sequence in σ, which would contradict that σ is alternating. So σ(i) is a swap in σ. Therefore, there exists a swap k in σ with σ(i) ≤ k < σ(j), and the relevant non-inversion (i, j) can be read from v σ k in the manner described above. Therefore, all relevant non-inversions in σ can be found as a non-adjacent 0 − 1 pair in a vertex in excl(σ). In particular, we can count the number of relevant non-inversions in σ from the vertices in excl(σ). Furthermore, if excl(σ) ⊂ vert(τ ), then all non-inversions in σ must also be non-inversions in τ , though τ can contain more non-inversions as well. So σ minimizes the number of non-inversions, and therefore maximizes the number of inversions, over all τ with excl(σ) ⊂ vert(τ ). Proposition 3.4. Let S ⊂ vert(O(Z n )) be contained in vert(σ) for some alternating σ. Then there exists a unique alternatingσ that maximizes inversion number over all alternating permutations whose vertex set contains S. Proof. Let S = {s 0 , s 1 , . . . , s r } ordered by decreasing coordinate sum. We can assume that S contains both the all zeroes and all ones vectors since those vectors belong to the simplex ∆ σ for any alternating permutation σ. Since S ⊂ vert(σ) for some alternating σ, if s i (j) = 0, then s k (j) = 0 for all k > i. For i = 1, . . . , r, let m i be the number of positions in s i that are equal to zero, and let n i = m i − m i−1 (with n 1 = m 1 ). Let τ be any alternating permutation such that S ⊆ vert(T ). The 0-pattern of each s i partitions the entries of all τ with S ⊂ vert(τ ) as follows: For 1 ≤ k ≤ r, the n k positions j such that s k (j) = 0 and s k−1 (j) = 1 are the positions of τ such that τ (j) ∈ {m k−1 + 1, . . . , m k }. The positions of inversions and non-inversions across these groups are fixed for all τ with S ⊂ vert(τ ). We can build an alternating permutationσ that maximizes the inversions within each group as follows. For 1 ≤ k ≤ r, let j k 1 , . . . , j k n k be the positions ofσ that must take values in {m k−1 + 1, . . . , m k }, as described above. We place these values in reverse; i.e. map j k l to m k − l + 1. The permutation obtained in this way need not be alternating, so we switch adjacent positions that need to contain non-descents in order to make the permutation alternating. Note that we never need to make such a switch between groups, since the partition given by S respects the structure of an alternating permutation. This permutation is unique because within the kth group, arranging the values in this way is equivalent to finding the permutation on n k elements with some fixed non-descent positions that maximizes inversion number. To obtain this permutation, we begin with the permutation (m k , m k − 1 . . . m k−1 + 1) and switch all the positions that must be non-descents. The alternating structure of the original permutation implies that none of these non-descent positions can be adjacent, so these transpositions commute and give a unique permutation.                           1 1 1 1 1 1 1          ,          0 1 0 1 1 1 0          ,          0 1 0 0 0 0 0          ,          0 0 0 0 0 0 0                           We will constructσ, the alternating permutation that maximizes inversion number overall alternating permuations whose vertex set contains S. The second and third vertices in S are the only one that gives information about the position of each character; we will denote them w 1 and w 2 , respectively. Since w 1 has 0's in exactly the first, third and seventh positions, we know that 1, 2 and 3 are in these positions. We insert them into these positions in decreasing order, so thatσ has the form 3 2 1. The zeros added in w 2 are in the fourth, fifth and sixth positions. Placing them in decreasing order yields the permutation However, this permutation cannot be alternating, since there must be an ascent from position 5 to position 6. To create this ascent, we switch the entries in these positions, yielding a permutation of the form 3 2 6 4 5 1. Finally, the only character missing is 7, which must go in the remaining space. This gives the permutation σ = 3 7 2 6 4 5 1. Proof of Theorem 3.2. It suffices to show that for any alternating permutations σ and τ , if inv(τ ) ≥ inv(σ) then excl(σ) ⊂ vert(τ ). If inv(τ ) > inv(σ), then since σ maximizes inversion number over all alternating permutations that contain the exclusion set of σ by Proposition 3.3, excl(σ) ⊂ vert(τ ). Furthermore, Proposition 3.4 implies that if inv(τ ) = inv(σ), then excl(σ) ⊂ vert(τ ) because σ is the unique permutation that maximizes inversion number of all alternating permutation that contain its exclusion set. Proof of Theorem 1.1. Let ∆ σ 1 , . . . , ∆ σ En be a shelling order as described in Theorem 3.2. Then by Proposition 3.1, each ∆ σ i is added in the shelling along exactly swap(σ i ) facets. Therefore, by Theorem 2.5, h * O(Z N ) (t) = σ t swap(σ) , where σ ranges overall alternating permutations of length n. The Swap Statistic Via Rank Selection An alternate proof of Theorem 1.1 relies heavily on the concepts of rank selection and flag f -vectors developed for general posets in Sections 3.13 and 3.15 of [7]. We will focus our attention to the zig-zag poset, Z n . Denote by J(Z n ) the distributive lattice of order ideals in Z n ordered by inclusion. Let S = {s 1 , . . . , s k } ⊂ [0, n], where [0, n] = {0, . . . , n}. We always assume that s 1 < s 2 < . . . < s k . Denote by α n (S) the number of chains of order ideals I 1 · · · I k in J(Z n ) such that #I j = s j for all j. Define In Section 3.13 of [7], the function α n : 2 [0,n] → Z is called the flag f-vector of Z n and β n : 2 [0,n] → Z is called the flag h-vector of Z n . For any poset P of size n, let ω : P → [n] be an order-preserving bijection that assigns a label to each element of P . Then for any linear extension σ : P → [n], we may define a permutation of the labels by ω(σ −1 (1)), . . . , ω(σ −1 (n)). The Jordan-Hölder set L(P, ω) is the set of all permutations obtained in this way. The following result for arbitrary finite posets can be found in chapter 3.13 of [7]. Recall that the Ehrhart polynomial of O(Z n ) evaluated at m is equal to the order polynomial of Z n evaluated at m+1 [5]. It follows from this fact and from Theorem 3.15.8 in [7] that the h * -polynomial of O(Z n ) is (1) h * O(Zn) (t) = S⊂[n−1] β n (S)t #S . So, Theorem 1.1 will follow from Equation 1 and the following theorem, which is analogous to Theorem 3.13.1 in [7]. To prove this theorem, for every S = {s 1 , . . . , s n } ⊂ [n − 1], we will find define a function φ S that maps chains of order ideals of sizes s 1 , . . . , s k to alternating permutations whose swap set is contained in S. Let I 1 , . . . , I k be a chain of order ideals in J(Z n ) with sizes #I j = s j . Let w i be the vertex of O(Z n ) that satisfies w i (j) = 0 if j ∈ I i 1 if j ∈ I i . Define φ S (I 1 , . . . , I k ) to be the unique alternating permutation that maximizes inversion number over all alternating permutations whose vertex set contains {w 1 , . . . , w k }. This map is well-defined by Proposition 3.4. Let ψ S be the map that sends an alternating permutation ω with Swap(ω) ⊂ S to the chain of order ideals (I 1 , . . . , I k ) where each I j = {ω −1 (1), . . . , ω −1 (s j )}. Since every alternating permutation ω is a linear extension of Z n , each I j obtained in this way is an order ideal. They form a chain by construction, so the map ψ S is well-defined. We will show that ψ S is the inverse of φ S in the proof of Theorem 4.2. w 1 =          0 1 0 1 1 1 0          and w 2 =          0 1 0 0 0 0 0          . Notice that these are the same vectors w 1 and w 2 as in Example 3.5. So the unique alternating permutation φ S (I 1 , I 2 ) that maximizes inversion number over all alternating permutations whose vertex set contains {w 1 , w 2 } is the same permutation as in Example 3.5, φ S (I 1 , I 2 ) = 3 7 2 6 4 5 1. Note that Swap(3726451) = {3} ⊂ {3, 6} = S. We will recover our original order ideals I 1 and I 2 by finding ψ S (ω). For clarity, we will treat ω as a map from {a, . . . , g} to {1, . . . , 7}. The first order ideal of ψ S (ω) consists of the inverse images of 1, 2, and 3 in ω. That is, I 1 = {ω −1 (1), ω −1 (2), ω −1 (3)} = {a, c, g}. The second order ideal of ψ S (ω) consists of the inverse images of 1 through 6 in ω. So we obtain I 2 = {ω −1 (1), . . . , ω −1 (6)} = {a, c, d, e, f, g}. Note that this is, in fact, the chain of order ideals with which we began. Proof of Theorem 4.2. Let S = {s 1 , . . . , s k } ⊂ [n − 1]. We will show that α n (S) is the number of alternating permutations whose swap set is contained in S by showing that the map φ S described above is a bijection. Let I 1 , . . . , I k be a chain of order ideals in J(Z n ) with sizes #I j = s j . It is clear from the definitions of φ S and ψ S that ψ S (φ S (I 1 , . . . , I k )) = (I 1 , . . . , I k ). Since φ S is injective, it suffices to show that ψ S is also injective. We will show that φ S (I 1 , . . . , I k ) is the only alternating permutation that maps to (I 1 , . . . , I k ) under ψ S . Since ω = φ S (I 1 , . . . , I k ) is the unique alternating permuation that maximizes inversion number over all alternating permutations with {w 1 , . . . , w k } in their vertex sets, any other alternating permutation σ that maps to (I 1 , . . . , I k ) under ψ S must have fewer inversions than ω. Let σ be such a permutation. Since each inversion between the sets I 1 , Z n − I k and I j − I j−1 for all 1 < j ≤ k are fixed, the additional non-inversion must be contained in one of these sets. Without loss of generality, let this be R = I j − I j−1 . Denote by σ\| R the restriction of σ to the domain R. Let (σ −1 (a), σ −1 (b)) be the non-inversion of σ\| R that is not required by the alternating structure. Then note that σ −1 (a) + 1 < σ −1 (b). We claim that σ\| R must have at least one swap position. To prove this, we will induct on b − a in a similar manner as in the proof of Proposition 3.3. If b − a = 1, then a is a swap in σ\| R since σ −1 (a) + 1 < σ −1 (b). Suppose b − a > 1. Consider the position of a + 1. If σ −1 (a + 1) < σ −1 (a), then (σ −1 (a + 1), σ −1 (b)) is a non-inversion of σ\| R that is not required by the alternating structure. Since b − (a + 1) < b − a, we are done by induction. If σ −1 (a + 1) > σ −1 (b), then a can be swapped in σ\| R . Consider the case where σ −1 (a) < σ −1 (a + 1) < σ −1 (b). If σ −1 (a) + 1 = σ −1 (a+1), then a can be swapped in σ\| R . Otherwise, if σ −1 (a)+1 = σ −1 (a+1), it cannot be the case that σ −1 (a+1)+1 = σ −1 (b) due to the alternating structure of σ. So (σ −1 (a + 1), σ −1 (b)) is a non-inversion of σ\| R that is not required by the alternating structure and we are done by induction. So σ must have a swap position that is not equal to s 1 , . . . , s k . Therefore, ω is the only alternating permutation that can map to (I 1 , . . . , I k ) under ψ S , and ψ S is the inverse map of φ S . So α n (S) is equal to the number of alternating permutations whose swap set is contained in S. By the Principle of Inclusion-Exclusion, β n (S) is the number of alternating permutations whose swap set is equal to S. β n (S)t #S . Theorem 4.2 tells us that β n (S) is the number of alternating permutations with swap set S. So the sum #S=k β n (S) is the number of alternating permutations σ with swap(σ) = k. So h * O(Zn) (t) = σ t swap(σ) , as needed. Combinatorial Properties of Swap Numbers Let s n (k) denote the number of alternating permutations on n letters such that have exactly k swaps. We call these numbers the swap numbers. Theorem 1.1 shows that the h * -polynomial of O(Z n ) is n−1 k=0 s n (k)t k . We are interested in interrogating these numbers. For example, it would be interesting to find an explicit formula for s n (k), though we have not been able to do this yet. One straightforward property that becomes apparent looking at examples is that s n (n−1) = 0. This is clear because it is not possible that every k ∈ [n−1] is a swap. Indeed, otherwise k is to the left of k + 1 for all k ∈ [n − 1] which implies that σ is the identity permutation, which is not alternating. Furthermore, s n (n − 2) = 1, since the unique alternating permutation with this many swaps is the one with 1, 2, . . . , ⌈ n 2 ⌉ in order in the odd numbered positions and ⌈ n 2 ⌉ + 1, . . . , n in order in the even numbered positions. Similarly, s n (0) = 1, because there is a unique alternating permutation with no swaps. It is the permutation (n − 1, n, n − 3, n − 2, n − 5, n − 4, . . .). Another property that is apparent from examples is summarized in the following: Theorem 5.1. The sequence s n (0), s n (1), . . . , s n (n − 2) is symmetric and unimodal. In fact, Theorem 5.1 and all the preceding properties will follow from the fact that O(Z n ) is a Gorenstein polytope of index 3. relative interior, and for each facet-defining inequality a T x ≤ b, we have that b − a T v = 1. The integer m is called the index of P . The following relevant theorem concerning the h * polynomials of Gorenstein polytopes with unimodular triangulations appears in [2]. Theorem 5.3. Suppose that P is a Gorenstein polytope of dimension d and index m. Then h * P (t) is a polynomial of degree d−m+1, whose coefficients form a symmetric sequence. Furthermore, the constant term of h * P (t) is 1. If, in addition, P has a regular unimodular triangulation, then the coefficient sequence is unimodal. Proof of Theorem 5.1. It suffices to show that O(Z n ) is a Gorenstein polytope of index three with a regular unimodular triangulation. The canonical triangulation of O(Z n ) is a regular unimodular triangulation. To see that it satisfies the Gorenstein property with respect to m = 3, note that the defining inequalities for 3O(Z n ) are that v i ≥ 0 for i odd, v i ≤ 3 for i even, v 2i−1 ≤ v 2i and v 2i+1 ≤ v 2i . The unique interior lattice point of 3O(Z n ) is the point v where v i = 1 for i odd, and v i = 2 for i even. Finally, this point has lattice distance 1 from each of the facet-defining inequalities. Hence O(Z n ) is a Gorenstein polytope of index three with a regular unimodular triangulation and Theorem 5.3 can be applied to deduce that the coefficient sequence is symmetric and unimodal. While general principles provide a proof of the symmetry and unimodality of the sequence s n (0), s n (1), . . . , s n (n − 2), it would be interesting to find explicit combinatorial arguments that would produce these results. In particular, we let A n,k denote the set of alternating permutations on n letters with exactly k swaps, then it would be interesting to solve the following problems. (1) Find a bijection between A n,k and A n,n−k−2 . (2) For each 0 ≤ k ≤ ⌊(n − 4)/2⌋ find an injective map from A n,k to A n,k+1 . Theorem 1 . 1 . 11The numerator of the Ehrhart series of O Let σ : P → [n] be a linear extension of P . Denote by e i the ith standard basis vector in R n . The simplex ∆ σ is the convex hull of v σ 0 , . . . , v σ n where v σ 0 is the all 1's vector and v σ i = v σ i−1 − e σ −1 (i) . Letting σ range over all linear extensions of P yields a unimodular triangulation of O(P ). Hence, the normalized volume of O(P ) is the number of linear extensions of P . In particular, this means that the volume of O(Z n ) is the Euler zig-zag number, E n . Example 2 . 3 . 23Consider the case when n = 4. The zig-zag poset Z 4 is pictured in Figure 2.1. The order polytope O(Z 4 ) has facet defining inequalities The alternating permutations on 4 elements, which correspond to linear extensions of Z 4 are 1324, 1423, 2314, 2413, and 3412. Note that there are E 4 = 5 such alternating permutations, so the normalized volume of O(Z 4 ) is 5. The simplex in the canonical triangulation of O(Z n ) The simplices ∆ σ and ∆ τ are joined along a facet if and only if σ swaps to τ or τ swaps to σ.Proof. Simplices ∆ σ and ∆ τ are joined along a facet if and only if vert(σ) and vert(τ ) differ by a single element. Since every simplex in the canonical triangulation of O(Z n ) has exactly one vertex with the sum of its components equal to i for 0 ≤ i ≤ n and the all 0's and all 1's vector are in every simplex in this triangulation, this occurs if and only if there exists an Proposition 3. 3 . 3An alternating permutation σ maximizes inversion number over all alternating permutations τ with excl(σ) ⊂ excl(τ ). Principle of Inclusion-Exclusion, or equivalently, via Möbius inversion on the Boolean lattice, α n (S) = T ⊂S β n (S). h * -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET 11 . Let S ⊂ [n − 1]. Then β n (S) is equal to the number of permutations τ ∈ L(P, ω) with descent set S. Theorem 4 . 2 . 42Let S ⊂ [n − 1]. Then β n (S) is the number of alternating permutations ω with Swap(ω) = S. Figure 4 . 1 . 41The zig-zag poset Z 7 Example 4.3. Consider the zig-zag poset on seven elements Z 7 pictured in Figure 4.1. Let S = {3, 6}, and let I 1 = {a, c, g} and I 2 = {a, c, d, e, f, g} be the given order ideals of sizes 3 and 6 respectively. Then the vectors w 1 and w 2 are Definition 5 . 2 . 52An integral polytope is Gorenstein if there is a positive integer m such that mP contains exactly one lattice point v in its h * -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET 15 Theorem 1.1 follows as a corollary of Theorem 4.2. Proof of Theorem 1.1. Equation 1 states thath * O(Zn) (t) = S⊂[n−1] h* -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET 7 h * -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET 9 2 6 5 4 1. h* -POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET 13 Integer-point enumeration in polyhedra. M Beck, S Robins, David AustinSpringerNew YorkComputing the continuous discretelyBeck, M., and Robins, S. Computing the continuous discretely, second ed. Undergraduate Texts in Mathematics. Springer, New York, 2015. Integer-point enumeration in polyhedra, With illustrations by David Austin. h-vectors of gorenstein polytopes. W Bruns, T Römer, J. Comb. Theory Ser. A. 114Bruns, W., and Römer, T. h-vectors of gorenstein polytopes. J. Comb. The- ory Ser. A 114, 1 (Jan. 2007), 65-76. The Cavender-Farris-Neyman model with a molecular clock. J I Coons, S Sullivant, arXiv:1805.04175arXiv preprintCoons, J. I., and Sullivant, S. The Cavender-Farris-Neyman model with a molecular clock. arXiv preprint arXiv:1805.04175 (2018). The online encyclopedia of integer sequences. N Sloane, Published electronically at http://oeis. orgSloane, N. The online encyclopedia of integer sequences, 2005. Published elec- tronically at http://oeis. org. Two poset polytopes. R P Stanley, Discrete & Computational Geometry. 1Stanley, R. P. Two poset polytopes. Discrete & Computational Geometry 1, 1 (1986), 9-23. A survey of alternating permuations. R P Stanley, Contemporary Mathematics. 531Stanley, R. P. A survey of alternating permuations. Contemporary Mathe- matics 531 (2010), 165-196. R P Stanley, Enumerative, of Cambridge Studies in Advanced Mathematics. CambridgeCambridge University Press1second ed.Stanley, R. P. Enumerative combinatorics. Volume 1, second ed., vol. 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012.	[]
[ "Learning to Represent Image and Text with Denotation Graph", "Learning to Represent Image and Text with Denotation Graph" ]	[ "Bowen Zhang \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Hexiang Hu [email protected] \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Vihan Jain [email protected] \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Google Research \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Eugene Ie [email protected] \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Google Research \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Sha Fei \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n", "Google Research \nUniversity of Southern California\nUniversity of Southern\nCalifornia\n" ]	[ "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia", "University of Southern California\nUniversity of Southern\nCalifornia" ]	[]	Learning to fuse vision and language information and representing them is an important research problem with many applications. Recent progresses have leveraged the ideas of pretraining (from language modeling) and attention layers in Transformers to learn representation from datasets containing images aligned with linguistic expressions that describe the images. In this paper, we propose learning representations from a set of implied, visually grounded expressions between image and text, automatically mined from those datasets. In particular, we use denotation graphs to represent how specific concepts (such as sentences describing images) can be linked to abstract and generic concepts (such as short phrases) that are also visually grounded. This type of generic-to-specific relations can be discovered using linguistic analysis tools. We propose methods to incorporate such relations into learning representation. We show that state-of-the-art multimodal learning models can be further improved by leveraging automatically harvested structural relations. The representations lead to stronger empirical results on downstream tasks of cross-modal image retrieval, referring expression, and compositional attribute-object recognition. Both our codes and the extracted denotation graphs on the Flickr30K and the COCO datasets are publically available on https://sha-lab.github.io/DG.A standard approach is to embed the visual and the language information as points in a (joint) visualsemantic embedding space (Frome et al., 2013; Kiros et al., 2014;. One can then infer whether the visual information is aligned with the text information by checking how these points are distributed.How do we embed visual and text information? Earlier approaches focus on embedding each stream of information independently, using models that are tailored to each modality. For example, for image, the embedding could be the features at the last fully-connected layer from a deep neural network trained for classifying the dominant objects in the image. For text, the embedding could be the last hidden outputs from a recurrent neural network.Recent approaches, however, have introduced several innovations (Lu et al., 2019; Li et al., 2019a;. The first is to contextualize	10.18653/v1/2020.emnlp-main.60	[ "https://arxiv.org/pdf/2010.02949v1.pdf" ]	222,178,107	2010.02949	73068d13d6e53876c374ebd4c862ec01351c9f39	Learning to Represent Image and Text with Denotation Graph Bowen Zhang University of Southern California University of Southern California Hexiang Hu [email protected] University of Southern California University of Southern California Vihan Jain [email protected] University of Southern California University of Southern California Google Research University of Southern California University of Southern California Eugene Ie [email protected] University of Southern California University of Southern California Google Research University of Southern California University of Southern California Sha Fei University of Southern California University of Southern California Google Research University of Southern California University of Southern California Learning to Represent Image and Text with Denotation Graph Learning to fuse vision and language information and representing them is an important research problem with many applications. Recent progresses have leveraged the ideas of pretraining (from language modeling) and attention layers in Transformers to learn representation from datasets containing images aligned with linguistic expressions that describe the images. In this paper, we propose learning representations from a set of implied, visually grounded expressions between image and text, automatically mined from those datasets. In particular, we use denotation graphs to represent how specific concepts (such as sentences describing images) can be linked to abstract and generic concepts (such as short phrases) that are also visually grounded. This type of generic-to-specific relations can be discovered using linguistic analysis tools. We propose methods to incorporate such relations into learning representation. We show that state-of-the-art multimodal learning models can be further improved by leveraging automatically harvested structural relations. The representations lead to stronger empirical results on downstream tasks of cross-modal image retrieval, referring expression, and compositional attribute-object recognition. Both our codes and the extracted denotation graphs on the Flickr30K and the COCO datasets are publically available on https://sha-lab.github.io/DG.A standard approach is to embed the visual and the language information as points in a (joint) visualsemantic embedding space (Frome et al., 2013; Kiros et al., 2014;. One can then infer whether the visual information is aligned with the text information by checking how these points are distributed.How do we embed visual and text information? Earlier approaches focus on embedding each stream of information independently, using models that are tailored to each modality. For example, for image, the embedding could be the features at the last fully-connected layer from a deep neural network trained for classifying the dominant objects in the image. For text, the embedding could be the last hidden outputs from a recurrent neural network.Recent approaches, however, have introduced several innovations (Lu et al., 2019; Li et al., 2019a;. The first is to contextualize Introduction There has been an abundant amount of aligned visual and language data such as text passages * Work done while at Google † Authors Contributed Equally ‡ On leave from USC ([email protected]) describing images, narrated videos, subtitles in movies, etc. Thus, learning how to represent visual and language information when they are semantically related has been a very actively studied topic. There are many vision + language applications: image retrieval with descriptive sentences or captions (Barnard and Forsyth, 2001;Barnard et al., 2003;Hodosh et al., 2013;Young et al., 2014), image captioning (Chen et al., 2015;Xu et al., 2015), visual question answering (Antol et al., 2015), visual navigation with language instructions (Anderson et al., 2018b), visual objects localization via short text phrases (Plummer et al., 2015), and others. A recurring theme is to learn the representation of these two streams of information so that they correspond to each other, highlighting the notion that many language expressions are visually grounded. the embeddings of one modality using information from the other one. This is achieved by using co-attention or cross-attention (in addition to selfattention) in Transformer layers. The second is to leverage the power of pre-training (Radford et al., 2019;Devlin et al., 2019): given a large number of parallel corpora of images and their descriptions, it is beneficial to identify pre-trained embeddings on these data such that they are useful for downstream vision + language tasks. Despite such progress, there is a missed opportunity of learning stronger representations from those parallel corpora. As a motivating example, suppose we have two paired examples: one is an image x 1 corresponding to the text y 1 of TWO DOGS SAT IN FRONT OF PORCH and the other is an image x 2 corresponding to the text y 2 of TWO DOGS RUNNING ON THE GRASS. Existing approaches treat the two pairs independently and compute the embeddings for each pair without acknowledging that both texts share the common phrase y 1 ∩ y 2 = TWO DOGS and the images have the same visual categories of two dogs. We hypothesize that learning the correspondence between the common phrase y 1 ∩ y 2 and the set of images {x 1 , x 2 }, though not explicitly annotated in the training data, is beneficial. Enforcing the alignment due to this additionally constructed pair introduces a form of structural constraint: the embeddings of x 1 and x 2 have to convey similar visual information that is congruent to the similar text information in the embeddings of y 1 and y 2 . In this paper, we validate this hypothesis and show that extracting additional and implied correspondences between the texts and the visual information, then using them for learning leads to better representation, which results in a stronger performance in downstream tasks. The additional alignment information forms a graph where the edges indicate how visually grounded concepts can be instantiated at both abstract levels (such as TWO DOGS) and specific levels (such as TWO DOGS SAT IN FRONT OF THE PORCH). These edges and the nodes that represent the concepts at different abstraction levels form a graph, known as denotation graph, previously studied in the NLP community (Young et al., 2014;Lai and Hockenmaier, 2017;Plummer et al., 2015) for grounding language expressions visually. Our contributions are to propose creating visuallygrounded denotation graphs to facilitate representation learning. Concretely, we apply the technique originally developed for the FLICKR30K dataset (Young et al., 2014) also to COCO dataset (Lin et al., 2014) to obtain denotation graphs that are grounded in each domain respectively ( § 3). We then show how the denotation graphs can be used to augment training samples for aligning text and image ( § 4). Finally, we show empirically that the representation learned with denotation graphs leads to stronger performance in downstream tasks ( § 5). Related Work Learning representation for image and text There is a large body of work that has been focusing on improving the visual or text embedding functions (Socher et al., 2014;Eisenschtat and Wolf, 2017;Nam et al., 2017;Gu et al., 2018). Another line of work, referred to as cross-stream methods infer fine-grained alignments between local patterns of visual (i.e., local regions) and linguistic inputs (i.e., words) between a pair of image and text, then use them to derive the similarity between the image and the text. uses cross-modal attention mechanism (Xu et al., 2015) to discover such latent alignments. Figure 1: (Left) A schematic example of denotation graph showing the hierarchical organization of linguistic expression (adapted from https://shannon.cs.illinois.edu/DenotationGraph/) (Right) A randomsubgraph from the denotation graph extracted from the FLICKR30K dataset, with images attached to concepts at different levels of hierarchy. deep Transformers (Lu et al., 2019;Li et al., 2019a;Su et al., 2020;Li et al., 2019c). The pre-training strategies of these methods typically involve many self-supervised learning tasks, including the image-text matching (Lu et al., 2019), masked language modeling (Devlin et al., 2019;Lu et al., 2019) and masked region modeling . In contrast to those work, we focus on exploiting additional correspondences between image and text that are not explicitly given in the many image and text datasets. By analyzing the linguistic structures of the texts in those datasets, we are able to discover more correspondences that can be used for learning representation. We show the learned representation is more powerful in downstream tasks. Vision + Language Tasks There has been a large collection of tasks combining vision and language, including image captioning (Chen and Lawrence Zitnick, 2015;Fang et al., 2015;Hodosh et al., 2013;Karpathy and Fei-Fei, 2015;Kulkarni et al., 2013), visual QA (Antol et al., 2015, text-based image verification (Suhr et al., 2017(Suhr et al., , 2018Hu et al., 2019), visual commonsense reasonin (Zellers et al., 2019), and so on. In the context of this paper, we focus on studying cross-modality retrieval (Barnard et al., 2003;Barnard and Forsyth, 2001;Gong et al., 2014;Hodosh et al., 2013;Young et al., 2014;Zhang et al., 2018), as well as transfer learning on downstream tasks, including compositional attributeobject recognition (Isola et al., 2015;Misra et al., 2017) and referring expressions (Dale and Reiter, 1995;Kazemzadeh et al., 2014;Kong et al., 2014;Mitchell et al., 2012). Please refer to § 5 for expla-nation of these tasks. Denotation Graph (DG) Visually grounded text expressions denote the images (or videos) they describe. When examined together, these expressions reveal structural relations that do not exhibit when each expression is studied in isolation. In particular, through linguistic analysis, these expressions can be grouped and partially ordered and thus form a relation graph, representing how (visually grounded) concepts are shared among different expressions and how different concepts are related. This insight was explored by Young et al. (2014) and the resulting graph is referred to as a denotation graph, schematically shown in the top part of Fig. 1. In this work, we focus on constructing denotation graphs from the FLICKR30K and the COCO datasets, where the text expressions are sentences describing images. Formally, a denotation graph G is a polytree where a node v i in the graph corresponds to a pair of a linguistic expression y i and a set of images X i = {x 1 , x 2 , · · · , x n i }. A directed edge e ij from a node v i to its child v j represents a subsumption relation between y i and y j . Semantically, y i is more abstract (generic) than y j , and the tokens in y i can be a subset of y j 's. For example, TWO DOGS describes all the images which TWO DOGS ARE RUNNING describes, though less specifically. Note that the subsumption relation is defined on the semantics of these expressions. Thus, the tokens do not have to be exactly matched on their surface forms. For instance, IN FRONT OF PERSON or IN FRONT OF CROWD are also generic concepts that Fig. 1 for another example. More formally, the set of images that correspond to v i is the union of all the images corresponding to v i 's children ch(v i ): X i = v j ∈ch(v i ) X j . We also use pa(v j ) to denote the set of v j 's parents. Denotation graphs (DG) can be seen as a hierarchical organization of semantic knowledge among concepts and their visual groundings. In this sense, they generalize the tree-structured object hierarchies that have been often used in computer vision. The nodes in the DG are composite phrases that are semantically richer than object names and the relationship among them is also richer. Constructing DG We used the publicly available tool 1 , following Young et al. (Young et al., 2014). For details, please refer to the Appendix and the reference therein. Once the graph is constructed, we attach the images to the proper nodes by setunion images of each node's children, starting from the sentence-level node. DG-FLICKR30K and DG-COCO 2 We regenerate a DG on the FLICKR30K dataset 3 (Young et al., 2014) and construct a new DG on the COCO (Lin et al., 2014) dataset. The two datasets come from different visual and text domains where the former contains more iconic social media photos and the latter focuses on photos with complex scenes and has more objects. Figure 1 shows a random subgraph of DG-FLICKR30K. note that in both graphs, a large number of internal nodes (more abstract concepts or phrases) are introduced. For such concepts, the linguistic expressions are much shorter and the number of images they correspond to is also larger. Learning with Denotation Graphs The denotation graphs, as described in the previous section, provide rich structures for learning representations of text and image. In what follows, we describe three learning objectives, starting from the most obvious one that matches images and their descriptions ( § 4.1), followed by learning to discriminate between general and specialized concepts ( § 4.2) and learning to predict concept relatedness ( § 4.3). We perform ablation studies of those objectives in § 5.4. Matching Texts with Images We suppose the image x and the text y are represented by (a set of) vectors φ(x) and ψ(y) respectively. A common choice for φ(·) is the last layer of a convolutional neural network (He et al., 2015;Xie et al., 2017) and for ψ(·) the contextualized word embeddings from a Transformer network (Vaswani et al., 2017). The embedding of the multimodal pair is a vector-valued function over φ(x) and ψ(y): v(x, y) = f (φ(x), ψ(y))(1) There are many choices of f (·, ·). The simplest one is to concatenate the two arguments. We can also use the element-wise product between the two if they have the same embedding dimension ( Matching Model We use the following probabilistic model to characterize the joint distribution p(x, y) ∝ exp(θ T v(x, y))(2) where the exponent s(x, y) = θ T v is referred as the matching score. To estimate θ, we use the maximum likelihood estimation θ * = arg max v i k log p(x ik , y i ) (3) where x ik is the kth element in the set X i . However, this probability is intractable to compute as it requires us to get all possible pairs of (x, y). To approximate, we use negative sampling. Negative Sampling For each (randomly selected) positive sample (x ij , y i ), we explore 4 types of negative examples and assemble them as a negative sample set D − ik : Visually mismatched pair We randomly sample an image x − / ∈ X i to pair with y i , i.e., (x − , y i ). Note that we automatically exclude the images from v i 's children. Semantically mismatched pair We randomly sample a text y j = y i to form the pair (x ik , y j ). Note that we constrain y j not to include concepts that could be more abstract than y i as the more abstract can certainly be used to describe the specific images x ik . Semantically hard pair We randomly sample a text y j that corresponds to an image x j that is visually similar to x ik to form (x ik , y j ). See (Lu et al., 2019) for details. DG Hard Negatives We randomly sample a sibling (but not cousin) node v j to v i such that x ik / ∈ X j to form (x ik , y j ) Note that the last 3 pairs have increasing degrees of semantic confusability. In particular, the 4th type of negative sampling is only possible with the help of a denotation graph. In that type of negative samples, y j is semantically very close to y i (from the construction) yet they denote different images. The "semantically hard pair", on the other end, is not as hard as the last type as y i and y j could be very different despite high visual similarity. With the negative samples, we estimate θ as the minimizer of the following negative log-likelihood MATCH = − v i k log e s(x ik ,y i ) (x,ŷ)∼D i e s(x,ŷ) (4) where D i = D − ik ∪ {(x ik ,y Learning to Be More Specific The hierarchy in the denotation graph introduces an opportunity for learning image and text representations that are sensitive to fine-grained distinctions. Concretely, consider a parent node v i with an edge to the child node v j . While the description y j matches any images in its children nodes, the parent node's description y i on a higher level is more abstract. For example, the concepts INSTRUMENT To incorporate this modeling notion, we introduce SPEC = e ij k [s(x jk , y i ) − s(x jk , y j )] + (5) as a specificity loss, where [h] + = max(0, h) denotes the hinge loss. The loss is to be minimized such that the matching score for the less specific description y i is smaller than that for the more specific description y j . Learning to Predict Structures Given the graph structure of the denotation graph, we can also improve the accuracy of image and text representation by modeling high-order relationships. Specifically, for a pair of nodes v i and v j , we want to predict whether there is an edge from v i to v j , based on each node's corresponding embedding of a pair of image and text. Concretely, this is achieved by minimizing the following negated likelihood EDGE = − e ij k,k log p(e ij = 1\| v(x ik , y i ), v(x jk , y j )) (6) We use a multi-layer perceptron with a binary output to parameterize the log-probability. The Final Learning Objective We combine the above loss functions as the final learning objective for learning on the DG DG = MATCH + λ 1 · SPEC + λ 2 · EDGE(7) where λ 1 , λ 2 are the hyper-parameters that tradeoff different losses. Setting them to 1.0 seems to work well. The performance under different λ 1 and λ 2 are reported in Table 12 and Table 13. We study how each component could affect the learning of representation in § 5.4. Experiments We examine the effectiveness of using denotation graphs to learn image and text representations. We first describe the experimental setup and key implementation details ( § 5.1). We then describe key image-text matching results in § 5.2, followed by studies about the transfer capability of our learned representation ( § 5.3). Next, we present ablation studies over different components of our model ( § 5.4). Finally, we validate how well abstract concepts can be used to retrieve images, using our model ( § 5.5). Experimental Setup We list major details in the following to provide context, with the full details documented in the Appendix for reproducibility. Embeddings and Matching Models Our aim is to show denotation graphs improve state-of-the-art methods. To this end, we experiment with two recently proposed state-of-the-art approaches and their variants for learning from multi-modal data: ViLBERT (Lu et al., 2019) and UNITER . The architecture diagrams and the implementation details are in the Appendix, with key elements summarized in the following. Both the approaches start with an image encoder, which obtains a set of embeddings of image patches, and a text encoder which obtains a sequence of word (or word-piece) embeddings. For ViLBERT, text tokens are processed with Transformer layers and fused with the image information with 6 layers of co-attention Transformers. The output of each stream is then element-wise multiplied to give the fused embedding of both streams. For UNITER, both streams are fed into 12 Transformer layers with cross-modal attention. A special token CLS is used, and its embedding is regarded as the fused embedding of both streams. For ablation studies, we use a smaller ViLBERT for and Fei-Fei, 2015). Key characteristics for the two DGs are reported in Table 1. Evaluation Tasks We evaluate the learned representations on three common vision + language tasks. In text-based image retrieval, we evaluate two settings: the text is either a sentence or a phrase from the test corpus. In the former setting, the sentence is a leaf node on the denotation graph, and in the latter case, the phrase is an inner node on the denotation graph, representing more general concepts. We evaluate the FLICKR30K and the COCO datasets, respectively. The main evaluation metrics we use are precisions at recall R@M where M = 1, 5 or 10 and RSUM which is the sum of the 3 precisions (Wu et al., 2019). Conversely, we also evaluate using the task of image-based text retrieval to retrieve the right descriptive text for an image. In addition to the above cross-modal retrieval, we also consider two downstream evaluation tasks, i.e., Referring Expression and Compositional Attribute-Object Recognition. (1) Referring Expression is a task where the goal is to localize the corresponding object in the image given an expression (Kazemzadeh et al., 2014). We evaluate on the dataset REFCOCO+, which contains 141,564 expressions with 19,992 images. We follow the previously established protocol to evaluate on the validation split, the TestA split, and the TestB split. We are primarily interested in zero-shot/few-shot learning performance. (2) Compositional Attribute-Object Recognition is a task that requires a model (7) are set to 1.0, unless specified (see the Appendix). Table 2 and Table 3 report the performances on cross-modal retrieval. On both datasets, models trained with denotation graphs considerably outperform the corresponding ones which are not. For the image-based text retrieval task, ViLBERT and UNITER on FLICKR30K suffers a small drop in R@10 when DG is used. On the same task, UNITER on COCO 5K Test Split decreases more when DG is used. However, note that on both splits of COCO, ViLBERT is a noticeably stronger model, and using DG improves its performance. Main Results Zero/Few-Shot and Transfer Learning Transfer across Datasets Table 4 illustrates that the learned representations assisted by the DG have better transferability when applied to another dataset (TARGET DOMAIN) that is different from the SOURCE DOMAIN dataset which the DG is based on. Note that the representations are not finetuned on the TARGET DOMAIN. The improvement on the direction COCO →FLICKR30K is stronger than the reverse one, presumably because the COCO dataset is bigger than FLICKR30K. (R@5 and R@10 are reported in the Appendix.) Zero/Few-shot Learning for Referring Expression We evaluate our model on the task of referring expression, a supervised learning task, in the setting of zero/few-shot transfer learning. In zero-shot learning, we didn't fine-tune the model on the referring expression dataset (i.e. REFCOCO+). Instead, we performed a "counterfactual" inference, where we measure the drop in the compatibility score (between a text describing the referring object and the image of all candidate regions) as we removed individual candidates results. The region that causes the biggest drop of compatibility score is selected. As a result, the selected region is most likely to correspond to the description. In the setting of fewshot learning, we fine-tune our COCO-pre-trained model on the task of referring expression in an endto-end fashion on the referring expression dataset (i.e. REFCOCO+). The results in Table 5 suggest that when the amount of labeled data is limited, training with DG performs better than training without. When the amount of data is sufficient for end-to-end training, the advantage of training with DG diminishes. Compositional Attribute-Object Recognition We evaluate our model for supervised compositional attribute-object recognition (Misra et al., 2017), and report results on recognizing UNSEEN attribute-object labels on the MIT-STATE test data (Isola et al., 2015). Specifically, we treat the text of image labels (i.e., attribute-object pairs as compound phrases) as the sentences to fine-tune the ViLBERT models, using the MATCH objective. Table 6 reports the results (in top-K accuracies) of both prior methods and variants of ViLBERT, which are trained from scratch (N/A), pre-trained on COCO and DG-COCO, respectively. ViLBERT models pre-trained with parallel pairs of images and texts (i.e., COCO and DG-COCO) improve sig- nificantly over the baseline that is trained on the MIT-STATE from scratch. The model pre-trained with DG-COCO achives the best results among ViL-BERT variants. It performs on par with the previous state-of-the-art method in top-1 accuracy and outperforms them in top-2 and top-3 accuracies. Ablation Studies The rich structures encoded in the DGs give rise to several components that can be incorporated into learning representations. We study whether they are beneficial to the performances on the downstream task of text-based image retrieval. In the notions of §4, those components are: (1) remove "DG HARD NEGATIVES" from the MATCH loss and only use the other 3 types of negative samples ( § 4.1); (2) align images with more specific text descriptions ( § 4.2); (3) predict the existences of edges between pairs of nodes ( § 4.3). Table 7 shows the results from the ablation studies. We report results on two versions of ViLBERT: In ViLBERT (reduced), the number of parameters in the model is significantly reduced by making the model less deep, and thus faster for development. Instead of being pre-trained, they are trained on the FLICKR30K dataset directly for 15 epochs with a minibatch size of 96 and a learning rate of 4e −5 . In ViLBERT (Full), we use the aforementioned settings. We report RSUM on the FLICKR30K dataset for the task of text-based image retrieval. All models with DG perform better than the mod- els without DG. Secondly, the components of DG HARD NEGATIVES, SPEC , and EDGE contribute positively and their gains are cumulative. Image Retrieval from Abstract Concepts The leaf nodes in a DG correspond to complete sentences describing images. The inner nodes are shorter phrases that describe more abstract concepts and correspond to a broader set of images, refer to Table 2 for some key statistics in this aspect. Fig. 2 contrasts how well abstract concepts can be used to retrieve images. The concepts are the language expressions corresponding to the leaf nodes, the nodes that are one level above (LEAF-1), or two levels above (LEAF-2) the leaf nodes from the DG-FLICKR30K. Since abstract concepts tend to correspond to multiple images, we use mean averaged precision (mAP) to measure the retrieval results. ViLBERT+DG outperforms ViLBERT significantly. The improvement is also stronger when the concepts are more abstract. It is interesting to note that while the MATCH used in ViLBERT w/ DG incorporates learning representations to align images at both specific and abstract levels, such learning benefits all levels. The improvement of retrieving at abstract levels does not sacrifice the retrieval at specific levels. Conclusion Image and text aligned data is rich in semantic correspondence. Besides treating text annotations as "categorical" labels, in this paper, we show that we can make full use of those labels. Concretely, denotation graphs (DGs) encode structural relations that can be automatically extracted from those texts with linguistic analysis tools. We proposed several ways to incorporate DGs into learning representation and validated the proposed approach on several tasks. We plan to investigate other automatic tools in curating more accurate denotation graphs with a complex composition of fine-grained concepts for future directions. and 7 maximum levels, respectively. The training hyper-parameters remain the same as ViLBERT + DG-FLICKR30K with 3 maximum layers. The aim is to check how much gain we could get from the additional annotations. We report the results in Table 8. It shows that actually, the model trained with 3 levels of DG achieves the best performance. This might be because those high-level layers of DG (counting from the sentences) contain very abstract text concepts, such as "entity" and "physical object", which is non-informative in learning the visual grounding. Once the graph is constructed, we attach the images to the proper nodes by set-union images of each node's children, starting from the sentence-level node. A.2 Model architectures of ViLBERT and UNITER A comparison of these models is schematically illustrate in Fig. 3 ViLBERT model contains 121 million parameters, while UNITER contains 111 million parameters. A.3 Training Details All models are optimized with the Adam optimizer (Kingma and Ba, 2015). The learning rate is initialized as 4e −5 . Following ViLBERT (Lu et al., 2019), a warm-up training session is employed, during which we linearly increase the learning rate from 0 to 4e −5 in the first 1.5% part of the training epochs. The learning rate is dropped to 4e −6 and 4e −7 at the 10th and the 15th epochs, respectively. For ViLBERT (Reduced), we randomly initialized the model parameters in the image stream. The text stream is initialized from the first 3 layers of the pre-trained BERT model, and its co-attention Transformer layers are randomly initialized. For ViLBERT (Full) and UNITER , we load the model's weights pre-trained on the Conceptual Caption dataset to initialize them. After tokenization, the tokens are transformed to 768 dimension features by a word embedding initialized from BERT pre-trained model. The 768dimensional position features are included in the input to represent the position of each token. A.5 Visual Pre-processing For both ViLBERT and UNITER, we use the image patch features generated by the bottom-up attention features, as suggested by the original papers (Anderson et al., 2018a). The image patch features contain up to 100 image patches with their dimensions to be 2048. Besides this, a positional feature is used to represent the spatial location of bounding boxes for both ViLBERT and UNITER. Specifically, ViLBERT uses 5-dimensional position feature that encodes the normalized coordinates of the upper-left and lower-right corner for the bounding boxes, as well as one additional dimension encoding the normalized patch size. UNITER uses two additional spatial features that encode the normalized width and height of the object bounding box. B Full Experimental Results In this section, we include additional experimental results referred to by the main text. Specifically, we include results from a variety of models (e.g., ViL-BERT, ViLBERT + DG, UNITER, and UNITER + DG) on COCO dataset 5K test split (Karpathy and Fei-Fei, 2015) in § B.1. Then we provide a comprehensive ablation study on the impact of λ 1 and λ 2 of Eq. 7 in the main text in § B.3. B.1 Complete Results on COCO Dataset We report the full results on COCO dataset (1K test split and 5K test split) in Table 9 and Table 10. Additionally, we contrast to other existing approaches on these tasks. It could be seen that ViLBERT + DG and UNITER + DG improves the performance over the counterparts without DG by a significant margin on both COCO 1K and 5K test split -the only exception is that on the task of imagebased text retrieval, UNITER performs better than UNITER+DG. These results support our claim that training with DG helps the model to learn better visual and lin- B.2 Complete Results on FLICKR30K Dataset We contrast to other existing approaches in Table 11 on the task of text-based image retrieval on the FLICKR30K dataset. B.3 Ablation Study on λ 1 and λ 2 We conduct an ablation study on the impact of the two hyper-parameters λ 1 and λ 2 in Eq. 7 of the main text. We conduct the study with two ViL-BERT variants: ViLBERT Reduced and ViLBERT. The results are reported in Table 12 and Table 13. As we have two hyper-parameters λ 1 and λ 2 , we analyze their impacts on the final results by fixing one λ to be 1. Fixing the λ 2 = 1 and changing λ 1 , we observe that ViLBERT prefers larger λ 1 , while ViLBERT Reduced achieves slightly worse performance when λ 1 is smaller or larger. Fixing the λ 1 = 1 and changing λ 2 , we observe that performance of both architectures slightly reduced when λ 2 = 0.5 and λ 2 = 2. B.4 Full Results on Zero/Few-Shot and Transfer Learning Implementation Details for Zero-shot Referring Expression Specifically, the learned ViL-BERT and ViLBERT w/DG models are used first to produce a base matching score s BASE between the expression to be referred and the whole image. We then compute the matching score s MASKED between the expression and the image with each region feature being replaced by a random feature in turn. As the masked image region might be a noisy region, s MASKED might be larger than s BASE . Therefore, the model's prediction of which region the expression refers to is the masked region which causes the largest score in s REGION , where s REGION = (s BASE −s MASKED )·I[s MASKED > s BASE ]. Here I[·] is an indicator function. Table 5 shows that ViLBERT + DG-COCO outperforms ViLBERT on this task. Table 14 reports the full set of evaluation metrics on transferring across datasets. Training with DG improves training without DG noticeably. Transfer Learning Results C Visualization of Model's Predictions on Denotation Graphs We show several qualitative examples of both success and failure cases of ViLBERT + DG, when retrieving the text matched images, in Fig. 4 and Fig. 5. The image and text correspondence is generated by the Denotation Graph, which are derived from the caption and image alignment. We observe that in the Fig.4, the ViLBERT + DG successfully Text-based Image Retrieval Image-based Text Retrieval Method recognizes the images that are aligned with the text: "man wear reflective vest", while the ViLBERT fails to retrieve the matched image. In the failure case in Fig. 5, although ViLBERT + DG fails to retrieve the images that are exactly matched to the text, it still retrieves very relevant images given the query. retrieve the text matched images. We mark the correct sample in green and incorrect one in red. Kiros et al., 2014), or complex mappings parameterized by layers of attention networks and convolutions (Lu et al., 2019; Chen et al., 2019) -we experimented some of them in our empirical studies. i )} contains both the positive and negative examples. and PLAY PERCUSSION INSTRUMENT in Fig 1 is a pair of examples showing the latter more accurately describes the image(s) at the lower-level. Figure 2 : 2Image Retrieval using Mid-level Linguistic Expression on FLICKR30K Denotation Graph. The results are reported in Mean Average Precision (Mean AP). Figure 3 : 3Architecture of (a) ViLBERT, (b) UNITER. The means element-wise product. The [CLS] represents the embedding of [CLS] token in the last UNITER layer. guistic features. Although ViLBERT and UNITER have different architectures, training with DG could improve the performance consistently. :Figure 4 :Figure 5 : 4512: Ablation studies on the impact of λ 1 and λ 2 of ViLBERT Reduced on Text-based Image Retrieval on FLICKR30K dataset (Higher is better) Ablation studies on the impact of λ 1 and λ 2 of ViLBERT on Text-based Image Retrieval on FLICKR30K dataset (Higher is better) R@10 RSUM R@1 R@5 R@10 RSUM ViLBERT 43.5 72.5 83.4 199.4 49.0 76.0 83.9 209.0 ViLBERT + SOURCE DG 44.9 72.7 83.0 200.5 52.8 79.2 86.2 218.2 a man wearing a reflective vest sits on the sidewalk and holds up pamphlets with bicycles on the FLICKR30K Denotation Graph: Given Text and Retrieve Image. Qualitative example of ViLBERT + DG successfully retrieves the text matched images. We mark the correct sample in green and incorrect one in red. a black and white dog is running through the grass FLICKR30K Denotation Graph: Given Text and Retrieve Image. Qualitative example of ViLBERT + DG fails to Single-stream methods learn each modality separately and align them together with a simple fusion model, often an inner product between the two representations. Frome et al.(Frome et al., 2013) learns the joint embedding space for images and labels and use the learned embeddings for zeroshot learning.Kiros et al. (Kiros et al., 2014) uses bi-directional LSTMs to encode sentences and then maps images and sentences into a joint embedding space for cross-modal retrieval and multi-modal language models. Li et al.(Li et al., 2019b) designs a high-level visual reasoning module to contextual- ize image entity features and obtain a more power- ful image representation. Vendrov et al. (Vendrov et al., 2016) improves image retrieval performance by exploiting the hypernym relations among words. Table 1 : 1Key statistics of the two DGs: averaged over the all nodes in the graph, internal nodes and leaf nodes (formated as all/internal/leaf)Dataset DG-FLICKR30K DG-COCO # of edges 1.94M 4.57M # of nodes 597K/452K/145K 1.41M /841K/566K # of tokens/node 6.78/4.45/14.04 5.88/4.07/8.58 # of images/node 4.46/5.57/1.00 5.06/7.79/1.00 subsume IN FRONT OF A CROWD OF PEOPLE, see the right-hand side of Table 1 1lists the key statistics of the two DGs. We The original DG, while publicly available at https:// shannon.cs.illinois.edu/DenotationGraph/ contains 1.75 million nodes which are significantly less than ours, due to the difference in the version of the NLP toolkit.1 Available online at https://github.com/aylai/ DenotationGraph 2 Both DGs are made publically available at https:// sha-lab.github.io/DG/ 3 Table 2 : 2Text-based Image Retrieval (Higher is better) Method R@1 R@5 R@10 RSUM FLICKR30K ViLBERT 59.1 85.7 92.0 236.7 ViLBERT + DG 63.8 87.3 92.2 243.3 UNITER 62.9 87.2 92.7 242.8 UNITER + DG 66.4 88.2 92.2 246.8 COCO 1K Test Split ViLBERT 62.3 89.5 95.0 246.8 ViLBERT + DG 65.9 91.4 95.5 252.7 UNITER 60.7 88.0 93.8 242.5 UNITER + DG 62.7 88.8 94.4 245.9 COCO 5K Test Split ViLBERT 38.6 68.2 79.0 185.7 ViLBERT + DG 41.8 71.5 81.5 194.8 UNITER 37.8 67.3 78.0 183.1 UNITER + DG 39.1 68.0 78.3 185.4 to learn from images of SEEN (attribute, object) label pairs, such that it can generalize to recog- nize images of UNSEEN (attribute, object) label pairs. We evaluate this task on the MIT-STATE dataset (Isola et al., 2015), following the protocol by Misra et al. (2017). The training split contains 34,562 images from 1,262 SEEN labels, and the test split contains 19,191 images from 700 UN- SEEN labels. We report the Top-1, 2, 3 accuracies on the UNSEEN test set as evaluation metrics. Training Details Both ViLBERT and UNITER models are pre-trained on the Conceptual Cap- tion dataset (Sharma et al., 2018) and the pre- trained models are released publicly 4 . On the DG- FLICKR30K, ViLBERT and UNITER are trained with a minibatch size of 64 and ViLBERT is trained for 17 epochs and UNITER for 15 epochs, with a learning rate of 0.00004. On the DG-COCO, ViL- BERT is trained for 17 epochs and UNITER for 15 epochs with a minibatch size of 64 and a learning rate of 0.00004. The hyperparameters in Eq. Table 3 : 3Image-based Text Retrieval (Higher is better) Method R@1 R@5 R@10 RSUM FLICKR30K ViLBERT 76.8 93.7 97.6 268.1 ViLBERT + DG 77.0 93.0 95.0 265.0 UNITER 78.3 93.3 96.5 268.1 UNITER + DG 78.2 93.0 95.9 267.1 COCO 1K Test Split ViLBERT 77.0 94.1 97.2 268.3 ViLBERT + DG 79.0 96.2 98.6 273.8 UNITER 74.4 93.9 97.1 265.4 UNITER + DG 77.7 95.0 97.5 270.2 COCO 5K Test Split ViLBERT 53.5 79.7 87.9 221.1 ViLBERT + DG 57.5 84.0 90.1 232.2 UNITER 52.8 79.7 87.8 220.3 UNITER + DG 51.4 78.7 87.0 217.1 Table 4 : 4Image Retrieval via Text (Transfer Learning)SOURCE FLICKR→COCO COCO →FLICKR →TARGET R@1 RSUM R@1 RSUM ViLBERT 43.5 199.5 49.0 209.0 + SOURCE DG 44.9 200.5 52.8 218.2 Table 5 : 5Zero/Few-shot Learning for Referring Expression (Reported in R@1 on validation, TestA and TestB data) Setting → 0% (Zero-shot) 25% 50% 100% Method Val TestA TestB Val TestA TestB Val TestA TestB Val TestA TestB ViLBERT 35.7 41.8 29.5 67.2 74.0 57.1 68.8 75.6 59.4 71.0 76.8 61.1 ViLBERT + DG-COCO 36.1 43.3 29.6 67.4 74.5 57.3 69.3 76.6 59.3 71.0 77.0 60.8 Table 6 : 6Image Recognition on UNSEEN Attribute-Object Pairs on the MIT-STATE DatasetMethod Top-1 Top-2 Top-3 VisProd (Misra et al., 2017) 13.6 16.1 20.6 RedWine (Misra et al., 2017) 12.1 21.2 27.6 SymNet (Li et al., 2020) 19.9 28.2 33.8 ViLBERT pre-trained on N/A 16.2 26.3 33.3 COCO 17.9 28.8 36.2 DG-COCO 19.4 30.4 37.6 Table 7 : 7Ablation Studies of Learning from DGViLBERT variants → Reduced Full w/o DG 215.4 236.7 w/ DG + MATCH 221.5 236.5 − DG HARD NEGATIVES + MATCH 228.4 241.7 + MATCH + SPEC 228.8 242.6 + MATCH + SPEC + EDGE 231.2 243.3 Table 8 : 8Text-based Image Retrieval Performance of ViLBERT trained with different number of DG levels # of DG levels R@1 R@5 R@10 RSUM 3 levels 65.9 91.4 95.5 252.7 5 levels 62.5 86.4 92.3 241.2 7 levels 62.8 86.3 91.6 240.7 .• ViLBERT. It has 6 basic Transformer layers for text and 8 layers for image. For all the Transformer layers on the text side, we use 12 attention heads and 256 feature dimensions, then linearly project down to 1024 feature dimensions. For all the Transformers on the image side, we use 8 attention heads and 128 feature dimensions, then combine into 1024 feature dimensions too. • UNITER. All the Transformer layers have 12 heads and 256 feature dimensions. The major difference between UNITER and ViL- BERT is how attentions are used. In ViLBERT, one modality is used as a query, and the other is used as value and key. In UNITER, however, both are used as query, key, and value. Additionally, UNITER is similar to another model Unicoder-VL (Li et al., 2019a). However, the latter has not provided pub- licly available code for experimenting. For ViLBERT model, each text and image co- attention Transformer layer contains 8 attention heads with 1024 dimensions in total. The text Transformer layer contains 12 attention heads with 3072 hidden dimensions in total. In contrast, the image Transformer layer has 8 attention heads with 1024 hidden dimensions in total. For UNITER model, each cross-attention Transformer layer con- tains 12 heads with 3072 hidden dimensions in total. Table 9 : 9Results on Cross-Modal Retrieval on COCO dataset 1K test split (Higher is better) Table 11 : 11Results on Text-based Image Retrieval on FLICKR30K test split (Higher is better) Models ran or implemented by us ViLBERT(Lu et al., 2019) 58.2 84.9 91.5 234.6 UNITER(Chen et al., 2019) 71.5 91.2 95.2 257.9Method R@1 R@5 R@10 RSUM ViLBERT 59.1 85.7 92.0 236.7 ViLBERT + DG 63.8 87.3 92.2 243.3 UNITER 62.9 87.2 92.7 242.8 UNITER + DG 66.4 88.2 92.2 246.8 Known results from literature VSE++(Faghri et al., 2018) 39.6 70.1 79.5 189.2 SCO(Huang et al., 2018) 41.1 70.5 80.1 191.7 SCAN(Lee et al., 2018) 48.6 77.7 85.2 211.5 VSRN(Li et al., 2019b) 54.7 81.8 88.2 224.7 Table The UNITER model performs an additional online hard-negative mining (which we did not) during the training of image-text matching to improve their results. This is computationally very costly. https://github.com/aylai/ DenotationGraph AcknowledgementWe appreciate the feedback from the reviewers. This work is partially supported by NSF Awards IIS-1513966/ 1632803/1833137, CCF-1139148, DARPA Award#: FA8750-18-2-0117, FA8750-19-1-0504 , DARPA-D3M -Award UCB-00009528, Google Research Awards, gifts from Facebook and Netflix, and ARO# W911NF-12-1-0241 and W911NF-15-1-0484. We particularly thank Haoshuo Huang for help in improving the efficiency of DG generation.AppendixIn the Appendix, we provide details omitted from the main text due to the limited space, including:• § A describes complete implementation details (cf. § 3 and § 5.1 of the main text). • § B provides complete experimental results (cf.§ 5.2 of the main text). • § C visualizes the model's predictions on denotation graphs.A Implementation DetailsA.1 Constructing Denotation GraphsWe summarize the procedures used to extract DG from vision + language datasets. For details, please refer to(Young et al., 2014). We used the publicly available tool 5 . The analysis consists of several steps: (1) spell-checking;(2) tokenize the sentences into words; (3) tag the words with Part-of-Speech labels and chunk works into phrases; (4) abstract semantics by using the WordNet(Miller, 1995)to construct a hypernym lexicon table to replace the nouns with more generic terms; (5) apply 6 types of templated rules to create fine-to-coarse (i.e., specific to generic) semantic concepts and connect the concepts with edges.We set 3 as the maximum levels (counting from the sentence level) to extract abstract semantic concepts. This is due to the computation budget we can afford, as the final graphs can be huge in both the number of nodes and the edges. Specifically, without the maximum level constraint, we have 2.83M concept nodes in total for Flickr dataset. If the training is run on all these nodes, we will consume 19 times more iterations than training on the original dataset, which has 145K sentences(Young et al., 2014). As a result, much more time would be required for every experiment. With the 3 layers of DG from the leaf concepts, we have in 597K nodes. In this case, the training time would be cut down to 4.1 times of the original dataset.Nonetheless, we experimented with more than 3 levels to train ViLBERT + DG-FLICKR30K with 5 . Vse++(faghri, 52VSE++(Faghri et al., 2018) 52 . Sco(huang, 56SCO(Huang et al., 2018) 56 . Scan(lee, 58SCAN(Lee et al., 2018) 58 . Vsrn(li, 62VSRN(Li et al., 2019b) 62 . Vse++(faghri, 64VSE++(Faghri et al., 2018) 64 . Sco(huang, 69SCO(Huang et al., 2018) 69 . Scan(lee, 72SCAN(Lee et al., 2018) 72 . Vsrn(li, 298VSRN(Li et al., 2019b) 76.2 94.8 98.2 269.2 Results on Cross-Modal Retrieval on COCO dataset 5K test split (Higher is better) Text-based Image Retrieval Image-based Text Retrieval. Table. 10Table 10: Results on Cross-Modal Retrieval on COCO dataset 5K test split (Higher is better) Text-based Image Retrieval Image-based Text Retrieval . Vse++(faghri, 30VSE++(Faghri et al., 2018) 30 . Sco(huang, SCO(Huang et al., 2018) . Scan(lee, 38SCAN(Lee et al., 2018) 38 . Vsrn(li, 405 70.6 81.1 192.2VSRN(Li et al., 2019b) 40.5 70.6 81.1 192.2 . Uniter(chen, 48UNITER(Chen et al., 2019) † 48 . Vse++(faghri, 41VSE++(Faghri et al., 2018) 41 . Sco(huang, SCO(Huang et al., 2018) . Scan(lee, 50.4 82.2 90.0 222.6SCAN(Lee et al., 2018) 50.4 82.2 90.0 222.6 . Vsrn(li, 53VSRN(Li et al., 2019b) 53 . Uniter (chen, 63UNITER (Chen et al., 2019) † 63 2019) model performs an additional online hard-negative mining (which we did not) during the training of image-text matching to improve their results. † : The Uniter( Chen, which is computationally very costly† : The UNITER(Chen et al., 2019) model performs an additional online hard-negative mining (which we did not) during the training of image-text matching to improve their results, which is computationally very costly.	[ "https://github.com/aylai/", "https://github.com/aylai/" ]
[ "On the sodium overabundance of giants in open clusters: The case of the Hyades ⋆", "On the sodium overabundance of giants in open clusters: The case of the Hyades ⋆" ]	[ "Rodolfo Smiljanic \nEuropean Southern Observatory\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany\n" ]	[ "European Southern Observatory\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany" ]	[ "Mon. Not. R. Astron. Soc" ]	Sodium abundances have been determined in a large number of giants of open clusters but conflicting results, ranging from solar values to overabundances of up to five orders of magnitude, have been found. The reasons for this disagreement are not wellunderstood. As these Na overabundances can be the result of deep mixing, their proper understanding has consequences for models of stellar evolution. As discussed in the literature, part of this disagreement comes from the adoption of different corrections for non-LTE effects and from the use of different atomic data for the same set of lines. However, a clear picture of the Na behaviour in giants is still missing. To contribute in this direction, this work presents a careful redetermination of the Na abundances of the Hyades giants, motivated by the recent measurement of their angular diameters. An average of [Na/Fe] = +0.30, in NLTE, has been found. This overabundance can be explained by hydrodynamical models with high initial rotation velocities. This result, and a trend of increasing Na with increasing stellar mass found in a previous work, suggests that there is no strong evidence of Na overabundances in red giants beyond those values expected by evolutionary models of stars with more than ∼ 2 M ⊙ .	10.1111/j.1365-2966.2012.20729.x	[ "https://arxiv.org/pdf/1202.2200v1.pdf" ]	119,184,600	1202.2200	8dc46e72d900df4e05af99de7e44174d6aca4aeb	On the sodium overabundance of giants in open clusters: The case of the Hyades ⋆ 1-?? (2011 Rodolfo Smiljanic European Southern Observatory Karl-Schwarzschild-Str. 285748Garching bei MünchenGermany On the sodium overabundance of giants in open clusters: The case of the Hyades ⋆ Mon. Not. R. Astron. Soc 0001-?? (2011Accepted . Received ; in original form(MN L A T E X style file v2.2)open clusters and associations: individual: Hyades -stars: abundances -stars: evolution -stars: fundamental parameters Sodium abundances have been determined in a large number of giants of open clusters but conflicting results, ranging from solar values to overabundances of up to five orders of magnitude, have been found. The reasons for this disagreement are not wellunderstood. As these Na overabundances can be the result of deep mixing, their proper understanding has consequences for models of stellar evolution. As discussed in the literature, part of this disagreement comes from the adoption of different corrections for non-LTE effects and from the use of different atomic data for the same set of lines. However, a clear picture of the Na behaviour in giants is still missing. To contribute in this direction, this work presents a careful redetermination of the Na abundances of the Hyades giants, motivated by the recent measurement of their angular diameters. An average of [Na/Fe] = +0.30, in NLTE, has been found. This overabundance can be explained by hydrodynamical models with high initial rotation velocities. This result, and a trend of increasing Na with increasing stellar mass found in a previous work, suggests that there is no strong evidence of Na overabundances in red giants beyond those values expected by evolutionary models of stars with more than ∼ 2 M ⊙ . INTRODUCTION In many stages of their evolution, low-and intermediatemass stars show signs of mixing between material of the surface with material of the interior that has been processed by nuclear reactions (Pinsonneault 1997;Charbonnel & Talon 2008;Smiljanic et al. 2009, and references therein). The standard model of stellar evolution, where convection is the only mixing mechanism, does not account for all the observational details. The introduction of non-standard physical processes, such as atomic diffusion, rotation-induced mixing, internal gravity waves, magnetic buoyancy and thermohaline mixing, is unavoidable (see e.g. Montalbán & Schatzman 2000;Young et al. 2003;Talon & Charbonnel 2005;Palacios et al. 2006;Denissenkov & Pinsonneault 2008;Denissenkov et al. 2009;Michaud et al. 2004Michaud et al. , 2010Angelou et al. 2011;Palmerini et al. 2011, and references therein). ⋆ Based on data obtained from the ESO Science Archive Facility. The observations were made with ESO Telescopes at the La Silla Paranal Observatory under programmes ID 070.D-0421, 072.C-0393, and 083.A-9011. † E-mail: [email protected] As a star leaves the main sequence towards the red giant branch (RGB), its convective envelope deepens, causing first a dilution of lithium, beryllium, and boron (Lèbre et al. 1999;Pasquini et al. 2004;Smiljanic et al. 2010;Canto Martins et al. 2011) and then the first dredgeup (Iben 1967), when material affected by hydrogen burning is mixed to the surface. The dredge-up causes an increase of the abundance of nitrogen and a decrease of carbon and of the 12 C/ 13 C ratio (Charbonnel et al. 1998;Gratton et al. 2000). An extensive literature has shown a further modification of the surface abundances after the bump in the luminosity function on the RGB. At this phase the abundances of Li, C and the ratio 12 C/ 13 C are further decreased while that of N increases. This effect has been detected in stars of the field and of both open and globular clusters (see e.g. Sneden et al. 1986;Gilroy 1989;Gilroy & Brown 1991;Charbonnel et al. 1998;Gratton et al. 2000;Tautvaišienė et al. 2000Tautvaišienė et al. , 2005Tautvaišienė et al. , 2010 The case of sodium During the first dredge-up, Na produced by the NeNa-cycle (Denisenkov & Denisenkova 1990) in H-burning regions can potentially be mixed to the surface. The observational behaviour of this element, however, is not clear. In open clusters, stars with Na overabundances as high as [Na/Fe] 1 = +0.50 or more have been reported (Bragaglia et al. 2001;Friel et al. 2003;Jacobson et al. 2007;Schuler et al. 2009), while others were found to have mild overabundances of [Na/Fe] ∼ +0.20 (Hamdani et al. 2000;Tautvaišienė et al. 2000;Pasquini et al. 2004;Friel et al. 2010) or abundances close to solar (Randich et al. 2006;Sestito et al. 2007;Smiljanic et al. 2009;Pancino et al. 2010). One of the factors behind this discrepancy is the adoption of different log gf s for the same Na lines by different authors. For example, the log gf s adopted in Smiljanic et al. (2009), from the NIST database (Ralchenko et al. 2010), are on average 0.22 dex higher than those adopted by Jacobson et al. (2007), derived with respect to Arcturus. Indeed, as discussed later by Jacobson et al. (2008), a revision of their gf -values resulted in an increase of about 0.20 dex with corresponding decrease in the Na abundances. The use of this revised scale now results in modest overabundances, [Na/Fe] = +0.10...+0.20 (see Table 14 of Friel et al. 2010) in comparison with the strong overabundances found in Jacobson et al. (2007), [Na/Fe] = +0.40...+0.60. Another issue are departures from the local thermodynamic equilibrium (LTE). Several authors provide non-LTE corrections for Na (Baumueller et al. 1998;Gratton et al. 1999;Mashonkina et al. 2000;Takeda et al. 2003;Shi et al. 2004;Andrievsky et al. 2007;Lind et al. 2011). As discussed by Lind et al. (2011), there is a scatter of 0.10 to 0.20 dex among the different corrections. Usually non-LTE abundances are smaller than the LTE ones. In the abundance analyses cited above, the different authors made different choices regarding these matters. It is thus difficult to make sense out of these results and understand whether there is a real problem with the sodium overabundances in giants. Dedicated studies aiming to better understand this issue are still lacking in the literature. The re-analysis of Na in the Hyades giants presented here is a step on this direction. It is motivated by the recent determination of angular diameters (and thus fundamental effective temperatures -T eff ) for the four Hyades giants from interferometric measurements using the CHARA array by Boyajian et al. (2009). Accurate Na abundances can help clarifying whether the strong overabundances are real and test whether there is an offset between observations and evolutionary models. This paper is divided as follows. Section 2 presents the observational data used in the analysis and the determination of the atmospheric parameters of the Hyades giants. Section 3 presents the determination of Na abundance from equivalent widths and spectrum synthesis while Sec. 4 presents a discussion of the results. Section 5 summarizes the conclusions. 1 [A/B] = log [N(A)/N(B)]⋆ − log [N(A)/N(B)] ⊙ THE HYADES The Hyades are the closest open cluster to the Sun (∼ 46.5 pc, van Leeuwen 2009). They have an age of 625 ± 50 Myr (Perryman et al. 1998) and a metallicity of [Fe/H] = +0.13 ± 0.01 (Paulson et al. 2003). There is no sign of Na overabundance in the dwarf stars, i.e. [Na/Fe] ∼ 0.00 (Paulson et al. 2003). For the giants an average of [Na/Fe] = +0.48 (in LTE) was found by Schuler et al. (2009), which the authors regard as "too large to be explained by any known self-enrichment scenario". The cluster has four evolved members, all He-burning clump giants: ǫ Tau (HR 1409 or HD28305), γ Tau (HR 1346 or HD27371), δ 1 Tau (HR 1373 or HD 27697), and θ 1 Tau (HR 1411 or HD 28307). A fifth suggested giant member, δ Ari (HR 951) is likely a non-member (de Bruijne et al. 2001). The stars ǫ Tau and γ Tau are single stars while δ 1 Tau and θ 1 are spectroscopic binaries (Griffin et al. 1988;Mermilliod et al. 2008). In addition, the star ǫ Tau was found to have a massive planetary companion (Sato et al. 2007). An HR diagram of the Hyades on the region around the turn-off and the clump is show in Fig. 1. An isochrone from Girardi et al. (2002) with 625 Myr and [Fe/H] = +0.13 indicates that a clump giant in the Hyades has ∼ 2.48 M⊙. To illustrate, the corresponding isochrone is also shown in Fig. 1 (no attempt to find the best fitting model was made). Observational data High-resolution spectra of three Hyades giants obtained with FEROS (Kaufer et al. 1999) at the 2.2m MPG/ESO telescope at La Silla and with UVES (Dekker et al. 2000) fed by the UT2 of the VLT at Paranal are analyzed here. FEROS is a fiber-fed echelle spectrograph that provides a full wavelength coverage of λλ350−920 nm over 39 orders at R = 48 000. The spectra were reduced using the FEROS Data Reduction System (DRS) as implemented within ESO-MIDAS. UVES is a cross-dispersed echelle spectrograph able to obtain spectra from the atmospheric cut-off at 300 nm to ∼ 1100 nm. Reduction was done with the ESO UVES pipeline within MIDAS (Ballester et al. 2000). The FEROS data of stars δ 1 Tau and γ Tau were made available to the author by Luca Pasquini (2010, private communication). The FEROS spectra of ǫ Tau and the UVES spectra of ǫ Tau and δ 1 Tau were retrieved from the ESO/ST-ECF science archive facility. The log book of the observations is given in Table 1. Effective temperatures As the only giants of the nearest open cluster, these stars have been analyzed many times. To obtain an idea of the range of temperatures found in the literature, previous determinations of the effective temperature (T eff ) of the four giants were queried at the PASTEL catalogue (Soubiran et al. 2010). A few selected and recent results, together with a "literature average", are given in Table 2 along with the corresponding reference. Note that the T eff calculated with the interferometric data by Boyajian et al. (2009) is not included in the PASTEL catalogue. (1) The effective temperatures adopted by Schuler et al. (2009) were calculated by Blackwell & Lynas-Gray (1994) using the infrared flux method for γ and δ 1 Tau and by Blackwell & Lynas-Gray (1998) for ǫ Tau. There is a range of about 300-350 K on the values of T eff determined for each star. As discussed in Boyajian et al. (2009), the values determined from recent angular diameters are on the low side of this range. Similarly, using the angular diameter of the Li-rich giant HD148 193, Baines et al. (2011) derived an T eff that is on the cooler side of the range of temperatures determined in the literature. Together these results might be indicating that the temperature of giants is usually overestimated. In the following discussion, Na abundances are calculated using two values of temperature for each star: The interferometric temperatures of Boyajian et al. (2009) and those used by Schuler et al. (2009) in their abundance analysis of the same stars. The temperatures of Schuler et al. (2009) are close to the average literature values (see Table 2) and differ from the interferometric ones by 90 to 150 K. Gravities Gravities for the Hyades giants were determined with the equation: log (g⋆/g ⊙ ) = log (M⋆/M ⊙ ) + 4 log (T eff⋆ /T eff⊙ ) − log (L⋆/L ⊙ ) The luminosities by de Bruijne et al. (2001), a mass of 2.48 M⊙, and the usual solar values (T eff⊙ = 5777 K and log g⊙ = 4.44) were adopted. For consistency, gravities were calculated using each of the T eff adopted for the analysis. In addition, to illustrate the effect of the gravity in the Na abundance, additional values were determined for γ Tau assuming masses of 2.0 and 3.0 M⊙. It can be seen that this change in the mass causes only a minor change on log g, arguing that this parameter is well constrained for these stars. Microturbulence Throughout the analysis, a fixed value of microturbulence, ξ = 1.30 km s −1 was always adopted. This is a typical value found for the giants analyzed in Smiljanic et al. (2009). This value was checked against empirical relations from the literature that calibrate ξ as a function of log g and/or T eff . Five calibrations from four references were investigated. It is interesting to note that some of these calibrations are given only as a function of log g: The values obtained from these calibrations for γ Tau are given in Table 4 for the parameters using the interferometric and IRFM temperatures. All these calibrations, apart from that of Carretta et al. (2004) give values that are very close to the one adopted here. What is more important, they show that the variation in T eff within the range considered here does not result in a large change in microturbulence. This argues that the choice of keeping ξ constant does not introduce systematic effects in the current analysis. (i) ξ = 2.22 − 0 Metallicity To estimate the metallicity of the stars ([Fe/H]), equivalent widths of a set of 15 selected Fe I lines were measured. The line list, atomic data, and equivalent widths are given in Table 5. The C6 broadening constants were taken from Coelho et al. (2005). The adopted solar iron abundance is A(Fe) = 7.50 (Grevesse & Sauval 1998 Table 4. Microturbulence velocities (in km s −1 ) derived with different calibrations for γ Tau, using both the interferometric and the IRFM temperature and the corresponding log g. Calibration Interf. T eff IRFM T eff Gratton et al. (1996) Line selection and atomic data All the 32 strong NaI and NaII lines with wavelengths between 4000 and 8200Å listed by Sansonetti & Martin (2005) were considered as possible features to be used in this analysis. The profiles of these lines were checked both in the UVES 2 and the Kurucz (2005) solar spectra and in the spectrum of γ Tau. All lines that were heavily blended or too strong for an abundance analysis were discarded (e.g. lines at λ 8183 and λ 8194Å).The atomic data of the remaining NaI lines are listed in Table 6. The excitation potential and the log gf of the lines were adopted from the NIST database (Ralchenko et al. 2010). The C6 broadening constants were adopted from Coelho et al. (2005) and Barbuy et al. (2006). An assessment of each line is given below: (1) 5 148.838Å: In the solar spectra the line has ∼ 14 mÅ. It is slightly blended on the blue wing with a line of similar strength. In the γ Tau spectrum the two lines are completely blended but their bottoms can be distinguished. Only possible to analyze with spectrum synthesis. (2) 5 682.633Å: It is clear in the solar spectra that the blue wing is blended with a weaker line. In the spectrum of γ Tau the blend can not be recognized, but the line is asymmetric. Equivalent widths would be affected by the blend and thus spectrum synthesis should be preferred. In the Sun the line has ∼ 100 mÅ. (3) 5 688.193 & 5 688.205Å: These lines are the fine structure components of the same transition. The feature is strong in the Sun ( 120 mÅ) and stronger in γ Tau ( 170 mÅ). Seems to be clean enough, but it is too strong for an analysis using equivalent widths. (4) 6 154.226Å: On the Sun there is a weak line close to the red wing at ∼ 6154.43Å. On γ Tau the same blend is apparent but less distinguishable. There is another weak line at ∼ 6154.6Å. The line seems to deviate from a Gaussian profile towards a Voigt one already in the Sun (where EW ∼ 37 mÅ). Analysis using equivalent widths should be possible. (5) 6 160.747Å: Both in the Sun and in γ Tau, the placement of the continuum is affected by the wing of the strong nearby CaI 6162.2Å line. It has closeby lines to the blue and red sides, but the profile seems clean. Analysis using equivalent widths is possible. It has ∼ 60 mÅ in the Sun. Equivalent widths Given the above assessment, only the equivalent widths of the lines at 6154 and 6160Å were used to determine Na abundances. However, other lines have been used in the literature. For example, Schuler et al. (2009) also use the line at 5682Å, which is clearly blended in the Sun and is asymmetric in γ Tau. The equivalent width of such line should be regarded as suspicious. Indeed, from the three lines adopted by Schuler et al. (2009), line 5682Å always results in an abundance that is higher by 0.10-0.14 dex than that obtained with the lines at 6154 and 6160Å. Without this line, the mean [Na/Fe] for the Hyades giants found by Schuler et al. (2009) is reduced by 0.05 dex. Equivalent widths were determined by fitting Gaussian profiles to the observed lines using IRAF 3 . For the Sun, lines were measured both in the UVES and in the Kurucz spectrum. The values obtained are listed in Table 7. Model atmospheres were computed using the Linux version (Sbordone et al. 2004;Sbordone 2005) of the AT-LAS9 code originally developed by Kurucz (see e.g. Kurucz 1993). For the calculations, the opacity distribution functions of Castelli & Kurucz (2003) without overshooting were adopted. These models assume local thermodynamic equilib-rium, plane-parallel geometry, and hydrostatic equilibrium. Abundances were derived using the WIDTH code, also in its Linux version. For the Hyades giants, abundances were calculated only with the equivalent widths measured in the FEROS spectra, as they have higher S/N. These Na abundances are listed in Table 8. Spectrum synthesis Abundances were also derived using spectrum synthesis and all the Na lines in Table 6. Synthetic spectra were calculated with the codes described in Coelho et al. (2005) and the model atmospheres described above. The line list is the one used to compute the spectrum library of Coelho et al. (2005). As mentioned before, the Na line at 5682Å is blended in its blue wing. In the line list used here, this blend is due to a Cr I line at 5682.495Å, with log gf = −0.609. It was modeled with the solar abundance recommended by Grevesse & Sauval (1998), A(Cr)⊙ = 5.67. The Na abundances obtained with spectrum synthesis are listed in Table 9. Uncertainties of the Na abundances The main source of error of the abundances are the errors of the atmospheric parameters. The uncertainty of the T eff derived by Boyajian et al. (2009) is of the order of ± 50 K. This corresponds to an average uncertainty of ± 0.04 dex in the Na abundance -A(Na) -and ± 0.03 in [Fe/H]. As shown for γ Tau (Table 3), gravities are well constrained and no significant impact on the Na abundance is expected. A change of ± 0.05 dex in log g results in a change of ∓ 0.015 dex in A(Na). The same uncertainty causes an effect of ∓ 0.005 in [Fe/H]. The discussion in Section 2.4 shows that the value of ξ is well constrained. The values given by the different calibrations have an rms of ± 0.08 km s −1 . A total change of ± 0.10 km s −1 in ξ results in a change of ∓ 0.03 dex in A(Na). This change causes no significant effect in [Fe/H]. Considering that there is no significant uncertainty in the Na and Fe abundances of the Sun, the final uncertainty in [Na/Fe] is of ± 0.04 dex. DISCUSSION The mean solar abundance using equivalent widths is A(Na)⊙ = 6.30 while using spectrum synthesis it is A(Na)⊙ = 6.33. The average [Na/Fe] values for each star are given in Tables 8 and 9, respectively for the cases using equivalent widths and spectrum synthesis. Although there is some difference in A(Na) between the different temperature scales, there is basically no noticeable effect on [Na/Fe]. However, at first glance the average values determined with equivalent widths seem to be about ∼0.12 dex higher than the values determined using spectrum synthesis. The difference is apparent in the giants but not in the Sun. As discussed below, this is not directly related to the use equivalent widths or spectrum synthesis to derive the abundances, but to the different choice of lines used in each case. There is however, a systematic difference of about ∼ 0.10 dex between the abundances derived with the λ 6154 line. This likely comes from an uncertainty in the equivalent width of this line. An error on the equivalent width of ± 6mÅ can produce a change of ± 0.10 on the abundance. There are three effects contributing to the difference. One, is the Na abundance given by the λ 5148 line. Excluding it from the average, the [Na/Fe] values increase by ∼ 0.05 dex. It is not clear why this line results systematically in smaller values. Nevertheless, it was decided to consider its abundance as suspicious and to exclude it from further discussion. The final average LTE [Na/Fe] values, without line λ 5148, are given in Table 10. A second effect discussed in the next subsection are the NLTE corrections. The last effect seems related to differences between the WIDTH code, used to calculate abundances from equivalent widths, and the PFANT code, used to compute the synthetic spectrum. Following a suggestion by the referee, a series of synthetic spectra with the parameters of γ Tau at the region around the line λ 6160 were calculated with different Na abundances. The equivalent widths were measured and the values used to recompute the abundances. The resulting Na abundances given by WIDTH are 0.03 to 0.04 dex higher than the values used to compute the synthetic spectra. At this point, it is not possible to say whether this is caused by some numerical effect, by some difference in other input data (e.g. partition functions, opacities...), or to some difference in the physics, like the treatment of broadening, for example. This should, of course, be further investigated. Nevertheless, as it is based in the analysis of more features and likely based in a more reliable way to deal with the effects of broadening, the abundances derived with spectrum synthesis are preferred here. NLTE abundances As mentioned in the introduction, the Na lines are affected by NLTE effects (see Lind et al. 2011, and references therein). This has been suggested as a likely reason behind the large Na overabundances found in giants of open clusters (see e.g. Randich et al. 2006;Sestito et al. 2007). To correct the abundances calculated here, the NLTE calculations of Lind et al. (2011) were adopted. Corrections were interpolated among the grid calculated in that work with an IDL routine kindly made available to the author by Karin Lind (2011, private communication). NLTE corrections were derived in a line-by-line basis, giving as input the atmospheric parameters and the LTE abundances 4 . Individual line corrections for all stars (giants and Sun) range from −0.03 (for line 5148Å) up to −0.15 (for line 6160 A). For the Sun, the average corrections are of −0.08 and −0.09 dex, respectively with the abundances using equivalent widths and spectrum synthesis. Thus, in NLTE, the solar Na abundances derived in this work are A(Na)⊙ = 6.22 with equivalent widths and A(Na)⊙ = 6.24 with spectrum synthesis. For the giants, the different selection of lines results on average corrections of −0.14 when using equivalent widths and around −0.11 when using spectrum synthesis. This difference in the average the NLTE corrections is another responsible for the difference among the LTE [Na/Fe] values given by equivalent widths and spectrum synthesis. With respect to the Sun, the correction when using equivalent widths (only lines 6154 and 6160Å) is of ∼ −0.06 dex. When using spectrum synthesis (4 lines), the correction is of −0.01 or −0.02 dex. This helps to explain why the different NLTE abundances in Table 11 are in better agreement than the LTE values in Table 10. The average absolute NLTE Na abundance of the Hyades giants, using the interferometric temperatures, is found to be [Na/Fe] = +0.30 5 . With the IRFM adopted by Schuler et al. (2009) . These values are in perfect agreement with the ones derived in this work. Comparison with evolutionary models An interesting question to look at now is whether the Na overabundances of the Hyades giants can be explained by evolutionary models. In standard models, no modification of the Na abundance is expected after the first dredge-up for stars below ∼ 2.0 M⊙ (Mowlavi 1999;. For stars of higher mass, an increase of up to +0.20 dex in the Na abundance is expected. When mixing induced by rotation is included in the models (transport of chemicals and angular momentum by shear turbulence and meridional circulation), larger overabundances are produced. These effects can also create a dispersion in the Na abundance among otherwise similar stars if they had different initial rotation velocities in the zero age main sequence . This happens because rotation affects the internal abundance profile of the elements involved in H-burning. In this way, the Na-rich region in rotating stars begins further out from the core, and more Na-rich material can be dredged-up to the surface. In Fig. 2 the Na abundance of the Hyades is shown in comparison with the models calculated by . Also shown in the figure are the Na abundances of 31 giants of 10 open clusters derived in Smiljanic et al. (2009). These clusters have turn-off masses between 1.7 and 3.1 M⊙. The recommended Na abundance of the Hyades ([Na/Fe] = +0.30) is in the upper part of the range expected by the models. Therefore, and contrary to the conclusion of Schuler et al. (2009), this comparison shows that the Na abundance in the Hyades can be explained by modern hydrodynamical models that include the effects of rotation. There is no need for an extra unknown mixing process. Although it is believed here that the absolute Na abundance of the Hyades was derived, this claim can not be extended to most of the results in the literature. One way to avoid such systematics and test whether the Na overabundances in giants conform with the prediction of the models, is to conduct a large homogeneous analysis of a sample including only giants but with different masses. In this way one can look for abundance trends with mass and test if they agree with the expectations of models. This is exemplified by the giants analyzed in Smiljanic et al. (2009). As seen in Fig. 2 and discussed also in Smiljanic et al. (2009) and, there is an off-set of about 0.10 dex between observations and models. However, there is an agreement in the increasing trend with mass. This suggests that we are indeed observing the effects of mixing in these stars, in spite of a possible systematic effect in the abundance scale. The Na abundance of the Hyades derived here and the trend of increasing Na with increasing mass found in Smiljanic et al. (2009) argue that there is no strong evidence for overabundances above those expected by the models, for stars above 2 M⊙. At this point, it is interesting to mention the results of Pasquini et al. (2004) for the cluster IC 4651 and of Randich et al. (2006) for M67. Pasquini et al. (2004) found a systematic difference at the level of 0.20 dex between the Na abundances of dwarfs and giants in IC 4651. A similar difference between dwarfs and giants was, however, not detected in stars of M67 by Randich et al. (2006). Although these results could be regarded as contradictory at first sight, they are not. Clump giants in M67 have ∼ 1.3 M⊙ while clump giants in IC 4651 have ∼ 1.8 M⊙. As can be seen in Fig. 2, according to the models of Charbonnel & Lagarde (2010) a star of 1.3 M⊙ is never expected to enrich itself in Na after the first dredge-up while stars of 1.8 M⊙ could be enriched by ∼ +0.15 dex. Although caution is needed in the comparison among dwarfs and giants, these results seem to support the idea that models with rotation can properly explain the behaviour of Na also in giants below 2 M⊙. Cluster versus field giants It is sometimes noticed that the overabundance of Na seen in giants of open clusters is not apparent in field stars (Friel 2006). This comparison is, however, usually made between giants in clusters and field dwarfs (see e.g. Jacobson et al. 2011). As also discussed by these authors, such an offset might be real and caused by mixing. Based on the discussion of the previous Section, this is a conclusion supported here. Jacobson et al. (2011) noticed indeed that their cluster abundances agree well with the Na abundances of field clump giants determined by Mishenina et al. (2006). Comparisons among dwarfs and giants need to be careful not just because of mixing, but also because systematic effects can cause biases that might be mistaken by real differences. As an example, according to Meléndez et al. (2008) and Alves-Brito et al. (2010) this seems to be the case behind previous claims of abundance differences among thick disc and bulge stars. A mismatch of Na abundances between stars in open clusters and in the field lead de Silva et al. (2009) to suggest that the dissolution of open clusters might not be the main contributor of stars for the Galactic disc. These authors compiled Na abundances in cluster giants from the literature, normalized them to a common solar scale, and did a comparison with Na abundances in field dwarfs from Soubiran & Girard (2005), field clump giants from Mishenina et al. (2006), and bulge giants from Fulbright, McWilliam & Rich (2007). Agreement was found between the cluster giants and the field giants of Mishenina et al. (2006), but an offset exists with respect to the dwarfs of Soubiran & Girard (2005) and the giants of Fulbright et al. (2007). Because of mixing, an offset between Na in giants and dwarfs might be expected, with the caveat that in samples of field stars one is adding together stars with different masses and metallicities, properties that affect the mixing of Na during the first dredge-up. In other words, depending on the mass range of the giants an offset between the Na abundances between giants and dwarfs and among giants themselves might be expected or not. A robust way to attempt a comparison such as the one done by de Silva et al. (2009) would be using dwarfs in the field and dwarfs in clusters, where mixing is not able to affect the Na abundances. SUMMARY Sodium abundances of three Hyades giants have been redetermined and an average value of [Na/Fe] = +0.30 in NLTE was found. This Na abundance was derived using the ab-solute T eff of the stars determined with the interferometric angular diameter measurements of Boyajian et al. (2009). This Na abundance agrees well with the ones predicted by the hydrodynamical models of for a star of 2.48 M⊙, after the first dredge-up, and taking into account rotation-induced mixing. This contradicts the conclusion of Schuler et al. (2009) that the Na overabundances of the Hyades could not be explained by any known mixing mechanism. The Na abundance of the Hyades giants are on the upper-limit of the range predicted by the models, implying that the stars had a rather high initial rotation. Absolute abundance values have always to be considered with care. Nevertheless, as a fundamental temperature was used, the Na abundance derived here should be quite accurate. In general, relative comparisons should be more robust. In this sense, the increasing trend of the Na abundance with increasing mass found in the giants analyzed by Smiljanic et al. (2009) is the same as the one expected by evolutionary models. Agreement with models is also seen in the results of Randich et al. (2006), that did not find a difference in Na among dwarfs and giants of M67 and in the results of Pasquini et al. (2004), that did find a difference in Na among dwarfs and giants of IC 4651. In addition, it should be noticed that when comparing field giants with cluster giants, similar Na overabundances are apparent . All these pieces of evidence seem to point to the conclusion that, so far, there seems to be no strong evidence for Na overabundances in giants of open clusters beyond those that can be well explained by the effects of evolutionary mixing, in stars more massive than ∼ 2.0 M⊙. A consistent and homogeneous reanalysis of Na abundances in a large sample of giants is still necessary to confirm (or refute) this conclusion. ACKNOWLEDGMENTS I thank the anonymous referee for the valuable suggestions and comments. I am also grateful to Luca Pasquini for making available the FEROS spectra of the Hyades giants and to Karin Lind for making available the code to interpolate among the grid of NLTE corrections. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007(FP7/ -2013 under grant agreement No 229517. This research has made use of the WEBDA database, operated at the Institute for Astronomy of the University of Vienna, of the Simbad database operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System. and Astrophysics, 3, 316 Talon S., Charbonnel C., 2005, A&A, 440, 981 Tautvaišienė G., Edvardsson B., Puzeras E., Barisevičius G., Ilyin I., 2010, MNRAS, 409, 1213Tautvaišienė G., Edvardsson B., Puzeras E., Ilyin I., 2005, A&A, 431, 933 Tautvaišienė G., Edvardsson B., Tuominen I., Ilyin I., 2000, A&A, 360, 499 van Leeuwen F., 2009 Young P. A., Knierman K. A., Rigby J. R., Arnett D., 2003, ApJ, 595, 1114 ; Smith et al. 2002; Pavlenko et al. 2003; Pilachowski et al. 2003; Geisler et al. 2005; Spite et al. 2006; Recio-Blanco & de Laverny 2007; Smiljanic et al. 2009; Mikolaitis et al. 2010, 2011; Suda et al. 2011, and references therein). Figure 1 . 1HR diagram of the Hyades on the region around the turn-off and the clump. The four giants are shown with different symbols (ǫ Tau as the blue square, γ Tau as the green circle, δ 1 Tau as the black triangle, and θ 1 Tau as the red starred symbol). Luminosities and temperatures are from deBruijne et al. (2001). .322 log g(Gratton et al. 1996, derived from giants with [Fe/H] between −1.00 to 0.00), (ii) ξ = 1.5 − 0.13 log g(Carretta et al. 2004, derived from open cluster giants with [Fe/H] between −0.37 to +0.24), (iii) ξ = 1.645 + (3.854 × 10 −4 (T eff − 6387)) + ( −0.64 (logg-4.373)) + (−3.427 × 10 −4 (T eff − 6387) (logg − 4.373)) (Allende Prieto et al. 2004, derived from solar neighborhood stars with [Fe/H] between −0.50 to +0.50), (iv) ξ = 3.40 − 4.41 × 10 −4 T eff and ξ = 1.84 − 0.202 log g (Alves-Brito et al. 2010, derived from bulge, thin and thick disc giants with [Fe/H] between −1.50 to +0.50). the value found here is [Na/Fe] = +0.31 6 . Using equivalent widths, Schuler et al. (2009) obtained [Na/Fe] = +0.48 in LTE. The question then is why the results are different. First, as pointed out in Section 3.2, Schuler et al. (2009) used the equivalent width of line λ 5682 to determine the Na abundance. This line, however, is blended and removing it from the average reduces [Na/Fe] by 0.05 dex. Second are the NLTE corrections for lines λ 6154 and 6160, causing another reduction by 0.05/0.06 dex. Last, Schuler et al. (2009) adopted [Fe/H] = +0.13 for the Hyades giants. The atmospheric parameters used inSchuler et al. (2009) were determined in a previous paper,Schuler et al. (2006), were FeI lines were also measured. Using these lines,Schuler et al. (2006) found an average of [Fe/H] = +0.16, although recalculating it with the Kurucz model atmospheres and codes used here a value of [Fe/H] 5 Using equivalent widths the value is [Na/Fe] = +0.33. 6 Using equivalent widths the value is [Na/Fe] = +0.34. Figure 2 . 2Sodium abundances, [Na/Fe], as function of the stellar mass at the clump. The circles indicate the open cluster giants analyzed in Smiljanic et al. (2009), the typical error bar of that work is shown in the lower right corner. The blue square corresponds to the recommended [Na/Fe] of the Hyades derived in this work. The lines represent the predicted [Na/Fe] as a function of initial stellar mass given by the models of Charbonnel & Lagarde (2010) for the standard case (solid line), for a model including thermohaline mixing only (blue dotted line), and models with thermohaline mixing and rotation-induced mixing with initial velocities in the ZAMS of 250 and 300 km s −1 (lower red dashed line and upper green dashed line, respectively). Table 1 . 1Observational data.Star Spectrograph V Date of Exp. Time S/N observation (s) @ 617 nm γ Tau FEROS 3.654 05 Mar. 2004 120 500 ǫ Tau FEROS 3.540 03 Oct. 2009 180 500 UVES RED 580 30 Nov. 2002 2 × 1 400 δ 1 Tau FEROS 3.764 31 Oct. 2000 120 700 UVES RED 580 30 Nov. 2002 2 × 1 250 Table 2. Effective temperatures of the Hyades giants taken from selected recent works from the literature. Star T eff Method Reference (K) γ Tau 4844 ± 47 Interferometry Boyajian et al. (2009) 4965 ± 75 IRFM 1 Schuler et al. (2009) 4960 ± 8.1 Line-depth ratio (average) Kovtyukh et al. (2006) 4981 ± 80 Average Selected results from the PASTEL catalogue δ 1 Tau 4826 ± 51 Interferometry Boyajian et al. (2009) 4938 ± 75 IRFM 1 Schuler et al. (2009) 4975 ± 7.6 Line-depth ratio (average) Kovtyukh et al. (2006) 5000 ± 80 FeI excitation equilibrium Hekker & Meléndez (2007) 4968 ± 82 Average Selected results from the PASTEL catalogue ǫ Tau 4827 ± 44 Interferometry Boyajian et al. (2009) 4911 ± 75 IRFM 1 Schuler et al. (2009) 4925 ± 8.7 Line-depth ratio (average) Kovtyukh et al. (2006) 4910 ± 80 FeI excitation equilibrium Hekker & Meléndez (2007) 4925 ± 84 Average Selected results from the PASTEL catalogue Table 3 . 3The different sets of atmospheric parameters calculated for the Hyades giants.Star T eff log g [Fe/H] Note γ Tau 4844 2.66 +0.14 ± 0.05 Interferometric T eff 4844 2.57 +0.13 ± 0.05 As above with 2.0 M ⊙ 4844 2.74 +0.15 ± 0.05 As above with 3.0 M ⊙ 4965 2.70 +0.23 ± 0.05 IRFM T eff δ 1 Tau 4826 2.69 +0.18 ± 0.06 Interferometric T eff 4938 2.73 +0.25 ± 0.06 IRFM T eff ǫ Tau 4827 2.60 +0.26 ± 0.08 Interferometric T eff 4911 2.63 +0.31 ± 0.08 IRFM T eff ). Using the interferometric temperatures, the mean metallicity of the Hyades giants is found to be [Fe/H] = +0.19 ± 0.06. Using the IRFM temperatures, this value increases to [Fe/H] = +0.26 ± 0.04. Table 5 . 5Atomic data and equivalent widths of the Fe I lines used to derive the metallicity of the stars.Table 6. Atomic data of the six selected NaI lines.λ χ log gf C 6 Sun γ Tau δ 1 Tau ǫ Tau (Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) 5054.642 3.64 −2.032 4.68E-32 40.2 80.6 80.2 84.5 5127.679 0.05 −6.005 7.38E-33 22.3 96.6 99.5 105.6 5223.185 3.64 −2.285 6.00E-32 29.7 65.8 68.8 76.0 5320.035 3.64 −2.542 8.91E-32 19.8 57.1 57.9 59.0 5483.098 4.15 −1.481 2.95E-31 46.5 78.8 80.8 83.1 5522.446 4.21 −1.432 3.02E-31 43.6 78.5 80.6 83.8 5778.453 2.59 −3.524 4.95E-32 22.2 73.6 74.6 81.1 5784.658 3.40 −2.626 3.57E-31 27.2 67.8 70.7 76.2 5814.807 4.28 −1.861 2.82E-31 23.4 56.5 58.1 59.7 6012.209 2.22 −3.843 3.35E-32 24.2 78.7 78.4 82.3 6079.008 4.65 −1.055 5.13E-31 46.2 77.8 80.9 81.2 6187.989 3.94 −1.712 4.90E-31 47.9 91.2 93.1 99.6 6271.278 3.33 −2.797 2.78E-31 23.6 68.3 74.2 79.5 6739.522 1.56 −4.942 2.10E-32 11.8 63.6 64.8 70.3 6837.006 4.59 −1.756 2.46E-32 17.5 47.1 50.2 52.1 λ (Å) log gf χ (eV) C 6 5 148.838 −2.044 2.102 1.01E-30 5 682.633 −0.706 2.102 3.38E-30 5 688.193* −1.406 2.104 2.03E-30 5 688.205* −0.452 2.104 3.38E-30 6 154.226 −1.547 2.102 0.90E-31 6 160.747 −1.246 2.104 0.30E-31 (*) Fine structure components. Table 7 . 7Equivalent widths of three NaI lines measured from the data collected in this work.Table 8. Sodium abundances calculated with equivalent widths and the different atmospheric parameters of each star.Line Sun Sun γ Tau δ 1 Tau δ 1 Tau ǫ Tau ǫ Tau Kurucz UVES FEROS FEROS UVES FEROS UVES (Å) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) 6154 37.9 37.7 105.4 104.1 103.1 108.0 107.4 6160 58.1 58.6 119.6 122.8 119.8 127.0 124.8 Line Sun Sun γ Tau γ Tau γ Tau γ Tau δ 1 Tau δ 1 Tau ǫ Tau ǫ Tau (Å) Kurucz UVES Interf. IRFM 2.0 M ⊙ 3.0 M ⊙ Interf. IRFM Interf. IRFM 6154 6.27 6.27 6.86 6.96 6.86 6.86 6.83 6.92 6.91 6.98 6160 6.32 6.33 6.81 6.92 6.81 6.81 6.86 6.96 6.95 7.02 Average 6.29 6.30 6.84 6.94 6.84 6.84 6.85 6.94 6.93 7.00 [Na/Fe] - - +0.40 +0.41 +0.41 +0.39 +0.37 +0.39 +0.37 +0.39 Table 9 . 9Sodium abundances with spectrum synthesis and the different atmospheric parameters of each star.Line Sun Sun γ Tau γ Tau δ 1 Tau δ 1 Tau ǫ Tau ǫ Tau (Å) Kurucz UVES Interf. IRFM Interf. IRFM Interf. IRFM 5148 6.24 6.24 6.48 6.56 6.49 6.58 6.53 6.62 5682 6.36 6.36 6.87 6.97 6.82 6.92 6.94 7.00 5688 6.42 6.42 6.88 6.98 6.83 6.94 6.98 7.05 6154 6.30 6.30 6.75 6.86 6.73 6.83 6.80 6.88 6160 6.32 6.32 6.81 6.91 6.85 6.96 6.93 7.00 Average 6.33 6.33 6.76 6.86 6.74 6.85 6.84 6.91 [Na/Fe] - - +0.29 +0.30 +0.23 +0.27 +0.25 +0.27 Table 10. Average [Na/Fe], in LTE, for each star and for each of the different analysis and parameters (excluding line λ 5148 for the synthesis values). Analysis γ Tau δ 1 Tau ǫ Tau EqW & Interf. +0.40 +0.37 +0.37 EqW & IRFM +0.41 +0.39 +0.39 Synthesis & Interf +0.34 +0.28 +0.30 Synthesis & IRFM +0.35 +0.31 +0.32 Table 11 . 11Average [Na/Fe] in NLTE.Analysis γ Tau δ 1 Tau ǫ Tau EqW & Interf. +0.34 +0.33 +0.32 EqW & IRFM +0.35 +0.33 +0.34 Synthesis & Interf +0.33 +0.27 +0.29 Synthesis & IRFM +0.33 +0.29 +0.31 = +0.19 is found. Taking into account this difference the final [Na/Fe] in NLTE found by Schuler et al. (2009) should be [Na/Fe] = +0.34 (in the Schuler et al. 2006, metallicity scale) or [Na/Fe] = +0.31 (in the metallicity scale recalcu- lated here) The spectrum is available for download at the ESO website: www.eso.org/observing/dfo/quality/UVES/pipeline/solar spectrum.html IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. Alternatively, the interpolation code can accept as input the equivalent width of the line and return both the LTE and the NLTE abundances. The first approach was preferred here because the LTE abundances calculated in Lind et al. (2011) for a given set of atmospheric parameters are slightly different from the ones derived in this work. The difference is caused by different choices made in the treatment of line broadening. . 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[ "Hyperbolicity of Divergence Cleaning and Vector Potential Formulations of GRMHD", "Hyperbolicity of Divergence Cleaning and Vector Potential Formulations of GRMHD" ]	[ "David Hilditch \nDepartamento de Física\nCENTRA\nInstituto Superior Técnico IST\nUniversidade de Lisboa UL\nAvenida Rovisco Pais 11049LisboaPortugal\n", "Andreas Schoepe \nFriedrich-Schiller-Universität Jena\n07743JenaGermany\n" ]	[ "Departamento de Física\nCENTRA\nInstituto Superior Técnico IST\nUniversidade de Lisboa UL\nAvenida Rovisco Pais 11049LisboaPortugal", "Friedrich-Schiller-Universität Jena\n07743JenaGermany" ]	[]	We examine hyperbolicity of general relativistic magnetohydrodynamics with divergence cleaning, a flux-balance law form of the model not covered by our earlier analysis. The calculations rely again on a dual-frame approach, which allows us to effectively exploit the structure present in the principal part. We find, in contrast to the standard flux-balance law form of the equations, that this formulation is strongly hyperbolic, and thus admits a well-posed initial value problem. Formulations involving the vector potential as an evolved quantity are then considered. Carefully reducing to first order, we find that such formulations can also be made strongly hyperbolic. Despite the unwieldy form of the characteristic variables we therefore conclude that of the free-evolution formulations of general relativistic magnetohydrodynamics presently used in numerical relativity, the divergence cleaning and vector potential formulations are preferred.	10.1103/physrevd.99.104034	[ "https://arxiv.org/pdf/1812.03485v2.pdf" ]	119,094,221	1812.03485	8525a3065dd547d6563e4b63a936d4d09b913b4e	Hyperbolicity of Divergence Cleaning and Vector Potential Formulations of GRMHD 9 Dec 2018 David Hilditch Departamento de Física CENTRA Instituto Superior Técnico IST Universidade de Lisboa UL Avenida Rovisco Pais 11049LisboaPortugal Andreas Schoepe Friedrich-Schiller-Universität Jena 07743JenaGermany Hyperbolicity of Divergence Cleaning and Vector Potential Formulations of GRMHD 9 Dec 2018(Dated: December 11, 2018) We examine hyperbolicity of general relativistic magnetohydrodynamics with divergence cleaning, a flux-balance law form of the model not covered by our earlier analysis. The calculations rely again on a dual-frame approach, which allows us to effectively exploit the structure present in the principal part. We find, in contrast to the standard flux-balance law form of the equations, that this formulation is strongly hyperbolic, and thus admits a well-posed initial value problem. Formulations involving the vector potential as an evolved quantity are then considered. Carefully reducing to first order, we find that such formulations can also be made strongly hyperbolic. Despite the unwieldy form of the characteristic variables we therefore conclude that of the free-evolution formulations of general relativistic magnetohydrodynamics presently used in numerical relativity, the divergence cleaning and vector potential formulations are preferred. We examine hyperbolicity of general relativistic magnetohydrodynamics with divergence cleaning, a flux-balance law form of the model not covered by our earlier analysis. The calculations rely again on a dual-frame approach, which allows us to effectively exploit the structure present in the principal part. We find, in contrast to the standard flux-balance law form of the equations, that this formulation is strongly hyperbolic, and thus admits a well-posed initial value problem. Formulations involving the vector potential as an evolved quantity are then considered. Carefully reducing to first order, we find that such formulations can also be made strongly hyperbolic. Despite the unwieldy form of the characteristic variables we therefore conclude that of the free-evolution formulations of general relativistic magnetohydrodynamics presently used in numerical relativity, the divergence cleaning and vector potential formulations are preferred. I. INTRODUCTION It is well appreciated [1,2] that the numerical modeling of binary neutron star spacetimes plays, and will continue to play, an important role in the new field of gravitational wave astronomy, particularly in the case of multi-messenger events. Such simulations are, however, hampered by relatively poor error behavior as compared with their vacuum, black hole counterparts. This is in part because the equations of motion of these models have a more complicated structure than those of pure general relativity, and are hence less well understood, but also because solutions naturally develop non-smooth features, not to mention the ever present complication of the stellar surface. In a recent contribution [3] we employed a new tool, the dual-frame formalism [4][5][6][7], to analyze well-posedness of various fluid models. Well-posedness is the weakest necessary condition to require of a set of evolution partial differential equations (PDE) so that numerical approximation to their solutions may be meaningfully sought. The formalism can be used to exploit structure in the field equations and hence simplifies earlier treatments. This should allow more sophisticated results to be shown in the future. One of the models treated in Ref. [3] was (ideal) general relativistic magnetohydrodynamics (GRMHD), taken in two different guises. In the Valencia flux-balance law form [8] we found that the field equations are only weakly hyperbolic, and therefore have an ill-posed initial value problem. Here we attend to two flavors of GRMHD untouched by our earlier study, namely the hyperbolic divergence cleaning (HDC) and vector potential (VP) formulations. Our main result is that both are strongly hyperbolic, provided suitable choices are made in the firstorder reduction of the latter. We work in 3+1 dimensions in geometric units with c = G = 1. Our calculations were performed primarily with xTensor for Mathematica [9]; our notebooks are available online in Ref. [10]. II. MATHEMATICAL BACKGROUND We start with a short overview of the relevant theory, definitions, and results to the PDE analysis and the dual-frame (DF) formalism. These are taken in a highly summarized form from Refs. [3,5,7]. Index notation. Latin letters a-e are used as abstract indices. We also use p as an abstract index, placing it always on the spatial derivative appearing on the righthand side of our first-order PDE system. The fourmetric g ab is the only object permitted to raise and lower indices. The symbol ∂ a stands for a flat covariant derivative. Indices u, S, s,× and × label contraction in that slot with u a or u a and so on, respectively. Capital Latin letters A-C are taken as abstract indices and denote appliance of the projection operators Q ⊥ and q ⊥, to be defined later. Similarly, we use indices -andˆ -ˆ to denote the application of the projection operator Õ ⊥ over a vector or dual-vector, respectively. DF formalism. We describe a region of spacetime in two different frames, namely the lowercase and the uppercase frame. We take the lowercase frame as an Eulerian frame, associated with a coordinate basis as is standard in numerical relativity. We denote the future pointing timelike unit normal vector to spatial slices of constant time, as usual, by n a . Additionally, we take any three linearly independent vector fields orthogonal to n a to form a basis of the four-dimensional spacetime. Tensors orthogonal to n a are called lowercase spatial, or just lowercase. The uppercase frame consists of a future pointing timelike unit vector N a , which is identified in the application below with the fluid four-velocity u a , plus any three linearly independent vector fields orthogonal to N a . Tensors orthogonal to N a are likewise called uppercase spatial, or just uppercase. The future pointing unit vectors of the lower-and uppercase frames can be mutually 3 + 1 decomposed as with the Lorentz factor W = (1 − V a V a ) −1/2 = (1 − v a v a ) −1/2 . The vectors v a =v a /W and V a are the boost vectors orthogonal to n a and N a , respectively. We define projection operators by n a = W (N a + V a ) , N a = W (n a + v a ) ,(1)Uppercase Lowercase Unit normal N a = W (n a + v a ) n a = W (N a + V a ) Boost vector V a v a =v a /W Lorentz factor W = (1 − V a Va) −1/2 W = (1 − v a va) −1/2 Projector (N) γ a b = g a b + N a N b γ a b = g a b + n a n b Boost metric (N) ab := (N) γ ab + W 2 VaV b ab := γ ab +vav b Inverse boost (N) ( −1 ) ab = (N) γ ab − V a V b ( −1 ) ab = γ ab − v a v bγ b a = δ b a + n b n a ,(N)γ b a = δ b a + N b N a ,(2) which are also denoted as the lowercase and uppercase spatial metrics, respectively. By definition, the relations γ b a n b = 0, (N) γ b a N b = 0 hold. We define furthermore the lowercase and uppercase boost metrics and their inverses, which are presented in Table I. PDE analysis. We consider a quasilinear system of first order evolution PDEs, in this case GRMHD with HDC, written in the form, ∇ u U = A p ∇ p U + S ,(3) with the covariant derivative along the streamlines of the fluid elements ∇ u ≡ u a ∇ a of the vector of evolved variables, called the state vector U, on the left-hand side. On the right-hand side, the covariant derivative of the state vector is contracted with the principal part A p , A a u a = 0. The symbol S stands for the source term which does not affect the level of hyperbolicity. We need only analyze the system of evolution equations for the matter variables, since they are minimally coupled to the Einstein equations for the components of the metric tensor. Strong hyperbolicity. For the hyperbolicity analysis, we have to perform a 2 + 1 decomposition against lowercase and/or uppercase spatial vectors and their respective orthogonal spatial projectors. The relevant quantities are defined in Table II. Taking an arbitrary uppercase unit spatial 1-form S a , we define the uppercase principal symbol of the system (3) as P S ≡ A p S p .(4) We call the system (3) weakly hyperbolic, if for each S a the eigenvalues of P S are real. We call the system (3) strongly hyperbolic, if the system is weakly hyperbolic and for each S a the principal symbol P S has a complete set of right eigenvectors written as columns in a matrix T S and there exists a constant K > 0, independent of S a , such that \|T S \| + \|T −1 S \| ≤ K. Similar definitions are made if we 3+1 decompose the system against n a rather than u a , and the initial value problem, where data is given at t = 0, can be well-posed only if it satisfies these lowercase strong hyperbolicity conditions [11][12][13]. Uppercase Lowercase Unit normal N a n a Spatial 1-form Sa ×a Spatial vector S a = (N) γ ab S b× Frame and variable independence of hyperbolicity. If the uppercase eigenvalues of the principal symbol fulfill the inequality \|λ N \|\|V \| < 1 then strong hyperbolicity is independent of the chosen frame [3]. By the form of the energy-momentum tensor of GRMHD, see below, a naturally preferred frame is the fluid rest frame. Therefore, in the PDE analysis in Sec. IV, we will work exclusively in the uppercase frame, taken to be the fluid rest frame, N a ≡ u a ; hence the 3 + 1 decomposition in Eq. (3), and in the following, of the equations against the fluid four-velocity u a and the orthogonal projector (u) γ a b . In numerical applications, particular sets of variables, such as the primitive or conservative sets are used. In our analysis, we make a choice of variables which differs slightly from those. Our variables are however related to the code variables by a regular transformation, across which hyperbolicity is unaffected. a = ( −1 ) ab × b Norm SaS a = 1 ×a( −1 ) ab × b = 1 Projector Q ⊥ b a = (N) γ b a − S b Sa Õ ⊥ b a = γ b a −× b ×a Index notation Q ⊥ B A Õ ⊥ ˆ III. BASICS OF GRMHD A brief review of the basic definitions, equations, and assumptions of GRMHD with HDC is now given, following Refs. [14][15][16]. Presently, the focus will lie on the mathematical structure of the equations, and thus we suppress some (important) physical insight and statements. We use Lorentz-Heaviside units for electromagnetic quantities with ε 0 = µ 0 = 1, where ε 0 and µ 0 are the vacuum permittivity (or electric constant) and permeability (or magnetic constant), respectively. Motivated by the arguments given in the previous section, we work exclusively in the uppercase (fluid) frame and thus, present the system of equations in a form so adjusted. The energy-momentum tensor of GRMHD consists of an ideal fluid part, T ab fluid = ρ 0 hu a u b + g ab p ,(5) with the four-velocity of the fluid elements u a , rest mass density ρ 0 , specific enthalpy h, and pressure p; plus the standard electromagnetic energy-momentum tensor T ab em = F ac F b c − 1 4 g ab F cd F cd ,(6) with the Faraday electromagnetic tensor field (or for short field strength tensor) F ab . The specific enthalpy h can be expressed in terms of ρ 0 , p, and the specific internal energy ε as h = 1 + ε + p ρ 0 .(7) The local speed of sound c s is defined by the relation c 2 s = 1 h χ + p ρ 2 0 κ , χ = ∂p ∂ρ 0 ε , κ = ∂p ∂ε ρ0 .(8) We assume an equation of state (EOS) of the form p = p(ρ 0 , ε),(9) with p > 0 is given satisfying furthermore that the local speed of sound lies in the range 0 < c s ≤ 1. Using the ideal MHD condition, where the electric conductivity tends to infinity while the electric four-current remains bounded, the field strength tensor and its dual become F ab = ǫ abcd u c b d ,(10)* F ab = u a b b − u b b a ,(11) respectively, where we introduced the uppercase magnetic field vector b a , satisfying u a b a = 0; and the Levi-Cività tensor ǫ abcd = − 1 √ −g [abcd] ,(12) where g is the determinant of the spacetime metric g ab , [abcd] is the completely antisymmetric Levi-Cività symbol, and 2 * F ab = −ǫ abcd F cd holds. Note that we use the sign convention of Ref. [17]. Taking the sum of Eqs. (5) and (6), and substituting the field strength tensor (10), the total energymomentum tensor of GRMHD may be written as T ab = ρ 0 h * u a u b + p * g ab − b a b b ,(13)with h * = h + b 2 /ρ 0 , p * = p + b 2 /2, and shorthand b 2 = b a b a . The covariant system of evolution equations is given by the conservation of the number of particles and the conservation of energy-momentum, ∇ a (ρ 0 u a ) = 0 ,(14)∇ b T ab = 0 ,(15) plus the relevant Maxwell equations ∇ b ( * F ab − g ab φ) = − 1 τ n a φ ,(16) which are already augmented by the terms with the scalar field φ to drive the Gauss constraint. Elsewhere the notation κ = τ −1 is employed. The constant τ is the timescale for the exponential driving toward the Gauss constraint for the magnetic field. Usually, φ is set to zero in the initial and boundary conditions. IV. HYPERBOLICITY ANALYSIS OF GRMHD WITH HDC Projecting Eqs. (14)- (16) along the four velocity of the fluid u a and perpendicular to it by (u) γ a b , the nine evolution equations which determine the time evolution of the GRMHD system with HDC are ∇ a (ρ 0 u a ) = 0 , (u) γ ab ∇ c T bc = 0 , (u) γ ab ∇ c ( * F bc − g bc φ) = − W τ V a φ , u b ∇ c T bc = 0 , u b ∇ c ( * F bc − g bc φ) = W τ φ ,(17) supplemented with an EOS (9). In the limit of φ → 0 we find the uppercase Gauss constraint: (u) γ bc ∇ b b c = u c ∇ b * F bc = 0. Taking Eq. (17) and performing algebraic manipulations similar to the investigation of other formulations of GRMHD in Ref. [3], we derive the evolution equations for the pressure, ∇ u p = − c 2 s ρ 0 h (u) γ p c ( −1 ) ce ∇ pve + κ ρ 0 b p ∇ p φ + S (p) ,(18) the boost vector, (u) γ ab ( −1 ) bc ∇ uvc = − b p b a ρ 2 0 hh * + (u) γ p a ρ 0 h * ∇ p p + 2 ρ 0 h * (u) γ [b a b p](u) γ bc + b a ρ 0 h (u) γ p c ( −1 ) ce ∇ p ⊥b e + S (v) a ,(19) the magnetic field, (u) γ ab ( −1 ) bc ∇ u ⊥b c =2 (u) γ ab (u) γ [b c b p] ( −1 ) ce ∇ pve − (u) γ p a ∇ p φ + S (⊥b) a ,(20) the specific internal energy, ∇ u ε = − p ρ 0 (u) γ p c ( −1 ) ce ∇ pve + b p ρ 0 ∇ p φ + S (ε) ,(21) and finally the scalar field variable, ∇ u φ = − (u) γ p c ( −1 ) ce ∇ p ⊥b e + S (φ) .(22) The sources are given by S (p) = −c 2 s W ρ 0 h (u) γ d c ( −1 ) ce ∇ d n e − κW τ ρ 0 (b a V a )φ , S (v) a = −W (u) γ ab ( −1 ) be ∇ u n e + 2W ρ 0 h * (u) γ [b a b e] V b b d ∇ d n e + 1 ρ 0 h b a W V d b e − W (b c V c ) (u) γ de ∇ d n e , S (⊥b) a = 2W (u) γ ab (u) γ [b c b d] ( −1 ) ce ∇ d n e + 2W (u) γ e [a V b] b b ∇ u n e + W τ V a φ , S (ε) = − W p ρ 0 (u) γ d c ( −1 ) ce ∇ d n e − W τ ρ 0 (b a V a )φ , S (φ) = − W V d b e − W (b c V c ) (u) γ de ∇ d n e − W φ τ . The auxiliary magnetic vector ⊥b c is defined by the relation (u) γ ac ( −1 ) cd ∇ b ⊥b d := (u) γ ac ( −1 ) cd ∇ bbd + V a b d ( −1 ) de ∇ bve .(23) As usual, square brackets around indices denote antisymmetrization, so that 2v [a b b] =v a b b −v b b a . We have shown explicitly that the set of equations (18)-(22) is, up to non-principal terms, which we have not carefully checked, simply a linear combination of the formulation of GRMHD with HDC used in numerical applications, see, for example, Ref. [16]. This verification can be found in the notebook that accompanies the paper [10]. Writing Eqs. (18)- (22) in a vectorial form with state vector U = (p,v a , ⊥b a , ε, φ) T , we obtain, in the notation of Ref. [3], the principal part in the form, B u ∇ u U = B p ∇ p U + S .(24) Let S a be an arbitrary unit spatial uppercase 1form, S a S a = 1, and Q ⊥ b a := (u) γ b a − S b S a be the associated orthogonal projector. Let × a and Õ ⊥ b a be their lowercase projected versions, × a = γ b a S b , Õ ⊥ b a := γ b a −( −1 ) bc × c × a . Decomposing (u) γ b a and γ b a against S a and × a , respectively, Eq. (24) can be written as (∇ u U)× ,ˆ ≃ P S (∇ S U)× ,ˆ ,(25) where ≃ denotes equality up to non-principal terms and uppercase spatial derivatives transverse to S a . The uppercase principal symbol is P S = B S =               0 −c 2 s ρ 0 h 0 B 0 0 B 0 κb S ρ0 − (b S ) 2 +ρ0h ρ 2 0 hh * 0 0 B b S ρ0h − b B ρ0h * 0 0 − b S bA ρ 2 0 hh * 0 A 0 B A bA ρ0h b S ρ0h * Q ⊥ B A 0 A 0 A 0 0 0 B 0 0 B 0 −1 0 A −b A b S Q ⊥ B A 0 A 0 B A 0 A 0 A 0 − p ρ0 0 B 0 0 B 0 b S ρ0 0 0 0 B −1 0 B 0 0              (26) with the state vector ordered as, (δU)× ,ˆ = (δp, (δv)×, (δv)ˆ , (δ⊥b)×, (δ⊥b)ˆ , δε, δφ) T . The characteristic polynomial P λ for the principal symbol (26) is calculated to P λ = λ (ρ 0 h * ) 2 (1 − λ 2 )P Alfvén P mgs ,(28) with the quadratic polynomial for Alfvén waves P Alfvén = − b S 2 + λ 2 ρ 0 h (29) and the quartic polynomial for the magnetosonic waves P mgs = λ 2 − 1 λ 2 b 2 − b S 2 c 2 s + λ 2 λ 2 − c 2 s ρ 0 h .(30) Comparing Eq. (30) with our earlier results for the fluxbalance law formulation of GRMHD in Ref. [3], we see that the linear polynomial associated with the Gauss constraint is replaced by the quadratic polynomial 1 − λ 2 . The entropy, Alfvén, and slow and fast magnetosonic uppercase eigenvalues remain the same, as before, and are given by λ (e) =0 , λ (a±) = ± b S √ ρ 0 h , λ (s±) = ± ζ S − ζ 2 S − ξ S , λ (f±) = ± ζ S + ζ 2 S − ξ S ,(31) respectively, with shorthands ζ S = b 2 + c 2 s b S 2 + ρ 0 h 2ρ 0 h * , ξ S = b S 2 c 2 s ρ 0 h * . (32) The remaining two speeds can be associated with the scalar field and the longitudinal magnetic field [16], and are given by λ ± = ±1.(33) Since all uppercase eigenvalues have absolute value smaller than or equal to one, the relation \|λ u \|\|V \| < 1 is satisfied, so we may analyze hyperbolicity independently of the frame [3]. Therefore, we analyze the characteristic structure of the principal symbol in the uppercase frame and the result of the analysis applies directly to the numerically used system (in the lowercase). Continuing the characteristic analysis, we find the left entropy, scalar field and longitudinal magnetic field, Alfvén, and magnetosonic eigenvectors being − p c 2 s ρ 2 0 h 0 0 A ρ 0 − κp c 2 s ρ0h b S ρ 2 0 0 A 1 0 , 0 0 0 A ±1 0 A 0 1 , 0 0 ∓ (S) ǫ AC b C √ ρ 0 h * 0 − (S) ǫ AC b C 0 0 , ρ0h * (λ(m±)) 2 −b 2 c 2 s ρ0h (b S ) 2 −ρ0h * (λ(m±)) 2 λ (m±) b S b A λ (m±) K b A L ,(34) respectively, where we defined the antisymmetric uppercase two-and three-Levi-Cività tensors as (S) ǫ AB = S d (u) ǫ dAB = u c S d Q ⊥ A a Q ⊥ B b ǫ cdab . We employ furthermore the shorthands K =(b 2 ⊥ c 2 s + ρ 0 h * (λ (m±) ) 2 − b 2 ) (κ + c 2 s ρ 0 )b S c 2 s ρ 2 0 h(1 − c 2 s ) ; L = ρ 0 h * λ (m±) 2 − b S 2 λ (m±) 2 κ λ (m±) 2 + c 2 s ρ 0 b S (1 − c 2 s )ρ 2 0 hλ (m±) .(35) The right eigenvectors can be computed and are presented in the same order,             0 0 0 B 0 0 B 1 0             ,             ∓ρ 0 h(κ + c 2 s ρ 0 )b S (κ + ρ 0 )b S (1 − c 2 s )ρ 0 b B ±(1 − c 2 s )ρ 2 0 h ∓(κ + c 2 s ρ 0 )b S b B ∓( κp ρ0 + p + (1 − c 2 s )ρ 0 h)b S −(1 − c 2 s )ρ 2 0 h             ,             0 0 ∓ (S) ǫBC √ ρ0h * b C 0 − (S) ǫ BC b C 0 0             ,              c 2 s ρ 2 0 h p − ρ0λ (m±) p ρ0λ (m±) pb S b 2 ⊥ b S 2 + ρ 0 h * (λ (m±) ) 2 − 2ζ S b B 0 ρ0 b 2 ⊥ p b 2 + ρ 0 h * (λ (m±) ) 2 − 2ζ S b B 1 0              .(36) We have introduced in the magnetosonic eigenvectors the orthogonal magnetic field vector as b a ⊥ = Q ⊥ a b b b with b 2 ⊥ = b a ⊥ b ⊥ a = b A b A . As for the prototype algebraic constraint free formulation treated in Ref. [3], rescaled versions of the left eigenvectors (34) and right eigenvectors (36) can be derived. They form complete sets of nine linearly independent eigenvectors under type I, type II, and type II ′ degeneracies [15,18]. The rescaling can be found in the notebook provided in Ref. [10]. Thus, as long as p = p(ρ 0 , ε) > 0 and 0 < c s < 1 hold, the formulation of GRMHD with HDC as given above forms a strongly hyperbolic system of equations. In the limit of c s → 1, it can be shown, that the fast magnetosonic waves collide pairwise with the waves associated to the scalar field and longitudinal magnetic field, in the case of which the system is only weakly hyperbolic. The limiting procedure can be found in the provided notebook. This is a consequence of taking the divergence cleaning to happen at the speed of light. By the simple replacement φ → c −2 φ φ, c φ > 0 in Eq. (22), the divergence cleaning speed becomes λ ± = ±c φ . For c φ > 1, strong hyperbolicity is also guaranteed in the limiting case c s = 1. This strategy does however place a nontrivial upper limit on the speed of flows that can be managed with the method, as strong hyperbolicity will break down for sufficiently fast flows. See Ref. [3] for details. By modifying the lowercase equations directly it may be possible to avoid this shortcoming, too. Finally, we want to present the uppercase rescaled characteristic variables for GRMHD with HDC. They are valid for all degeneracies, and are given bŷ U e =δε − p c 2 s ρ 2 0 h δp + ρ 0 − κp c 2 s ρ 0 h b S ρ 2 0 (δ⊥b)× , U ± =δφ ± (δ⊥b)× , U a± = ± (S) ǫ AC ρ 0 h * b ⊥ C \|b ⊥ \| (δv)ˆ + (S) ǫ AC b ⊥ C \|b ⊥ \| (δ⊥b)ˆ , U m 1 ± = H(λ 2 − 1) ρ 0 h δp + (1 − c 2 s )Hλ(δv)× + b S λ b A ⊥ \|b ⊥ \| (δv)ˆ − H(κ + c 2 s ρ 0 )b S ρ 2 0 h (δ⊥b)× + b A ⊥ \|b ⊥ \| (δ⊥b)ˆ + b S λ H(κλ 2 + c 2 s ρ 0 ) ρ 2 0 h δφ , U m 2 ± = 1 c 2 s ρ 0 h δp + (1 − c 2 s )λ c 2 s (λ 2 − 1) (δv)× + b S λ F A (δv)ˆ + b S λ λ(κ + c 2 s ρ 0 ) c 2 s (1 − λ 2 )ρ 2 0 h (δ⊥b)× + F A (δ⊥b)ˆ − b S λ (κλ 2 + c 2 s ρ 0 ) c 2 s (1 − λ 2 )ρ 2 0 h δφ ,(37) with {m 1 , m 2 } equal to {s, f} or {f, s}. The abbreviations in Eq. (37) are given by H = \|b ⊥ \| c 2 s − λ 2 ,(38)F A = b A ⊥ (ρ 0 h * λ 2 − b 2 ) ,(39) where for type II and even for type II ′ degeneracy we take Q 1 a and Q 2 a such that in the degenerate limit we have b ⊥ C \|b ⊥ \| = 1 √ 2 (Q 1 C + Q 2 C ) ,(40)H =0 ,(41)F A =0 A .(42) For further explanations concerning degeneracies and rescaling, see also Ref. [18]. Using the recovery procedure given in Ref. [3], the lowercase characteristic quantities such as eigenvalues and eigenvectors can be derived. The calculation can be found in the notebook [10], but results in rather long expressions which we suppress here. Both the lowercase left magnetosonic eigenvectors and the lowercase right eigenvectors associated with the scalar field and longitudinal magnetic field eigenvalues have a particularly complicated structure, for which a useful simplification seems difficult. In applications it may therefore be appropriate to compute the characteristics numerically. V. DISCUSSION OF FORMULATIONS OF GRMHD WITH VP The formulations of GRMHD we have thus far considered use the magnetic field as an evolved variable. Another possibility is to introduce the four-vector potential instead [19][20][21]. In practice, the potential is then 3 + 1 decomposed. Such formulations have the advantage that the Gauss constraint is satisfied by construction, and in this sense can be considered a type of constrainedrather than free-evolution. On the other hand one obtains a system of equations which is a priori not, from the PDE point of view, minimally coupled to the gravitational field equations. The resulting evolution equations for the GRMHD variables are moreover themselves not in first-order form, but rather first order in time and second order in space, and there is an additional gauge degree of freedom. Different choices in this freedom may have different PDE properties as the principal part of the evolution system is altered. We follow Ref. [20] and focus on the Lorenz gauge, but similar comments will hold elsewhere. Strong hyperbolicity of first-order in time, second order in space systems can be defined [22,23] by the requirement that there exists a first order reduction which satisfies the definition given for first order PDEs in Sec. II. Therefore, we must reduce the governing system of equations as in Eq. (3), by introducing reduction variables. There are two natural ways to go about this. The first, naive, possibility is to introduce reduction constraints c ab = d ab − γ c a γ d b ∂ c A d , which should vanish, for the lowercase spatial derivatives of the lowercase spatial part of the vector potential A a , and likewise for the electric potential. The reduction variables d ab should satisfy also the ordering constraint, c abc = γ d a γ e b γ f c ∂ [d c e]f = γ d a γ e b γ f c ∂ [d d e]f = 0 , (43) and similarly for the electric potential reduction variables. The reduction constraints must then be added to the equations of motion to remove all second spatial derivatives. Besides that, both the reduction and ordering constraints can be added freely to try and find a hyperbolic reduction. Such a reduction does not use the special structure of the Maxwell equations, does not utilize the fact that the original system satisfies the Gauss constraint by construction, and is not minimally coupled to the evolution equations for the geometric variables. Worse, the resulting principal symbol does not have a clear structure, which makes the analysis very difficult. The less obvious option is to bring back the magnetic field as a reduction variable for the curl of the spatial vector potential by defining a reduction constraint, C a = ǫ a bc D b A c − B a .(44) In this reduction we need not introduce a reduction variable to the electric potential as it appears with at most one spatial derivative. Part of the analog of the ordering constraint in such a reduction turns out to be simply the Gauss constraint, C = −D a C a = D a B a .(45) A generic PDE system does not allow a reduction of this type, in which new variables that only capture part of the spatial derivatives are introduced. Due to the gauge freedom of the Maxwell equations however the 'longitudinal' part of the vector potential does not appear elsewhere in the remaining equations of motion, and so we can close the evolution system using only B a . Note that such a restricted reduction does have consequences on the norms in which rigorous estimates would be demonstrated, and also that as usual first derivatives of the metric here are non-principal. Ultimately we end up with evolution equations for the matter variables which are minimally coupled to the Einstein equations. Naively writing out the lowercase principal symbol of the matter variables we can obtain moreover a block-diagonal structure, P s = A 0 0 B ,(46) where block A denotes the principal symbol of the system of evolution equations of the spatial part of the vector potential and the electric potential, whereas B can be rendered identical to the principal symbol of the prototype algebraic constraint free formulation of GRMHD investigated in Ref. [3]. Here, crucially, we rely on the fact that, as it is not to be used in applications, this formal first order reduction need not be of a flux-balance form, and therefore we can add the ordering constraint C as desired. The upper right block vanishes trivially and the lower left block vanishes by appropriate choice of reduction. We showed already that prototype algebraic constraint free formulation of GRMHD is strongly hyperbolic in the lowercase frame, with an EOS of the form (9) and 0 < c s ≤ 1, so all that remains is to show that the block A satisfies the conditions for strong hyperbolicity. This was done already in Ref. [20], but with the use of the reduction variable B a we can give a slightly cleaner treatment. The lowercase principal symbol can be read off from, ∇ n Φ ≃ −γ pe ∇ p A e ,(47)γ b a ∇ n A b ≃ −γ p a ∇ p Φ .(48) Note that in Eq. (48) the term D a A b − D b A a is written in terms of the reduction variable B a and does not contribute to the principal part. Let s a , s a s a = 1, be an unit spatial lowercase vector and be Õ ⊥ a b the orthogonal projector. The characteristic variables associated with this block are hence, δΦ ∓ (δA) s ,(49) with speeds ±1, respectively, and, (δA) A ,(50) with speed 0 for the two orthogonal directions to s a . The calculation is provided in a notebook that accompanies the paper [10]. VI. CONCLUSION In previous work [3] we examined two formulations of ideal GRMHD, and showed that a formulation similar to that studied in Refs. [14,24], which we call the prototype algebraic constraint free formulation is strongly hyperbolic. Unfortunately, this formulation is not in the flux-balance law form desirable for the application of standard numerical methods. Turning to GRMHD in flux-conservative form, we found the system to be only weakly hyperbolic. This formulation of GRMHD hence has an ill-posed initial value problem. Fortunately, two popular, applicable, alternative formulations of GRMHD were left untreated by that analysis. Presently, we have addressed this shortcoming with the outcome first, that formulations of GRMHD with HDC [16,25,26] are indeed strongly hyperbolic as long as the sound speed is suitably bounded 0 < c s < 1. In fact, it is straightforward to achieve hyperbolicity also in the case c s = 1 by changing the speed of the cleaning in the formulation. Second, we have shown that by a careful reduction to first order, formulations of GRMHD with VP [20] can also be rendered strongly hyperbolic whenever 0 < c s ≤ 1. The latter result is a corollary of strong hyperbolicity of the prototype algebraic constraint free formulation. Here we have discussed only the Lorenz gauge choice, but our results carry over trivially to generalized Lorenz-gauge, in which there is a modification by source terms, and a natural treatment will be very similar in other cases, too. Both HDC and the VP formulations were introduced as strategies to control Gauss-constraint violation in applications. Another popular approach, called constrained transport (CT) [27][28][29], uses a carefully constructed discretization so that in a particular approximation the constraint is identically satisfied. There is some subtlety in precisely what continuum PDE should be analyzed given such a constrained evolution, but supposing that the constraints are identically satisfied, they may again be added arbitrarily to the evolution equations, and stronghyperbolicity can again be achieved, in the restricted, constraint-satisfying phase space, as a corollary of hyperbolicity of the prototype algebraic constraint free formulation. In Ref. [3] we discussed two minimally coupled formulations of resistive GRMHD with HDC, one with and one without the evolution of the charge density q. Both were found to be only weakly hyperbolic. A natural question is therefore whether the use of the VP approach could cure this problem. Replacing the divergence cleaning variables by A a and Φ, and making a minimally coupled first order reduction as we did for GRMHD, one arrives with a lower block triangular structure in the principal symbol, with the lower-right block C being precisely a subblock of the principal symbol of the original formulation of RGRMHD. Neither of the original two formulations were strongly hyperbolic because C was not diagonalizable. Consequently, the vector potential formulations are also not strongly hyperbolic. Thus, at least if we insist on taking only minimally coupled first order reductions, use of a VP reformulation of RGRMHD does nothing to circumvent weak hyperbolicity of RGRMHD. For numerical applications we therefore have the clear conclusion that, by the fundamental requirement of wellposedness, HDC and VP formulations (and likely also CT schemes) are preferred over their older variant which should henceforth be avoided. From the PDEs point of view it is, at this stage, difficult to choose between the favored formulations. One might be tempted to argue in favor of the vector potential formulation, as indeed it is true that there the characteristic structure, inherited from the prototype algebraic constraint free formulation, is simpler, but this is not a principle advantage. In the future it is hoped that the characteristic structure uncovered by our analysis can be put to good use in numerical work in both systems. TABLE I : IOverview of the uppercase and lowercase quantities. TABLE II : IISummary of the various unit spatial vectors ap- pearing in our 2 + 1 decomposed equations, plus their associ- ated projection operators. AcknowledgmentsWe are grateful to Sebastiano Bernuzzi and Bruno Giacomazzo for useful discussions and comments. This work was partially supported by the FCT (Portugal) IF Program No. IF/00577/2015 and the GWverse COST action Grant No. CA16104. 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[]	[ "Olga I Piskounova \nP.N.Lebedev Physics Institute of Russian Academy of Science\nLeninski prosp. 53119991MoscowRussia\n" ]	[ "P.N.Lebedev Physics Institute of Russian Academy of Science\nLeninski prosp. 53119991MoscowRussia" ]	[]	This paper examines the transverse momentum spectra of baryons in the multi particle production at modern colliders using Quark-Gluon String Model (QGSM). It discusses 1) the difference in Λ 0 hyperon spectra at proton-antiproton vs. protonproton reactions; 2) the growth of average transverse momenta of Λ hyperon with proton-proton collision energies and 3) the dependence of average p t on the masses of mesons and baryons at the LHC energy 7 TeV. This analysis of baryon spectra led to the following conclusions. First, the fragmentation of antidiquark-diquark side of pomeron diagram makes the major contribution to baryon production spectra in the asymmetric p-p reaction. Second, the average p t 's of hyperons steadily grow with energy on the range from √ s= 53 GeV to 7 TeV. Since no dramatic changes were seen in the characteristics of baryon production, the hadroproduction processes do not cause the "knee" in the cosmic ray proton spectra at the energies between Tevatron and LHC. Third, the average transverse momentum analysis, through examining the different mass of hadrons, reveals the regularity in the mass gaps between baryon-meson generations. This observation gives the possibility for more hadron states with the masses: 13.7, 37.3, 101.5, 276, 750... GeV, which are produced by geometrical progression with the mass factor of order δ(lnM)=1. These hadrons may possess new quantum numbers or consist of heavy multi quarks.	null	[ "https://arxiv.org/pdf/1706.07648v7.pdf" ]	119,433,781	1706.07648	a47a5a84afe72a449fba983d41e58e282aeec676	23 Mar 2018 Olga I Piskounova P.N.Lebedev Physics Institute of Russian Academy of Science Leninski prosp. 53119991MoscowRussia 23 Mar 2018Baryon Production at p-p Collider Experiments: Average p t vs. Energy and Mass This paper examines the transverse momentum spectra of baryons in the multi particle production at modern colliders using Quark-Gluon String Model (QGSM). It discusses 1) the difference in Λ 0 hyperon spectra at proton-antiproton vs. protonproton reactions; 2) the growth of average transverse momenta of Λ hyperon with proton-proton collision energies and 3) the dependence of average p t on the masses of mesons and baryons at the LHC energy 7 TeV. This analysis of baryon spectra led to the following conclusions. First, the fragmentation of antidiquark-diquark side of pomeron diagram makes the major contribution to baryon production spectra in the asymmetric p-p reaction. Second, the average p t 's of hyperons steadily grow with energy on the range from √ s= 53 GeV to 7 TeV. Since no dramatic changes were seen in the characteristics of baryon production, the hadroproduction processes do not cause the "knee" in the cosmic ray proton spectra at the energies between Tevatron and LHC. Third, the average transverse momentum analysis, through examining the different mass of hadrons, reveals the regularity in the mass gaps between baryon-meson generations. This observation gives the possibility for more hadron states with the masses: 13.7, 37.3, 101.5, 276, 750... GeV, which are produced by geometrical progression with the mass factor of order δ(lnM)=1. These hadrons may possess new quantum numbers or consist of heavy multi quarks. Introduction The aim of this paper is to analyze the transverse momentum spectra of hadrons from the modern collider experiments (ISR 1 ,STAR 2 ,UA5 4 , UA1 3 , CDF 5 , AL-ICE 6 , ATLAS 7 and CMS 8 ). A number of reasons warrant this study. First of all, the preliminary compilation of data on Λ 0 hyperon transverse momentum distributions 2,4 demonstrates a difference in the dynamics of multi particle production in proton-proton vs. antiproton-proton collisions, which suggests that the baryon spectra are sensitive to the asymmetrical reactions where the fragmentation of diquark-antidiquark chain plays an important role. Secondly, the detailed study of characteristics of baryon spectra is necessary at the energies between Tevatron and LHC, because the cosmic ray proton spectrum has a "knee" in this range of energies 9 . The change in the slope of spectrum of protons, produced in the space, either have an astrophysical origin or can be explained by the substantial change in the dynamics of hadron production at this energy gap. Finally, the course of average transverse momenta with the growth of hadron masses can demonstrate the availability of higher mass generations behind beauty hadrons, which supply the further increase of average transverse momenta. Interpretation of these distributions in the up-to-date phenomenological model can help resolve these issues. The Quark-Gluon String Model [QGSM] approach is applied here to the description of p t spectra for all available flavors of baryons 10,11 . The Model has successfully described the large volume of data from previous generation of colliders up to the energies √ s= 53 GeV in the area of low p t 's 12 . Recently, Λ 0 hyperon production have been studied in in updated version of QGSM 13 . The figure 1 presents the compilation of the data, dN ( Λ 0 )/dp t , in the region 0.1 GeV/c < p t < 5 GeV/c from the following experiments (ISR 1 , STAR 2 , UA1 3 , UA5 4 and CDF 5 ). It illustrates the changes in hyperon transverse distributions in the energy distance from ISR to Tevatron experiments. Since to calculate the average p t we do not need only the absolute values of distributions, those are chosen arbitrarily. The range of low p t 0.3 GeV/c < p t < 4 GeV/c has the most impact on the value of average p t . The figure clearly shows that average transverse momenta grow with energy. Figure 1: Transverse momentum distributions of Λ 0 hyperons from colliders that preceded LHC. The data are from ISR 1 p − p at √ s = 53GeV -empty triangles, STAR 2 p − p at √ s=200 GeV -black stars; UA5 4p − p energies : 546 GeV(empty squares) and 900 GeV(black squares); UA1 3p − p (630 GeV)black circles and CDF 5p − p at 1.8 TeV -black triangles. Preliminary Comparison of Hyperon Transverse Momentum Spectra from LHC Experiments The recent data on Λ 0 hyperon distributions are obtained in the following LHC groups: ALICE 6 at 900 GeV, ATLAS 7 and CMS 8 at 900 GeV and 7 TeV. We are going to compare the results of these LHC experiments with the data of lower energy p − p colliders, ISR 1 ( √ s = 53GeV ) and STAR 2 ( √ s = 200GeV ). In the figure 2 the B values, the slopes of spectra, change, if we fit the data with a simple exponential function: exp(-Bp t ). We can conclude that transverse momentum spectra are harder with the energy growth that provides the change in the slopes, beginning from B=4,2 for ISR data, B=2,6 for STAR and to B=2,0 at 900 GeV in ALICE. The slope is flatter at √ s = Both LHC experiments at 7 TeV, ATLAS and CMS, have presented the hyperon spectra with the same slopes as expected (see the figure 3). The different forms of the distributions at low p t region might be caused by efficiency specifics of ATLAS experiment. Baryon Transverse Momentum Distributions in QGSM The QGS Model has been devised for the description of rapidity distributions and hadron spectra in x F 10 . The Model operates with pomeron diagrams (see the figure 6), which help calculate the rapidity spectra. These spectra are presented as the convolutions of constituent quark structure functions with the diquark-antidiquark pair distributions at the pomeron cylinder fragmentation into baryons. This approach took into account mostly the average p t values for given energy. The early QGSM study 12 on the hadron transverse momentum distributions has shown that the spectra of baryons in proton-proton collisions can be described with the following p t -dependence: E d 3 σ H dx F d 2 p t = dσ H dx F A 0 * exp[−B 0 * (m t − m 0 )],(1) where m 0 is the mass of produced hadron, m t = p 2 t + m 2 0 . The slope parameter, B 0 , used to bring the dependence on x F in previous research 12 . The values of B 0 for the spectra of many types of hadrons (π, K, p) were estimated for the data of proton-proton collisions up to the energies of ISR experiment. The value of the slopes of baryon spectra for the data in central region of rapidities was universal and equal to B 0 = 6,0. As discussed above, the slopes of spectra, B 0 , at the modern collider experiments depend on energy. Moreover, the form of spectra at LHC and RHIC indicates that the value of m 0 is not the mass of proton or hyperon. A better description of hyperon spectra can be achieved with m 0 = 0,5 GeV that is actually the mass of kaon, see the figure 4. This effect can be provisionally explained as the minimal transverse momentum of hyperon at the fragmentation of diquark-quark chain (see the QGSM pomeron diagram for p-p collisions in the figure 6). The value of m 0 should be equal to the kaon mass, because the minimal diquark-quark chain fragmentation produces only two hadrons: Λ 0 +K. The Difference between Distributions in Proton-Proton and Antiproton-Proton Collisions Here we consider the difference in the spectra of baryon production in symmetric (p − p) and asymmetric (p − p) reactions. The Data of UA5 Experiment This subsection discusses the influence of quark composition of beam particles on the shape of transverse momentum spectra of Λ 0 hyperon production. The data from p-p experiments UA5 4 of energies, √ s= 200 GeV and 546 GeV, are studied (see figure 5). The sharp exponential contribution to spectra is seen inp − p reaction at √ s = Figure 4: The description of proton-proton experiment data ISR 1 (53 GeV), STAR 2 (200 GeV), ALICE 6 (900GeV) and CMS 8 (7 TeV) data on hyperon production with the QGSM fit. This exponential component might exist in other antiproton-reaction spectra as well, but it is not seen because of the absence of measurements in low p t 's. The form of the spectrum at low p t has a strong impact on the value of cross section, if the experimental distributions are integrated beginning from low momenta, p t = 0.3 GeV/c. The resulting cross section from antiproton-proton reaction should be smaller than the cross section, obtained in proto-proton collision of the same energy, if there are no data points at p t < 0.5 GeV/c. The complex form of spectra in UA5 can be explained by two component spectra for antiproton-proton collisions. The QGSM diagrams explain the nature of two components in the asymmetrical reaction. The Difference in Distributions from antiproton-proton and proton-proton reactions in QGSM The difference in p t -spectra of Λ 0 's produced in high energy p − p andp − p collisions cannot be explained in the perturbative QCD models, since both interactions should give the mutiparticle production in central rapidity region due to dominating one Pomeron exchange. The total cross section and the spectra in p − p and p − p collisions should be equal because they depend only on the parameters of the Pomeron exchange between two interacting hadrons and should not be sensitive to the quark contents of colliding particles. The pomeron diagrams of p − p andp − p collisions are shown in the figure 6. In the framework of QGSM 10,11 , which is based on the Regge theory and on the phenomenology of pomeron exchange, the spectra of produced particles are the results of the cut of one-pomeron diagrams. The comparison of diagrams shows that the most important contribution to hadron production spectra inp − p reaction is brought by the fragmentation of antidiquark-diquark chain of pomeron cylinder, because this side of diagram takes the greater part of energy of colliding particles. Otherwise, the p − p collision diagram is symmetric and built from two similar quark-diquark chains. Therefore, the incorrect description of the region of small transverse momenta at antiproton-proton reaction leads to underestimated values of cross section. The figure 7 shows that baryon production at low energy goes with the quark-antiquark annihilation. The resulting spectra consist of the contribution from only diquark-antidiquark chain that allows us to see the pure form of baryon transverse momentum distributions in asymmetric case of fragmentation. The form of baryon spectra in antiproton-proton reaction of low energy was planned to be studied in experiment TAPAS 14 . Average Baryon Transverse Momenta vs. Energy and "Knee" in Cosmic Ray Spectra As the data on baryon distributions in antiproton-proton reactions are irrelevant for consideration, the spectra from LHC can be compared only with measurements of proton-proton collision experiments : ISR and STAR. We consider the transverse momentum spectra in the wide range of energies: beginning from 53 GeV to the LHC energy 7 TeV. The cosmic ray proton spectrum shows the "knee" (see figure 8) at the energy gap between Tevatron and LHC colliders 9 . The change in the slope of proton spectrum at E lab ≈ 3* 10 15 eV might be a manifestation of a new regime in hadronic interactions. Otherwise, the change in the spectrum may have an astrophysical origin. The analysis of average baryon transverse momenta in the framework of QGSM clearly demonstrates that the average transverse momentum of baryons grows steadily with energy (see figure 9). The < p t > values grow in the range of energy from 53 GeV (ISR) up to √ s = 200 GeV (STAR) and then they go with the asymptote s 0.05 . This behavior cannot be considered as substantial change in hadroproduction processes. Since no spesific points exist in baryon production up to √ s = 7 TeV, which corresponds to E lab = 2, 5 * 10 16 eV, it is reasonable to conclude that "knee" has an astrophysical explanation. For instance, the "knee" may indicate the maximal energy of protons that are produced in a nearest galaxy. The growth of average transverse momenta was calculated in the framework of QGSM on the energy distance up to LHC. This result should be used for the interpretation of futher data of LHC groups. However, the energy dependence of < p t > is not yet included into MC generators calculations. In such a way, the results of this research will help improve LUND, Pythia and other MC models. Average Transverse Momenta vs. Mass of Hadrons The previously published analysis of transverse momentum spectra of baryons from LHC experiments (ALICE, ATLAS, CMS) 15 provided only partial data on hadron spectra. In order to get a full understanding of average transverse dependence on hadron mass, we supplement here the data with the spectra of kaons, D-mesons and B-mesons from LHC 16 at 7 TeV. The heavy quark meson spectra were fitted with the same formula (1) as the baryon spectra (see the figure 10). As the result of the fit, the average p t of B-mesons is equal to 4,46 GeV at the energy (s)=7 TeV. The dependence of average transverse momenta on the mass of hadron in the figure 11 shows that < p t >'s grow with masses. If we imagine symmetric point between meson and baryon masses of a given quark flavor, the mass distance between points of one hadron generation and the next one can be estimated with the mass factor δlnM H = 1. It means that we have the geometric progression for the masses of hypotetical hadrons. The extension of this sequence provides the hadron states with the following masses: 13.7, 37.3, 101.5, 276, 750 GeV and so forth. The plot is expanded with average p t dependence < p t >∝ M 0.1 H above the bottom hadron masses. These hypothetical hadrons may represent heavy multi quark states that are of neutral baryon and lepton charge. This hypothesis attempts to predict new particles and helps achieve the "supersymmetric" unification of baryon and meson features at heavy masses revisited recently in 18 . It should be noted, however, that the mass of top quark does not match the suggested collection of new hadron states. But top quark mass is to be released from the lower mass multiquark state with applying the energy. Conclusions The overview of results in transverse momentum distributions of hyperons produced in proton-proton collisions of various energies 13,21 has revealed a significant change in the slopes of baryon spectra in the region of p t = 0,3 -8 GeV/c. The spectra of baryons become harder with the energy growth from ISR( √ s=53 GeV) and RHIC (200 GeV) up to LHC (0,9 and 7 TeV). The detailed analysis of hyperon spectra in the framework of QGSM demonstrates the change of slopes from B 0 = 4,6 (ISR at 53 GeV) to B 0 = 2,1 (LHC at 7 TeV). The transverse momentum baryon spectra in antiproton-proton collisions (UA1, UA5, CDF) differ from the p t distributions of baryons in proton-proton collisions (ISR, STAR, LHC). QGSM explains this phenomenon as a difference in the splitting of transverse energy between two sides of pomeron diagram. The cut diagram for antiproton-proton case includes the unusual side with the diquark-antidiquark ends, which accumulates more energy than another quark-antiquark side of cylinder. It is reasonable to suggest that the difference in spectra disappears with the growth of energy due to the increasing multipomeron contributions into the differential cross section that are similar for bothp−p and p−p collisions. The average p t values in proton-proton collisions grow steadily as the power of energy, s 0.05 , up to highest LHC energy 7 TeV. Therefore, the baryon production processes at the energies of LHC are not totally random. This conclusion has crucial implications for cosmic ray physics, since it suggests that the "knee" at E lab ≈ 4* 10 15 eV in cosmic proton spectra does not originate in hadronic interactions. There is no significant change in baryon spectra up to √ s = 7 TeV corresponding to E lab = 2, 5 * 10 16 eV in the cosmic ray physics. I suggest that the "knee" may indicate the maximal energy of protons from a nearest galaxy. The idea of proton production in space warrants a further detailed investigation of the hadroproduction dynamics in the framework of QGSM. The average transverse momentum analysis, through examining the different mass of hadrons, indicates a regularity in the mass gaps between heavy quark Figure 2 : 2Transverse momentum distributions from p − p collider experiments: ISR 1 (53 GeV) -empty triangles, STAR 2 (200 GeV) -black stars, ALICE 6 (900GeV) -empty circles and ATLAS 7 (7 TeV) -black triangles, as fitted with the exponents. Figure 3 : 3Transverse momentum distributions at √ s= 7 TeV: ATLAS 7 -black triangles and CMS]citecms -black squares Figure 5 : 5The forms of spectra at antiproton-proton reaction in UA5 4 -black squares at the energy √ s=200 GeV. The UA5 4 data of higher energy ( √ s=546 GeV) are shown with empty squares. As I mentioned above, the absolute values of spectra are chosen arbitrarily. The fit for UA5(546) with solid and dashed lines demonstrates two different components at the asymmetric one-pomeron cut. Figure 6 : 6QGSM multiparticle production diagrams for a) proton-proton and b) antiproton-proton reactions. Figure 7 : 7The low energy diagram with quark-antiquark annihilation and diquark-antidiquark chain fragmentation. Figure 8 : 8The cosmic proton spectrum with the "knee" between Tevatron and LHC energies. Figure 9 : 9Average transverse momenta of hyperons increase with the energy, as s 0.05 . Figure 10 : 10The QGSM fit of B-meson spectra at √ s=7 TeV in LHCb 16 experiment. Figure 11 : 11The mass dependence of average momenta of hadrons (mesons -empty triangles, baryonsblack circles). The extensions for the hypothetical heavy hadron states are shown with the black stars. TeV, B = 1,5. 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